9th Math full book

UNIT 1

Q. Who did give the idea of matrices?

09301001

Q. Define matrix. 09301002

Q. Define Row and column of a matrix.

09301003

Q. Define order of a matrix. Give example.

09301004

Q. Define equal matrices. 09301005

1. Find the order of the following matrices. 09301006

2. Which of the following matrices are equal? 09301007

A = [3] ,

3. Find the values of a, b, c and d which satisfy the matrix equation. 09301008

Q. Define row matrix. 09301009

Q. Define column matrix. 09301010

Q. Define rectangular matrix. 09301011

Q. Define square matrix. Give example.

09301012

Q. Define null or zero matrix. (Board 2014)

09301013

Q. Define transpose of a matrix. 09301014

Q. Define negative of a matrix. 09301015

Q. Define symmetric matrix. 09301016

Q. Define skew-symmetric matrix.09301017

Q. Show that is skew symmetric matrix. 09301018

Q. Define diagonal matrix. (Board 2013)

09301019

Q. Define Scalar Matrix. (Board 2014) 09301020

Q. Define Identity Matrix. 09301021

Q.1 From the following matrices, identify unit matrices, row matrices, column matrices and null matrices. 09301022

Q.2 From the following matrices, identify

(a) Square matrices 09301023

(b) Rectangular matrices 09301024

(d) Column matrices 09301026

(e) Identity matrices 09301027

(f) Null matrices

3. From the following matrices, identify diagonal, scalar and unit (identity) matrices. 09301029

4. Find negative of matrices A, B, C, D and E when: 09301030

5. Find the transpose of each of the following matrices: 09301031

Q.6 Verify that if then 09301032

(i) (At)t = A 09301033

(ii) (Bt)t = B 09301034

Q. What is condition for addition and subtraction for matrices? 09301035

Q. How can we add matrices? 09301036

Q. How can we subtract a matrix from other matrix? 09301037

Q. How can we multiply a matrix by a real number? 09301038

Q. If A then find

Q. Define Commutative Law of addition of Matrices. 09301040

Q. If and

Then verify that A + B = B + A 09301041

Q. Define Associative Law of addition.

09301042

Q. If and 09301043

Then verify that (A+B) + C = A+ (B+C)

Q. Define additive identity of a matrix.

09301044

Q. If and 09301045

then show that O is additive identity of A

Q. Define additive inverse of a matrix.

09301046

Q. If the find its additive inverse and verify. 09301047

Q.1 Which of the following matrices are conformable for addition? 09301048

09301049 09301050

09301051 , 09301052

, 09301053 09301054

Q.2 Find the additive inverse of following matrices.

(Board 2013) 09301055

09301056

09301057

09301058

Q.3 If

, then find,

(i) 09301059 (ii) 09301060

(iii) C+ 09301061

(iv) 09301062 (v) 2A 09301063

(vi) (1)B 09301064 (vii) (2)C 09301065

(viii) 3D 09301066 (ix) 3C 09301067

Q.4 Perform the indicated operations and simplify the following.

(i) 09301068

(ii) 09301069

(iii) 09301070

(iv) 09301071

(v) 09301072

(vi) 09301073

Q.5 For the matrices and verify the following rules. 09301074

(i) 09301075

(ii) 09301076

(iii) 09301077

(iv) 09301078

(v) 09301079

(vi) 09301080

(vii) 09301081

(viii) 09301082

(ix) 09301083

(x) 09301084

Q.6 If and , find

(i) 3A2B 09301085 (ii) 2At  3Bt. 09301086

Q. 7 If then find a and b. 09301087

Q.8 If then verify that 09301088

(i) (A+B) t = At + Bt 09301089

(ii) 09301090

(iii) is symmetric 09301091

(iv) is skew symmetric 09301092

(v) is symmetric 09301093

(vi) is skew symmetric 09301094

Q. Write condition for multiplication of matrices. 09301095

Examples

(i) If and then find AB 09301096

(ii)If and then find AB 09301097

Q. Define associative law of multiplication.

09301098

Q. If

Then verified that (AB)C=A(BC). 09301099

Q. Define distributive laws of multiplication over addition. 09301100

Q. If and then verify that 09301101

Q. Define distributive laws of multiplic-ation over subtraction. 09301102

Q. If and then verify that

09301103

Q. If and

then verify that

(A – B) C = AC – BC 09301104

Q. If and then

Show that AB BA 09301105

Q. In general which property does not exist in matrices? 09301106

Q. If and then show that AB = BA 09301107

Q. Define multiplicative identity of a matrix. 09301108

Q. If , , then show that AB = A = BA 09301109

Q. Define Law of Transpose of product of matrices. 09301110

Q. If and

then show that (AB)t = Bt At 09301111

Q.1 Which of the following product of matrices is conformable for multiplication?

09301112

(i)

09301113

09301114

09301115

09301116

Q.2 If , , find

09301117

(i) AB 09301118

(ii) BA (if possible). 09301119

Q.3 Find the following products.

09301120

09301121

09301122

09301123

09301124

Q.4 Multiply the following matrices.

09301125

09301126

09301127

(Board 2015)

09301128

09301129

Q.5 Let , and . Verify whether 09301130

(i) AB = BA. 09301131

(ii) A(BC) = (AB)C 09301132

(iii) A(B+C)=AB+AC 09301133

(iv) A(BC)=ABAC 09301134

Q.6 For the matrices. 09301135

, ,

Verify that

(i)(AB)t= BtAt 09301136 (ii)(BC)t=CtBt. 09301137

Q. Define determinant of a matrix.

09301138

Q. If . Then find |B| 09301139

Q. If , then find det M 09301140

Q. Define singular and non-singular matrix. (Board 2014) 09301141

Q. How can we find adjoint of a matrix?

09301142

Q. What do you know about multiplicative inverse of a matrix? 09301143

Q. If be a square matrix. Then find . 09301144

Q. If then find A–1 and check.

09301145

Q. Define law of inverse of product of matrices. 09301146

Q. If and

Then verify that (AB)1 = B1 A1 09301147

Q.1 Find the determinant of the following matrices. 09301148

09301149

(iii) (Board 2015) 09301150

(iv) 09301151

Q.2 Find which of the following matrices are singular or non-singular? 09301152

09301153

09301154

09301155

Q.3 Find the multiplicative inverse

(if it exists) of each. 09301156

09301157

09301158

09301159

Q.4 If and , then

(i) A(Adj A) = (Adj A) A = (det A )I 09301160

(ii) BB–1 = I = B1 B 09301161

Q.5 Determine whether the given matrices are multiplicative inverses of each other.

and 09301162

Q.6 If , , , then verify that

(i) (AB)1 = B1 A1 09301163

(ii) (DA)1 = A1 D1 09301164

a. Q. Define System of Simultaneous Linear Equation.

b. Q. Write different methods to solve the simultaneous linear equation by using matrices. 09301166

Q. By using matrices inverse method find the solution of system of linear equations. 09301167

Example 1 09301168

Solve the following system by using matrix inversion method.

4x  2y = 8

3x + y = 4

Example 2 09301169

Solve the following system of linear equations by using Cramer’s rule.

3x  2y = 1, 2x +3y = 2

Example 3 09301170

The length of a rectangle is 6 cm less than three times its width. The perimeter of the rectangle is 140 cm. Find the dimensions of the rectangle.

(by using matrix inversion method)

Exercise 1.6

Q.1 Use matrices, if possible, to solve the following systems of linear equations by:

(i) the matrix inverse method 09301171

(ii) the Cramer’s rule. 09301172

Solution by Matrix inverse method

(i) (Board 2013) 09301173

(ii) 09301174

(iii) 09301175

(iv) (Board 2014) 09301176

(v) 09301177

(vi) 09301178

(vii) 09301179

(viii) 09301180

Solution by Cramer’s rule

(i) (Board 2013, 14) 09301181

(ii) 09301182

(iii) (Board 2015) 09301183

(iv) 09301184

(v) 09301185

(vi) (Board 2014) 09301186

(vii) (Board 2015) 09301187

(viii) 9301188

Solve the following word problems by using:

(i) Matrix inversion method 09301189

(ii) Cramer’s Rule 09301190

Q.2 The length of a rectangle is 4 times its width. The perimeter of the rectangle is 150cm. Find dimensions of the rectangle.

Let width of rectangle = x. 09301191

and length of rectangle = y

Q.3 Two sides of a rectangle differ by 3.5cm. Find the dimensions of the rectangle if its perimeter is 67cm. 09301192

(i) By Matrix inversion method 09301193

x – y = 3.5

x + y = 33.5

(ii) By Cramer’s Rule: 09301194

x – y = 3.5 ……………… (i)

x + y = 33.5 ……………...(ii)

Q.4 The third angle of an isosceles triangle is 16o less than the sum of the two equal angles. Find three angles of the triangle. 09301195

Let third angle of triangle = y

Two equal angles of triangle = x

(i) By Matrix Inversion Method: 09301196

In matrices form

Q.5 One acute angle of a right triangle is 12o more than twice the other acute angle. Find the acute angles of the right triangle. 09301198

Let acute angles of right angled triangle are x and y

According to given condition

We know that

…………(ii)

Q.6 Two cars that are 600 km apart are moving towards each other. Their speeds differ by 6km per hour and the cars are 123 km apart after hours. Find the speed of each car. 09301200

OBJECTIVE

Q.1 Select the correct answer in each of the following.

1. The order of matrix [2 1] is … 09301201

(a) 2-by-1 (b) 1-by-2 (Board 2014)

2. is called ……. Matrix. 09301202

(a) zero (b) unit (Board 2014)

3. Which is order of a square matrix?09301203

(a) 2-by-2 (b) 1-by-2

4. Which is order of a rectangular matrix? 09301204

(a) 2-by-2 (b) 4-by-4

5. Order of transpose of is …

09301205

(a) 3-by-2 (b) 2-by-3 (Board 2014)

6. Adjoint of is ……… 09301206

(a) (b)

7. If , then x is equal to 09301207

(a) 9 (b) –6 (Board 2013)

8. Product of [x y] is ……..

(Board 2013, 15) 09301208

(a)

(b)

(c)

(d)

9. 09301209

(a) (b)

10. The idea of a matrices was given by:__

(a) Arthur Cayley 09301210

(b) Leonard Euler

(d) John Napier

11. The matrix M=[ 2 – 1 7 ] is a--matrix.

(a) Row (b) Column 09301211

12. The matrix is a ____ matrix.

(a) Row (b) Column 09301212

13. The matrix is a _______ matrix.

(a) Rectangular (b) Square 09301213

14. The matrix is a __ matrix.

(a) Rectangular (b) Square 09301214

15. If A is a matrix then its transpose is denoted by: 09301215

(a) A-1 (b) At

16. If then A = ______ 09301216

(a) (b)

17. A square matrix is symmetric if ___

(a) At = A (b) A-1 = A 09301217

18. A square matrix is skew-symmetric if:

(a) At = A (b) A-1 = A 09301218

19. The matrix is a _ matrix.

(a) Diagonal (b) Scalar 09301219

20. The matrix is a__matrix.

(a) Diagonal (b) Scalar 09301220

21. The matrix is a _ matrix.

(a) Diagonal (b) Identity 09301221

22. The scalar matrix and identity matrix are ____ matrices. 09301222

(a) Diagonal (b) Rectangular

23.Every diagonal matrix is not a _ matrix.

(a) Square 09301223

(b) non-singular

(d) None

24. If A, B are two matrices and At, Bt are their respective transpose, then: 09301224

(a) (AB)t = Bt At (b) (AB)t = At Bt

25. If then the det. A is: 09301225

(a) ad – bc (b) bc – ad

26. A square matrix A is called singular if

(a) |A|  0 (b) |A| = 0 09301226

27. A square matrix A is called

non-singular if: 09301227

(a) |A| = 0 (b) A = 0

28. Inverse of identity matrix is _matrix.

(a) Identity (b) Zero 09301228

29. AA1 = A1 A = _____ 09301229

(a) Identity matrix

(b) Rectangular matrix

(d) Singular matrix

30. (AB)1 = ____ 09301230

(a) A1 B1 (b) B1 A 1

31. Additive inverse of is ____

(a) (b) 09301231

32. Which of the following is commutative property of addition of matrices?

(a) 09301232

(b)

(c)

(d)

33.Which of the following is associative property of multiplication of matrices?

09301233

(a) (b)

34. Which of the following is commutative property of multiplication of matrices?

(a)

(b) 09301234

(c)

(d)

35. Which of the following is associative property of addition of matrices?

(a) 09301235

(b)

(c)

(d)

36. Which of the following does not exist in matrices in general? 09301236

(a) (b)

(c)

(d)

37. Which of the following is true for matrices in general? 09301237

(a)

(b)

(c)

(d)

38. Which of the following is distributive property of multiplication over addition?

(a) 09301238

(b)

(c)

(d)

39. Which of the following is singular matrix? 09301239

(a) (b)

Q.2 Complete the following:

i. is called……matrix. 09301240

(Null / Zero)

ii. is called…….matrix. 09301241

(Identity /Unit)

iii. Additive inverse of is… 09301242

iv. In matrix multiplication, in general,

AB …… BA. 09301243

v. Matrix A + B may be found if order of A and B is …… (Same) 09301244

vi. A matrix is called …. matrix if number of rows and columns are equal. 09301245 (Square)

Q.3 If , then find a and b. (Board 2013, 14) 09301246

Q.4 If , , then find the following. 09301247

(i) 2A + 3B

(ii) 3A +2B

09301248

(iii) – 3 (A+2B) 09301249

(iv) 09301250

Q.5 Find the value of X, if . 09301251

Q.6 If , , then prove that AB  BA 09301252

Q.7 If and , then verify that 09301253

(i) (AB)t = Bt At 09301254

(ii) (AB)1 = B1 A1 09301255

Q8. If , then find x.

09301256

Q9. Find the product .

09301257

Q10. If find X. 09301258

UNIT 2

Q. Define Natural Numbers. 09302001

Q. Define Whole Numbers. (Board 2013) 09302002

Q. Define the set of Integers? 09302003

Q. Define Rational Numbers. 09302004

Q. Define Irrational Numbers. 09302005

Q. Define Set of Real Numbers. 09302006

Q. What is one to one correspondence?

09302007

Q. What is convention of number line?

09302008

Q. Define Terminating Decimal Fractions.

09302009

Q. Define Recurring and Non-terminating Decimal Fractions. 09302010

Q. What is decimal representation for Irrational Numbers? 09302011

Q. How can we represent a rational number on number line? 09302012

09302013

Express the following decimals in the form , where p, q  Z and q  0

(a) = 0.333 …. (b) = 0.232323

09302014

Represent the following numbers on the number line.

(i)

(ii) (Board 2015)

(iii)

Q. How can we represent an Irrational Number on a Number Line? 09302015

3. Exercise 2.1:

Q.1 Identify which of the following are rational and irrational numbers. 09302016

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Q.2 Convert the following fractions into decimal fraction. 09302017

(i) 09302018

(ii) 09302019

(iii) 09302020

(iv) 09302021

(v) 09302022

(vi) 09302023

Q.3 Which of the following statements are true and which are false? 09302024

(i) is an irrational number. 09302025

(ii) is an irrational number. 09302026

(iii) is a terminating fraction. 09302027

(iv) is a terminating fraction. 09302028

(v) is a recurring fraction. (09302029

Q.4 Represent the following numbers on the number line. 09302030

(i) 09302031

(ii) 9302032

(iii) 9302033

iv) 09302034

(v) 09302035

(vi) 09302036

Q.5 Give a rational number between and .

Q6. Express the following recurring decimals as the rational number where p, q are integers and q  0

(i) (Board 2013) 09302038

(ii) 09302039

(iii) 09302040

4. Exercise 2.2:

Q.1 Identify the property used in the following. 09302041

(i) 09302042

(ii) (ab)c = a(bc) 09302043

(iii) 71=7 09302044

(iv) x > y or x=y or x<y 0930204

(v) ab = ba 09302046

(vi) 09302047

(vii) 09302048

(viii) 09302049

(ix) 09302050

Q.2 Fill in the following blanks by stating the properties of real numbers used.

09302051

Q.3 Give the name of property used in the following.

(i) 09302052

(ii) 09302053

(iii) 09302054

(iv) 09302055

(v)

5. Exercise 2.3:

Q1. Write each radical expression in exponential notation and each exponential expression in radical notation. Do not simplify.

09302057

(i) 09302058

(ii) 09302059

(iii) 09302060

(iv) 09302061

Q2. Tell whether the following statements are true or false? 09302062

(i) 09302063

(ii) 09302064

(iii) 09302065

(iv) 09302066

Q3. Simplify the following radical expressions. 09302067

(i) 09302068

(ii) 09302069

(iii) 09302070

(iv) (Board 2013) 9302071

Q. Define Base and Exponent. 09302072

Q. Write laws of exponents. 09302073

Example 1: 09302074

Use rules of exponents to simplify each expression and write the answer in terms of positive exponents.

(i)

Example 2: 09302075

Simplify the following by using laws of indices:

(i) (Board 2015)

(ii) (Board 2014)

Exercise 2.4

Q.1 Use laws of exponents to simplify 09302076

(i) (Board 2013) 09302077

(ii) 09302078

(iii) 09302079

(iv) (Board 2014)

09302080

Q.2 Show that 09302081

Q.3 Simplify (Board 2014) 09302082

(i) 09302083

(ii) 09302084

(iii) 09302085

iv) (Board 2013, 14) 09302086

Q. Define a Complex Number. 09302087

Q. Define Pure Imaginary Number. 09302088

Q. Define Set of Complex Numbers. 09302089

Q. Define Conjugate of a Complex Number. 09302090

Q. What do you mean by equality of Complex Number? 09302091

Q. If 2x + y2i = 4 + 9i then find the value of x and y. 09302092

Q. Which properties of Real Numbers are valid for Complex Number? 09302093

8. Exercise 2.5

b. Q.1 Evaluate 09302094

(i) 09302095

(ii) 09302096

(iii) 09302097

(iv) 09302098

(v) 09302099

(vi) (Board 2014) 09302100

Q.2 Write the conjugate of the following numbers. 09302101

(i) 09302102

(ii) 09302103

(iii) 09302104

(iv) 09302105

(v) 09302106

(vi) 09302107

Q3. Write the real and imaginary part of the following numbers. 09302108

(i) 09302109

(ii) 09302110

(iii) (Board 2014) 09302111

(iv) 09302112

(v) 09302113

(vi) (Board 2015) 09302114

Q4. Find the value of and if 09302115

Q. How can we add two complex numbers?

09302116

Q. How can we multiply two complex numbers? 09302117

Q. How can we multiply a complex number with scalar? 09302118

Q. How can we get difference of two complex numbers? 9302119

Q. Explain division of complex numbers?

Example 1: 09302121

Separate the real and imaginary parts of

Example 2 09302122

Express in the standard form a+ bi.

Example 3: 09302123

Express in the standard form a + bi.

Example 4 09302124

Solve (3  4i)(x + yi) = 1 + 0 i for real numbers x and y, where i =

10. Exercise 2.6:

Q.1 Identify the following statements as true or false. 09302125

(i) 09302126

(ii) 09302127

(iii) 09302128

(iv) Complex conjugate of

is (-1+6i) 09302129

(v) Difference of a complex number and its conjugate is a real number.

09302130

(vi) If then and . 09302131

(vii) Product of a complex number and its conjugate is always a non-negative real number. 09302132

Q.2 Express each complex number in the standard form where ‘ ’ and ‘ ’ are real numbers. 09302133

(i) 09302134

(ii) 09302135

(iii) 09302136

(iv) 09302137

Q.3 Simplify and write your answer in the form 09302138

(i) 09302139

(ii) 09302140

(iii) 09302141

(iv) 09302142

Q.4 Simplify and write your answer in the form of 09302143

(i) 09302144

(ii) 09302145

(iii) 09302146

(iv) 09302147

(v) (Board 2013) 09302148

(vi) 09302149

Q.5 Calculate (a) (b) (c) (d) for each of the following. 09302150

(i) 09302151

(ii) (G.B 2014) 09302152

(iii) 09302153

(iv) 09302154

Q.6 If and , show that: 09302155

(i) 09302156

(ii) 09302157

(iii) 09302158

(iv) , where 09302159

(v) is the real part of 09302160

(vi) is the imaginary part of z. 09302161

Q.7 Solve the following equation for real and (Board 2014) 09302162

(i) (Board 2013) 09302163

(ii) 09302164

(iii) 09302165

11. Review Exercise 2 OBJECTIVE

Q.1 Multiple Choice Questions. Choose the correct answer.

1. 09302166

(a) (b)

2. Write in exponential form 09302167

(a) x (b) x7 (Board 2013)

3. Write with radical sign…. 09302168

(a) (b) (Board 2014)

4. In the radicand is 09302169

(a) 3 (b)

5. (Board 2014) 09302170

(a) (b)

6. The conjugate of 5 + 4i is _____ 09302171

(a) – 5 + 4i (b) – 5 – 4i

7. The value of i9 is ____ 09302172

(a) 1 (b) –1

8. Every real number is ____ 09302173

(a) A positive integer

(b) A rational number

(d) A complex number

9. Real part of 2ab is ____ 09302174

(a) 2ab (b) 2ab (Board 2014)

10. Imaginary part of (3 +2) is_ 09302175

(a) 2 (b) 2 (Board 2014)

11. Which of the following sets have the closure property w.r.t. addition_____ 09302176

(a) {0} (b) {0, 1}

12. Name the property of real numbers used in 09302177

(a) Additive identity

(b) Additive Inverse

(d) Multiplicative Inverse

13. If x, y, z R z < 0 then 09302178

(a) x z < y z (b) x z > y z

14. If a, then only one of a = b or a < b or a > b holds is called… 09302179

(a) Trichotomy property

(b) Transitive property

(d) Multiplicative property

15. A non-terminating, non-recurring decimal represents: 09302180

(a) A natural number

(b) A rational number

(d) A prime number

16. The union of the set of rational numbers and irrational numbers is known as set of ___ 09302181

(a) Rational number (b) Irrational

17. For each prime number A, is an (a) Irrational (b) Rational 09302182

18. Square roots of all positive non-square integers are ____ 09302183

(a) Irrational (b) Rational

19.  is an _____ number. 09302184

(a) Irrational (b) Rational

20.  then a < b and b < c  a < c is ____ property. 09302185

(a) Transitive

(b) Trichotomy

(d) Multiplicative

21. Name the property of real numbers used in x > y or x = y or x < y. 09302186

(a) Trichotomy (b) Transitive

22. Name the property of real numbers used in  + () = 0. 09302187

(a) Additive inverse

(b) Multiplicative inverse

(d) Multiplicative identity

23. is a ___ number. 09302188

(a) Rational (b) Irrational

24. 09302189

(a) (b)

25. 09302190

(a) (b) (8)5

26. The value of i 10 is: 09302191

(a) 1 (b) 1

27. The solution set of x2 +1 = 0 is: 09302192

(a) {i, i} (b) {i, i}

28. The conjugate of 2 + 3i is ___ 09302193

(a) 2  3i (b) 2 3i

29. Real part of is: 09302194

(a) 1 (b)

30. Imaginary part of is 09302195

(a) 1 (b)

31. Product of a complex number and its conjugate is always a non-negative____ number. 09302196

(a) Real (b) Irrational

32. is a/an………. number 09302197

(a) irrational (b) rational

33. Q and Qare ______sets 09302198

(a) disjoint set (b) over lapping

(d) Q is equal to Q

34. Additive identity of real number is 09302199

(a) 0 (b) –a

35. Additive inverse of (a) is…….. 09302200

(a) –a (b) a

36. The value of i (iota) is_______ 09302201

(a) (b) –1

37. In –2+3i, 3 is called _______ 09302202

(a) imaginary part (b) real part

38. In , the symbol is called…….

09302203

(a) radical sign (b) index

39. In ‘n’ is called…….. 09302204

(a) base (b) radical sign

40. The set of natural numbers is…… 09302205

(a) {0,1,2,3….} (b) {2,4,6….}

41. The set of whole numbers is….. 09302206

(a) {1,3,5….} (b) {01, 2,….}

42.  , e, , and are called…

(a) irrational numbers 09302207

(b) rational number

43. 09302208

(a) (b) Q

44. “For all” is represented by symbol 09302209

(a) (b)

45. is usually written as 09302210

(a) a1/4 (b) a

46. The roots of y2+1 = 0 are 09302211

(a) {i, –1} (b) {1, –i}

47. If Z = -1-i then is equal to……. 09302212

(a) 1–i (b) 1+ i

48. The real part of 3i +2 is…….. 09302213

(a) –2 (b) 2

49. A pure imaginary number is the _____ of a negative real number 09302214

(a) square root (b) square

50. Number like , etc. are called

(a) real numbers 09302215

(b) pure imaginary numbers

(d) irrational numbers

51. If (a-1)-(b+3)i = 5+8i then 09302216

(a) a = 6, b = –11 (b) a= 6, b= 11

52. The conjugate of real is the _____ real number. 09302217

(a) negative of (b) same

53. If then 09302218

(a)

(b)

(c)

(d)

54. form of is _______. 09302219

(a) (b)

55. On the number line lies between ___.

(a) 1 and 2 (b) 2 and 3 09302220

56. form of is ________. 09302221

(a) (b)

Q.2 True or False? Identify

(i) Division is not associative operation. 09302222

(ii) Every whole number is a natural number. 09302223

(iii) Multiplicative inverse of 0.02 is 50. 09302224

(iv) is a rational number. 09302225

(v) Every integer is a rational number. 09302226

(vi) Subtraction is a commutative operation. 09302227

(vii) Every real number is a rational number. 09302228

(viii) Decimal representation of a rational number is either terminating or recurring 09302229

(ix.) 1. = 1 + 09302230

Q.3 Simplify: (i) 09302231

(ii) 09302232

(iii) 09302233

(iv) 09302234

Q.4 Simplify: (Board 2015)

09302235

Q.5 Simplify: 09302236

Q.6 Simplify:

09302237

Q.7 Simplify: 009302238

Q8. Simplify: 09302239

Q9. Simplify: 09302240

Q10. If then find the value of a and b. 09302241

UNIT 3

Example 1: 09303001

Write each of the following ordinary numbers in scientific notation

(i) 30600 (ii) 0.000058 09303003

Example 2 09303004

Change each of the following numbers from scientific notation to ordinary notation.

(i) 6.35 106 09303005 (ii)7.61 104 09303006

12.

13.

14. Exercise 3.1:

Q.1 Express each of the following numbers in scientific notation. 09303007

i) 5700 09303008

ii) 49,800,000 09303009

iii) 96,000,000 09303010

iv) 416.9 09303011

v) 83,000 09303012

vi) 0.00643 09303013

vii) 0.0074 09303014

viii) 60,000,000 09303015

ix) 0.00000000395 09303016

x) (Board 2013) 09303017

Q.2 Express the following numbers in ordinary notation. 09303018

i) 09303019

ii) 09303020

iii) 09303021

iv) 09303022

Q. Define Logarithm of a Real Number.

09303023

Example 3: 09303024

Find , i.e., find log of 2 to the base 4.

Q. Define Common or Brigg’s Logarithm.

09303025

Q. Define Natural Logarithm. 09303026

Q. Define characteristic of Logarithm.

09303027

Q. Define characteristic of Logarithm of a number > 1. 09303028

Q. Define characteristic of Logarithm of a number < 1. 09303030

Example : 09303031

Write the characteristic of the log of following numbers by expressing them in scientific notation and noting the power of 10. 0.872, 0.02 , 0.00345

Q. Define Mantissa. 09303032

Example 1: 09303033

Find the mantissa of the logarithm of 43.254

Example 2: 09303034

Find the mantissa of the logarithm of 0.002347

Example 3: 09303035

Find (i) log 278.23 (ii) log 0.07058

Q. Define Antilogarithm. 09303036

Example: 09303037

Find the numbers whose logarithms are

(i) 1.3247 09303038 (ii) .1324 09303039

Exercise 3.2:

Q.1 Find the common logarithm of the following numbers. 09303040

i) 232.92

ii) 29.326 09303041

iii) 0.00032 09303042

iv) 0.3206 09303043

Q.2 If log 31.09 = 1.4926, find the values of following: 09303044

i) log 3.109 09303045

ii) log 310.9 09303046

iii) log 0.003109 09303047

iv) log 0.3109 09303048

Q.3 Find the numbers whose common logarithms are: 09303049

i) 3.5621 09303050

ii) 09303051

Q.4 What replacement for the unknown in each of following will make the statement true? 09303052

i) 09303053

ii) 09303054

iii) 09303055

iv) 09303056

1. Q.5 Evaluate

i) 09303058

ii) log 512 to the base 09303059

Q.6 Evaluate the value of ‘x’ from the following statements. 09303060

i) 09303061

ii) 09303062

iii) 09303063

iv) (Board 2014, 15) 09303064

v) (Board 2013,14) 09303065

Q. Prove that loga(mn) = logam + logan

09303067

Example 1 09303068

Evaluate 291.3 42.36

Example 2 : 09303069

Evaluate 0.2913 0.004236.

Q. Prove that

09303070

Example 1: 09303071

Evaluate

Example 2: 09303072

Evaluate

Q. Prove that loga(mn) = nlogam 09303073

Example 1: 09303074

Evaluate

Q. Prove that 09303075

or =

Q. How can we convert natural log into common log. 09303076

Example:

Calculate 09303077

Exercise 3.3:

Q.1 Write the following into sum or difference. 09303078

i) 09303079

ii) 09303080

iii) 09303081

iv) (Board 2013) 09303082

v) (Board 2015) 09303083

vi) 09303084

a. Q.2 Express 09303085

as a single logarithm

Q.3 Write the following in the form of a single logarithm. 09303086

i) log 21 + log 5 (Board 2013) 09303087

ii) log 25 – 2 log3 (Board 2015) 09303088

iii) (Board 2014) 09303089

iv) 09303090

Q.4 Calculate the following: 09303091

i) 09303092

ii) 09303093

Q.5 If

, then find the values of the following. 09303094

i) log 32 (Board 2014) 09303095

ii) log 24 09303096

iii) 09303097

iv) 09303098

v) (Board 2015) 09303099

Applications of logarithm

Example 1 : 09303100

Show that

Example 2: 09303101

Evaluate:

Example 3 09303102

Given A = Aoekd. If k = 2, what should be the value of d to make ?

Exercise 3.4:

Q.1 Use log tables to find the values of 09303103

i) (Board 2013) 09303104

ii) 09303105

iii) 09303106

iv) (Board 2013,14) 09303107

v) 09303108

vi) (Board 2015) 09303109

vii) 09303110

viii) (Board 2015) 09303111

Q.2 A gas is expanding according to the law . Find C when P=80, V=3.1 and . 09303112

Q.3 The formula applies to the demand of a product, where ‘q’ is the number of units and p is the price of one unit. How many units will be demanded if the price is Rs. 18.00? 09303113

Q.4 If

09303114

Q.5 If , find when and 09303115

Review Exercise 3 OBJECTIVE

Q.1 Multiple Choice Questions. Choose the correct answer.

1. If ax = n, then _____ 09303116

(a) a = (b) x = logn a

2. The relation of y = logz x implies 09303117

(a) (b) (Board 2014)

3. The logarithm of unity to any base is

(a) 1 (b) 10 (Board 2014,15) 09303118

4. The logarithm of any number to itself as base is___ 09303119

(a) 1 (b) 0

5. log e = ____ where e 2. 718 09303120

(a) 0 (b) 0.4343 (Board 2015)

6. The value of log is ___ 09303121

(a) log p log q (b)

7. logp – logq is same as: 09303122

(Board 2014, 15)

(a) (b)

8. log can be written as 09303123

(a) (log m)n (b) m log n

9. can be written as___

(a) (b) 09303124

10. Logy x will be equal to___ 09303125

(a) (b)

11. For common logarithm, the base is_

(a) 2 (b) 10 09303126

12. For natural logarithm, the base is__

(a) 10 (b) e 09303127

13. The integral part of the common logarithm of a number is called the_

(a) Characteristic (b) Mantissa 09303128

14. The decimal part of the common logarithm of a number is called

the _____: 09303129

(a) Characteristic (b) Mantissa

15. If x = log y, then y is called the _______ of x. 09303130

(a) Antilogarithm

(b) Logarithm

(d) None

16. If the characteristic of the logarithm of a number is , that number will have zero (s) immediately after the decimal point. 09303131

(a) One (b) Two

17. If the characteristic of the logarithm of a number is 1, that number will have ____ digits in its integral part 09303132

(a) 2 (b) 3

18. The value of x in log3 x = 5 is____

(a) 243 (b) 143 09303133

19. The value of x in log x = 2.4543 is

09303134

(a) 284.6 (b) 1.521

20. The number corresponding to a given logarithm is known as ___ 09303135

(a) Logarithm (b)Antilogarithm

21. 30600 in scientific notation is __ 09303136

(a) 3.06 x 104 (b) 3.006 x 104

22. 6.35 x 106 in ordinary notation is___

(a) 6350000 (b) 635000 09303137

23. A number written in the form

a x 10n, where and n is an integer is called ____ 09303138

(a) Scientific notation

(b) Ordinary notation

(d) None

24. 09303139

(a) log1 (b) log n

25. 09303140

(a) 0 (b) 1

26. 09303141

(a) 0 (b) 1

27. The characteristic of is ________. 09303142

(a) 0 (b) 2

28. The characteristic of is:

(a) 0 (b) 3 09303143

29. If , then what is the mantissa of ? 09303144

(a) 0.3705 (b) – 0.6294

30. Common logarithm is also known as ______ logarithm. 09303145

(a) natural (b) simple

31. is same as: 09303146

(a)

(b)

(c)

(d)

32. John Napier prepared the logarithms tables to the base _______. 09303147

(a) 0 (b) 1

33. in common logarithm is written as _________. 09303148

(a) (b)

34. in single logarithm can be written as _____. 09303149

(a) (b)

35. in single logarithm is written as: 09303150

(a) (b)

36. 09303151

(a) 2.3026 (b) 0.4343

37. If then x is: 09303152

(a) 25 (b) 32

38. If , then x = ____ 09303153

(a) 2 (b) 9

Q.2 Complete the following: 09303154

(i) For common logarithm, the base is ……………. 09303155

(ii) The integral part of the common logarithm of a number is called the…. 09303156

(iii) The decimal part of the common logarithm of a number is called the ……… 09303157

(iv) If x = log y, then y is called the ………… of x. 09303158

(v) If the characteristic of the logarithm of a number is , that number will have …….. zero (s) immediately after the decimal point. 09303159

(vi) If the characteristic of the logarithm of a number is 1, that number will have ……… digits in its integral part. 09303160

Q.3 Find the value of ‘x’ in the following.

i) log3 x = 5 09303161

ii) log4 256 = x 09303162

iii) log625 (Board 2014) 09303163

iv) 09303164

Q.4 Find the value of ‘x’ in the following.

i) log x = 2.4543 09303165

ii) log x = 0.1821 09303166

iii) log x = 0.0044 (Board 2014) 09303167

iv) log x = 09303168

Q.5 If log2 = 0.3010, log3 = 0.4771 and

log 5 = 0.6990, then find the values of the following. 09303169

i) log45 09303170

ii) 09303171

iii) log 0.048 09303172

Q.6 Simplify the following:

i) 09303173

ii) 09303174

iii) (Board 2014) 09303175

UNIT 4

Q. Define the Algebraic Expressions. 09304001

Q. Define Polynomials. 09304002

Q. Define Degree of Polynomials. 09304003

Q. Define leading coefficient. 09304004

Q. Define Rational Expression. 09304005

_____________________________________________________________________

Example 1: 09304006

Simplify(i)

(ii)

Example 2: 09304007

Find the product

Example:

Simplify

Q. What do you mean by Value of Algebraic Expression? 09304008

Example: 09304009

Evaluate if x = 4 and y=9

Exercise 4.1:

Q.1 Identify whether the following algebraic expression are polynomials (yes or no). 09304010

(i) 09304011

(ii) 09304012

(iii) 09304013

(iv) 09304014

Q.2 State whether each of the following expression is a rational expression or not. 09304015

(i) 09304016

(ii) 09304017

(iii) 09304018

(iv) 09304019

Q.3 Reduce the following rational expression to the lowest forms. 09304020

(i) 09304021

(ii) 09304022

(iii) 09304023

(iv) 09304024

(v) 09304025

(vi) (Board 2013) 09304026

(vii) 09304027

Q.4 Evaluate (a) for 09304028

(i) x = 3,y = 1, z = 2. 09304029

(ii) x = -1, y = -9, z = 4 09304030

Q.5 Perform the indicated operation and simplify: 09304032

(i) 09304033

(ii) 09304034

(iii) 09304035

(iv) 09304036

(v) 09304037

(vi) 09304038

Q.6 Perform the indicated operation and simplify: 09304039

(i) 09304040

(ii) 09304041

(iii) 09304042

(iv) 09304043

(v) 09304044

Example: 09304045

If a + b = 7 and a  b = 3, then find the value of (a) (b)

Example 1: 09304046

If and then find the value of .

Example 2: 09304047

If and then find the value of .

Example 3: 09304048

If and then find the value of

Example 1: 09304049

If 2x and , then find the value of .

Example 2: 09304050

If , then find the value of

Example 3: 09304051

If then find

Example 1: 09304052

Factorize

Example 2: 09304053

Factorize

Example 3: 09304054

Factorize

Example 4: 09304055

Find the product

Example 5: 09304056

Find the continued product of

Exercise 4.2:

Q.1(i) If a + b = 10 and a  b = 6 then find the value of a2 + b2. (Board 2013) 09304057

(ii) If a + b = 5, a  b = then find the value of ab. 09304058

Q.2 If a2 + b2 + c2 = 45 and a + b + c = 1 find the value of ab + bc + ca. 09304059

Q3. If m+n+p = 10, mn + np + pm = 27 find the value of m2+n2+p2. 09304060

Q.4 If x2 +y2 + z2 = 78 and xy+yz+zx=59 find the value of x + y + z. 09304061

Q5. If x + y + z = 12 and x2 +y2+z2=64 find the value of xy+yz+zx. 09304062

Q.6 If x + y = 7 and xy = 12 then find the value of x3 + y3. 09304063

Q.7 If 3x + 4y = 11 and xy = 12 then find the value of 27x3 + 64 y3. 09304064

Q8. If x – y = 4 and xy = 21 then find the value of x3 – y3. 09304065

Q.9 If 5x  6y = 13 and xy = 6 then find the value of 125x3 – 216y3. 09304066

Q.10If then find the value of .

09304067

Q11. If , then find the value of

Q.12 If then find the value of 09304069

Q.13 If , then find the value of . 09304070

Q.14 Factorize (i) x3 – y3 – x +y 09304071

(i) x3 – y3 – x + y 09304072

(ii) 09304073

Q.15 Find products, using formulae 09304074

(i) (x2+y2)(x4–x2y2+y4) 09304075

(ii) 09304076

(iii)

09304077

(iv).(2x2 –1)(2x2+1)(4x4 + 2x2+1) (4x4 – 2x2 + 1)

09304078

Q. Define Surd. 09304079

Example: 09304080

Simplify by combining similar terms.

(i) (ii)

Multiplication and Division of Surds

Example: 09304081

Simplify and express the answer in the simplest form.

(i) (ii)

Exercise 4.3:

Q.1 Express each of the following surd in the simplest form. 09304082

(i) 09304083

(ii) 09304084

(iii) 09304085

(iv) 09304086

Q.2 Simplify 09304087

(i) 09304088

(ii) (Board 2013) 09304089

(ii) 09304090

(iii) 09304091

(iv) 09304092

Q.3 Simplify by combining similar terms: 09304093

(i)

(ii) 09304094

(iii) 09304095

(iv) 09304096

Q.4 Simplify: 09304097

(i) 09304098

(ii) 09304099

(iii) 09304100

(Board 2014)

09304101

(iv)

09304102

Define Monomial Surd. 09304103

Q. Define Binomial Surd. 09304104

Q. Define Conjugate of Surds. 09304105

Q. What is Rationalization of Surds?

09304106

Example 1:

Rationalize the denominator 09304107

Example 2: 09304108

Rationalize the denominator

Example 3:

Simplify 09304109

Example 4:

Find rational numbers x and y such that 09304110

Example 5:

If , then evaluate

(i) and (ii) 09304111

Exercise 4.4:

Q.1 Rationalize the denominator

(i) 09304112

(ii) 09304113

(iii) 09304114

(iv) 09304115

(v) 09304116

(vi) 09304117

(vii) 09304118

(viii) 09304119

Q.2 Find conjugate of : 09304120

(i) 09304121

(ii) 09304122

(iii) 09304123

(iv) 09304124

(v) 09304125

(vi) 09304126

(vii) 09304127

(viii) 09304128

Q.3 If find 09304129

(ii) If find 09304130

(iii) If , find 09304131

Q.4 Simplify 09304132

(ii) Simplify 09304133

(iii) Simplify

Q.5(i) If , find value of and (Board 2014) 09304135

(ii) If find the value of and 09304136

Q.6 Determine the rational numbers a and b. If

(Board 2014)

09304137

Review Exercise 4 OBJECTIVE

Q.1 Multiple Choice Questions. Choose the correct answer.

1. 4x + 3y  2 is an algebraic____09304138

(a) Expression (b) Sentence

2. The degree of polynomial 4x4+2x2y is ____ 09304139

(a)1 (b)2

3. a3 + b3 is equal to____ 09304140

(a) (ab) (a2+ab+b2)

(b) (a+b) (a2ab + b2)

(d) (ab) (a2 + abb2)

4. is equal to:___09304141

(a) 7 (b) –7 (Board 2013,14,1 5)

5. Conjugate of Surd is_ 09304142

(a) (b) (Board 2013)

(d) (d)

6. is equal to 09304143

(a) (b) (Board 2015)

7. is equal to: 09304144

(a) (ab)2 (b) (a+b)2

8. is equal to:__ 09304145

(a)a2 + b2 (b) a2 b2 (Board 2014)

(c)a  b (d) a + b

9. The degree of the polynomial x2y2+3xy+y3 is ___ 09304146

(a)4 (b)5

10. = ………………… 09304147

(a) (x2) (x+2) (b) (x2) (x2)

11. (……………) 09304148

(a) (b)

12. 2(a2 + b2) = ____ 09304149

(a) (a+b)2 + (ab)2(b)(a+b)2

(b) (a+b)2 (ab)2 (d) 4ab

13. Order of surd is ____ 09304150

(a)3 (b)

14. 09304151

(a) (b)

(d) (d)

15. (a+b)2 (ab)2 = ________ 09304152

(a)2(a2 + b2) (b)4ab

16. 09304153

(a) (b)

17. A surd which contains a single term is called _______surd. 09304154

(a) Monomial (b) Binomial

18. What is the leading coefficient of polynomial ? 09304155

(a) 2 (b) 3

19. A surd which contains two terms is called _______surd. 09304156

(a) Monomial (b) Binomial

20. If H.C.F of p(x) and q(x) is ____ then expression is in lowest form.

09304157

(a) 0 (b) 1

21. Which of the following is polynomial?

(a) (b) 09304158

22. If and , then value of a is: 09304159

(a) 2 (b) 3

23. If and , then value of b is _______. 09304160

(a) 2 (b) 3

24. 09304161

(a) 12 (b) 9

25. 09304162

(a) 34 (b) 21

26. If is a surd of order n, then “a” is ____ number. 09304163

(a) rational (b) irrational

27. Which of the following is not surd?

(a) (b) 09304164

28. 09304165

(a) 3 (b) 7

29. Rationalizing factor of is:

09304166

(a) (b)

30. In the polynomial with the variable x, all the powers of x are------ integers. 09304167

(a) non-negative (b) negative

31. If the product of two surds is a rational number, then each surd is called the _____ of the other. 09304168

(a) monomial surd (b) binomial surd

(d) rationalizing factor

32. Polynomial means an expression with:

(a) one term (b) two terms 09304169

Q.2 Fill in the blanks. 09304170

(i) The degree of the polynomial x2 y2 + 3xy + y3 is ……..….. 09304171

(ii) x2 – 4 = ……..….. 09304172

(iii) x3 + = ………………… 09304173

(iv) 2(a2 + b2) = (a + b)2 + ……………. 09304174

(v) = …………… 09304175

(vi) Order of surd is ………… 09304176

(vii) = …………… 09304177

____________________________________________________________________

Q.3 If find 09304178

(i) 09304179 (ii) 09304180

Q.4 If find 09304181

(i) 09304182 (ii) 09304183

Q.5 Find value of and if and 09304184

Q.6 If find 09304185

(i) 09304186 (ii) 09304187

(iii) 09304188 (iv) 09304189

Q.7 If q = + 2 Find 09304190

(i) q + 09304191 (ii) q – 09304192

(iii) q2 + 09304193 (iv) q2 – 09304194

Q.8 Simplify 09304195

(ii) Simplify

09304196

Q9. Simplify:

09304197

Q10. Simplify:

09304198

Q11. Simplify:

09304199

Q12. Factorize:

09304200

Q13. Rationalize the denominator of:

09304201

UNIT 5

Q. Define Factorization. 09305001

(i) Factorization of the Expression of the type ka + kb + kc.

Example 1 09305002

Factorize 5a-5b+5c

Example 2 09305003

Factorize 5a – 5b – 15c

(ii) Factorization of the Expression of the type ac + ad + bc + bd

Example 1 09305004

Factorize 3x  3a + xy ay

Example 2 (Board 2013) 09305005

Factorize pqr + qr2pr2r3

(iii) Factorization of the Expression of the type .

Example 1 09305006

Factorize 25x2 + 16 + 40x.

Solution:

25x2+40x+16 =(5x)2+2(5x)(4) + (4)2

= (5x+4)2

= (5x+4) (5x+4)

Example 2 09305007

Factorize 12x2–36x+27

(iv) Factorization of the Expression of the type a2 – b2.

Example 1 Factorize 09305008

(i) 4x2 –(2y  z)2 (ii)6x4 – 96

(v)Factorization of the Expression of the types a2 2ab + b2 – c2.

Example 09305009

Factorize (i)

(ii)

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

15. Exercise 5.1

Q.1 Factorize: 09305010

09305011

09305012

09305013

09305014

09305015

09305016

Q.2 Factorize:

09305017

09305018

09305019

09305020

Q.3 Factorize:

09305021

09305022

09305023

09305024

Q.4 Factorize:

09305025

09305026

09305027

09305028

Q.5 Factorize:

09305029

09305030

09305031

(Board 2014)09305032

(Board 2014)09305033

09305034

(a) Factorization of the Expression of type a4+a2b2 + b4 or a4 + 4b4 09305035

Example 1 09305036

Factorize

Example 2

Factorize 09305037

(b) Factorization of the Expression of the type .

Example 09305038

Factorize (i) (ii)

Example Factorize (i) 09305039

(ii) (iii)

(d) Factorization of the following types of Expressions.

Example 1 09305040

Factorize

Example 2 Factorize 09305041

Example 3 09305042

Factorize

(e) Factorization of Expressions of the following Types

Example: 09305043

Factorize

(f) Factorization of Expressions of the following types

We recall the formulas,

Example 1 09305044

Factorize

Example 2 (Board 2014) 09305045

Factorize

Exercise 5.2

Q.1 Factorize: 09305046

09305047

09305048

09305049

09305050

(Board 2013) 09305051

09305052

Q.2 Factorize:

09305053

09305054

09305055

09305056

Q.3 Factorize:

09305057

(L.B. 2014) 09305058

09305059

09305060

09305061

09305062

09305063

09305064

Q.4 Factorize:

09305065

09305066

09305067

09305068

09305069

Q.5 Factorize:

(Board 2014) 09305070

09305071

09305072

093050073

Q.6 Factorize:

09305074

09305075

09305076

(Board 2014) 09305077

Q. Define Remainder Theorem. 09305078

Application of reminder theorem:

Example 1 09305079

Find the remainder when is divided by

(i) x – 3 (ii) x + 3 (iii) 3x + 1 (iv) x=0

Example 2 09305080

Find the value of k if the expression leaves a remainder of 2 when divided by .

Q. Define Zero of a polynomial. 09305081

Q. Define Factor Theorem. 09305082

Example 1 09305083

Determine if is a factor of

Example 2 09305084

Find a polynomial of degree 3 that has 2, 1, and 3 as zeros (i.e., roots).

16. Exercise 5.3

Q.1 Use the remainder theorem to find the remainder, when. 09305085

09305086

09305087

is divided by (x + 2)

09305088

is divided by 2x+1 09305089

is divided by x + 2

09305090

Q.2(i) If (x+2) is a factor of then find the value(s) of k. 09305091

(ii) If (x 1) is factor of then find the value of k. 09305092

Q.3 Without actual long division determine whether 09305093

(i) (x  2) and (x  3) are factors of 09305094

(ii)(x – 2), (x + 3) and (x  4) are factors of 09305095

Q.4 For what value of m is the polynomial exactly divisible by x+2? 09305096

Q.5 Determine the value of k if and q (x)= x3 – 4x + k. Leaves the same remainder when divided by x3. 09305097

Q.6 The remainder of dividing the polynomial by (x + 1) is 2b. Calculate the value of ‘a’ and ‘b’ if this expression leaves a remainder of (b + 5) on being divided by (x  2) 09305098

Q.7 The polynomial has a factor (x + 4) and it leaves a remainder of 36 when divided by (x2). Find the value of and m. (Board 2013) 09305099

Q.8 The Expression leaves remainder of 3 and 12 when divided by (x1) and (x+2) respectively. Calculate the values of and m. 09305100

Q.9 The expression is exactly divisible by . Find the values of a and b. 09305101

Q. Define Rational Root Theorem.09305102

Example 09305103

Factorize the polynomial , by using Factor Theorem.

is a zero of P(x).

Hence is the third factor of P(x).

Thus the factorized form of

Exercise 5.4

Factorize each of the following cubic polynomials by factor theorem. 09305104

Q.1 09305105

Q.2 09305106

Q.3 09305107

Solution:

Let

Expected zeros of P(x) are  1,  2, 5

So, x = 1 is not a zero of P(x)

So, x = 1 is a zero of P(x).

So, x = 2 is a zero of P(x).

Q.4 09305108

Q.5 09305109

Q.6 09305110

Q. 7 09305111

Q.8 (Board 2015) 09305112

Review Exercise 5 OBJECTIVE

Q.1 Multiple Choice Questions. Chose the correct answer.

33. The factor of x25x+6 are: __ 09305113

(a) x +1, x  6 (b) x 2, x3 (Board 2014)

34. Factors of 8x3 + 27y3 are:___ 09305114

(a) (2x+3y) (4x29y2)

(b) (2x-3y) (4x2 – 9y2)

(d) (2x3y) (4x2 + 6xy + 9y2)

35. Factors of 3x2 x2 are:(Board 2013,14)09305115

(a) (x+1) (3x2) (b) (x+1) (3x+2)

36. Factors of a4 4b4 are: ___ 09305116

(a) (ab) (a+b) (a2+4b2) (Board 2014)

(b) (a22b2) (a2 + 2b2)

(d) (a2b) (a2+ 2b2)

37. What will be added to complete the square of 9a212ab?___ 09305117

(a) –16 b2 (b) 16 b2 (Board 2013, 15)

38. Find m so that x2 + 4x+m is a complete square: 09305118

(a) 8 (b) 8

39. Factors of 5x2 – 17xy 12y2 are___09305119

(a) (x+4y) (5x+3y) (b) (x4y) (5x – 3y)

40. Factors of are___ 09305120

(a)

(b)

(c)

(d)

41. If x–2 is a factor of

p(x) = x2+2kx+8, then k = __ 09305121

(a) –3 (b) 3

42. 4a2+4ab+(…..) is a complete square

(a) b2 (b) 2b 09305122

43. 09305123

(a) (b)

44. (x+y) (x2 – xy + y2) = ___ 09305124

(a) x3 y3 (b) x3 + y3

45. Factors of x4 – 16 is ___ 09305125

(a) (x2)2

(b) (x2) (x+2) (x2+4)

(d) (x+2)2

46. Factors of 3x – 3a + xy – ay. 09305126

(a) (3+y) (xa) (b) (3y) (x+a)

47. Factors of pqr + qr2 –pr2 – r3 is: 09305127

(a) r(p+r) (qr)

(b) r(pr) (q + r)

(d) r(p+r) (q+r)

48. If is a factor of , then remaider is: 09305128

(a) (b)

49. What is the value of at ? 09305129

(a) (b) 3

50. What is the value of at ? 09305130

(a) 9 (b) 8

51. 09305131

(a) (b)

52. 09305132

(a)

(b)

(c)

(d)

53. How many factors of a cubic expression are there? 09305133

(a) zero (b) 1

54. If a polynomial is divided by a linear divisor , then the remainder is: 09305134

(a) zero (b) a

55. If a polynomial can be expressed as , then each of the polynomial and is called a _________ of . 09305135

(a) remainder (b) factor

56. (x – y) (x2 + xy + y2) = ___ 09305136

(a) x3 y3 (b) x3 + y3

Q.2. Completion items fill in the blanks. 09305137

i. x2 + 5x + 6 = …………. 09305138

ii. 4a2 – 16 = ………….. 09305139

iii. 4a2 + 4ab + (………) is a complete square. 09305140

iv. = ……….. 09305141

v. (x + y) (x2 – xy + y2) = ………. 09305142

vi. Factorized form of x4 – 16 is 09305143

vii. If x – 2 is a factor of p(x) = x2+2kx + 8, then k = ……. 09305144

Q.3.Factorize the following:

(i) x2 + 8x + 16 – 4y2 09305145

(ii) 4x2 – 16y2 09305146

(iii) 9x2 + 27 x + 8 09305147

(iv) 1 – 64 09305148

(v) 8x3 – 09305149

(vi) 2y2 + 5y – 3 09305150

(vii) x3 + x2 – 4x – 4 09305151

(viii) 25m2 n2 + 10mn + 1 09305152

(ix) 1 – 12 pq + 36 p2 q2 09305153

Q.4.Factorize the following:

(i) 09305154

(ii) 09305155

(iii) 09305156

(iv) 09305157

(v) 09305158

Q5. What will be added to complete the square of ? 09305159

Q6. Find m so that x2 + 4x+m is a complete square. 09305160

UNIT 6

Q. Define Factorization. 09305001

(i) Factorization of the Expression of the type ka + kb + kc.

Example 1 09305002

Factorize 5a-5b+5c

Example 2 09305003

Factorize 5a – 5b – 15c

(ii) Factorization of the Expression of the type ac + ad + bc + bd

Example 1 09305004

Factorize 3x  3a + xy ay

Example 2 (Board 2013) 09305005

Factorize pqr + qr2pr2r3

(iii) Factorization of the Expression of the type .

Example 1 09305006

Factorize 25x2 + 16 + 40x.

Solution:

25x2+40x+16 =(5x)2+2(5x)(4) + (4)2

= (5x+4)2

= (5x+4) (5x+4)

Example 2 09305007

Factorize 12x2–36x+27

(iv) Factorization of the Expression of the type a2 – b2.

Example 1 Factorize 09305008

(i) 4x2 –(2y  z)2 (ii)6x4 – 96

(v)Factorization of the Expression of the types a2 2ab + b2 – c2.

Example 09305009

Factorize (i)

(ii)

17. Exercise 5.1

Q.1 Factorize: 09305010

09305011

09305012

09305013

09305014

09305015

09305016

Q.2 Factorize:

09305017

09305018

09305019

09305020

Q.3 Factorize:

09305021

09305022

09305023

09305024

Q.4 Factorize:

09305025

09305026

09305027

09305028

Q.5 Factorize:

09305029

09305030

09305031

(Board 2014)09305032

(Board 2014)09305033

09305034

(a) Factorization of the Expression of type a4+a2b2 + b4 or a4 + 4b4 09305035

Example 1 09305036

Factorize

Example 2

Factorize 09305037

(b) Factorization of the Expression of the type .

Example 09305038

Factorize (i) (ii)

Example Factorize (i) 09305039

(ii) (iii)

(d) Factorization of the following types of Expressions.

Example 1 09305040

Factorize

Example 2 Factorize 09305041

Example 3 09305042

Factorize

(e) Factorization of Expressions of the following Types

Example: 09305043

Factorize

(f) Factorization of Expressions of the following types

We recall the formulas,

Example 1 09305044

Factorize

Example 2 (Board 2014) 09305045

Factorize

Exercise 5.2

Q.1 Factorize: 09305046

09305047

09305048

09305049

09305050

(Board 2013) 09305051

09305052

Q.2 Factorize:

09305053

09305054

09305055

09305056

Q.3 Factorize:

09305057

(L.B. 2014) 09305058

09305059

09305060

09305061

09305062

09305063

09305064

Q.4 Factorize:

09305065

09305066

09305067

09305068

09305069

Q.5 Factorize:

(Board 2014) 09305070

09305071

09305072

093050073

Q.6 Factorize:

09305074

09305075

09305076

(Board 2014) 09305077

Q. Define Remainder Theorem. 09305078

Application of reminder theorem:

Example 1 09305079

Find the remainder when is divided by

(i) x – 3 (ii) x + 3 (iii) 3x + 1 (iv) x=0

Example 2 09305080

Find the value of k if the expression leaves a remainder of 2 when divided by .

Q. Define Zero of a polynomial. 09305081

Q. Define Factor Theorem. 09305082

Example 1 09305083

Determine if is a factor of

Example 2 09305084

Find a polynomial of degree 3 that has 2, 1, and 3 as zeros (i.e., roots).

18. Exercise 5.3

Q.1 Use the remainder theorem to find the remainder, when. 09305085

09305086

09305087

is divided by (x + 2)

09305088

is divided by 2x+1 09305089

is divided by x + 2

09305090

Q.2(i) If (x+2) is a factor of then find the value(s) of k. 09305091

(ii) If (x 1) is factor of then find the value of k. 09305092

Q.3 Without actual long division determine whether 09305093

(i) (x  2) and (x  3) are factors of 09305094

(ii)(x – 2), (x + 3) and (x  4) are factors of 09305095

Q.4 For what value of m is the polynomial exactly divisible by x+2? 09305096

Q.5 Determine the value of k if and q (x)= x3 – 4x + k. Leaves the same remainder when divided by x3. 09305097

Q.6 The remainder of dividing the polynomial by (x + 1) is 2b. Calculate the value of ‘a’ and ‘b’ if this expression leaves a remainder of (b + 5) on being divided by (x  2) 09305098

Q.7 The polynomial has a factor (x + 4) and it leaves a remainder of 36 when divided by (x2). Find the value of and m. (Board 2013) 09305099

Q.8 The Expression leaves remainder of 3 and 12 when divided by (x1) and (x+2) respectively. Calculate the values of and m. 09305100

Q.9 The expression is exactly divisible by . Find the values of a and b. 09305101

Q. Define Rational Root Theorem.09305102

Example 09305103

Factorize the polynomial , by using Factor Theorem.

is a zero of P(x).

Hence is the third factor of P(x).

Thus the factorized form of

Exercise 5.4

Factorize each of the following cubic polynomials by factor theorem. 09305104

Q.1 09305105

Q.2 09305106

Q.3 09305107

Solution:

Let

Expected zeros of P(x) are  1,  2, 5

So, x = 1 is not a zero of P(x)

So, x = 1 is a zero of P(x).

So, x = 2 is a zero of P(x).

Q.4 09305108

Q.5 09305109

Q.6 09305110

Q. 7 09305111

Q.8 (Board 2015) 09305112

Review Exercise 5 OBJECTIVE

Q.1 Multiple Choice Questions. Chose the correct answer.

57. The factor of x25x+6 are: __ 09305113

(a) x +1, x  6 (b) x 2, x3 (Board 2014)

58. Factors of 8x3 + 27y3 are:___ 09305114

(a) (2x+3y) (4x29y2)

(b) (2x-3y) (4x2 – 9y2)

(d) (2x3y) (4x2 + 6xy + 9y2)

59. Factors of 3x2 x2 are:(Board 2013,14)09305115

(a) (x+1) (3x2) (b) (x+1) (3x+2)

60. Factors of a4 4b4 are: ___ 09305116

(a) (ab) (a+b) (a2+4b2) (Board 2014)

(b) (a22b2) (a2 + 2b2)

(d) (a2b) (a2+ 2b2)

61. What will be added to complete the square of 9a212ab?___ 09305117

(a) –16 b2 (b) 16 b2 (Board 2013, 15)

62. Find m so that x2 + 4x+m is a complete square: 09305118

(a) 8 (b) 8

63. Factors of 5x2 – 17xy 12y2 are___09305119

(a) (x+4y) (5x+3y) (b) (x4y) (5x – 3y)

64. Factors of are___ 09305120

(a)

(b)

(c)

(d)

65. If x–2 is a factor of

p(x) = x2+2kx+8, then k = __ 09305121

(a) –3 (b) 3

66. 4a2+4ab+(…..) is a complete square

(a) b2 (b) 2b 09305122

67. 09305123

(a) (b)

68. (x+y) (x2 – xy + y2) = ___ 09305124

(a) x3 y3 (b) x3 + y3

69. Factors of x4 – 16 is ___ 09305125

(a) (x2)2

(b) (x2) (x+2) (x2+4)

(d) (x+2)2

70. Factors of 3x – 3a + xy – ay. 09305126

(a) (3+y) (xa) (b) (3y) (x+a)

71. Factors of pqr + qr2 –pr2 – r3 is: 09305127

(a) r(p+r) (qr)

(b) r(pr) (q + r)

(d) r(p+r) (q+r)

72. If is a factor of , then remaider is: 09305128

(a) (b)

73. What is the value of at ? 09305129

(a) (b) 3

74. What is the value of at ? 09305130

(a) 9 (b) 8

75. 09305131

(a) (b)

76. 09305132

(a)

(b)

(c)

(d)

77. How many factors of a cubic expression are there? 09305133

(a) zero (b) 1

78. If a polynomial is divided by a linear divisor , then the remainder is: 09305134

(a) zero (b) a

79. If a polynomial can be expressed as , then each of the polynomial and is called a _________ of . 09305135

(a) remainder (b) factor

80. (x – y) (x2 + xy + y2) = ___ 09305136

(a) x3 y3 (b) x3 + y3

Q.2. Completion items fill in the blanks. 09305137

i. x2 + 5x + 6 = …………. 09305138

ii. 4a2 – 16 = ………….. 09305139

iii. 4a2 + 4ab + (………) is a complete square. 09305140

iv. = ……….. 09305141

v. (x + y) (x2 – xy + y2) = ………. 09305142

vi. Factorized form of x4 – 16 is 09305143

vii. If x – 2 is a factor of p(x) = x2+2kx + 8, then k = ……. 09305144

Q.3.Factorize the following:

(i) x2 + 8x + 16 – 4y2 09305145

(ii) 4x2 – 16y2 09305146

(iii) 9x2 + 27 x + 8 09305147

(iv) 1 – 64 09305148

(v) 8x3 – 09305149

(vi) 2y2 + 5y – 3 09305150

(vii) x3 + x2 – 4x – 4 09305151

(viii) 25m2 n2 + 10mn + 1 09305152

(ix) 1 – 12 pq + 36 p2 q2 09305153

Q.4.Factorize the following:

(i) 09305154

(ii) 09305155

(iii) 09305156

(iv) 09305157

(v) 09305158

Q5. What will be added to complete the square of ? 09305159

Q6. Find m so that x2 + 4x+m is a complete square. 09305160

UNIT 7

Q. Define linear equation in one variable and write down its standard form. 09307001

Q. What is Solution of Equation? 09307002

Q. Define Equivalent Equations. 09307003

Q. Define Identity. 09307004

Q. Define inconsistent equation.

Example 1:

Solve the equation 09307005

Example 22 09307006

Solve

Example 3 09307007

Solve

Q. Define Radical Equation. 09307008

Q. What is Extraneous Solution? 09307009

Example 1 Solve the equations 09307010

(a) (b)

Example 2 09307011

Solve and check:  = 0

Example 3 09307012

Solve

19. Exercise 7.1

20.

Q.1 Solve the following equations. 09307013

(i) 09307014

(ii) (Board 2014) 09307015

(iii) 09307016

(iv) 09307017

(v) 09307018

(vi) , 09307019

(vii)

09307020

(viii) 09307021

(Board 2014)

(ix) 09307022

(x)

09307023 (Board 2014)

Q.2 Solve each equation and check for extraneous solution, if any. 09307024

(i) 09307025

(ii) (Board 2015) 09307026

(Board 2014) 09307027

(iv) 09307028

(Board 2015)

(v) 09307029

(vi) 09307030

(vii) or 09307031

(viii) 09307032

Q. Define Absolute Valued Equation.

09307033

Example 1: 09307034

Solve and check |2x + 3| = 11

Example 2: 09307035

Solve |8x  3| = |4x + 5| (Board 2015)

Example 3:

Solve and check |3x + 10| = 5x + 6 09307036

___________________________________________________________________________________________________________________________________________________________________

21. Exercise 7.2

Q.1 Identify the following statements as True or False. 09307037

(i) has only one solution. 09307038

(ii) All absolute value equations

have two solutions. 09307039

(iii) The equation is 09307040

equivalent to .

(iv) The equation

has no solution. 09307041

(v) The equation is

equivalent to

or 09307042

Q.2 Solve for ‘ ’. 09307043

(i) 09307044

(ii) 09307045

(iii) (Board 2013) 09307046

(iv) 09307047

(v) 09307048

(vi) 09307049

(vii) 09307050

(viii) (Board 2015) 09307051

Example 1: 09307052

Solve

Example 2: 09307053

Solve , where .

Example 3 Solve the double inequality

2 < , where . 09307054

Example 4: 09307055

Solve the inequality

i. Exercise 7.3

Q.1 Solve the following inequalities. 09307056

(i) 09307057

(ii) 09307058

(iii) 09307059

(iv) 09307060

(v) 09307061

(vi)

09307062

(vii) 09307063

(viii) 09307064

Q.2 Solve the following inequalities. 09307065

(i) 09307066

(ii) 09307067

(iii) 09307068

(iv) 09307069

(v) (Board 2014) 09307070

(vi) 09307071

(vii) 09307072

(viii) 09307073

22. Review Exercise 1 OBJECTIVE

Q.1 Choose the correct answer:

81. Which of the following is the solution of the inequality 3 – 4x  11?

(Board 2014) 09307074

(a) 8 (b) 2

82. A statement involving any of the symbols <, > or  or  is called:

09307075

(a) Equation

(b) Identity

(d) Linear equation

83. x = ________ is a solution of the inequality 2 < x < 09307076

(a) 5 (b) 3 (c) 0 (d)

84. If x is no larger than 10, then:

(Board 2015) 09307077

(a) (b)

85. If the capacity c of an elevator is at most 1600 pounds, then_ 09307078

(Board 2013, 15)

(a) c < 1600 (b)

86. x=0 is a solution of the inequality:

(Board 2014) 09307079

(a) x > 0

(b) 3x + 5 < 0

(d) x  2 < 0

87. The linear equation in one variable x is: 09307080

(a) ax + b = 0

(b) ax2 + bx + c = 0

(d) ax2 + by2 + c = 0

88. An inconsistent equation is that whose solution set is: 09307081

(a) Empty (b) Not empty

89. Absolute value of a real number a is defined as: 09307082

(a)

(b)

(c)

(d) None of these

90. is equivalent to: 09307083

(a) x = a or x = a

(b)

(c)

(d) None of these

91. A linear inequality in one variable x is: 09307084

(a) a x + b > 0, a  0

(b) ax2 + bx + c < 0, a  0

(d) ax2 + by2 + c < 0, a  0

92. Law of Trichotomy is …

09307085

(a) a < b or a = b or a > b

(b) a < b or a = b

(d) None of these

93. Transitive law is____ 09307086

(a) a < b and b < c, then a < c

(b) a > b and b < c, then a > c

(d) None of these

94. If a > b, c > 0 then: 09307087

(a) a c < bc (b) ac > bc

95. If a > b, c > 0 then: 09307088

(a) (b)

96. If a > b, c < 0, then: 09307089

(a) (b)

97. If a, b  R then: b 0 09307090

(a) (b)

(c)

(d)

98. When the variable in an equation occurs under a radical, the equation is called a _______ equation. 09307091

(a) Radical (b) Absolute value

99. has only ___ solution. 09307092

(a) one (b) two

100. The equation is equivalent to:

(a) 09307093

(b) x = –2 or x = 2

(d) x = 2 or x =

101. An __ is equation that is satisfied by every number for which both sides are defined: 09307094

(a) Identity (b) Conditional

102. An__ equation is an equation whose solution set is the empty set: 09307095

(a) Identity (b) Conditional

103. A _ equation is an equation that is satisfied by atleast one number but is not an identity: 09307096

(a) Identity (b) Conditional

104. x + 4 = 4 + x is _ equation: 09307097

(a) Identity (b) Conditional

105. 2x + 1 = 9 is ___ equation: 09307098

(a) Identity (b) Conditional

106. x = x + 5 is ___ equation: 09307199

(a) Identity (b) Conditional

107. Equations having exactly the same solution are called ___ equations.

(a) equivalent (b) Linear 09307100

108. A solution that does not satisfy the original equation is called ____ solution: 09307101

(a) Extraneous (b) Root

109. If is positive, then: 09307102

(a) (b)

110. If is negative, then: 09307103

(a) (b)

111. If is zero, then: 09307104

(a) (b)

112. If then is: 09307105

(a) Positive (b) Negative

113. If , then is: 09307106

(a) Positive (b) Negative

114. Which of the following inequality is strict? 09307107

(a) (b)

115. Which of the following inequality is non-strict? 09307108

(a) (b)

116. If and , then 09307119

(a) (b)

117. If and , then 09307110

(a) (b)

118. The sign of inequality is reversed if each side is multiplied by _____ real number. 09307111

(a) zero (b) positive

119. If and , then: 09307112

(a) (b)

Q.2 Identify the following statements as True or False. 09307113

(i) The equation 3x – 5 = 7 – x is a linear equation. 09307114

(ii) The equation x – 0.3x = 0.7x is an identity. 09307115

(iii) The equation -2x + 3 = 8 is equivalent to –2x = 11. 09307116

(iv) To eliminate fractions, we multiply each side of an equation by the L.C.M. of

denominators 09307117

(v) 4(x + 3) = x + 3 is a conditional equation. 09307118

(vi) The equation 2(3x + 5) = 6x + 12 is an inconsistent equation. 09307119

(vii) To solve x = 12, we should multiply each side by . 09307120

(viii) Equations having exactly the same solution are called equivalent equations. 09307121

(ix) A solution that does not satisfy the original equation is called extraneous solution. 09307122

_________________________________________________________________________

Q.3 Answer the following short questions. 09307123

(i) Define a linear inequality in one variable. 09307124

(ii) State the trichotomy and transitive properties of inequality. 09307125

(iii) The formula relating degrees Fahrenheit to degrees Celsius is

. For what value of C is F < 0?

(iv) Seven times the sum of an integer and 12 is at least 50 and at most 60. Write and solve the inequality that expresses this relationship. 09307126

Q.4 Solve each of the following and check for extraneous solution, if any. 09307118

(i) 09307127

Squaring both sides

(ii) 09307128

Q.5 Solve for x 09307129

(i) 09307130

(ii) 09307131

Q.6 Solve the following inequality. 09307132 (i) 09307133

(ii) 09307134

Q7. Solve: 09307135

Q.8 What are strict and non strict inequalities?

UNIT 8

Q. Define Ordered Pair of Real Numbers.

09308001

Q. Define Cartesian Plane. 09308002

Q. What do you mean by coordinates of a point ?

Q Define abscissa and ordinate.

Q. What is a line segment?

Drawing different geometrical Shapes in Cartesian Plane

(a) Line-Segment

Example 1: 09308003

Let P(2, 2) and Q(6, 6) be two points.

Example 2: 09308004

Plot points P(2, 2) and Q(6, 2). By joining them, we get a line segment PQ parallel to x-axis,

Example 3: 09308005

Plot points P(3, 2) and Q(3, 7). By joining them, we get a line segment PQ parallel to

y-axis. In this graph abscissas of both points are equal.

(b) Triangle

Example1: 09308006

Plot the points P(3, 2), Q(6, 7) and R(9, 3).

Example 2: 09308007

For points O(0, 0), P(3, 0) and

R(3, 3), the triangle OPR is constructed.

Example 1 09308008

Plot the points P(2, 3), Q(2, 0), S(2, 0) and R (2, 3). Joining the points P, Q ,S and R, we get a rectangle PQSR.

Exercise 8.1:

Q.7 Determine the quadrant of the coordinate plane in which the following points lie. 09308009

Ans.

(i) P (4, 3) IIquadrant(Board 2013)

(ii) Q (5, 2) IIIquadrant

(iii) P (2, 2) Iquadrant

(iv) S(2, 6) IVquadrant(Board 2013)

2.Draw the graph of each of the following.09308010

(i) x = 2 09308011 (ii) x = – 3 09308012

(iii) y = – 1 09308013 (iv) y = 3 09308014

(v) y = 0 09308015 (vi) x = 0 09308016

(vii) y = 3x 09308017 (viii) – y = 2x0930818

(ix) 09308019 (x) 3y = 5x09308020

(xi) 2x – y = 009308021 (xii)2x – y = 209308022

xiii)x – 3y + 1=009308023(xiv) 3x – 2y + 1 = 0

09308024

(i) x = 2

(ii)

(iii) (Board 2015)

(iv)

(v)

(vi)

(vii) y = 3x(Board 2014)

Table for y = 3x

x –1 0 1

y –3 0 3

(viii) y = 2x

Table for –y = 2x

x –1 0 1

y 2 0 –2

(ix) x = 12

(x) 3y = 5x

Table for 3y = 5x

x –3 0 3

y –5 0 5

(xi) 2x–y = 0 (Board 2014)

Table for 2x – y = 0

x –1 0 1

y –2 0 2

(xii) 2x  y = 2

Table for 2x – y = 2

or 2x – 2 = y

y = 2x -2

x 0 1 2 3

y –2 0 2 4

(xiii) x  3y + 1 = 0

Table for x – 3y + 1 = 0

or x + 1 = 3y

3y = x +1

y =

x –1 2 5

y 0 1 2

(xiv) 3x2y + 1 = 0

or 3x + 1 = 2y

2y = 3x +1 y =

Table for 3x2y + 1 = 0

x –1 1 3

y –1 2 5

Q.3 Are the following lines: 09308025

(i) Parallel to xaxis

(ii) Parallel to yaxis

(i) 2x  1 = 3 09308026

2x = 3 + 1x = = 2

Parallel to yaxis

(ii) x + 2 = 1 09308027

 x =12

x = 3

Parallel to yaxis

(iii) 2y + 3 = 2 09308028

 2y = 2  3y =

Parallel to xaxis

(iv) x + y = 0 09308029

 x = y

Graph of x = y is neither parallel to x-axis nor parallel to y-axis but passes through the origin.

(v) 2x  2y = 0 09308030

2x = 2y

x = y

Graph of x = y is neither parallel to x-axis nor parallel to y-axis but passes through the origin.

Q.4 Find the value of m and c of the following lines by expressing them in the form y = mx + c 09308031

(a) 2x + 3y  1 = 0 09308032

(b) x 2y = 2 09308033

(d) 2x  y = 7 09308035

(e) 3  2x + y = 0 09308036

(f) 2x = y + 3 09308037

Q.5Verify whether the following points lies on the line 2x  y + 1 = 0 or not. 09308038

(i) (2, 3)  x = 2, y = 3 09308039

(ii) (0, 0)  x = 0, y = 0 09308040

(iii) (1, 1)  x = 1, y = 1 09308041

(iv) (2, 5)  x = 2, y = 5 09308042

(v) (5, 3)  x = 5, y = 3 09308043

Conversion Graphs

(a) Example: (Kilometre (Km) and Mile (M) Graphs) 09308044

To draw the graph between kilometre (Km) and Miles (M), we use the following relation:

One kilometre = 0.62 miles,

(approximately)

And One mile = 1.6 km (approximately)

(ii) The conversion graph of kilometer against mile is given by

y = 1.6x (approximately)

If y represents kilometers and x a mile, then the values x and y are tabulated as:

x 0 1 2 3 4 ….

y 0 1.6 3.2 4.8 6.4 …

(b)Conversion Graph of Hectares andAcres

(i)The relation between Hectare and Acre is defined as: Hectare = Acres

= 2.5 Acres (approximately)

In case when hectare = x and

acre= y, then relation between them is given by the equation, y = 2.5x

If x is represented as hectare along the horizontal axis and y as Acre along

y-axis, the values are tabulated below:

x 0 1 2 3 4 ….

Y 0 2.5 5.0 7.5 10 ….

(i) The relation between Celsius (C) and degree Fahrenheit (F) is given by

F C + 32 09308046

The value of Fat C = 0 is obtained as

F = 0 + 32 = 0 + 32 = 32

Similarly,

F = 10 + 32 = 18 + 32 = 50,

F = 20 + 32 = 36 + 32 = 68,

F = 100 + 32 = 180 + 32 = 212

We tabulate the values of C and F.

C 0o 10o 20o 50o 100o…

F 32o 50o 68o 122o 212o…

(d) Conversion graph of US$ and Pakistani Currency 09308047

The daily News, on a particular day informed the conversion rate of Pakistani currency to the US$ currency as.

1 US$ = 66.46 Rupees

If the Pakistani currency y is an expression of US$ x, expressed under the rule

y = 66.46 x  66x (approximately)

Then draw the conversion graph.

x 1 2 3 4 …

y 66 132 198 264 …

Exercise 8.2:

Q.1Draw the conversion graph between 1 litre and gallons using the relation 9 litres = 2 gallons (approximately) and taking litres along horizontal axis and gallons along vertical axis.From the graph,read:09308048

(i) The number of gallons in 18 litres09308049

(ii) The number of litres in 8 gallons09308050

Q.2 On 15.03.2008 the exchange rate of Pakistani currency and Saudi Riyal was as, under 1 S. Riyal = 16.70 rupees.

If Pakistani currency y is an expression of S.Riyal x, expressed under the rule y = 16.70x then draw conversion graph between two currencies by taking S. Riyal along xaxis. 09308051

Q.3 Sketch the graph of each of the following lines: 09308052

(a) x3y + 2 = 0 09308053

(b) 3x  2y  1 = 0 09308054

(d) y2x = 0 y = 2x 09308056

(e) 3y – 1 = 0 09308057

(f) y + 3x = 0 09308058

(g) 2x + 6 = 0 09308059

Q.4 Draw the graph for following relations: 09308060

(i) One mile=1.6 km 09308061

Let mile be represented by x and km by y:

y =1.6 x

(ii) One acre =0.4 Hectare 09308062

Let acres = x

Hectare = y

(iii) (Board 2014) 09308063

The value of F at C = 0 is obtained as

We tabulate the values of C and F

C 0o 5 o 10o 15o 20o 25o

F 32o 41o 50o 59o 68o 77o

iv. One Rupee = $ Or 86 Rupees = 1 $ Or 1$ = 86 Rupees 09308064

Let dollarsare x and rupees are y

y = 86x.

x 0 1 2 3 4

y 0 86 172 258 344

Example 09308065

Solve graphically, the following linear system of two equations in two variables andy;

................(i)

…………..(ii)

Exercise 8.3:

Solve the following pair of equations in x and y graphically. 09308066

Q.1 x + y = 0 and 2x  y + 3 = 009308067

Q.2 x  y + 1 = 0 and x  2y = 1

09308068

Q.3 2x + y = 0 and x + 2y = 2 09308069

Q.4 x + y  1 = 0 09308070

x y + 1 = 0

Q.5 2x + y  1 = 0, x = y 09308071

23. Review Exercise 8OBJECTIVE

Q.1 Chose the correct answers.

120. If (x–1, y+1) = (0, 0), then (x, y) is:09308072

(a) (1, 1) (b) (1, 1)(Board 2013, 14)

121. If (x, 0) = (0, y), then (x, y) is:09308073

(a) (0, 1) (b)(1, 0) (Board 2013)

122. Point (2 3) lies in quadrant: 09308074

(a) I (b) II(Board 2014, 15)

123. Point (3, 3) lies in quadrant:09308075

(a) I (b) II(Board 2014,15)

124. If y = 2x + 1, x = 2 then y is: 09308076

(a) 2 (b) 3(Board 2013)

125. Which ordered pair satisfy the equation y = 2x: 09308077

(a) (1, 2) (b) (2, 1)

126. The real numbers x, y of the ordered pair (x, y) are called _____ of point P(x,y) in a plane. 09308078

(a) co-ordinates(b) x co-ordinates

127. Cartesian plane is divided into __ quadrants. 09308079

(a) Two (b) Three

128. The point of intersection of two coordinate axes is called: 09308080

(a) Origin (b) Centre

129. The x-coordinate of a point is called__ 09308081

(a) Origin (b) abcissa

130. The y-coordinate of a point is called:

(a) Origin (b) x-coordinate

131. The set of points which lie on the same line are called ___ points. 09308083

(a) Collinear

(b) Similar

(d) None of these

132. The plane formed by two straight lines perpendicular to each other is called: (a) Cartesian plane 09308084

(b) Coordinate axes

(d) None of these

133. An ordered pair is a pair of elements in which elements are written in specific:

(a) Order (b) Array 09308085

134. Point lies in quadrant.

(a) I (b) II 09308086

135. Point lies in quadrant.

(a) I (b) II 09308087

136. Point lies in quadrant.

(a) I (b) II 09308088

137. Which of the following lines is horizontal?

(a) (b) 09308089

138. Which of the following is vertical line?

(a) (b) 09308090

139. ______ is a line on which origin lies.

(a) (b) 09308091

140. Which of the following points is on the x-axis? 09308092

(a) (b)

141. Which of the following points is on the y-axis? 09308093

(a) (b)

142. Which of the following points is on the origin? 09308094

(a) (b)

143. Which of the following lines is parallel to x-axis? 09308095

(a) (b)

144. Which of the following lines is parallel to y-axis? 09308096

(a) (b)

145. If two lines do not intersect, their solution set will be: 09308097

(a) Singleton Set (b) Empty Set

146. y = x is a line on which ___ lies.

(a) (b) 09308098

Q.2 Identify the following statements as True or False. 09308099

(i) The point O(0, 0) is in quadrant II. 09308100

(ii) The point P(2, 0) lies on x –axis. 09308101

(iii) The graph of x = –2 is a vertical line. 09308102

(iv) 3 – y = 0 is a horizontal line. 09308103

(v) The point Q(–1, 2) is in quadrant III. 09308104

(vi) The point R(–1, –2) is in quadrant IV. 09308105

(vii) y = x is a line on which origin lies. 09308106

(viii) The point P(1, 1) lies on the line x + y = 0. 09308107

(ix) The S(1, –3) lies in quadrant III. 09308108

(x) The point R(0, 1) lies on the x-axis. 09308109

Q. No.3 Draw the following points on the graph paper: (–3, –3), (–6, 4), (4, –5), (5, 3)

09308110

Q. No.4 Draw the graph of the following:

(i) x= – 6 09308111

(ii) y = 7 09308112

(iii) x = 09308113

(iv) y = 09308114

(v) y = 4x 09308115

(vi) y = – 2x + 1 09308116

Q. No. 5 Draw the graph of the following:

(i) y = 0.62x 09308117

(ii) y = 2.5x 09308118

x 0 1 2 3

y 0 2.5 5 7.5

Q. No. 6 Solve the following pair of equations graphically.

(i) x – y =1 09308119

x + y =

(ii) x = 3y 2x – 3y = – 6 09308120

(iii) (x + y) = 2 09308121

(iv) (x – y) = - 1 09308122

Q7. If y = 2x +1, then find the value of y for x = 2. 09308123

UNIT 9

24. Q. Define Plane Geometry and Coordinate Geometry. 09309001

Q. Derive Distance Formula. 09309002

25. Use of Distance Formula

26. Example 1: 09309003

Using the distance formula, find the distance between the points.

(i) P(1, 2) and Q(0,3)

(ii) S(1, 3) and R(3, 2)

(iii) U (0, 2) and V(3, 0)

(iv)

Q. Define Collinear or Non-Collinear Points.

(Board 2013)09309004

27. Use of Distance Formula to show the Collinearity of Three or more Points in the Plane:

28. Examples: 09309005

Using distance formula show that the points.

(i) P(2,1), Q(0, 3) and R(1, 5) are collinear.

(ii) The above P,Q,R and S (1,–1) are non collinear

Q. Define Triangle. 09309006

29. Q. Define Equilateral Triangle. 09309007

30. Example: 09309008

The triangle OPQ is an equilateral triangle since the points O(0,0), and

Q. Define Isosceles Triangle. 09309009

Ans. Isosceles Triangle:

An isosceles triangle is a triangle which has two of its sides with equal length while the third side has a different length.

31. Example: 09309010

The triangle PQR is an isosceles triangle as for the non-collinear points P(1,0), Q(1, 0) and R(0, 1) shown in the following figure.

Q. Define Right Angle Triangle. 09309011

32. Example: 09309012

Let O(0, 0), P(3, 0) and Q(0, 2) be three non-collinear points. Verify that triangle OPQ is right-angled.

Q. Define Scalene Triangle. 09309013

33. Example: 09309014

Show that the points P(1, 2), Q(2, 1) and R(2, 1) in the plane form a scalene triangle.

Q. Define Square. 09309015

34. Example: 09309016

If A(2, 2), B(2, 2), C(2, 2) and D(2, 2) be four non-collinear points in the plane, then verify that they form a square ABCD.

Q. Define Rectangle. 09309017

Example: Show that the points A (–2, 0), B (–2, 3), C(2, 3) and D (2,0) form a rectangle.

Using distance formula. 09309018

Q. Define Parallelogram. 09309019

35. Example: 09309020

Show that the points A(2, 1), B(2, 1), C(3, 3) and D(1, 3) form a parallelogram.

Q. Derive Mid-Point Formula. 09309021

36. Example 1: 09309022

Find the mid-point of the line segment joining A(2,5) and B(1, 1).

Example 2 09309023

Let P(2, 3) and Q (x, y) be two points plane in the plane such that R(1, –1) is the mid-point of the points P and Q. Find x and y.

37. Example 3: 09309024

Let ABC be a triangle as shown below. If and are the middle points of the line-segments AB, BC and CA respectively, find the coordinates of M1, M2 and M3. Also determine the type of the triangle M1M2M2.

38. Example 4 09309025

Let O(0,0), A(3,0) and B(3,5) be three points in the plane. If is the mid point of AB and M2 of OB, then show that .

39.

40. Exercise 9.1

Q.1 Find the distance between the following pairs of points 09309026

(a) 09309027

(b) 09309028

(d) (Board 2013)

09309030

(e) (Board 2013, 14) 09309031

(f) A(0,0) , B(0,-5) (Board 2013) 09309032

Q2. Let P be the point on x-axis with x-coordinate a and Q be the point on y-axis with y-coordinate b, as given below. Find distance between P and Q. 09309033

ii) 09309035

iii) 09309036

iv) 09309037

v) 09309038

vi) 09309039

41. Exercise 9.2

Q.1 Show whether the points with vertices and are vertices of an equilateral triangle or an isosceles triangle?

09309040

Q.2 Show whether or not the points with vertices and form a square? 09309041

Q.3 Show whether or not the points with coordinates and are vertices of a right triangle? 09309042

Q.4 Use the distance formula to prove whether or not the points and lie on a straight line. 09309043

Q.5 Find K given that the point is equidistance from and . 09309044

Q.6 Use distance formula to verify that the points are collinear. 09309045

Q.7 Verify whether or not the points are vertices of an equilateral triangle. 09309046

Q.8 Show that the points

A( 6, 5), B(5, 5), C(5, 8), are vertices of a rectangle. Find the lengths of its diagonals. Are they equal? 09309047

Q.9 Show that the points

and are the vertices of a parallelogram. 09309048

Q.10 Find the length of the diameter of the circle having centre at and passing through . 09309049

42. Exercise 9.3

Q.1 Find the mid-point of the line segment joining each of the following pairs of points.

09309050

(a) 09309051

(b) 09309052

(d) (Board 2014) 09309054

(e) 09309055

(f) 09309056

Q.2 The end point P of a line segment PQ is (–3,6) and its mid point is (5,8). Find the

co-ordinates of the end point Q. 09309057

Q.3 Prove that midpoint of the hypotenuse of a right triangle is equidistant from its three vertices and 093090358

Q.4 O (0, 0), A(3, 0) and B(3, 5) are three points in the plane, find M1 and M2 as midpoints of the line segments and respectively. Find 09309059

Q.5 Show that the diagonals of the parallelogram having vertices , bisect each other. 09309060

Q.6 The vertices of a triangle are P(4,6), Q(–2,–4) and R(–8, 2) show that the length of line segment joining the mid points of line segment PR, QR is PQ. 09309061

Review Exercise 9 OBJECTIVE

Q.1 Choose the correct answer

147. Distance between points (0, 0) and (1, 1) is: (Board 2014) 09309062

(a) 0 (b) 1

148. Distance between the points (1, 0) and (0, 1) is: 09309063

(a) 0 (b) 1

149. Mid-point of the points (2, 2) and (0,0) is: (Board 2015) 09309064

(a) (1, 1) (b) (1, 0)

150. Mid-point of the points (2, 2) and (2, 2) is: (Board 2013, 15) 09309065

(a) (2, 2) (b) (2, 2)

151. A triangle having all sides equal is called: (Board 2013) 09309066

(a) Isosceles (b) Scalene

152. A triangle having all sides different is called: 09309067

(a) Isosceles (b) Scalene

153. The points P, Q and R are collinear if: 09309068

(a)

(b)

(c)

(d) None of these

154. The distance between two points P(x1, y1) and Q (x2, y2) in the coordinate plane is: d > 0 09309069

(a)

(b)

(c)

(d)

155. A triangle having two sides equal is called: 09309070

(a) Isosceles (b) Scalene

156. A right angled triangle is that in which one of the angles has measure equal to: 09309071

(a) 80o (b) 90o

157. In a right angled triangle ABC, where m ACB = 900. 09309072

(a)

(b)

(c)

(d)

158. If M is the mid-point of a line segment , which of the following is true?

09309073

(a)

(b)

(c)

(d)

159. In a if the triangle will be: 09309074

(a) isosceles (b) scalene

160. If three or more than three points lie on the same line then points are called ______.

(a) non-collinear (b) collinear 09309075

161. A ________ has two end points. 09309076

(a) line (b) line segment

162. A line segment has __ midpoint. 09309077

(a) one (b) two

163. Each side of triangle has ____ collinear vertices. 09309078

(a) one (b) two

164. A triangle is formed by _____ non-collinear points. 09309079

(a) one (b) two

165. All points on x-axis are ____. 09309080

(a) collinear (b) non-collinear

166. All points on y-axis are ____. 09309081

(a) collinear (b) non-collinear

Q.2 Answer the following, which is true and which is false. 09309082

(i) A line has two end points. 09309083

(ii) A line segment has one end point. 09309084

(i) A triangle is formed by three collinear points. 09309085

(ii) Each side of a triangle has two collinear vertices. 09309086

(iii) The end points of each side of a rectangle are collinear. 09309087

(iv) All the points that lie on the x-axis are collinear. 09309088

(v) Origin is the only point collinear with the points of both the axes separately. 09309089

Q.3 Find distance between pairs of points

09309090

(i) 09309091

(ii) 09309092

(iii) 09309093

Q.4 Find the midpoint between the following pairs of points. 09309094

Solution: (i) 09309095

(ii) 09309096

(iii) 09309197

Q.5 Define the following: 09309198

(i) Co-ordinate Geometry 09309199

(ii) Collinear points 09309100

(iii) Non-collinear points 09309101

(iv) Equilateral Triangle 09309102

(v) Scalene Triangle 09309103

(vi) Isosceles Triangle 09309104

(vii) Right Triangle 09309105

(viii) Square 09309106

Q6. Find distance between the points and . 09309107

Q7. Find distance between the points (1,0) and (0,1). 09309108

Q8. Find midpoint of the points (2,2) and (0,0). 09309109

Q9. Find midpoint of the points and . 09309110

43.

UNIT 10

Q. What is correspondence of Triangles?

093010001

Q. Define Congruency of Triangles. 093010002

Q. Define S.A.S Postulate. 093010003

Theorem 10.1.1: 093010004

In any correspondence of two triangles, if one side and any two angles of one triangle are congruent to the corresponding, side and angles of the other, then the triangles are congruent.

Given:

Example: 093010005

In any correspondence of two triangles, if one side and any two angles of one triangle are congruent to the correspondence side and angles of the other, then the triangles are congruent. (S.A.A  S.A.A.)

Given:

In ABC  DEF

, ,

Example: 093010006

If ABC and DCB are on the opposite sides of common base such that

, , , then bisects .

Given:

ABC and DCB are on the opposite sides of such that

Exercise 10.1:

Q.1 In the given figure. 093010007

Prove that

ABD CBE

Given:

1= 2

Q.2 From a point on the bisector of an angle, perpendiculars are drawn to the arms of the angle. Prove that these perpendiculars are equal in measure. 093010008

Given: ABC, the bisector of ABC, M any point on , perpendicular on , .

Q.3 In a triangle ABC, the bisectors of B and C meet in a point I. Prove that I is equidistant from the three sides of ABC. 093010009

Given:

In ABC, are the bisectors of the angles B and C respectively.

Theorem: 10.1.2 093010010

If two angles of a triangle are congruent, then the sides opposite to them are also congruent.

Given:

In ABC, B  C

Example 1: 093010011

If one angle of a right triangle is of 30o, the hypotenuse is twice as long as the side opposite to the angle.

Given:

In ABC, mB = 90o and mC = 30o

Example 2: 093010012

If the bisector of an angle of a triangle bisects the side opposite to it, the triangle is isosceles.

Given: In ABC, bisects A and 

Exercise: 10.2

Q.1 Prove that any two medians of an equilateral triangle are equal in measure. 093010013

Given:

An equilateral ABC, and two medians and

Q. 2 Prove that a point, which is equidistant from the end points of a line segment, is on the right bisector of the line segment. 093010014

Theorem 10.1.3: 093010015

In a correspondence of two triangles, if three sides of one triangle are congruent to the corresponding three sides of the other, then the two triangles are congruent. (S.S.S.  S.S.S.)

Given:

In C  DEF

Corollary::

If two isosceles triangles are formed on the same side of their common base, the line through their vertices would be the right bisector of their common base.

Given: ABC and DBC are formed on the same side of such that

Exercise 10.3:

Q.1 In the figure, , .

Prove that A  C, ABC  ADC. 093010016

Given:

Q.2 In the figure, , .

Prove that NP,NML PLM.

Given: 093010017

Q.3 Prove that the median bisecting the base of an isosceles triangle bisects the vertex and it is perpendicular to the base. 093010018

Given: In isosceles ABC, is the base and A is the vertex angle such that . Median meets side at point D.

Theorem 10.1.4 093010019

If in the correspondence of the two right-angled triangles, the hypotenuse and one side of one triangle are congruent to the hypotenuse and the corresponding side of the other, then the triangles are congruent. (H.S  H.S)

Given:

In ABC  DEF

B E (right angles)

Example: 093010020

If perpendiculars from two vertices of a triangle to the opposite sides are congruent, then the triangle is isosceles.

Given:

In ABC,

Such that 

Exercise 10.4

Q.1 In PAB of figure, , prove that  and APQ BPQ. 093010021

Given:

In PAB, and

Q.2 In the figure, mC = mD = 90o and  . Prove that 

and BAC ABD. 093010022

Given:

mC = mD = 90o

Q.3 In the figure, mB = mD = 90o and  . Prove that ABCD is a rectangle. 093010023

Given:

m B = m D = 90o,

To Prove:

ABCD is a rectangle

Review Exercise 10 OBJECTIVE

Choose the correct answer.

167. ________ triangle is an equiangular triangle. 093010024

(a) A scalene (b) An isosceles

168. A _______ has two end points 093010025

(a) line (b) line segment

169. A ________ has no end point 093010026

(a) line (b) line segment

170. A _______ has one end point 093010027

(a) line (b) line segment

171. In a triangle there can be only one:

(Board 2014) 093010028

(a) acute angle (b) right angle

172. Three points are said to be collinear, if they lie on the same: 093010029

(a) plane (b) line

173. Two lines can intersect at: 093010030

(a) one point (b) two points

174. Two ________ lines cannot intersect each other: 093010031

(a) perpendicular (b) parallel

175. If perpendiculars from two vertices of a triangle to the opposite sides are congruent, then the triangle is.

(a) scalene (b) isosceles 093010032

176. All the medians of _______ triangle are equal in measure. 093010033

(a) a scalene

(b) an isosceles

(d) a right angled

177. If the bisectors of the angles of a triangle bisect the sides opposite to them, the triangle is 093010034

(a) scalene (b) isosceles

178. If one angle of a right triangle is of ____ then hypotenuse is twice as long as the side opposite to this angle

(a) 60o (b) 45o 093010035

179. Symbol for congruent is: 093010036

(a) (b) N

180. Symbol for correspondence is 093010037

(a) (b) N

181. How many end points has a ray? 093010038

(a) 1 (b) 2 (Board 2015)

182. Symbolically two congruent triangles ABC and PQR are written as: 093010039

(a)

(b)

(c)

(d)

183. Which of the following is postulate?

(a) 093010040

(b)

(c)

(d)

184. If sum of measures of two angles is 180o then angles are ____ angles. 093010041

(a) Complementary (b) Supplementary

185. If sum of measure of two angles is 90o then angles are _____ angles. 093010042

(a) Complementary (b) Supplementary

186. Hypotenuse is a side opposite to _____ in right angled triangle. 093010043

(a) 30o (b) 60o

187. In equilateral triangle each angle is of ______. 093010044

(a) 30o (b) 60o

188. Corresponding sides of congruent triangles are: 093010045

(a) equal (b) different

189. Median bisecting the base angle of an isosceles triangle bisects the _____ angle. 093010046

(a) base (b) vertical

190. The median bisecting the base of an isosceles triangle is ___ to the base.

093010047

(a) parallel (b) perpendicular

191. Corresponding angles of congruent triangles are: 093010048

(a) congruent (b) non-congruent

192. Any two medians of an ____ triangle equal is measure. 093010049

(a) isosceles (b) equilateral

193. An equilateral triangle is ____ triangle.

093010050

(a) acute (b) obtuse

Q.1 Which of the following are true and which are false? 093010051

(i) A ray has two end points. 093010052

(ii) In a triangle, there can be only one right angle. 093010053

(iii) Three points are said to be collinear if they lie on same line. 093010054

(iv) Two parallel lines intersect at a point. 093010055

(v) Two lines can intersect only in one point. 093010056

(vi) A triangle of congruent sides has non-congruent angles. 093010057

Q.2 If ABC LMN, then 093010058

(i) mM 093010059

(ii) mN 093010060

(iii) mA 093010061

Q.3 If ABC LMN, then find the unknown x. 093010062

Q.4 Find the value of unknowns for the given congruent triangles. 093010063

ABD  ACD

Q.5 If PQR ABC, then find the unknowns. 093010064

UNIT 11

Theorem 11.1.1: 093011001

In a parallelogram

(i) Opposite sides are congruent.

(ii) Opposite angles are congruent.

(iii) The diagonals bisect each other.

Given: In a quadrilateral ABCD, and the diagonals , meet each other at point O.

Example: 093011002

The bisectors of two angles on the same side of a parallelogram cut each other at right angles.

Given:

A parallelogram ABCD, in which

The bisectors of A and B cut each other at E.

EXERCISE 11.1:

Q.1 One angle of a parallelogram is 130o. Find the measures of its remaining angles. 093011003

Given:

ABCD is a parallelogram that

mA = 130o

To Find

The measures of B, C, D

Q.2 One exterior angle formed on producing one side of a parallelogram is 40o. Find the measures of its interior angles.

Given: 093011004

ABCD is a parallelogram, side AB has been produced to p to form exterior angle mCBP = 40o and name the interior angles as 1, C, D, A.

Required:

To find the degree measures of 1, C, D, A

Theorem 11.1.2: 093011005

If two opposite sides of a quadrilateral are congruent and parallel, it is a parallelogram.

Given:

In a quadrilateral ABCD,

EXERCISE 11.2:

Q. 1 Prove that a quadrilateral is a parallelogram if its 093011006

(a) Opposite angles are congruent. 093011007

(b) Diagonals bisect each other. 093011008

Given: Given ABCD is a quadrilateral.

mA = mC,

mB = mD

To prove: ABCD is a parallelogram.

Q. 2 prove that a quadrilateral is a parallelogram if its opposite sides are congruent.

Given 093011009

In quadrilateral

ABCD, ,

Required:

ABCD is a || gm

Construction:

Join point B to D and name the angles 1, 2, 3 and  4

Theorem 11.1.3:

The line segment, joining the mid-points of two sides of a triangle, is parallel to the third side and is equal to one half of its length. 093011010

Given: In ABC, the mid-points of and are L and M respectively.

To Prove:

and

Construction:

Join M to L and produce to N such that . Join N to B and in the figures name the angles 1, 2, 3 and  4 as shown.

Example: 093011011

The line segments, joining the mid-points of the sides of a quadrilateral, taken in order, form a parallelogram.

Given:

A quadrilateral ABCD, in which P is the mid-point of , Q is the mid-point of , R is the mid-point of , S is the mid-point of .

P is joined to Q, Q is joined to R. R is joined to S and S is joined to P.

To prove:

PQRS is a parallelogram.

Construction:

Join A to C.

EXERCISE 11.3:

Q.1 Prove that the line-segments joining the mid-points of the opposite sides of a quadrilateral bisect each other. 093011012

Given:

ABCD is a quadrilateral.

P, Q, R, S are the mid-points of respectively.

P is joined to R, Q is joined to S. intersect at point “O”

Q.2 Prove that the line-segments joining the mid-points of the opposite sides of a rectangle are the right-bisectors of each other. 093011013

Given:

ABCD is a rectangle.

and P, Q, R, S are the mid-points of sides , , respectively.

P is joined to R, S to Q These intersect at “O”

To Prove:

and

Note: Diagonals of a rectangle are congruent.

Q.3 Prove that the line-segment passing through the mid-point of one side and parallel to another side of a triangle also bisects the third side. 093011014

Given: In ABC, D is mid-point of which meets at E.

Required:

E is mid-point of

Theorem 11.1.4: 093011015

The medians of a triangle are concurrent and their point of concurrency is the point of trisection of each median.

Given:

ABC

To Prove:

The medians of the ABC are concurrent and the point of concurrency is the point of trisection of each median.

EXERCISE 11.4:

Q. 1 The distances of the point of concurrency of the medians of a triangle from its vertices are respectively 1.2cm; 1.4 cm and 1.5 cm. Find the lengths of its medians. 093011016

Given: Let ABC be a triangle with center of gravity at G where , ,

Required: To find the length of ,

Q. 2 Prove that the point of concurrency of the medians of a triangle and the triangle which is made by joining the mid-points of its sides is the same. 093011017

Given:

In ABC, are its medians that are concurrent at point G.

is formed by joining mid-points of

To Prove:

Point G is point of concurrency of triangle PQR.

Theorem 11.1.5: 093011018

If three or more parallel lines make congruent segments on a transversal, they also intercept congruent segments on any other line that cuts them.

Given:

The transversal intersects at the

points M, N and P respectively, such that .

The transversal intersects them at points R, S

and T respectively.

To Prove:

Corollaries: (i) A line, through the mid-point of one side, parallel to another side of a triangle, bisects the third side. 093011019

Given: In ABC, D is the mid-point of . which cuts at E.

To prove:

Exercise 11.5:

Q. 1 In the given figure. and if m =1cm then find the length of and 093011020

Given: In given figure ,

Required:: To find

Q. 2 Take a line segment of length 5cm and divide it into five congruent parts. 093011021

[Hint: Draw an acute angle BAX. On take .

Joint T to B. Draw line parallel to from the points P, Q, R and S.]

Review Exercise 11 OBJECTIVE

Choose the correct answer.

194. In a parallelogram opposite sides are…

(Board 2014) 093011022

(a) different (b) perpendicular

195. In a parallelogram opposite angles are ……………. : 093011023

(a) parallel (b) congruent

196. Diagonals of a parallelogram …….. each other at a point. 093011024

(a) perpendicular to (b) intersect

197. Medians of triangle are………. 093011025

(a) equal (Board 2015)

(b) concurrent

(d) parallel

198. Diagonal of a parallelogram divides the parallelogram into ……. triangles.

(Board 2013)093011026

(a) two equal (b) two different

199. In a parallelogram shown in fig. yo = …… 093011027

(a) 115o (b) 90o

200. In a parallelogram shown in fig. xo = …… 093011028

(a) 115o (b) 90o

201. In a parallelogram shown in fig. xo………… 093011029

(a) 55o (b) 5o

202. In a parallelogram shown in fig. m=……… 093011030

(a) 8 (b) 10

203. In ABC E and D are midpoints of the sides and respectively. Find the value of m . 093011031

(a) 6cm (b) 9cm

204. In parallelogram congruent parts are:

(Board 2015) 093011032

(a) Opposite sides

(b) Diagonals

(d) Opposite sides and angles

205. Alternate angles on parallel lines intersected by a transversal are_____. 093011033

(a) Congruent

(b) Non-congruent

(d) Supplementary

206. Corresponding angles on parallel lines intersected by a transversal are ____. 093011034

(a) Congruent

(b) Non-congruent

(d) Supplementary

207. If two lines intersect each other, then vertical angles so formed are ______. 093011035

(a) Congruent

(b) Non-congruent

(d) Supplementary

208. Diagonals of a rectangle are ____. 093011036

(a) Congruent

(b) Non-congruent

(d) Parallel

209. Which of the following is true for a parallelogram ABCD? 093011037

(a) (b)

210. Symbolically two parallel lines AB and PQ written as: 093011038

(a) (b)

211. The point of concurrency of median is the point of _____ of each median. 093011039

(a) bisection (b) trisection

Q.1. Fill in the blanks.

(i) In a parallelogram opposite sides are….. 093011040

(ii) In a parallelogram opposite angles are ……. 093011041

(iii) Diagonals of a parallelogram ….. each other at a point. 093011042

(iv) Medians of a triangle are …………. 093011043

(v) Diagonal of a parallelogram divides the parallelogram into two ……….. triangles. 093011044

2. In parallelogram ABCD 093011045

(i) 093011046

(ii) 093011047

(iii) m1 m3 093011048

(iv) m2 m4 093011049

Q. 3 Find the unknowns in the given figure. 093011050

Given: Let ABCD be the given figure with

To Find: The values of mo, no, xo and yo

Q.4 If the given figure ABCD is a parallelogram, then find x, m. 093011051

Given: ABCD is a parallelogram with angles as shown in figure.

To Find: The value of xo and mo

Q. 5 The given figure LMNP is a parallelogram.

Find the value of m, n. 093011052

Given: The parallelogram LMNP with lengths and angles as shown in figure.

To find: The values of mo and no

Q.6 In the question 5, sum of the opposite angles of the parallelogram is 110o, find the remaining angles. 093011053

Given: LMNP is a parallelogram with angles 55o, 55o as shown

To Find: All angles

44.

UNIT 12

Q. Define Right Bisector of a Line Segment. 093012001

Q. Define Angle Bisector. (Board 2014) 093012002

Theorem: 12.1.1: 093012003

Any point on the right bisector of a line segment is equidistant from its end points.

Given::

A line LM intersects the line segment AB at the point C such that and P is a point on

To Prove::

Theorem 12.1.2: (Board 2014) 093012004

Any point equidistant from the end points of a line segment is on the right bisector of it.

Given: is a line segment. Point P is such that .

To Prove: The Point P is on the right bisector of .

Construction:: Joint P to C, the midpoint of

Exercise 12.1:

Q. 1 Prove that the centre of a circle is on the right bisectors of each of its chords. 093012005

Given: Circle with centre O. Draw any chord .

To Prove:Centre of the circle is on right bisectors of each of its chords

Construction:Draw Join O with A and B.

Q. 2 Where will be the centre of a circle passing through three non-collinear points and why? (Board 2014) 093012006

Q. 3 Three villages P, Q and P are not on the same line. The people of these villages want to make a Children Park at such a place which is equidistant from these three villages. After fixing the place of children Park, prove that the Park is equidistant from the three villages.

093012007

Given: Three villages P, Q, and R not on the same line and a park O

To Prove:

A place for park O is equidistant from villages P,Q and R.

Theorem 12.1.3: (Board 2013, 15) 093012008

The right bisectors of the sides of a triangle are concurrent.

Given: ABC

To Prove:

The right bisectors of and are concurrent.

Theorem 12.1.4 (Board 2013) 093012009

Any point on the bisector of an angle is equidistant from its arms.

Given:

A point P is on , the bisectors of AOB.

To Prove

i.e., P is equidistant from and

Theorem 12.1.5: Any point inside an angle, equidistant from its arms, is on the bisector of it. (Board 2015) 093012010

Given: Any point P lies inside AOB such that where and

To Prove: Point P is on the bisector of AOB.

Construction: Join P to O.

Exercise 12.2:

Q.1 In a quadrilateral ABCD, and the right bisectors of meet each other at point N. prove that is a bisector of ABC. 093012011

Given: Quadrilateral ABCD in which . Right bisectors of meet each other at point N.

To prove: is a bisector of ABC

Q.2 The bisectors of A, B and C of a quadrilateral ABCP meet each other at point O. Prove that the bisectors of P will also pass through the point O. 093012012

Given:Bisector of the angles A, B, C meet at O.

To Prove:

Bisector of P will also pass through O.

Theorem 12.1.6: (Board 2014) 093012013

The bisectors of the angles of a triangle are concurrent.

Given: ABC

To Prove:

The bisectors of A, B and C are concurrent.

Construction:

Draw the bisectors of B and C which intersect at point I. From I, draw , and .

Exercise 12.3:

Q.1 Prove that the bisector of the angles of base of an isosceles triangle intersect each other on its altitude. 093012014

Given:An isosceles ABC and A and B are its base angles. and intersecting at O are angle bisectors of A and B respectively .

To prove: point O is on the altitude of ABC

Construction:Draw perpendicular (altitude) from the vertex C to the base

Q.2 Prove that the bisector of two exterior and third interior angles of a triangle are concurrent: 093012015

Given:A ABC, angle bisector of its interior A,

and are angle bisectors of its two exterior angles

B and C respectively which intersect each other at point I.

To prove: and are concurrent

Construction: Draw perpendiculars and from point I to the produced side and respectively

45. Review Exercise 12 OBJECTIVE

Choose the correct answers

212. Bisection means to divide into ___ equal parts 093012016

(a) Two (b) Three

213. __ of line segment means to draw perpendicular which passes through the midpoint of line segment.093012017

(a) Right bisection

(b) Bisection

(d) Mid-point

214. Any point on the _____ of a line segment is equidistant from its end points: 093012018

(a) Right bisector (b) Median

(b) Angle bisector (d) Altitude

215. Any point equidistant from the end points of line segment is on the ____ of it: 093012019

(a) Right bisector (b) Median

(b) Angle bisector (d) Altitude

216. The bisectors of the angles of a triangle are: (Board 2015) 093012020

(a) Concurrent (b) Congruent

217. Bisection of an angle means to draw a ray to divide the given angle into ___ equal parts: 093012021

(a) Four (b) Three

218. If is right bisector of line segment then: (i) 093012022

(a) (b)

219. If is right bisector of line segment , then =____ 093012023

(a) (b)

220. The right bisectors of the sides of an acute triangle intersects each other ___ the triangle. 093012024

(a) Inside (b) Outside

221. The right bisectors of the sides of a right triangle intersect each other on the ___ 093012025

(a) Vertex (b) Midpoint

222. The right bisectors of the sides of an obtuse triangle intersect each other ___ the triangle. 093012026

(a) Outside (b) Inside

223. The point of line segment through which the right bisector passes is called its _____ point. 093012027

(a) end (b) mid

224. The ______ of a circle is on the right bisectors of each of its chords. 093012028

(a) radius (b) centre

225. The point of intersection of right bisectors of sides of a triangle is equidistant from the ____ of triangle.

(a) sides (b) vertices 093012029

226. The point of intersection of right bisectors of sides of a triangle is equidistant from the _______ of triangle. 093012030

(a) one vertex (b) two vertices

227. The altitudes of a triangle are _____.

093012031

(a) congruent (b) concurrent

228. The bisectors of two exterior angles and third interior angle of a triangle are __________. 093012032

(a) congruent (b) concurrent

229. The bisectors of the base angles of an isosceles triangle intersect each other on its ______. 093012033

(a) base (b) vertex

Q.1 Which of the following are true and which are false?

(i) Bisection means to divide into two equal parts. 093012034

(ii) Right bisection of line segment means to draw perpendicular which passes

through the mid-point of line segment 093012035

(iii) Any point on the right bisector of a line segment is not equidistant from its

end points. 093012036

(iv) Any point equidistant from the end points of a line segment is on the right

bisector of it. 093012037

(v) The right bisectors of the sides of a triangle are not concurrent. 093012038

(vi) The bisectors of the angles of a triangle are concurrent. 093012039

(vii) Any point on the bisector of an angle is not equidistant from its arms 093012040

(viii) Any point inside an angle, equidistant from its arms, is on the bisector of it.

093012041

Q.2 If is right bisector of line segment , then: 093012042

Q.3 Define the following 093012043

(i) Bisector of a line segment

(ii) Bisector of an angle

Q.4 The given triangle ABC is equilateral triangle and is bisector of angle A , then find the values of unknowns xo, yo and zo. (Board 2013) 093012044

Q. 5In the given congruent triangles LMO and LNO find the unknown x and m. 093012045

is right bisector of the line segment . 093012046

(i) If then find the and 093012047

(ii) If , then find . 093012048

Given: CD is a right bisector on the line segment AB.

To find(i)

(ii)

UNIT 13

Theorem 13.1.1: If two sides of a triangle are unequal in length, the longer side has an angle of greater measure opposite to it. 093013001

Given: In ABC,

To Prove: mABC > mACB

Construction:On take a point D such that Join B to D so that ADB is an isosceles triangle. Label 1 and 2 as shown in the given figure.

Example 1: Prove that in a scalene triangle, the angle opposite to the largest side is of measure greater than 60o. (i.e., two-third of a right-angle). 093013002

Given:InABC, > , > .

To Prove:

mB > 60o.

Example 2: In a quadrilateral ABCD, is the longest side and is the shortest side. Prove that mBCD > mBAD. 093013003

Given: In quad. ABCD, is the longest side and is the shortest side.

To Prove: mBCD > mBAD

Theorem: 13.1.2: (Converse of theorem 13.1.1) 093013004

If two angles of a triangle are unequal in measure, the side opposite to the greater angle is longer than the side opposite to the smaller angle.

Given: In ABC, mA > mB

Example: 093013005

ABC is an isosceles triangle with base . On a point D is taken away from C. A line segment though D cuts at L and at M. Prove that .

Given:

In ABC, .

D is a point on away from C.

A line segment through D cuts and L and at M.

Theorem: 13.1.3: 093013006

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Given:

ABC

To Prove:

(i)

(iii)

Construction:

Take a point D on such that . Join B to D and name the angles. 1, 2 as shown in the given figure.

Example 1: 093013007

Which of the following sets of lengths can be the lengths of the sides of a triangle.

(a) 2cm, 3cm, 5cm

(b) 3cm, 4cm, 5 cm

Example 2:Prove that the sum of the measures of two sides of a triangle is greater than twice the measure of the median which bisects the third side. 093013008

Given:

In ABC,

median AD bisects side at D.

To Prove:

Example 3: 093013009

Prove that the difference of measures of two sides of a triangle is less than the measure of the third side.

Given:ABC

To Prove:

Exercise 13.1:

Q. 1 Two sides of a triangle measure 10 cm and 15 cm. Which of the following measure is possible for the third side?

(a) 5 cm 093013010 (b) 20 cm 093013011

Ans. 20cm.

Q. 2 O is an interior point of the ABC. Show that

093013014

Given: O is the interior point of ABC

To Prove::

Q. 3 In the ABC, mB = 70o and mC = 45o. Which of the sides of the triangle is longest and which is the shortest? 093013015

Q. 4 Prove that in a right-angled triangle, the hypotenuse is longer than each of the other two sides. 093013016

Given:A right angled triangle ABC and its hypotenuse .

Q. 5 In the triangular figure, and are the bisectors of B and C respectively. Prove that 093013017

Given: m , and are the bisectors of the angles B and C

To Prove::

Theorem: 13.1.4:From a point, outside a line, perpendicular is the shortest distance from the point to the line. 093013018

Given: A line AB and a point C (not lying on ) and a point D on such that

To Prove:

is the shortest distance from the point C to .

Construction:

Take a point E on . Join C and E to form a CDE

Exercise 13.2:

Q. 1 In the figure, P is any point and AB is a line. Which of the following is the shortest distance between the point P and the line AB. 093013019

(a) (b)

Q. 2 In the figure, P is any point lying away from the line AB. Then will be the shortest distance if: 093013020

(a) mPLA = 80o 093013021

(b) mPLB = 100o 093013022

Q. 3 In the figure, is perpendicular to the line AB and > . Prove that . 093013024

Proof::

Here

As is the shortest distance from P to line AB. So

As we go away from point L, the distance from points to L increases Hence

46. Review Exercise 13 OBJECTIVE

Choose the correct answer:

230. Which of the following sets of lengths can be the lengths of the sides of a triangle: 093013025

(a) 2cm, 3cm, 5cm

(b) 3cm, 4cm, 5cm

(d) 1cm, 2cm, 3cm

231. Two sides of a triangle measure 10cm and 15cm. Which of the following measure is possible for the third side! 093013026

(a) 5cm (b) 20cm

232. In the figure, P is any point and AB is a line. Which of the following is the short distance between the point P and line AB. 093013027

(a) (b)

233. In the figure, P is any point lying away from the line AB. Then will be shortest distance if: 093013028

(a) m< PLA = 80o

(b) m < PLB = 100o

(d) None

234. The angle opposite to the longer side is: 093013029

(a) Greater (b) Shorter

235. In right angle triangle greater angle of: 093013030

(a) 60o (b) 30o

236. In an isosceles right-angled triangle angles other than right angle are each of: 093013031

(a) 40o (b) 45o

237. A triangle having two congruent sides is called ___ triangle. 093013032

(a) Equilateral

(b) Isosceles

(d) None

238. Perpendicular to line form an angle of __ 093013033

(a) 30o (b) 60o

239. Sum of two sides of triangle is ___ than the third. 093013034

(a) Greater (b) Smaller

240. The distance between a line and a point on it is ___ 093013035

(a) Zero (b) One

241. The difference of two sides of a triangle is ___ the third side.093013036

(a) greater than (b) smaller than

242. In a triangle, the side opposite to greater angle is_____. 093013037

(a) smaller (b) greater

243. In a triangle the angles opposite to congruent sides are ____. 093013038

(a) congruent (b) concurrent

244. In a triangle, the side opposite to smaller angle is ____. 093013039

(a) smaller (b) greater

245. An exterior angle of a triangle is ___ non-adjacent interior angle.093013040

(a) equal to (b) smaller than

246. For a , which of the following is true? 093013041

(a)

(b)

(c)

(d)

247. For a , which of the following is true? 093013042

(a)

(b)

(c)

(d)

248. What is the supplement of a right angle? 093013043

(a) (b)

249. The sum of the measures of two sides of a triangle is greater than_____ the measure of the median which bisects the third side. 093013044

(a) twice (b) thrice

250. In an obtuse angled triangle, the side opposite to the obtuse angle is ____ than each of the other two sides.

093013045

(a) smaller (b) longer

Q. 1 Which of the following are true and which are false?

(i) The angle opposite to the longer side is greater. 093013046

(ii) In a right-angled triangle greater angle is of 60o. 093013047

(iii) In an isosceles right-angled triangle, angles other than right angle are each of 45o.

093013048

(iv) A triangle having two congruent sides is called equilateral triangle.

(Board 2015)093013049

(v) A perpendicular from a point to the line is shortest distance. 093013050

(vi) Perpendicular to line form an angle of 90o. 093013051

(vii) A point out-side the line is collinear. 093013052

(viii) Sum of two sides of triangle is greater than the third. 093013053

(ix) The distance between a line and a point on it is zero. 093013054

(x) Triangle can be formed of lengths 2 cm, 3 cm and 5 cm. 093013055

Q.2 What will be angle for shortest distance from an outside point to the line? 093013056

Q.3 If 13 cm, 12 cm, and 5 cm are the lengths of a triangle, then verify that difference of measures of any two sides of a triangle is less than the measure of the third side. 093013057

Q.4 If 10 cm,6 cm and 8 cm are the lengths of a triangle, then verify that sum of measures of two sides of a triangle is greater than the third side. 093013058

Q.5 3 cm, 4 cm and 7cm are not the lengths of the triangle. Give the reason. 093013059

Q. 6 If 3 cm and 4 cm are lengths of two sides of a right angle triangle then what should be the third length of the triangle. 093013060

Q7. 3 cm,6cm and 9 cm are not lengths of triangle. Why? (Board 2015) 093013061

Q8. Define acute angled triangle.(Board 2015)

093013062

Q8. Define obtuse angled triangle. 093013063

Q9. Which side is longer in a triangle?

093013064

Q10. Which side is longer in right triangle?

093013065

Q11. Which side is longer in obtuse angled triangle? 093013066

Q12. What is the distance between a line and a point not on it? 093013067

Q13. What is the distance between a line and a point on it? 093013068

47.

UNIT 14

Theorem: 14.1.1

A line parallel to one side of a triangle and intersecting the other two sides divides them proportionally. 093014001

Given

In the line is intersecting the sides and at points E and D respectively such that .

To Prove

Theorem 14.1.2 093014002

(Converse of Theorem)

If a line segment intersects the two sides of a triangle in the same ratio then it is parallel to the third side.

Given In intersects and such that

To Prove

Construction If , then draw to meet produced at F.

Exercise 14.1

Q. 1 In 093014003

i) cm, cm,

cm then find . 093014004

ii) If cm, cm,

cm, find 093014005

iii) If , cm, find 093014006

iv) If cm, cm, (L.B.2014)

cm, cm, find

093014007

v) If 093014008

, and find the value of 093014009

Q. 2 If is an isosceles triangle, is vertex angle and intersects the sides and as shown in the figure so that. 093014010

Prove that is also an isosceles triangle.

Given: An isosceles triangle ABC in which and A is its vertical angle

To Prove:

ADE is an isosceles triangle.

Q. 3 In an equilateral triangle ABC shown in the figure. 093014011

Find all three angles of and name it also.

Given: is an equilateral triangle.

To Prove: is isosceles triangle.

Q. 4 Prove that the line segment drawn through the mid-point of

one side of a triangle and parallel to another side bisects the third side.

Given in is such that and 093014012

To Prove:

Q. 5 Prove that the line segment joining the mid-points of any two sides of a triangle is parallel to the third side. 093014013

Given: In , points D, E are such that

and

To Prove:

Construction: Let and take a point k on produced

such that .

Theorem: 14.1.3

The internal bisector of an angle of a triangle divides the side opposite to it in the ratio of the lengths of the sides containing the angle. 093014014

Given: In internal angle bisector of

meets at the point D.

To Prove:

Construction:

Draw a line segment to meet produced at E.

Theorem: 14.1.4 If two triangles are similar, then the measures of their corresponding sides are proportional. 093014015

Given:

i.e., and

To Prove:

Construction:

(i) Suppose that (ii)

On take a point L such that

On take a point M such that . Join L and M by the line segment LM.

a. Exercise 14.2

Q. 1 In as shown in the figure, bisects and meets at D, is equal to

(a) 5 093014016 (b) 16 093014017

Q. 2 In as shown in the figure, bisects . If and then find and .

093014020

Q. 3 Show that in any correspondence of two triangles if two angles of one triangle are congruent to the corresponding angles of the other, then the triangles are similar. 093014021

Given: In

and

To Prove:

Q. 4 If line segments and are intersecting at point X and then show that and are similar. 093014022

Given:

intersect each other at point x and

To Prove:

48. Review Exercise 14 OBJECTIVE

Choose the correct answers

251. In ABC as shown in figure, bisects C and meets at D, m is equal to: 093014023

(a) 5

(b) 16

(d) 18

252. In ABC shown in figure, bisects C, if , and then = 093014024

(a) (b)

253. In ABC shown in figure, bisects

C, if , and then 093014025

(a) (b)

254. One and only one line can be drawn through ___ points. 093014026

(a) Two (b) Three

255. The ratio between two alike quantities is defined as: 093014027

(a) a : b (b) b - a

256. If a line segment intersects the two sides of a triangle in the same ratio then it is parallel to the __ side.

(a) Third (b) Fourth 093014028

257. Two triangles are said to be similar if these are equiangular and their corresponding sides are 093014029

(a) Proportional (b) congruent

258. In LMN shown in the figure if = 5cm, , then : 093014030

(a) 4.6cm (b) 4.5cm

9. A line segment has ________mid-point

(a) only one (b) only two 093014031

10. Ratio has no (Board 2014) 093014032

(a) value (b) symbol

11. Statement of equality of two ratios is called ……. 093014033

(a) double ratio (b) simple ratios

12. The symbol used for similarity is……

(a) = (b) 093014034

13. The symbol used for congruency is …..

(a) = (b) 093014035

14. The symbol used for ratio is …….

(a) (b) 093014036

15. The ratio between two alike quantities has no…… 093014037

(a) value (b) symbol

16. The symbol used for line AB is ……

(a) AB (b) 093014038

17. The symbol used for ray AB is …….

(a) AB (b) 093014039

18. The symbol used for line segment AB is ……. 093014040

(a) AB (b)

19. stands for …….. 093014041

(a) line AB (b) Ray AB

20. stands for …….. 093014042

(a) line AB (b) Ray AB

21. stands for …….. 093014043

(a) line AB (b) Ray AB

22. The symbol used for parallel is …….

(a) = (b) || 093014044

23. The symbol used for perpendicular is…

(a) = (b) || 093014045

24. Which of the following show that and are parallel? 093014046

(a) (b)

25. Which of the following show that and are perpendicular? 093014047

(a) (b)

26. Two congruent triangles ABC and DEF are symbolically written as……. 093014048

(a)

(b)

(c)

(d)

27. Two similar triangles ABC and DEF are symbolically written as …… 093014049

(a)

(b)

(c)

(d)

28. Correspondence between two triangles ABC and DEF are symbolically written as…….. 093014050

(a)

(b)

(c)

(d)

29. Symbol used for proportion is….

(a) (b) 093014051

30. Proportion is a equality of …… ratios.

(a) Two (b) Three 093014052

31. Similar triangles are of the same shape but …… in sizes. (Board 2015) 093014053

(a) The same (b) Different

(d) None of these

32. is the symbol of: (Board 2015) 093014054

(a) equal (b) parallel

33. Development of prints of different sizes from the same negative of a photograph is an example of _______. 093014055

(a) Congruency (b) Similarity

34. In a ratio elements must be expressed in ______ units. 093014056

(a) Different (b) Similar

35. At least how many points determine a line? 093014057

(a) one (b) two

36. At least how many non-collinear points determine a plane? 093014058

(a) one (b) two

37. If two intersecting lines form equal adjacent angles, the lines are ___. 093014059

(a) parallel (b) perpendicular

38. The line segment obtained by joining the midpoints of any two sides of a triangle is ______ third side. 093014060

(a) parallel to (b) perpendicular to

39. Congruent Triangles are also: 093014061

(a) similar

(b) different shapes

(d) none

40. Triangles are of same size and shape:

(a) similar (b) congruent 093014062

Q. 1 Which of the following are true and which are false?

i. Congruent triangles are of same size and shape. 093014063

ii. Similar triangles are of same shape but different sizes. 093014064

iii. Symbol used for congruent is ‘ ’. 093014065

iv. Symbol used for similarity is ‘~’. 093014066

v. Congruent triangles are similar. 093014067

vi. Similar triangles are congruent. 093014068

vii. A line segment has only one mid-point. 093014069

viii. One and only one line can be drawn through two points. 093014070

ix. Proportion is non-equality of two ratios. 093014071

x. Ratio has no unit. 093014072

Q. 2 093014073

(i) Define Ratio.

(ii) Define Proportion. (Board 2014, 15)

(iii)Define Congruency of Triangles.

(iv) Define Similar Triangles.

Q. Write to practical applications of similar triangles in daily life. 093014074

Q. How many midpoint a line segment has? 093014075

Q. How many lines can be drawn through two points? 093014076

Q. Why does ratio has no unit? 093014077

Q. 3 In show in the figure, .

i) If 5cm, 2.5cm, 2.3cm, then find .

093014078

(ii) If 6cm, , 5cm, then find

Given: In 093014079

To Find: = ?

___________________________________________________________________________________________________________________________________________

(ii)

Given:

To Find:

Q.4 In the shown figure, let , 093014080

. Find the value of if .

If || then

Q.5 In LMN shown in the figure bisects . If , then find and .

(Board 2015) 093014081

Given:

In is angle bisector of

To Find:

_________________________________________________________________________________

Q.6 In Isosceles shown in the figure, find the value of and . 093014082

Given:

In and .

To Find: x = ? y = ?

49.

UNIT 15

Pythagoras Theorem 15.1.1: 093015001

In a right angled triangle, the square of the length of hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Given:ACB is a right angled triangle in which m C = 90o and =a, = b and = c.

To Prove:

Construction:

Draw perpendicular from C on .

Let = h, = x and = y. Line segment CD

splits ABC into two s ADC and BDC.

Corollary:

In a right angled ABC, right angled at A.

(i)

(ii)

Converse of Pythagoras’ Theorem:

Q. What is meant by Converse of Pythagoras’ Theorem? (Board 2015) 093015002

Given:

In a ABC, and = b such that a2 + b2 = c2.

To Prove:

ACB is a right angled triangle.

Construction:

Draw perpendicular to such that  . Join the points B and D.

Exercise 15:

Q.1 Verify that the s having the following measures of sides are right-angled.

(i) a = 5 cm, b = 12 cm, c = 13 cm 093015003

(ii) a = 1.5 cm, b = 2 cm, c = 2.5 cm 093015004

(iii) a = 9 cm, b = 12 cm, c = 15 cm 093015005

(iv) a = 16 cm, b = 30 cm, c = 34 cm 093015006

Q.2 Verify that a2 + b2, a2 b2 and 2ab are the measures of the sides of a right angled triangle where a and b are any two real numbers (a > b). 093015007

Q.3 The three sides of a triangle are of measure 8, x and 17 respectively. For what value of x will it become base of a right angled triangle? (Board 2014) 093015008

Q.4 In an isosceles , the base = 28 cm, and = 093015009

If , then find:

(i) Length of 093015010

(ii) Area of ABC 093015011

Given

Q.5 In a quadrilateral ABCD, the diagonals and are perpendicular to each other. Prove that: 093015012

Given: Quadrilateral ABCD diagonal and

are perpendicular to each other.

To Prove: 2+ 2 = 2+ 2

Q.6 (i) In the ABC as shown in the figure, mACB = 90o and . Find the lengths a, h and b if = 5 units and units. 093015013

Given: A  ABC as shown in figure ,m ACB = 900and

To Find The value of a, h and b.

(ii)Find the value of x in the shown figure. 093015014

Q.7 A plane is at a height of 300 m and is 500 m away from the airport as shown in the figure. How much distance will it travel to land at the airport?

093015015

Q.8 A ladder 17 m long rests against a vertical wall. The foot of the ladder is 8m away from the base of the wall. How high up the wall will the ladder reach? 093015016

Q.9 A student travels to his school by the route as shown in the figure. Find the direct distance from his house to school. 093015017

__________________________________________________________________________

Review Exercise 15 OBJECTIVE

Choose the correct answer:

259. In a right angled triangle, the square of the length of hypotenuse is equal to the ____ of the squares of the lengths of the other two sides. 093015018

(a) Sum (b) Difference

260. If the square of one side of a triangle is equal to the sum of the squares of the other two sides then the triangle is a ____ triangle. 093015019

(a) Right angled (b) Acute angled

(d) None of these

261. Let c be the longest of the sides a, b and c of a triangle. If a2 +b2 = c2, then the triangle is ___: 093015020

(a) Right (b) Acute

262. Let c be the longest of the sides a, b and c of a triangle. If a2 + b2> c2 then triangle is: 093015021

(a) Acute (b) Right

263. Let c be the longest of the sides a, b and c of a triangle. If a2+b2< c2, then the triangle is: 093015022

(a) Acute (b) Right

264. If 3cm and 4cm are two sides of a right angled triangle, then hypotenuse is; 093015023

(a) 5cm (b) 3cm

265. In right triangle ____ is a side opposite to right angle. 093015024

(a) Base (b) Perpendicular

266. In the fig. 093015025

(a) x = 6cm (b) x = 8cm

267. In the fig. 093015026

(a) x = 5cm (b) x = 8cm

268. In the fig. 093015027

(a) x = 2cm (b) x = 1cm

269. In right angled triangle greater angle is ________. 093015028

(a) (b)

270. In right angled triangle on angle is and other two angles are_____.093015029

(a) obtuse (b) acute

271. If hypotenuse of an isosceles right angled triangle is then each of other side is: 093015030

(a) 1cm (b) 2cm

272. In right angled triangle which side is the longest side? 093015031

(a) perpendicular

(b) base

(d) none of these

273. In right angled triangle if then which of the following is true?

093015032

(a) (b)

274. In a Isosceles right angled triangle two acute angles are equal to: 093015033

(a) 30o (b) 45o

Q.1 Which of the following are true and which are false?

(i) In a right angled triangle greater angle is 90o. 093015034

(ii) In a right angled triangle right angle is 60o. 093015035

(iii) In a right triangle hypotenuse is a side opposite to right angle. 093015036

(iv) If a, b, c are sides of right angled triangle with c as longer side then c2 = a2 + b2. 093015037

(v) If 3 cm and 4 cm are two sides of a right angled triangle, then hypotenuse is 5 cm. 093015038

(vi) If hypotenuse of an isosceles right triangle is cm then each of other side

is of length 2 cm. 093015039

Q. 2 Find the unknown value in each of the following figures.

(Board 2014) 093015040

093015041

(iii) If two sides of a triangle are 5cm and 13cm, then find perpendicular of triangle. (Board 2015) 093015042

(Board 2013, 15) 093015043

Q. If a2 + b2 = c2 what kind of triangle it is?

093015044

Q. If a2 + b2 > c2 what kind of triangle it is?

093015045

Q. Define hypotenuse of right angled triangle.

093015046

Q. If a2 + b2 < c2 what kind of triangle it is? 093015047

Q. Which angle is greater angle in right angled triangle? 093015048

UNIT 16

Q. Define Area of a Figure. 093016001

Q. Define interior of triangle.

Q. Define Triangular Region. 093016002

Q. What is Congruent Area Axiom? 093016003

Q. Define interior of a rectangle. 093016004

Q. Define Rectangular Region. 093016005

Q. In how many triangular regions a rectangular region can be divided? 093016006

Q. What is meant by ||gm between the same parallels? 093016007

Q. What is meant by the triangles between the same parallels? 093016008

Q. What is meant by the triangle and parallelogram between the same parallels?

093016009

Q. Define Altitude of Parallelogram.

093016010

Q. Define Altitude of the Triangle. 093016011

Q. Under what condition areas of a parallelogram and rectangle are equal?

093016012

__________________________________________________________________________

Theorem 16.1.1: Parallelograms on the same base and between the same parallel lines (or of the same altitude) are equal in area. 093016013

Given: Two parallelograms ABCD and ABEF having the

same base and between the same parallel lines and .

To Prove:

Area of parallelogram ABCD = area of parallelogram ABEF

Theorem 16.1.2: (Board 2013, 14) 093016014

Parallelograms on equal bases and having the same (or equal) altitude are equal in area.

Given: Parallelograms ABCD, EFGH are on the equal bases , , having equal altitudes.

To Prove: Area of (parallelogram ABCD) = area of (parallelogram EFGH)

Construction: Place the parallelograms ABCD and EFGH so that their equal bases , are in the straight line BCFG. Join and .

Exercise 16.1:

Q. 1 Show that the line segment joining the mid-points of opposite sides of a parallelogram, divides it into two equal parallelograms. 093016015

Given: ABCD is parallelogram. Point p is midpoint of side i.e. and point Q is midpoint of side i.e. .

To Prove: ||gm AQPD ||gm QBCP

Construction:

Join P to Q and Q to C.

Q. 2 In a parallelogram ABCD, = 10cm. The altitudes corresponding to sides AB and AD are respectively 7 cm and 8cm. Find . 093016016

Given: Parallelogram ABCD, m =10cm. The altitudes. Corresponding to the sides and arc 7cm and 8cm.

To Find: m =?

Construction: Make ||gm ABCD and show the given altitudes = 7cm, = 8cm.

Q. 3 If two parallelograms of equal areas have the same or equal bases, their altitudes are equal. 093016017

Given: Two parallelograms of same or equal bases and same areas.

To Prove: Their altitudes are equal.

Construction: Make the ||gm ABCD and EFGH. Draw and

Theorem 16.1.3: Triangles on the same base and (Board 2014)

of the same (i.e., equal) altitudes are equal in area. 093016018

Given: s ABC, DBC on the same base

and having equal altitudes.

To Prove: Area of (ABC) = area of (DBC) (Board 2013, 15)

Construction: Draw || to , || to meeting produced in M, N.

Theorem 16.1.4: Triangles on equal bases and of equal altitudes are equal in area. (Board 2014)

093016019

Given: s ABC, DEF on equal bases , and having altitudes equal.

To Prove:

Area ( ABC) = Area ( DEF)

Construction

Place the s ABC and DEF so that their equal bases and are in the same straight line BCEF and their vertices on the same side of it. Draw BX || to CA and FY || to ED meeting AD produced in X, Y respectively

Corollaries:

(1) Triangles on equal bases and between the same parallels are equal in area.

(2) Triangles having a common vertex and equal bases in the same straight line, are equal in area.

Exercise 16.2:

Q. 1 Show that a median of a triangle divides it into two triangles

of equal area. 093016020

Given: In is a median i.e.

To Prove: Median divides the triangle into two triangles of equal area.

i.e.

Construction: Draw altitude

Q. 2 Prove that a parallelogram is divided by its diagonals into four triangles of equal area. 093016021

Given: ||gm divided by its diagonals into four triangles

To Prove: Areas of the four triangles are equal

Construction: Make the ||gm ABCD with diagonals intersecting each other at O to make four triangle, AOB BOC, COD and DOA.

Q.3 Divide a triangle into six equal triangular parts. 093016022

Review Exercise 16 OBJECTIVE

Choose the correct Answers:

275. The region enclosed by the bounding lines of a closed figure is called the __ of the figure: 093016023

(a) Area (b) Circle

276. Base × altitude = 093016024

(a) Area of parallelogram

(b) Area of square

(d) Area of Triangle

277. The union of a rectangle and its interior is called: 093016025

(a) Circle region

(b) Rectangular region

278. If a is the side of a square, its area will be equal to… 093016026

(a) a square unit (b) a2 square units

279. The union of a triangle and its interior is called as: 093016027

(a) Triangular region

(b) Rectangular region

280. Altitude of a triangle means perpendicular distance to base from its opposite___ 093016028

(a) Vertex (b) Side

281. Area of given figure is……. 093016029

(a) 18cm

(b) 9cm

(d) 9cm2

282. Area of given figure is…… 093016030

(a) 4cm

(b) 8cm2

(d) 16cm2

283. Area of given figure is…… 093016031

(a) 4cm2

(b) 12cm2

(d) 32cm2

284. Area of given figure is….. 093016026

(a) 160cm2

(b) 80cm2

(d) 160cm

285. Area of triangle is …… 093016032

(a) A = Base  Height

(b) A = Base  Height

(d) A = L2

286. Area of square is …… 093016033

(a) A = Base  Height

(b) A = Base  Height

(d) A = L2

287. Area of rectangle is …… 093016034

(a) A = Base  Height

(b) A = Base  Height

(d) A = L2

288. Area of parallelogram is … 093016035

(a) A = Base  Height

(b) A = Base  Height

(d) A = L2

289. If the length and breadth of a rectangle are ‘a’ and ‘b’ then its area will be: 093016036

(a) a + b (b)

290. In most cases similar figures have _____ areas. 093016037

(a) same (b) different

291. All congruent figures have _____ areas. 093016038

(a) same (b) different

292. Area of a geometrical figure is always ___ real number. 093016039

(a) zero (b) positive

Q.1 Which of the following are true and which are false?

(i) Area of a figure means region enclosed by bounding lines of closed figure. 093016040

(ii) Similar figures have same area. 093016041

(iii) Congruent figures have same area. 093016042

(iv) A diagonal of a parallelogram divides it into two non-congruent triangles. 093016043

(v) Altitude of a triangle means perpendicular from vertex to the opposite side (base). 093016044

(vi) Area of a parallelogram is equal to the product of base and height. 093016045

Q.2 Find the area of the following.

(i) 093016046 (ii) (Board 2013) 093016047

(iii) 093016048 (iv) 093016049

Q.3 Define the following: (i) Area of figure 093016050 (ii) Triangular region 093016051

(iii) Rectangular region 093016052 (iv) Altitude or height of triangle 093016053

UNIT 17

50. Exercise 17.1

Q.1 Construct aABC, in which:

(i) = 3.2cm, (Board 2013) 093017001

51. Given

The sides = 3.2cm,

ofABC

(ii) 093017002

52. Given

The sides

ofABC

(iii) , 093017003

53. Given

The sides and of ABC

(iv) , 093017004

54. Given

The sides and of ABC. (Board 2013, 15)

(v) m =3.5cm, 093017005

55. Given

The sides

m =3.5cm and of ABC

(vi) 093017006

56. Given

The side and angles of ABC

(vii)

093017007

57. Given

The side and angles , of ABC

Q.2 Construct a XYZ in which

(i) 093017008

58. Given

The sides and of XYZ.

59.

(ii) m =6.4cm,

m Y = 90o 093017009

60. Given

The sides and of XYZ.

(iii) 093017010

61. Given

The sides ofXYZ.

Q.3 Construct a right angled  measure of whose hypotenuse is 5cm and one side is 3.2cm. 093017011

62. Given

In right angled  hypotenuse is 5cm and one side is 3.2cm

Q.4 Construct a right angled isosceles

triangle. Whose hypotenuse is: 093017012

i) 5.2cm long 093017013

63. Given

In right angled isosceles triangle hypotenuse is 5.2 cm.

(ii) 4.8 cm 093017014

64. Given

In right angled isosceles triangle hypotenuse is 4.8 cm.

(iii) 6.2 cm 093017015

65. Given

In right angled isosceles triangle hypotenuse is 6.2 cm.

(iv) 5.4 cm 093017016

66. Given

In right angled isosceles triangle hypotenuse is 5.4 cm.

Q.5 (Ambiguous case) construct a ABC in which

(i) , (Board 2014) 093017017

67. Given

In ABC ,

(ii) 093017018

68. Given

In ABC

(iii) 093017019

69. Given

(a) Draw angle bisectors of a given triangle and verify their concurrency.

70. Example 093017020

(i) Construct a ABC having given

= 4.6 cm, =5 cm and = 5.1 cm.

(ii) Draw its angle bisectors and verify that they are concurrent.

71. Given

The side =4.6 cm, = 5 cm and = 5.1 cm of a ABC.

(b) Draw altitudes of a given triangle and verify their concurrency.

72. Example 093017021

(i) Construct a triangle ABC in which = 5.9 cm, =56oand m =44o

(ii) Draw the altitudes of the triangle and verify that they are concurrent.

i. Given

The side = 5.9 cm, and =56o,

m =44o

(c). Draw perpendicular bisectors of the sides of a given triangle and verify their concurrency.

73. Example 093017022

(i) Construct a ABC having given

=4cm, = 4.8cm and =3.6cm.

(ii) Draw perpendicular bisectors of its sides and verify that they are concurrent.

i. Given

Three sides = 4cm, = 4.8cm and =3.6cm of a ABC.

(d)Draw medians of a given triangle and verify their concurrency.

74. Example 093017023

(i) Construct a ABC in which = 4.8 cm, = 3.5cm and = 4cm.

(ii) Draw medians of ABC and verify that they are concurrent at a point within the triangle. By measurement show that the medians divide each other in the ratio 2:1.

75. Given

Three sides =4.8 cm, = 3.5cm and = 4cm of a ABC.

76.

77. Exercise 17.2

Q.1Construct the following sABC. Draw the bisectors of their angles and verify their concurrency.

(i) 093017024

78. Given

The sides , and

(ii)

093017025

79. Given

The sides

of a ABC.

(iii) 093017026

80. Given

The sides and

Q.2 Construct s PQR. Draw their altitudes and show that they are concurrent.

(i) (Board 2015) 093017027

81. Given

The sides and of a PQR.

(ii) 093017028

82. Given

and

(iii) , 093017029

83. Given

Q.3 Construct the following triangles ABC. Draw the perpendicular bisectors of their sides and verify their concurrency. Do they meet inside the triangle?

(i) , 093017030

84. Given

Side , of a ABC.

(ii)

m B=60o (Board 2014) 093017031

85. Given

The side and

(iii) (Board 2014) 093017032

86. Given

The sides

Q.4 Construct following ’s XYZ.

Draw their three medians and show that they are concurrent.

(i) (Board 2014) 093017033

87. Given

The side

(ii) 093017034

88. Given

The sides and

(iii) , .

093017035

89. Given

The side and of XYZ.

Figure with Equal Areas

(i) Construct a triangle equal in area to a given quadrilateral. 093017036

Given:A quadrilateral ABCD.

90. Exercise 17.3

Q.1 (i) Construct a quadrilateral ABCD, having

and 093017037

(ii) On the side BC construct a equal in area to the quadrilateral ABCD.

093017038

91.

92. Given

Sides of quadrilateral ABCD

Q.2 Construct a  equal in area to the quadrilateral PQRS, having , , , and . 093017039

93. Given

Parts of the quadrilateral PQRS are given.

Q.3 Construct a equal in area to the quadrilateral ABCD, having , , , and . 093017040

94. Given

Parts of the quadrilateral ABCD are given

Q.4 Construct a right-angled triangle equal in area to a given square. 093017041

95. Given

Square ABCD

Construct a rectangle equal in area to a given triangle.

Given: ABC

96. Exercise 17.4

Q.1 Construct a  with sides 4cm, 5cm and 6 cm and construct a rectangle having its area equal to that of the . Measure its diagonals. Are they equal?

093017042

97. Given

4cm, 5cm, 6cm the sides of the triangle .

Q.2Transform an isosceles  into a rectangle. 093017043

Q.3 Construct a ABC such that , , . Construct a rectangle equal in area to the ABC, and measure its sides. 093017044

98. Given

Three sides of the ABC

(iii) Construct a square equal in area to a given rectangle.

Given: A rectangle ABCD

99. Exercise 17.5

Q.1 Construct a rectangle whose adjacent sides are 2.5 cm and 5cm respectively. Construct a square having area equal to the given rectangle. 093017045

Q.2 Construct a square equal in area to a rectangle whose adjacent sides are 4.5 cm and 2.2 cm respectively. Measure the sides of the square and find its area and compare with the area of the rectangle. 093017046

Q.3 In Q.2 above verify by measurement that the perimeter of the square is less than that of the rectangle.

100. 093017047

Q.4 Construct a square equal in area to the sum of two squares having sides 3cm and 4 cm respectively. 093017048

101.

Q.5 Construct a  having base 3.5 cm and other two sides equal to 3.4 cm and 3.8 cm respectively. Transform it into a square of equal area. 093017049

Q. 6 Construct ahaving base 5 cm and other sides equal to 5 cm and 6 cm. Construct a square equal in area to given. 093017050

Let

REVIEW EXCERSIE 17

Q.1 Fill in the blanks to make the statement true. 093017051

(i) The side of a right angled triangle opposite to 90o is called _________.

093017052

(ii) The line segment joining a vertex of a triangle to the mid-point of its

opposite side is called a____________. 093017053

(iii) A line drawn from a vertex of a triangle which is _________to its

opposite side is called an altitude of the triangle. 093017054

(iv) The bisectors of the three angles of a triangle are __________. 093017055

(v) The point of concurrency of the right bisectors of the three sides of

the triangle is ___________ from its vertices. 093017056

(vi) Two or more triangles are said to be similar if they are equiangular and measures of their corresponding sides are_______________. 093017057

(vi) The altitudes of a right triangle are concurrent at the ____ of the right angle. 093017058

OBJECTIVE

Q.2 Choose the correct answer:

293. A triangle having two sides congruent is called: ___ 093017059

(a) Scalene (b) Right angled

294. A quadrilateral having each angle equal to 90o is called ____ 093017060

(a) Parallelogram (b)Rectangle (Board 2014)

295. The right bisectors of the three sides of a triangle are ___ 093017061

(a) Congruent (b) Collinear

296. The __ altitudes of an isosceles triangle are congruent: 093017062

(a) Two (b) Three (Board 2015)

297. A point equidistant from the end points of a line segment is on its __ 093017063

(a) Bisector

(b) Right bisector

(d) Median

298. ___ congruent triangles can be made by joining the mid points of the sides of a triangle: (Board 2015) 093017064

(a) Three (b) Four

299. The diagonals of a parallelogram ___ each other: (Board 2014) 093017065

(a) Bisect

(b) Trisect

(d) None of these

300. The medians of a triangle cut each other in the ratio: (Board 2013, 15) 093017066

(a) 4:1 (b) 3:1

301. One angle on the base of an isosceles triangle is 30o. What is the measure of its vertical angle: (Board 2014) 093017067

(a) 30o (b) 60o

302. If the three altitudes of a triangle are congruent then the triangle is _ 093017068

(a) Equilateral (b) Right angled

303. If two medians of a triangle are congruent then the triangle will be: 093017069

(a) Isosceles (b) Equilateral

304. A line segment joining a vertex of a triangle to the midpoint of its opposite side is called a ___ of the triangle: (a) Altitude (b)Median 093017070

305. A line segment from a vertex of triangle perpendicular to the line containing the opposite side, is called an __ of the triangle: 093017071

(a) Altitude (b) Median

306. The point of concurrency of the three altitudes of a  is called its __ 093017072

(a) Ortho centre (b) In centre

307. The internal bisectors of the angles of a triangle meet at a point called the _______ of the triangle: 093017073

(a) In centre (b) Ortho centre

308. The point of concurrency of the three perpendicular bisectors of the sides of a triangle is called the ____ of the triangle. 093017074

(a) Circumcentre (b) In centre

309. Point of concurrency of three medians of a triangle is called. 093017075

(a) In centre three (b) Ortho centre

310. Sum of interior angles of a triangle is …… (Board 2013, 14) 093017076

(a) 60o (b) 120o

311. Sum of four interior angle of a rectangle is ……. 093017077

(a) 90o (b) 180o

312. Sum of four interior angles of a parallelogram is ………… 093017078

(a) 90o (b) 180o

313. Sum of four interior angles of a square is……. 093017079

(a) 360o (b) 270o

314. Sum of four internal angles of a quadrilateral is ….. 093017080

(a) 60o (b) 120o

315. The side opposite to right angle in right angled triangle is called….. 093017081

(a) Base (b) Perpendicular

316. The altitudes of a right angled triangle are concurrent at the …….. 093017082

(a) Midpoint of hypotenuse

(b) Vertex of right angle

317. The triangles are said to be ….. if they are equiangular. 093017083

(a) Congruent (b) Similar

318. All the ….. right bisectors of sides of triangle are concurrent. 093017084

(a) One (b) Two

319. All the three bisectors of angles of triangle are…… (Board 2014) 093017085

(a) Congruent (b) Concurrent

320. All the three medians of a triangle are…….. 093017086

(a) Congruent (b) Concurrent

321. All the three altitudes of a triangle are……… 093017087

(a) Congruent

(b) Concurrent

(c)Parallel

(d) Perpendicular

322. In-centre is the point of concurrency of three….. of triangle. 093017088

(a) Right bisectors (b) Angle bisectors

323. Circumcentre is point of concurrency of three of three….. of triangle. 093017089

(a) right bisectors (b) angle bisectors

324. Ortho centre is the point of concurrency of three….. of triangle.

(a) right bisectors 093017090

(b) angle bisectors

(d) medians

325. Centroid is the point of concurrency of three….. of triangle. 093017091

(a) right bisectors (b) angle bisectors

326. In ambiguous case of triangle how many maximum triangles are constructed? 093017092

(a) one (b) two

(c)three (d) four

327. Three or more than three lines passing through the same point are called …… Lines. 093017093

(a) congruent

(b) concurrent

(d) perpendicular

328. The common point of three or more than three lines is called…… 093017094

(a) central point

(b) point of concurrency

(d) centroid

329. Which of the following can be constructed by compass? 093017095

(a) 15o (b) 25o

330. Which of the following cannot be constructed with compass? 093017096

(a) 30o (b) 45o

331. Which of the following is used to measure the angle? 093017097

(a) compass (b) protractor

332. In right-angled triangle if one angle is 30o, then other angle will be…..:

(a) 15o (b) 30o 093017098

333. In right-angled triangle if one angle is 60o, then other angle will be…..:

(a) 15o (b) 30o 093017099

334. In right-angled triangle if one angle is

45o, then other angle will be…..:

(a) 15o (b) 30o 093017100

335. By drawing the right bisector of a line segment we can find its ____ point. 093017101

(a) end (b) midpoint

336. By drawing the right bisectors of sides of a triangle we can find its _________. 093017102

(a) incentive (b) circumcentre

337. By drawing the angle bisectors of a triangle we can find its_____. 093017103

(a) incentre (b) circumcentre

338. By drawing the medians of a triangle we can find its _______. 093017104

(a) incentre (b) circumcentre

339. By drawing the altitudes of a triangle we can find its ____. 093017105

(a) incentre (b) circumcentre

Q.3. Define the following:

i) Define in Centre. 093017106

ii) Define Circumcentre. (Board 2015) 093017107

iii) Define Orthocentre. (Board 2014) 093017108

iv) Define Centroid. (Board 2013) 093017109

v) Define Concurrent lines. / What do you mean by Point of concurrency? 093017110

Vi ) Define median of triangle . 093017111

vii) Define Altitude of a triangle.

viii) What is the ambiguous case of triangle?

Causative

9th Math full book

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