12th Math Subjective + Exercises

Unit Functions And Limits

Function:

When two variables x and y are so related with each other that the value of y depends upon x then y is called function of x. Correspondence to each value of x there is unique value of y. It is denoted by

It is also written as

Note:

Domain of f = X

Range of f = Y

1. EXERCISE 1.1

i. Q.1 Given that 12301001

(a) 12301002

(b) 12301003

Find (i) 12301004

(ii) 12301005

(iii) 12301006

(iv) 12301007

Q.2 Find and simplify where, 12301008

(i) 12301009

(ii) (Board 2008) 12301010

(iii) 12301011

(iv) 12301012

Q.3 Express the following:

(a) The perimeter P of square as a function of its area A (Board 2009, 10) 12301013

(b) The area A of a circle as a function of its circumference C. 12301014

1. Domain of the Function: 12301016

2. Range of the Function: 12301017

Q.4 Find the domain and the range of the function g defined below and sketch the graph of g. 12301018

(i) 12301019

(ii) (Board 2008) 12301020

(iii) (Board 2011) 12301021

(iv) 12301022

(v) 12301023

(vi) 12301024

(vii) 12301025

(viii) 12301026

Q.5 Given . If and find the values of a and b.

(Board 2008) 12301027

Q.6 A stone falls from a height of 60m on the ground, the height h after x seconds is approximately given by .

12301028

(i) What is height of the stone when: 12301029

(ii) When does the stone strike the ground?

12301030

Q.7 Show that the parametric equations.

(i) represents the equation of parabola . 12301031

(ii) represent the equation of ellipse . 12301032

(iii) represent the equation of hyperbola .= 12301033

Q.8 Prove the identities:

(i) (Board 2006, 11)

12301034

(ii) 12301035

(iii) 12301036

Hence

Even Function: (Board 2012)

A function f is an even function if . 12301037

Odd Function: (Board 2012)

A function f is said to be an odd if . 12301038

Q.9 Determine whether the given function f is even or odd. 12301039

(i) 12301040

(ii) 12301041

(iii) 12301042

(iv) (Board 2009)

12301043

(v) (Board 2007) 12301044

(vi) 12301045

Composition of Functions: 12301046

Inverse Of a Function (Board 2010) 12301047

2. EXERCISE 1.2

i. Q.1 The real valued function and are defined below. Find 12301048

(a) (b)

(i) and

12301049

(ii) and

12301050

(iii) and

12301051

(iv) and

12301052

Q.2 For the real valued function, f defined below, find (a) (b) and verify . 12301053

(Board 2009)

(ii) 12301054

Verification:

(iii) (Board 2010) 12301055

(iv) 12301056

Q.3 Without finding the inverse, state the domain and range of . (Board 2009) 12301057

(i) 12301058

(ii) 12301059

(iii) 12301060

(iv) (Board 2007)

12301061

Limit of a Function and Theorems on Limits: 12301062

1. Meaning of

2. Meaning of

3. Meaning of

4. Limit of a function: (Board 2010) 12301063

Theorems on Limits: 12301064

Note: 12301065

1. Where n is an integer and a > 0

Prove that: (Board 2011) 12301066

Note: 12301067

1. (1 + x)1/x = e

3. Sandwitch Theorem:

Prove that: (Board 2012) 12301068

Example 1: (Board 2006, 11) 12301069

Example 2: 12301070

Example 3: 12301071

Evaluate (Board 2009)

(i)

(ii)

(Board 2007, 09)

3. EXERCISE 1.3

i. Q.1 Evaluate each limit by using theorem of limits. 12301072

ii. (i) 12301073

(ii) 12301074

(iii) 12301075

(iv) 12301076

(v) 12301077

(vi) 12301078

Q.2 Evaluate each limit by using algebraic techniques. 12301079

(i) 12301080

(ii) 12301081

(iii) (Board 2008) 12301082

(iv) (Board 2009) 12301083

(v) 12301084

(vi) 12301085

(vii) 12301086

(viii) (Board 2012)

12301087

(ix) 12301088

Q.3 Evaluate the following limits. )

(i) 12301089

(ii) 12301090

(iii) (Board 2007) 12301091

(iv) 12301092

(v) (Board 2007) 12301093

(vi) (Board 2007, 08) 12301094

(vii) 12301095

(viii) (Board 2009)

12301096

(ix) 12301097

(x) 12301098

(xi) (Board 2012) 12301099

(xii) (Board 2012) 12301100

Q.4 Express each limit in terms of e:

(i) 12301101

(ii) 12301102

(iii) 12301103

(iv) 12301104

(v) 12301105

(vi) 12301106

(vii) 12301107

(viii) 12301108

(ix) (Board 2008, 09) 12301109

(x) 12301110

(xi) 12301111

4. EXERCISE 1.4

i. Q.1 Determine the left hand and right hand limit and then find the limit of the following functions: when x  c.

ii. (i) 12301112

(ii) 12301113

(iii) 12301114

Q.2 Discuss the continuity of f(x) at x = c.

(i)

(Board 2008, 09, 11) 12301115

(ii) 12301116

Q.3 If

Discuss continuity at and .

(Board 2011) 12301117

Q.4 If 12301118

Find ‘c’ so that exists.

Q.5 Find the values m and n, so that given function f is continuous at x = 3. 12301119

(Board 2009)

(i) 12301120

(ii) 12301121

1. Q.6 If

Find the value of k so that f is continuous at x = 2. (Board 2008) 12301122

EXERCISE 1.5

Q.1 Draw the graphs of the following equations:

(i) 12301123

(ii) 12301124

(iii) 12301125

(iv) 12301126

Q.2 Graph the curves that has the parametric equations given below:

(i) where “t” is a parameter. 12301127

(ii) where “t” is a parameter. 12301128

(iii) where is a parameter. 12301129

Q.3 Draw the graphs of the function defined below and find whether they are continuous.

12301130

(i) 12301131

(ii) 12301132

(iii) 12301133

(iv) 12301134

Continuity of graph:

Graph is not continuous at

Q.4 Find the graphical solution of the following equations:

(i) 12301135

(ii) 12301136

(iii)

Unit Differentiation

Example:

Find derivative of the following by definition.

(a) (Board 2010) 12302001

(b) (Board 2010) 12302002

1. EXERCISE 2.1

Q.1 Find by definition, the derivatives w.r.t. ‘x’ of the following functions defined as: 12302003

(i) (Board 2011, 12) 12302004

(ii) 12302005 (iii) 12302006

(iv) 12302007 (v) 12302008 (vi) 12302009 (vii) 12302010 (viii) (Board 2008) 12302011 (ix) 12302012

(x) 12302013

(xi) 12302014 (xii) 12302015 (xiii) x40 12302016

(xiv) x–100. 12302017

Q.2 Find from first principles if 12302018

(i) 12302019 (ii) (Board 2010) 12302020

2. EXERCISE 2.2

Q.1 Find from first principles, derivatives of the following expressions w.r.t. their respective independent variables. 12302021

(i) (ax + b)3 (Board 2006) 12302022 (ii) (2x + 3)5 (Board 2012) 12302023

(iii) (3t + 2)2 12302024 (iv) (ax + b)–5 12302025 (v) (Board 2010) 12302026

Q.2 If y = cf(x), prove that

(Board 2011) 12302027

Basic Formulas for Differentiation

(i) 12302028

(ii) 12302029

(iii) where, 12302030

(iv) The product rule or u into v formula. 12302031

(v) The quotient rule or over v formula. 12302032

(vi) General power rule. 12302033

(vii) Power rule. 12302034

(viii) 12302035

3. EXERCISE 2.3

Differentiate w.r.t. x

Q.1 x4 + 2x3 + x2 (L.B. 2009) 12302036

Q.2 12302037

Q.3 (Board 2008, 10, 11) 12302038

Q.4 (Board 2008) 12302039

Q.5 (x  5) (3  x) 12302040

Q.6 (Board 2006, 10) 12302041

Q.7 12302042

Q.8 12302043

Q.9 (Board 2009) 12302044

Q.10 12302045

Q.11 (Board 2011) 12302046

Q.12 12302047

Q.13 12302048

Q.14 (Board 2006) 12302049

Q.15 12302050

Q.16 If Show that 12302041

Q.17 If y = x4 + 2x2+2, prove that

= 4x (Board 2007) 12302052

Note:

(i) The chain rule. 12302053

(ii) (Board 2009) 12302054

Example 1:

Find the derivative of (x3 + 1)9 with respect to x. (Board 2011) 12302055

Example 2:

Find if x = 1 – t2 and y =3t2 – 2t3. (Board 2010) 12302056

Example 3:

Find If (Board 2012) 12302057

Example 4: 12302058

Find , if y2 + x2 – 4x = 5. (Board 2007)

Example 5:

Find if y2 – xy – x2 + 4=0. 12302059

(Board 2009, 11)

EXERCISE 2.4

Q.1 Find by making suitable substitutions in the following functions defined as:

(i) 12302060

(ii) y = (Board 2007) 12302061 (iii) 12302062 (iv) y = (3x2 – 2x + 7)6

12302063

(v) y = 12302064

Q.2 Find , if:

(i) 3x + 4y + 7 = 0 12302065

(ii) xy + y2 = 2 12302066

(iii) x2 – 4yx – 5y = 0 12302067

(iv) 4x2 + 2hxy + by2 + 2gx + 2fy + c =0 12302068

(v) x 12302069

(iv) y (x2 – 1) = x 12302070

Q.3 Find of the following parametric functions. 12302070(a)

(i) x =  + and y =  + 1 12302071

(ii) x = (Board 2009) 12302072

Q.4 Prove that y + x = 0 if

x = 12302073

Q.5 Differentiate: (i) x2 – w.r.t. x4 12302074

(ii) (1+x2)n w.r.t. x2 12302075

(iii) w.r.t. 12302076

(iv) 12302077

(v) w.r.t. x3 12302078

Formulas for Derivatives of Trigonometric Functions:

(i) ; 12302079

(ii) 12302080

(iii) 12302081

(iv) 12302082

(v) 12302083

(vi) 12302084

Where u = f(x)

Example 1:

If y = cos x, find . (Board 2009) 12302085

Formulas for Derivatives of Inverse Trigonometric Function:

1. ; [Sin-1 x] = , x (-1 , 1) or – 1< x <1 12302086

2. [Cos-1 u] = – ; [Cos-1 x] = - , x (-1 , 1) 12302087

3. ; 12302088

4. (Cosec-1 u)= – ; (Cosec-1 x)= - ,x 12302089

5. [Sec-1 u] = ; [Sec-1 x] = , x 12302090

6. [cot-1 u] = ; [cot-1 x] = 12302091

Where u = f(x)

Example 2:

If y = tan Show that (Board 2007) 12302092

EXERCISE 2.5

Q.1 Differentiate the following trigonometric function from the first principles.

(i) sin 2x 12302093 (ii) tan 3x 12302094

(iii) sin 2x + cos 2x (Board 2009) 12302095 (iv) cos x2 12302096 (v) tan2 x 12302097

(vi) 12302098 (vii) cos 12302099

Q.2 Differentiate the following w.r.t. the variable involved.

(i) x2 sec 4x 12302100 (ii) tan3  sec2  (Board 2009) 12302101 (iii) (sin 2 - cos 3)2 12302102 (iv) cos (Board 2008) 12302103

Q.3 Find if:

(i) y = x cos y (Board 2009) 12302104 (ii) x = y sin y (Board 2009) 12302105

Q.4 Find the derivative w.r.t ‘x’

(i) cos 12302106

(ii) sin 12302107

Q.5 Differentiate:

(i) sin x w.r.t. cot x (Board 2006, 09) 12302108

(ii) sin2 x w.r.t. cos4 x (Board 2008, 09) 12302109

Q.6 If tan y (1 + tan x) = 1 – tan x, show that (Board 2009) 12302110

Q.7 If y = Prove that (2y – 1) = sec2 x. 12302111

Q.8 If x = a cos3 , y b sin3  Show that + b tan  = 0 12302112

Q.9 Find if x = a (cost + sin t)

y = a (sin t – t cos t) 12302113

Q.10 Differentiate w.r.t. x

(i) 12302114

(ii) cot-1  (1) 12302115

(iii) (Board 2010) 12302116

(iv) sin-1 12302117 (v) sec-1 12302118 (vi) cot-1 12302119

(vii) cos-1 12302120

Q.11 Show that if 12302121

Q.12 If y = tan (p tan-1 x), show that (1+x2)y1 – p(1+y2) = 0. 12302122

Formulas for Derivative of Exponential Function:

(i) ; 12302123

(ii) ; 12302124

Where u = f(x)

Formulas for Derivatives of the Logarithmic Function:

(i) ; 12302125

(ii) ; 12302126

Where u = f(x)

Example 1:

Find dydx if y = log10 (ax2 + bx + c)

(Board 2010) 12302127

Example 2:

Differentiate ln (x2 + 2x) w.r.t. ‘x.

(Board 2009, 10, 11) 12302128

Example 3:

Differentiate (ln x)x w.r.t. ‘x’ 12302129

Formula For Derivatives of Hyperbolic Functions:

(i) ; 12302130

(ii) ; 12302131

(iii) ; 12302132

(iv) ; 12302133

(v) ; 12302134

(vi) ; 12302135

Formula For Derivatives of the Inverse Hyperbolic Functions:

(i) ; 12302136

(ii) ; 12302137

(iii) ; 12302138

(iv) ; 12302139

(v) ; 12302140

(vi) ; 12302141

Example:

Find dydx if y = sinh1 (ax + b)

(Board 2009) 12302142

EXERCISE 2.6

Q.1 Find if

(i) = (Board 2009 ) 12302143

(ii) = 12302144 (iii) = 12302145

(iv) = 12302146 (v) 12302147

(vi) = 12302148

(vii) =

(Board 2007) 12302149

(viii) = 12302150

Q.2 Find if

(i) y = (Board 2010) 12302151 (ii) y = (Board 2009) 12302152 (iii) y = 12302153

(iv) y = 12302154

(v) y = 12302155 (vi) y = 12302156

(vii) y = 12302157

(viii) y = 12302158

(ix) y = 12302159

(x) y = (Board 2007, 09) 12302160 (xi) y = 12302161 (xii) y = 12302162

(xiii) y = (Board 2009) 12302163

(xiv) y = 12302164

Q.3 Find if

(i) y = cosh 2x (Board 2008) 12302165 (ii) y = sinh 3x 12302166

(iii) y = tanh-1 (sin x)- 12302167 (iv) y = sinh-1 (x3) (Board 2008) 12302168 (v) y = ln tan h x (Board 2006) 12302169

(vi) y = sinh-1 12302170

EXERCISE 2.7

Q.1 (i) Find y2 if y = 2x5 – 3x4 + 4x3+x-2

12302171

(ii) y = (2x + 5)3/2 12302172

(iii) y = 12302173

Q.2 Find if; (i) y = x2 . e-x

(Board 2008, 11) 12302174

(ii) y = In 12302175

Q.3 Find if; (i) x2 + y2 = a2 12302176 (ii) x3 – y3 = a3 12302177 (iii) x = a cos , y = a sin  12302178

Q.3(i)

(iv) x = at2 , y = bt4 12302179 (v) x2 + y2 + 2gx + 2fy + c = 0 12302180

Q.4 Find if;

(i) y = sin 3x (Board 2008, 10, 11) 12302181

(ii) y = cos3 x (Board 2008) 12302182

(iii) y = ln (x2 – 9) 12302183

Q.5 If x = sin , y = sin m, show that

(1-x2) y2 – xy1 + m2 y = 0 12302184

(1 – x2) y2 –xy1 + m2 y = 0

Q.6 y = ex sin x, show that 12302185

Q.7 If y = eax sin bx 12302186

Q.8 If y = (cos-1x)2 , prove that

(Board 2008) 12302187

Q.9 If y = a cos (In x) + b sin (In x), prove that 12302188

Example 1:: (Board 2011)

Expand ax in the Maclaurin series. 12302189

EXERCISE 2.8

Q.1 Apply Maclaurin series expansion to prove that: (Board 2006)

(i) 12302190

(ii) =

(Board 2011) 12302191

(iii) = 12302192

(iv) =

(Board 2011) 12302193

(v) = 12302194

Q.2 Show that cos(x+h)

and evaluate cos 61. (Board 2010) 12302195

Q.3 Show that

12302196

Examples 1:

Examine the function defined as

f(x) = 1 + x3 for extreme values 12302197

EXERCISE 2.9

Q.1 Determine the interval in which f is increasing or decreasing

(i) f (x) = sin x , x [- , ] 12302198

(ii) f (x) = cos x , x

(Board 2005) 12302199

(iii) (Board 2008) 12302200

(iv)

(Board 2007) 12302201

Q.2 Find the extreme values for the following functions defined as:

(i) f(x) = 1  x3 (Board 2008, 09) 12302202

(ii) f(x) = x2  x  2 12302203 (iii) f (x) = 5x2  6x + 2 12302204

(iv) f(x) = 3x2 12302205 (v) f(x) = 3x2 –4x+5 (Board 2009) 12302206 (vi) f(x) = 2x3  2x2  36 x + 3

(Board 2005) 12302207

(vii) f (x) = x4 – 4x2 12302208 (viii) f (x) = (x – 2)2 (x – 1) 12302209

(ix) f (x) = 5 + 3x – x3 12302210

(i) At x = 1, (1) = - 6 (1) = – 6 < 0

Q.3 Find the maximum and minimum values of the function defined by the equation occurring in the interval [0, 2];

f (x) = sin x + cos x 12302211

Q.4 Show that y = is maximum at x=e (Board 2011) 12302212

Q.5 Show that y = xx is minimum at x =

(Board 2009) 12302213

Example 1:

Find two positive integers whose sum is 9 and the product of one with the square of the other will be maximum. 12302214

EXERCISE 2.10

Q.1 Find two positive integers whose sum is 30 and their product will be maximum.

12302215

Q.2 Divide 20 into two parts so that sum of their square will be Minimum.

(Board 2012) 12302216

Q.3 Find two positive integers whose sum is 12 and the product of one with square of the other will be Maximum:(Board 2009) 12302217

Q.4 The perimeter of a triangle is 16cm if one side is of length 6cm, what are lengths of other sides for maximum area of the triangle. 12302218

Q.5 Find dimensions of a rectangle of largest area having perimeter 120cm.

12302219

Q.6 Find length of the sides of a variable rectangle having area 36cm2 when perimeter is minimum. 12302220

Q.7 A box with a square base and open top is to have a volume of 4 cubic dm. Find the dimensions of the box which will require the least material. 12302221

Q.8 Find the dimensions of a rectangular garden having perimeter 80m if its area is to be maximum. 12302222

Q.9 An open tank of square base of side x and vertical sides to be constructed to contain a given quantity of water, Find the depth in terms of x if the expense of lining the inside of the tank with lead will be least.

12302223

Q.10 Find the dimensions of the rectangle of maximum area which fits inside the

semi-circle of radius 8 cm as shown in the figure. 12302224

Q.11 Find the point on the curve y=x2 – 1 that is closest to the point (3, -1). 12302225

Q.12 Find the point of the curve y = that is closest the point (18, 1). 12302226

Unit Integration

Integration:

The technique to find a function when its derivative is given is called anti-derivation or integration.

Symbol for integration w.r.t. x.

Differential of a Variable:

Let y = f(x)  a function.

Increment in the independent variable.

Increment in the dependent variable.

derivative of f(x) w. r. t. x.

Then the product is called differential of the dependent variable. It is denoted by dy or df.

Put in (1)

, differential of the independent variable is equal to its increment.

Put = dx in (1)

Note:

(1)

(2)

(3)

(4)

Example: (Board 2005) 12303001

Find and dy of the function defined as = when x = 2, dx = 0.01

EXERCISE 3.1

Q.1 Find y and dy in the following cases:

(i) y = when x changes from 3 to 3.02

(Board 2008, 10,11,12) 12303002

(ii) y = when x changes from 2 to 1.8 12303003 (iii) y = when x changes from 4 to 4.41

(Board 2005) 12303004

Q.2 Using differentials find

in the following equations

(i) 12303005

(ii) 12303006

(iii) 12303007

(iv) 12303008

(ii) (Board 2006, 08) 12303009

(iii) 12303010

(iv) = c 12303011

Taking differentials of both sides

Q.3 Use differentials to approximate the values of

(i) (Board 2007) 12303012

(ii) 12303013

(iii) 12303014

(iv) 12303015

(ii) = 12303016

So let x = 32, dx = −1

(iii) = 12303017

or let x = , dx = = −0.017453

(iv) = 12303018

or let x = , dx = = 0.017453 radian

Q.4 Find the approximate increase in the volume of a cube if the length of its each edge changes from 5 to 5.02.

(Board 2011) 12303019 03(013)

Q.5 Find the approximate increase in the area of a circular disc if its diameter is increased from 44 cm to 44.4 cm. 12303020

Anti-Derivative / Integral of a Function: 12303021

Let f(x) and (x) Two functions such that is called anti-derivative or integral of f(x) w.r.t. x i.e. .

Note:

(i) symbol for integration w.r.t. x

(ii) Integrand i.e. the function to be integrated is called integrand.

(iii) Integral / anti-derivative of f(x).

(iv) Constant of integration.

(v)

(vi)

(vii) cancel each other when they come together i.e.

Some Standard Formulae for Integration:

1. 12303022

12303023

3. = 12303024

called power adding formula.

4. =

12303025

5. 12303026

6. =

12303027

7. 12303028 =

12303029

8. 12303030

9. =

12303031

10. 12303032

11. =

12303033

12. 12303034

13.

= 12303035

14.

12303036

15.

= 12303037

16. = 12303038

17. = 12303039

18. = 12303040

19. = 12303041

20. dx = n |x| + c 12303042

21. = 12303043

22. = 12303044

23. 12303045

24. =

12303046

25. 12303047

26. = = 12303048

27. =

12303049

28. = 12303050

29. =

12303051

30.

12303052

31. = 12303053

32. 12303054

33. = 12303055

34. =

12303056

35. =

12303057

36.

12303058

37. 03(054)

38. =

12303059

Example:

Evaluate the following:

(i) (Board 2005) 12303060

(ii) (Board 2010) 12303061

(iii) 12303062

(iv) (Board 2009,10)12303063

(v) 12303064

(vi) 12303065

(vii) 12303066

(Board 2010)

EXERCISE 3.2

Q.1 Evaluate the following indefinite integrals

(i) 12303067 (ii) 12303068

(iii) 12303069

(iv) (Board 2009, 10) 12303070

(v) 12303071 (vi) 12303072 (vii) 12303073 (viii) (Board 2010) 12303074 (ix) 12303075

(x) (Board 2006, 08) 12303076 (xi) (Board 2009) 12303077

Q.2 Evaluate

(i) 12303078

Rationalizing

(ii) (Board 2008) 12303079 (iii) 12303080

Rationalizing

(iv) (Board 2006) 12303081

(v) 12303082 (vi) 12303083

(vii) (Board 2008, 09) 12303084 (viii) (Board 2010) 12303085

(ix) 12303086

(x) 12303087

(xi) 12303088

(xii) (Board 2010) 12303089

(xiii) 12303090

(xiv) (Board 2005, 07, 09, 11) 12303091

Note:

Some times an integrand is not integrible by an ordinary method. It can be easily made integrible by suitable substitution.

Some Useful Substitutions:

Expressions Involving Suitable Substitution

(i) x = a

(ii) x = a

(iii) x = a

(iv) = t

(v) = t

(vi)

(vii)

Example 1:

(Board 2011) 12303092

Example 2:

(i) (Board 2009) 123030933

(ii)

Example 3:

(Board 2010) 12303094

Example 4:

(Board 2008) 12303095

Example 5:

(i) (Board 2010) 12303096

(ii) 12303097

Let x =  dx =

Example 6:

Evaluate: (Board 2010)12303098

Example 7: 12303099

Evaluate: (Board 2009)

EXERCISE 3.3

Evaluate the following integrals:

Q.1 12303100

Q.2 12303101

Q.3 (Board 2009, 12) 12303102

Q.4 (Board 2008) 12303103

Q.5 12303104

Q.6 12303105

Q.7 (Board 2008) 12303106

Q.8(a) Show that: 12303107

(b) Show that:

12303108

Evaluate the following integrals:

Q.9 12303109

Q.10 (Board 2009) 12303110

Q.11 12303111

Multiplying and dividing by

Q.12 12303112

Q.13 12303113

Q.14 12303114

Q.15 12303115

Q.16 12303116

Q.17 12303117

Q.18 12303118

Q.19 12303119

Q.20 (Board 2011) 12303120

Q.21 (Board 2007, 08)12303121

Q.22

(Board 2012) 12303122

Integration By Parts: (Board 2010) 12303123

Let be two functions then, dx=

i.e. 1st function  integral of II function

 (integral of IInd function  derivative of 1st function) dx is called integration by parts.

Note:

(i) Generally an algebraic function is taken as 1st function.

(ii) In case of logarithmic or inverse trigonometric function it is always taken as 1st function.

(iii) If only one function is given and we are to apply integration by parts, then second function is always one.

See Q.1 (i), (iii), (ii) Ex.3.4

Example 1:

Evaluate: (Board 2009, 11) 12303124

Example 2:

Evaluate: (Board 2005, 10) 12303125

Example 3:

Evaluate: (Board 2009, 10) 12303126

Example 4:

Evaluate: (Board 2009) 12303127

Example 5:

Evaluate (Board 2009) 12303128

Example 6: (Board 2009)

Evaluate 12303129

Example 7: (Board 2009, 11)

Evaluate: 12303130

Example 8: (Board 2007)

Show that

= 12303131

EXERCISE 3.4

Q.1 Evaluate the following integrals by parts: (Board 2012)

(i) 12303132

(ii) (Board 2008, 10) 12303133

(iii) (Board 2006, 08, 09) 12303134

(iv) 12303135

(v) (Board 2007) 12303136

(vi) 12303137

(vii) (Board 2011) 12303138

(viii) 12303139

(ix) 12303140

(x) 12303141

(xi) 12303142

(xii) 12303143

(xiii) (Board 2011) 12303144

Integrating by parts

(xiv) 12303145

Integrating by parts

(xv) 12303146

(xvi) 12303147

(xvii) 12303148

(xviii) 12303149

(xix) (Board 2010) 12303150

Integrating by parts

(xx) 12303151

(xxi) 12303152

Q.2 Evaluate the following integrals:

(i) 12303153

(ii) 12303154

(iii) 12303155

(iv) 12303156

(v) 12303157

(vi) 12303158

(vii) (Board 2008) 12303159

(viii) 12303160

Q.3 Show that: 12303161

Q.4 Evaluate the following integrals:

(i) (Board 2009) 12303162

(ii)

(iii) (Board 2008, 09) 12303164

(iv) 12303165

(v)

(vi) 12303167

Q.5 Evaluate the following integrals

(i) (Board 2012) 12303168

(ii) 12303169

(Board 2009, 10)

(iii) 12303170

(Board 2008, 09)

(iv) 12303171

(v) 12303172

(vi) 12303173

(vii)

(Board 2005) 12303174

(viii) 03(164)

(ix) 12303175

(x) 12303176

(xi) 12303177

Integration Involving Partial Fractions:

Example 1:

Evaluate:

(Board 2010) 12303178

Example 2:

(i) Evaluate: (Board 2008) 12303179

(ii) Evaluate: (Board 2009) 12303180

Example 3:

Evaluate (Board 2008) 12303181

EXERCISE 3.5

Q.1 Evaluate: 12303182

Q.2 Evaluate: 12303183

→ (1)

Q.3 Evaluate 12303184

Q.4 Evaluate

(Board 2010) 12303185

Q.5 Evaluate 12303186

Q.6 Evaluate (Board 2012) 12303187

Q.7 Evaluate 12303188

Q.8 Evaluate 12303189

Q.9 Evaluate 12303190

Q.10 Evaluate 12303191

Q.11 Evaluate 12303192

Q.12 Evaluate 12303193

Q.13 Evaluate 12303194

Q.14 Evaluate 12303195

Q.15 Evaluate 12303196

Q.16 Evaluate 12303197

Q.17 Evaluate 12303198

Q.18 Evaluate 12303199

Q.19 Evaluate 12303200

Q.20 Evaluate 12303201

Q.21 Evaluate 12303202

Q.22 Evaluate 12303203

Q.23 Evaluate 12303204

Q.24 Evaluate 12303205

Q.25 Evaluate 12303206

Q.26 Evaluate 12303207

Q.27 Evaluate 12303208

Q.28 Evaluate 12303209

Q.29 Evaluate 12303210

Q.30 Evaluate 12303211

Q.31 Evaluate 12303212

Properties of Definite Integrals:

(i) = 12303214

(Board 2011, 12)

(ii) =

where (Board 2011) 12303215

(iii)

12303216

(iv)

Example 1:

Evaluate (Board 2011) 12303217

Example 2:

Evaluate:

(Board 2009, 10) 12303218

Example 3:

Evaluate (Board 2009) 12303219

Example 4:

Evaluate (Board 2008, 11) 12303220

Example 5:

Evaluate: (Board 2009) 12303221

Example 6:

If , ,

Then evaluate the following definite integrals.

(i) (Board 2008, 2010) 12303222

(ii) 12303223

(iii) 12303224

EXERCISE 3.6

Evaluate the following definite integrals:

Q.1 (Board 2008, 11) 12303225

Q.2 (Board 2009, 10) 12303226

Q.3 12303227

Q.4 12303228

Q.5 12303229

Q.6 12303230

Q.7 12303231

Q.8 12303232

Q.9 12303233

Q.10 (Board 2009) 12303234

Q.11 = (Board 2008) 12303235

Q.12 (Board 2011) 12303236

Q.13 12303237

Q.14 12303238

Q.15 12303239

Q.16 (Board 2012) 12303240

Q.17 12303241

Q.18 (Board 2012) 12303242

Q.19 12303243

Q.20 12303244

Q.21 12303245

Q.22 (Board 2007) 12303246

Q.23 12303247

Q.24 12303248

Q.25 12303249

Q.26 (Board 2010) 12303250

Q.27 (Board 2009, 10) 12303251

Q.28 (Board 2008) 12303252

Q.29 12303253

Q.30 12303254

Example 1:

Find the area bounded by the curve.

y = and x-axis. (Board 2009) 12303255

Example 2:

Find the area bounded by y = and the x-axis. 12303256

Example 3:

Find the area between x-axis and the curve y2=4−x in the first quad from x = 0 to x = 3. (Board 2008) 12303257

EXERCISE 3.7

Q.1 Find the area between the x-axis and the curve y = x 2 + 1 from x = 1 to x = 2. (Board 2008) 12303258

Q.2 Find the area, above the x-axis and under the curve y = 5 – x2 from x = –1 to

x = 2. (Board 2007, 11) 12303259

Q.3 Find the area below the curve

y = 3x and above the x-axis between x = 1 and x = 4. (Board 2009) 12303260

Q.4 Find the area bounded by cos function from x = – to x= . (Board 2008) 12303261

Q.5 Find the area between the x-axis and the curve y = 4x – x2.(Board 2005, 09,10) 12303262

Q.6 Determine the area bounded by the parabola y = x2 + 2x – 3 and the x-axis.

12303263

Q.7 Find the area bounded by the curve

y = x3 + 1, the x-axis and line x = 2. 12303264

Q.8 Find the area bounded by the curve

y = x3 – 4x and the x-axis. 12303265

Q.9 Find the area between the curve

y = x(x – 1)(x + 1) and the x-axis. 12303266

Q.10 Find the area above the x-axis, bounded by the curve y2 = 3  x from

x = 1 to x = 2. 12303267

Q.11 Find the area between the x-axis and the curve y = cos 12 x from x = −  to x = 

(Board 2011) 12303268

Q.12 Find the area between the x-axis and the curve y = sin 2x from x = 0 to x = 12303269

Q.13 Find the area between the x-axis and the curve y = when a > 0. 12303270

Examples:

(i) = 0 12303271

(ii) 12303272

Example: 12303273

Example 1:

Solve = (Board 2006) 12303274

Example 2:

Solve the differential equation.

= 0 or (Board 2007) 12303275

Example 3:

Solve the differential equation.

= (Board 2008) 12303276

If y = 0 when x = 2

EXERCISE 3.8

Q.1 Check that each of the following equations written against the differential equation is its solution.

(i) = 12303277

y = cx − 1

= → (1) y = cx − 1 → (2)

(ii) = 0 12303278

= 0 → (1)

= → (2)

(iii) = 1 12303279

y2 =

= 1 → (1)

y2 = → (2)

(iv) = 0 (Board 2008) 12303280

y =

= 0 → (1)

y = → (2)

(v) = 12303281

y =

= → (1)

y = → (2)

Solve the following differential equations:

12303282

Q.2 = − y 12303283

Q.3 = 0 12303284

ydx = −xdy

Q.4 = (Board 2009) 12303285

Q.5 = 12303286

Q.6 = 1 (Board 2010) 12303287

Q.7 = 0 12303288

Q.8 = 12303289

Q.9 = 12303290

Q.10 = 12303291

Q.11 = x 12303292

Q.12 = 0 12303293

Q.13 = 0 12303294

Q.14 = 12303295

Q.15 = 0 12303296

Q.16 = (Board 2009) 12303297

Q.17 = 0 12303298

Q.18 = 12303299

(Board 2011, 12)

Q.19 Find the general solution of the equation = . Also find particular solution if y = 1 when x = 0. 12303300

Q.20 Solve the differential equation

= 2x given that x = 4 when t = 0. 12303301

Q.21 Solve the differential equation = 0. Also find particular solution if s = 4e, when t = 0. 12303302

Q.22 In a culture, bacteria increases at the rate proportional to the number of bacteria present. If bacteria are 200 initially and are doubled in 2 hours, find the number of bacteria present four hours later. 12303303

Q.23 A ball is thrown vertically upward with a velocity of 2450 cm/sec. Neglecting air resistance find 12303304

(i) Velocity of ball at any time t 12303305

(ii) Distance traveled in any time t 12303306

(iii) Maximum height attained by the ball.

Unit Introduction to Analytic Geometry

The Distance Formula:

Let and be two points in the plane. Let ‘d’ be the distance between A and B. Then,

Distance = d = 12304001

Proof: A(x1, y1) and B(x2, y2) are two given points.

Example 1:

Show that the points A(1, 2), B(7, 5) and C(2, 6) are the vertices of a right triangle. (Board 2010) 12304002

5. Point Dividing The Join Of Two Points (Line Segment) In A Given Ratio:

Theorem:

Let and be the two points in a plane. The coordinates of the point dividing the line segment AB in the ratio k1 : k2 are

12304003

External Ratio: (Board 2010) 12304004

If the directed distances AP and PB have the opposite sign, i.e., P is beyond AB, then their ratio is negative and P is said to divide externally, then their ratio is:

Example:

Find the coordinates of the point that divides the join of A(6, 3) and B(5,2) in the ratio 2:3 (i) internally (ii) and externally. (Board 2009, 10) 12304005

Theorem: (Board 2010)

The centroid of a ∆ABC is a point that divides each median in the ratio 2:1 and that medians of a triangle are concurrent. 12304006

Theorem: (Board 2010, 11)

Prove that bisectors of angles of a triangle are concurrent. 12304007

What are the coordinates of the in centre of a triangle whose vertices are

EXERCISE 4.1

Q.1 Describe the location in the plane of the point for which

(i) x > 0 12304008

(ii) x > 0 and y > 0 12304009

(iii) x = 0 12304010

(iv) y = 0 12304011

(v) x < 0 and y  0 12304012

(vi) x = y 12304013

(vii) 12304014

(viii) 12304015

(ix) x > 2 and y = 2 12304016

(x) x and y have opposite signs. 12304017

Q.2 Find in each of the following: 12304018

(i) The distance between the two given points

12304018(a)

(ii) Midpoint of the line segment joining two points. (Board 2007, 10) 12304018(b)

(a) 12304019

(b) 12304020

12304021

Q.3 Which of the following points are at a distance of 15 units from the origin?

(a) 12304022

(b) 12304023

(d) (Board 2011) 12304025

Q.4 Show that; (Board 2009)

(i) The points and are vertices of a right triangle.

12304026

(ii) The points and are vertices of an isosceles triangle.

If two of its sides are equal then it will be an isosceles triangle; (Board 2010) 12304027

(iii) The points and are the vertices of a parallelogram. Is that parallelogram a square? 12304028

Q.5 The midpoints of the sides of a triangle are and . Find the coordinates of the vertices of the triangle.

(Board 2008) 12304029

Q.6 Find h such that the vertices , and are vertices of a right triangle with right angle at the vertex A. 12304030

Q.7 Find h such that and are collinear (Board 2008, 12) 12304031

Q.8 The points and are ends of diameter of a circle. Find centre and radius of the circle. (Board 2009) 12304032

Q.9 Find h such that the vertices and are the vertices of a right triangle with right angle at vertex A. 12304033

Q.10 A quadrilateral has the points and as its vertices. Find the midpoints of the sides. Show that the figure formed by joining the midpoints consecutively is a parallelogram. 12304034

Q.11 Find h such that the quadrilateral with its vertices and is parallelogram. Is it a square? 12304035

Q.12 If two vertices of an equilateral triangle are , find the third vertex. How many of these triangles are possible? 12304036

(i) = 12304037

(ii) = 12304038

Q.13 Find the point trisecting the join of and . 12304039

(Board 2008, 09, 11)

(i) C divides AB in the ratio 1:2. 12304040

(ii) D divides AB is the ration 2:1. 12304041

Q.14 Find the point three fifth of the way along the line segment from

. (Board 2008) 12304042

Q.15 Find the point P on the join of and that is twice as far from A as B is from A and lies (Board 2009) 12304043

(i) on the same side of A as B does. 12304044

(ii) On the opposite side of A as B does.

12304045

Q.16 Find the point which is equidistant from the points and . What is the radius of the circum-circle of ∆ABC? 12304046

Q.17 The points and are the vertices of triangle. Find

in-centre of the triangle. 12304047

Q.18 Find the points that divide the line segment joining and in four equal parts. 12304048

Translation of Axes: (Board 2010) 12304049

Let, xy-coordinates system, be given with as the origin. The axes be translated through the point and be the new axes.

Rotation of Axes: 12304050

Let, xy-coordinate system be given. We rotate ox and oy about the origin through an angle . The new axes are OX and OY.

Let P be a point having coordinates .

EXERCISE 4.2

Q.1 The two points P and O are given in

xy-coordinate system. Find the

XY-coordinates of P referred to the translated axes OX and OY. 12304051

(Board 2010, 11)

(i) 12304052

(ii) 12304053

(iii) 12304054

(iv) 12304055

Q.2 The xy-coordinate axes are translated through the point O whose coordinates are given in xy-coordinate system. The coordinates of P are given in the

XY-coordinate system. Find the coordinates of P in xy-coordinate system. 12304056

(i) 12304057

(ii) (Board 2011) 12304058

(iii) 12304059

(iv) 12304060

Q.3 The xy-coordinate axes are rotated about the origin through the indicated angle. The new axes are OX and OY. Find the XY-coordinates of the point P with the given xy-coordinates. 12304061

(i) (Board 2012) 12304062

(ii) (Board 2011) 12304063

(iii) 12304064

(iv) 12304065

Q.4 The xy-coordinate axes are rotated about the origin through the indicated angle and the new axes are OX and OY. Find the xy-coordinates of P with the given XY-coordinates. 12304066

(i) 12304067

(ii) 12304068

Inclination of a Line: (Board 2008) 12304069

The angle α (0o <  < 180o) measured

anti-clockwise from positive x-axis to a

non-horizontal line ‘ ’ is called the inclination of .

(i) If is parallel to x-axis, then α = 0

(ii) If is parallel to y-axis, then α = 90

Slope or Gradient of a Line: 12304070

Slope of a line is m, m = tan α

(i) If α = 0, m = tan 0 = 0

(ii) If α = 90, m = tan 90 =  = undefined.

(i) If a line with inclination α passes through and then slope or Gradient of is m = .

Example1:

Show that the points A(3, 6), B(3,2) and C(6,0) are collinear. (Board 2006) 12304071

1. EQUATIONS OF A STRAIGHT LINE IN STANDARD FORMS

Q.1 Equation of a line in slope-intercept form is y = mx + c

where, m = slope and c = y-intercept.

(Board 2011) 12304072

2. EQUATION OF A LINE IN POINT SLOPE FORM:

Equation of a non-vertical line with slope m and passes through a point is . 12304073

Proof:

Example 1: (Board 2010)

Find an equation of the straight line if

(a) its slope is 2 and y-intercept is 5

12304074

(b) it is perpendicular to a line with slope 6 and its y-intercept is . 12304075

3. EQUATION OF A LINE IN TWO POINTS FORM:

Equation of a line passing through two points and is

= 12304076

4. EQUATION OF A LINE IN INTERCEPTS FORM:

(also known as two intercepts form).

Equation of a line ‘ ’ having x-intercept is ‘a’ and y-intercept is ‘b’ is = 1.

(Board 2009, 11) 12304077

Example 1:

Write down an equation of the line which cuts the x-axis at (2, 0) and y-axis at (0,4). (Board 2009) 12304078

5. SYMMETRIC FORM OF EQUATION OF A STRAIGHT LINE. 12304079

6. EQUATION OF A LINE IN NORMAL FORM:

Equation of a line ‘ ’ when p is the length of perpendicular from origin to ‘ ’ and ‘α’ is the inclination of perpendicular is = p. (Board 2011) 12304080

A LINEAR EQUATION IN TWO VARIABLES REPRESENTS A STRAIGHT LINE:

6. Theorem:

7. A linear equation in two variables of the form ax + by + c = 0 always represents a straight line. 12304081

8. Transformation of the General Linear Equation ax + by + c = 0 to Standard forms:

9. 12304082

Proof of Normal Form: 12304083

(vi) Symmetric Form:

Example 1:

Transform the equation

into (Board 2008)

(i) Slope intercept form 12304084

(ii) Two-intercept form 12304085

(iii) Normal form 12304086

(iv) Point-slope form 12304087

(v) Two-point form 12304088

(vi) Symmetric form. 12304089

10. POINT OF INTERSECTION OF TWO LINES 12304090

CONDITION OF CONCURRENCY OF THREE STRAIGHT LINES 12304091

a. The Equation of Lines through the point of intersection of two Lines: 12304092

11. Example 1:

12. Find the family of lines through the point of intersection of the lines. (Board 2008) 12304093

Find the member of the family, which is

(a) parallel to a line with slope 12304094

(b) perpendicular to the line 12304095

Theorem: (Board 2008)

The distance d from the point to the line : = 0 is given by

d = 12304096

CONDITION OF PARALLELISM AND PERPENDICULARITY OF TWO LINES

13. 12304097

14.

POSITION OF A POINT WITH RESPECT TO A LINE

Example 1:

Check whether the point (2, 4) lies above or below the line (Board 2008, 09) 12304099

4x + 5 y  3 = 0

Example 2:

Check whether the origin and the point P(5, 8) lies on the same side or the opposite sides of the line: 12304100

3x + 7 y + 15 = 0

AREA OF A TRIANGULAR REGION:

Draw s AA, BB and CC from points A, B and C on X-axis. 12304101

Example 1:

Find the area of the region bounded by the triangle with vertices (a, b + c), (a, b  c) and (a, c). (Board 2008) 12304102

Example 2:

By considering the area of the region bounded by the triangle with vertices

A(1, 4), B(2, 3) and C(3, 10).

Check whether the three points are collinear or not. (Board 2008) 12304102

EXERCISE 4.3

Q.1 Find the slope and inclination of the line joining the points. Sketch each line in the plane.

(i) (Board 2005, 09) 12304104

(ii) 12304105

(iii) 12304106

Q.2 In the triangle

find slope of

(i) each side of the triangle. (Board 2009)12304107

(ii) Each median of the triangle. 12304108

(iii) Each altitude of the triangle. 12304109

Q.3 By means of slopes, show that the following points lie on the same line.

(a) (Board 2005) 12304110

(b) 12304111

(d) 12304113

Q.4 Find k so that the line joining and the line joining are:

(Board 2008, 09) 12304114

Q.5 Using slopes, show that the triangle with its vertices and is a right triangle.

(Board 2009) 12304115

Q.6 The three points and are consecutive vertices of a parallelogram. Find the fourth vertex.

(Board 2008) 12304116

Q.7 The points and are consecutive vertices of a rhom-bus. Find the fourth vertex and show that the diagonals of the rhombus are perpendicular to each other. 12304117

Q.8 Two pairs of points are given. Find whether the two lines determined by these points are: (i) parallel (ii) perpendicular (iii) none (Board 2006) 12304118

(a) 12304119

(b) 12304120

Q.9 Find an equation of

(a) the horizontal line through

(Board 2007, 10) 12304121

(b) the vertical line through

(Board 2011, 12) 12304122

(d) the line bisecting the second and fourth quadrants. (Board 2010) 12304124

Q.10 Find an equation of the line through having slope 7 12304125

(b) through having slope O 12304126

12304127

(d) through and

(Board 2009) 12304128

(e) y-intercept: –7 and slope: –5 12304129

(f) x-intercept: –3 and y-intercept: 4 12304130

(g) x-intercept: –9 and slope: –4

(Board 2008, 11) 12304131

Q.11 Find an equation of the perpendicular bisector of the segment joining points and . (Board 2010, 12) 12304132

Q.12 Find equations of the sides, altitudes and medians of the triangle whose vertices are , and .

(Board 2011) 12304133

(i) Equations of sides; 12304134

(ii) Equations of Altitudes; 12304135

(iii) Equations of Medians. 12304136

Q.13 Find an equation of the line through and perpendicular to a line having slope . (Board 2006) 12304137

Q.14 Find an equation of the line through and parallel to a line with

slope –24. 12304138

Q.15 The points are vertices of a triangle. Show that the line joining the midpoint D of AB and the midpoint E of AC is parallel to BC and

DE = BC. (Board 2008, 11) 12304139

Q.16 A milkman can sell 560 liters of milk at Rs.12.50 per liter and 700 liters of milk at Rs.12.00 per liter. Assuming the graph of the sale price and the milk sold to be a straight line, find the number of liters of milk that the milkman can sell at Rs.12.25 per liter. 12304140

Q.17 The population of Pakistan to the nearest million was 60 million in 1961 and 95 million in 1981. Using t as the number of years after 1961, find an equation of the line that gives the population in terms of t. Use this equation to find the population in (a) 1947 (b) 1997. 12304141

Q.18 A house was purchased for Rs.1 million in 1980. It is worth Rs.4 million 1996. Assuming that the value increased by the same amount each year, find an equation that gives the value of the house after t years of the date of purchase. What was its value in 1990? 12304142

Q.19 Plot the Celsius (C) and Fahrenheit (F) temperature scales on the horizontal axis and the vertical axis respectively. Draw the line joining the freezing point and the boiling point of water. Find an equation giving F temperature in terms of C. 12304143

Q.20 The average entry test score of engineering candidates was 592 in the year 1998 while the score was 564 in 2002. Assuming that the relationship between time and score is linear, find the average score for 2006. 12304144

Q.21 Convert each of the following equation into (Board 2009) 12304145

(i) Slope intercept form 12304146

(ii) Two intercept form 12304147

(iii) Normal form 12304148

(a) = 0 12304149

(i) Slope intercept form;

(b) = 0

(i) Slope intercept form; 12304150

(ii) Two-intercept form; 12304151

(iii) Normal form; (Board 2009) 12304152

(i) Slope intercept form; 12304153

(ii) Two intercept form; 12304154

(iii) Normal form;

Q.22 In each of the following check whether the two lines are; 12304155

(i) Parallel 12304156

(ii) Perpendicular 12304157

(iii) Neither parallel nor perpendicular 12304158

(a) = 0, = 0 12304159

(b) 3y = 2x + 5, 3x + 2y – 8 = 0 12304160

(d) 4x – y + 2 = 0, 12x – 3y + 1 = 0 12304162

(e) 12x + 35y – 7 = 0, 105x – 36y + 11 = 0

12304163

Q.23 Find the distance between the given parallel lines. Also find an equation of the line parallel lying midway between them.

(a) 3x – 4y + 3 = 0 , 3x – 4y + 7 = 0

(Board 2010) 12304164

(b) 12x + 5y – 6 = 0, 12x + 5y + 13 = 0

(Board 2010) 12304165

Q.24 Find an equation of the line through and parallel to line 2x – 7y + 4 = 0.

12304167

Q.25 Find the equation of line through and perpendicular to join of and .

(Board 2011) 12304168

Q.26 Find the equations of two parallel lines perpendicular to 2x – y + 3 = 0 such that the product of the x and y intercepts of each is 3. (Board 2012) 12304169

Q.27 One vertex of a parallelogram is

(1, 4). The diagonals intersect at (2, 1) and the sides have slopes 1 and . Find the other three vertices. 12304170

Q.28 Find whether the given points lie above or below the given line. 12304171

(a) (5, 8) ; 2x – 3y + 6 = 0 12304172

(b) (–7, 6) ; 4x + 3y – 9 = 0 12304173

Q.29 Check whether the given points are on the same or opposite sides of the given line. 12304174

(a) (0, 0) and (–4, 7) ; 4x – 7y + 70 = 0 12304175

(b) (2, 3) and (–2, 3); 3x – 5y + 8 = 0 12304176

Q.30 Find the distance from the point

P(6, –1) to the line 6x – 4y + 9 = 0. 12304177

Q.31 Find the area of the triangle region whose vertices are A(5, 3), B(–2, 2),

C(4, 2). (Board 2008) 12304178

Q.32 The coordinates of these points are

A(2, 3), B(–1, 1) and C(4, –5). By computing the area bounded by ABC. Check whether the points are collinear. 12304179

Theorem:

Let and be two non-vertical lines such that they are not perpendicular to each other. If m1 and m2 are the slopes of and respectively, then the angle  from to is given by: tan  =

and the angle  from to is given by tan  = . 12304180

Example 1:

Find the angle from the line with slope to the line with slope .

(Board 2010, 11, 12) 12304181

Example 1: (Board 2009)

Express the system

in matrix form and check whether the three lines are concurrent. 12304182

EXERCISE 4.4

Q.1 Find the point of intersection of the lines. 12304183

(i) x – 2y + 1 = 0, 2x – y + 2 = 0

(Board 2009, 11) 12304184

Q.2 Find an equation of the line through

(i) The point (2, –9) and the intersection of the lines; 12304187

2x + 5y – 8 = 0 and 3x – 4y – 6 = 0

(ii) The intersection of the lines; (Board 2009)

x – y – 4 = 0 ; 7x + y + 20 = 0 12304187

(a) Parallel 12304188

(b) Perpendicular to the line 6x + y – 14 = 0

12304189

(iii) Through the intersection of the lines

x + 2y + 3 = 0 and 3x + 4y + 7 = 0 and making equal intercepts on axes. 12304190

Q.3 Find the equation of the line through the intersection of 16x – 10y – 33 = 0;

12x + 14y + 29 = 0 and the intersection of

x – y+ 4 = 0 ; x – 7y + 2 = 0 12304191

Q.4 Find the condition that the lines

(Board 2011)

y1 = m1x + c1

y2 = m2x + c2

y3 = m3x + c3 are concurrent. 12304192

Q.5 Determine the value of p such that the lines (Board 2009)

2x – 3y – 1 = 0

3x – y – 5 = 0

3x + py + 8 = 0

meet at a point. 12304193

Q.6 Show that the lines 4x – 3y – 8 = 0,

3x – 4y – 6 = 0 and x – y – 2 = 0 are concurrent and the third line bisects the angle formed by the first two lines.

(Board 2011) 12304194

Q.7 The vertices of a triangle are A(–2, 3), B(–4, 1) and C(3, 5). Find coordinates of the (i) Centroid (ii) Orthocentre 12304195

(iii) Circumcenter of the triangle.

Are these three points collinear?

1. Centroid: 12304196

2. Orthocentre: 12304197

3. Circumcentre: 12304198

Q.8 Check whether the lines are concurrent. If so, find the point where they meet. (Board 2011) 12304199

Q.9 Find the coordinates of vertices of triangle formed by the lines x – 2y – 6 = 0, 3x – y + 3 = 0, 2x + y – 4 = 0. Also find measure of angles of triangle. 12304200

Q.10 Find the angle measured from the line 1 to the line where; (Board 2010) 12304201

(i) 1: joining (2, 7) and (7, 10) 12304202

2: joining (1, 1) and (–5, 3)

(b) 1: joining (3, –1) and (5, 7) 12304203

2: joining (2, 4) and (-8, 2)

2: joining (–1, 2) and (–6, –1)

(d) 1: joining (–9, –1) and (3, –5) 12304205

2: joining (2, 7) and (–6, –7)

Q.11 Find the interior angles of the triangle whose vertices are; (Board 2008, 09) 12304206

(a) A(–2, 11), B(–6, –3) and C(4, –9) 12304207

(b) A(6, 1), B(2, 7), C(–6, –7) 12304208

(d) A(2, 8), B(–5, 4), C(4, –9) 12304210

Q.12 Find the interior angles of the quadrilateral whose vertices are A(5, 2),

B(–2, 3), C(–3, –4) and D(4, –5). 12304211

Q.13 Show that the points A(0, 0), B(2, 1), C(3, 3) and D(1, 2) are the vertices of a rhombus. Find its interior angles. 12304212

Q.14 Find the area bounded by the triangle whose sides are; 12304213

7x – y – 10 = 0

10x + y – 41 = 0

3x + 2y + 3 = 0

Q.15 The vertices of a triangle are A (–2, 3), B(–4, 1) and C(3, 5). Find the circumcentre of the triangle. (S.B. 2010) 12304214

Q.16 Express the given system of equations in matrix form. Find in each case whether the lines are concurrent. 12304215

(a) x + 3y – 2 = 0 12304216

2x – y + 4 = 0

x – 11y + 14 = 0

(b) 2x + 3y + 4 = 0 12304217

x – 2y – 3 = 0

3x + y – 8 = 0

x + 2y – 4 = 0

3x – 2y + 5 = 0

Q.17 Find a system of linear equations corresponding to the given matrix form. Check whether the lines are concurrent.

(a) = 12304219

(b) = 12304220

Example: (Board 2007, 09, 11)

Find an equation of each of the lines represented by 12304221

20x2 + 17xy – 24y2 = 0

Example 1:

Find measure of the angle between the lines represented by x2– xy – 6y2 = 0.

(Board 2008, 09) 12304222

Example 2:

Find a joint equation of the straight lines through the origin perpendicular to the lines represented by (Board 2010) 12304223

x2 + xy – 6y2 = 0  (1)

EXERCISE 4.5

Find the lines represented by each of the following and also find measure of the angle between them (1 – 6):

(Board 2010)

Q.1 10x2 – 23xy – 5y2 = 0 12304224

Q.2 3x2 + 7xy + 2y2 = 0 04(138)

(Board 2006, 07, 09) 12304225

Q.3 9x2 + 24xy + 16y2 = 0

(Board 2009) 12304226

Q.4 2x2 + 3xy – 5y2 = 0

(Board 2009, 10, 11) 12304227

Q.5 6x2 – 19xy + 15y2 = 0 (Board 2009) 12304228

Q.6 x2 + 2xy sec α + y2 = 0 12304229

Q.7 Find a joint equation of the line through the origin and perpendicular to the lines; x2 – 2xy tan α + y2 = 0 12304230

Q.8 Find a joint equation of the lines through the origin and perpendicular to the lines; ax2 + 2h xy + by2 = 0 12304231

Q.9 Find the area of the region bounded by; 10x2 – xy – 21y2 = 0 and x + y + 1 = 0

12304232

Unit Linear Inequalities & Linear Programming

ii.

iii. LINEAR INEQUALITIES IN ONE VARIABLE: (BOARD 2008) 12305001

A linear inequality in one variable x is an inequality which can be written in the form

ax + b > c (or  c, < c,  c).

For example,

(i) 2x + 3 < 5 (ii) 7x + 8  6x + 1

(iii) 5 x3 + 12  2 x4 + 1 (iv) x + 12 + x + 23  16

Example: (Board 2006) 12305002

A solution of an inequality in one variable such as 2x  3 < 0  (a)

is a real number which satisfies the inequality (a).

The inequality (a) is true if x = 1, that is, 2(1)  3 < 0  1 < 0 which is true.

But if we put x = 2 in (a), then 2(2)  3 < 0  1 < 0 which is not true, so 2 is not a solution of (a). To solve (a), we add 3 to both sides of (a), that is,

Definition: (Board 2008) 12305003

(i) A half-plane is said to be a closed half-plane if all the points on the line separating the two half-planes are also included in the half-plane.

(ii) A half-plane is said to be an open half-plane if the points on the line separating the two half-planes are not included in the half-plane.

2. Example:

Find which of the following points lie on the graph of the closed half-plane 3x + 4y  14

(i) (2, 3) (ii) (4, 6) (iii) 223 2 (iv) (0, 0).

Example 1:

Graph the system of inequalities: (Board 2008, 12) 12305004

x  2y  6

2x + y  2

Example 2:

Graph of the solution region for the following system of inequalities: (Board 2010) 12305005

x  2y  6, 2x + y  2, x + 2y  10

Definition: (Board 2009) 12305006

A point of a solution region where two of its boundary lines intersect is called a corner point or vertex of the solution region.

Such points play a useful role while solving linear programming problems. In example 2, the following three corner points are obtained by solving the corresponding equations (of linear inequalities given in the example 2) in pairs.

Corresponding lines of inequalities: Corner Points

x  2y = 6, 2x + y = 2 P(2, 2)

x  2y = 6, x + 2y = 10 Q(8, 1)

2x + y = 2, x + 2y = 10 R(2, 6)

Example 3:

Graph the following systems of inequalities.

(i) 2x + y  2 12305007 (ii) 2x + y  2 12305008 (iii) 2x + y  2 12305009

x + 2y  10 x + 2y  10 x + 2y  10

y  0 x  0 x  0, y  0.

EXERCISE 5.1

Q.1 Graph the solution set of each of the following linear inequality in xy-plane.

(i) 2x + y  6 (Board 2011) 12305010

(ii) 3x + 7y  21 12305011

(iii) 3x  2y  6 (Board 2010) 12305012

(iv) 5x  4y  20 (Board 2008, 10) 12305013

(v) 2x + 1  0 (Board 2010, 11) 12305014

(vi) 3y  4  0 (Board 2011) 12305015

Q.2 Indicate the solution set of the following systems of linear inequalities by shading.

(i) 2x  3y  6 (Board 2011) 12305016

2x + 3y  12

(ii) x + y  5 12305017

 y + x  1

(iii) 3x + 7y  21 (Board 2008) 12305018

x  y  2

(iv) 4x  3y  12 (Board 2011, 12) 12305019

x   32

(v) 3x + 7y  21 (Board 2011, 12) 12305020

y  4

Q.3 Indicate the solution set of the following systems of linear inequalities by shading. (Board 2011)

(i) 2x  3y  6 12305021

2x + 3y  12

y  0

(ii) x + y  5 12305022

y  2x  2

x  0

(iii) x + y  5 12305023

x  y  1

y  0

(iv) 3x + 7y  21 (Board 2008) 12305024

x  y  2

x  0

(v) 3x + 7y  21 12305025

x  y  2

y  0

(vi) 3x + 7y  21 (Board 2012) 12305026

2x  y   3

x  0

Q.4 Graph the solution set of the following systems of linear inequalities and find the corner points in each case.

(i) 2x  3y  6 12305027

2x + 3y  12

x  0

(ii) x + y  5 (Board 2011) 12305028

 2x + y  2

y  0

(iii) 3x + 7y  21 12305029

2x  y   3

y  0

(iv) 3x + 2y  6 12305030

x + 3y  6

y  0

(v) 5x + 7y  35 (Board 2011) 12305031

 x + 3y  3

x  0

(vi) 5x + 7y  35 12305032

x  2y  2

x  0

Q.5 Graph the solution set of the following systems of linear inequalities by shading.

(i) 3x  4y  12 12305033

3x + 2y  3

x + 2y  9

(ii) 3x  4y  12 12305034

x + 2y  6

x + y  1

(iii) 2x + y  4 12305035

2x  3y  12

x + 2y  6

(iv) 2x + y  10 12305036

x + y  7

 2x + y  4

(v) 2x + 3y  18 12305037

2x + y  10

 2x + y  2

(vi) 3x  2y  3 12305038

x + 4y  12

3x + y  12

Problem Constraints: (Board 2008, 12) 12305039

The systems of linear inequalities involved in the problem concerned are called problem constraints. The variables used in the system of linear inequalities relating to the problems of every day life are non-negative and are called non-negative constraints. These non-negative constraints play an important role for taking decision. So these variables are called decision variables.

Feasible Solution Set:d (Board 2007) 12305040

A region (which is restricted to the first quadrant) is referred to as a feasible region for the set of given constraints. Each point of the feasible region is called a feasible solution of the system of linear inequalities (or for the set of a given constraints). A set consisting of all the feasible solutions of the system of linear inequalities is called a feasible solution set. (Board 2008, 10) 12305041

The feasible region is unbounded as it cannot be enclosed in any circle and feasible region is bounded as it can be enclosed within a circle. If the line segment obtained by joining any two points of a region lies entirely within the region, then the region is called convex. (Board 2008) 12305042

Example 1:

ii. Graph the feasible region and find the corner points for the following system of inequalities

(or subject to the following constraints). 12305043

x  y  3, x + 2y  6 , x  0, y  0

iii. Example 2:

iv. A manufacturer wants to make two types of concrete. Each bag of A-grade concrete contains 8 kilograms of gravel (small pebbles with coarse sand) and 4 kilograms of cement while each bag of B-grade concrete contains 12 kilograms of gravel and two kilograms of cement. If there are 1920 kilograms of gravel and 480 kilograms of cement, then graph the feasible region under the given restrictions and find corner points of the feasible region. 12305044

Example 3:

Graph the feasible regions subject to the following constraints.

(a) 2x  3y  6 12305045 (b) 2x  3y  6 12305046

2x + y  2 2x + y  2

x  0, y  0 x + 2y  8, x  0, y  0

EXERCISE 5.2

Q.1 Graph the feasible region of the following system of linear inequalities and find the corner points in each case. (Board 2011) 12305047

(i) 2x  3y  6 12305048

2x + 3y  12

x  0, y  0

(ii) x + y  5 (Board 2011) 12305049

 2x + y  2

x  0, y  0

(iii) x + y  5 12305050

 2x + y  2

x  0

(iv) 3x + 7y  21 12305051

x  y  3

x  0, y  0

(v) 3x + 2y  6 12305052

x + y  4

x  0, y  0

(vi) 5x + 7y  35 12305053

x  2y  4

x  0, y  0

Q.2 Graph the feasible region of the following system of linear inequalities and find the corner points in each case. 12305054

(i) 2x + y  10 12305055

x + 4y  12

x + 2y  10

x  0, y  0

(ii) 2x + 3y  18 12305056

2x + y  10

x + 4y  12

x  0, y  0

(iii) 2x + 3y  18 12305057

x + 4y  12

3x + y  12

x  0, y  0

(iv) x + 2y  14 12305058

3x + 4y  36

2x + y  10

x  0, y  0

(v) x + 3y  15 12305059

4x + 3y  24

2x + y  12

x  0, y  0

(vi) 2x + y  20 12305060

8x + 15y  120

x + y  11

x  0, y  0

i. Linear Programming: (Board 2007, 08)

Definitions Related to Linear Programming

1. Objective Function:

In linear programming, our object is always to maximize or minimize a linear function of decision variables. This function is called the objective function.

2. Standard form of linear programming problem (L.P.P.)

Maximize/Minimize P = ax + by

Subject to a1 x + b1 y (, =, ) c1

a2 x + b2 y (, =, ) c2

a3 x + b3 y (, =, ) c3

x, y  0

3. Optimal Feasible Solution: (Board 2008)

A feasible solution of a linear programming problem is said to be an optimal feasible solution

(or optimal solution), if it also optimizes (maximizes or minimizes) the objective functions.

Theorem 2: (Board 2010)

Fundamental theorem of linear programming.

(a) If feasible region is bounded, the objective function will have both a maximum and minimum values and these will occur at corner points.

(b) If the feasible regions is unbounded, the objective function may not have a maximum or minimum, but if there is, then it must occur at a corner point.

Convex Set: (Board 2008)

A set of points, S, is called convex if, for any two points P and Q in S, the entire segment PQ is in S (see fig. 9.6).

Theorem 3:

Fundamental extreme point theorem.

An optimal solution of a linear programming problem (LPP), if it exists, occurs at one of the extreme (corner) points of the convex polygon of the set of all feasible solutions.

Convex sets

Figure 9.6

Sets that are not convex.

In linear programming, the boundaries of the sets will be lines or line segments, so we will be dealing with what mathematicians call polygonal convex sets. Parts (b) and (c) of figure 9.6 are polygonal convex sets.

Note that the polygonal convex set in part (c) of figure 9.6 is unbounded.

Theorem 4:

The set of all feasible solutions of a linear programming problem is a convex set.

Procedure to Solve a Linear Programming Problem:

(i) Find the objective function (the quantity to be maximized or minimized).

(ii) Find and graph the constraints defined by a system of linear inequalities; the simultaneous solution is called the feasible region.

(iii) Find the corner points of the feasible region; this may require the solution of a system of two equations with two unknowns, one for each corner point.

(iv) Find the value of the objective function for the coordinates of each corner point.

The largest value is the maximum; the smallest value is the minimum.

Example 1:

Find the maximum and minimum values of the function defined as:

f(x, y) = 2x + 3y

subject to the constraints; 12305061

x  y  2

x + y  4

2x  y  6, x  0

Example 2:

Find the minimum and maximum values of f and  defined as:

f(x, y) = 4x + 5y, (x, y) = 4x + 6y

under the constraints (Board 2010, 12) 12305062

2x  3y  6

2x + y  2

2x + 3y  12

x  0, y  0

Example 1:

A farmer possesses 100 canals of land and wants to grow corn and wheat. Cultivation of corn requires 3 hours per canal while cultivation of wheat requires 2 hours per canal. Working hours cannot exceed 240. If he gets a profit of Rs. 20 per canal for corn and Rs.15/- per canal for wheat, how many canals of each he should cultivate to maximize his profit?

(Board 2010)12305063

Example 2:

A factory produces bicycles and motorcycles by using two machines A and B. Machine A has at most 120 hours available and machine B has a maximum of 144 hours available. Manufacturing a bicycle requires 5 hours in machine A and 4 hours in machine B while manufacturing of a motorcycle requires 4 hours in machine A and 8 hours in machine B. If he gets profit of Rs.40 per bicycle and profit of Rs.50 per motorcycle, how many bicycles and motorcycles should be manufactured to get maximum profit? 12305064

EXERCISE 5.3

Q.1 Maximize f (x, y) = 2x + 5y subject to the constraints (Board 2010) 12305065

2y  x  8 ; x  y  4 ; x  0 ; y  0 (Board 2009)

Q.2 Maximize f (x, y) = x + 3y subject to the constraints 12305066

2x + 5y  30 ; 5x + 4y  20 ; x  0 ; y  0

Q.3 Maximize z = 2x + 3y ; subject to the constraints: 12305067

3x + 4y  12 ; 2x + y  4 ; 4x y  4; x  0; y  0

Q.4 Minimize z = 2x + y: subject to the constraints: 12305068

x + y  3 ; 7x + 5y  35 ; x  0 ; y  0

Q.5 Maximize the function defined as; f (x, y) = 2x + 3y subject to the constraints:

2x + y  8 ; x + 2y  14 ; x  0 ; y  0 (Board 2008, 10, 11) 12305069

Q.6 Minimize z = 3x + y; subject to the constraints: (Board 2010, 11) 12305070

3x + 5y  15 ; x + 3y  9 ; x  0 ; y  0

Q.7 Each unit of food X costs Rs.25 and contains 2 units of protein and 4 units of iron while each unit of food Y costs Rs.30 and contains 3 units of protein and 2 unit of iron. Each animal must receive at least 12 units of protein and 16 units of iron each day. How many units of each food should be fed to each animal at the smallest possible cost? 12305071

Q.8 A dealer wishes to purchase a number of fans and sewing machines. He had only

Rs. 5760 to invest and has space at most for 20 items. A fan costs him Rs. 360 and a sewing machine costs Rs.240 His expectation is that he can sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. 18 Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? 12305072

Q.9 A machine can produce product A by using 2 units of chemical and 1 unit of a compound or can produce product B by using 1 unit of chemical and 2 units of the compound. Only 800 units of chemical and 1000 units of the compound are available. The profits per unit of A and B are Rs. 30 and Rs. 20 respectively, maximize the profit function. 12305073

Unit Conic Section

THE CIRCLE

Definition: (Board 2008) 12306001

A circle is the set of points in a plane whose distance from a fixed point remains constant. The fixed point is called the centre and the constant distance is called the radius of the circle.

Theorem: (Board 2011)

Find the equation of a circle whose centre is the point (h, k) and the radius is 'r'. 12306002

Theorem: (Board 2012) 12306003

Find the equation of a circle whose centre is at the origin and whose radius is equal to 'r'.

Example:

Write an equation of the circle with center (3, 5) and radius 7. 12306004

General Form Of An Equation Of A Circle

15. Theorem: (Board 2009, 12) 12306005

16. The equation

x2 + y2 + 2gx + 2fy + c = 0

represents a circle g, f and c being constants.

Properties of General Form Of A Circle:

(i) Due to the terms x2 and y2, it is second degree equation in x and y and the coefficients of x2 and y2 are equal to 1.

(ii) There is no term involving the product xy.

1. A Circle Passing Through Two Points As Ends of a Diameter:

Example:

Find an equation of the circle having the join of A(x1, y1) and B(x2 , y2) as a diameter.

12306006

2. A Circle Passing Through Two Points And Equation of Tangent at One of These Points is Known.

[Example:

Find an equation of the circle passing through the point (2, 5) and touching the line 3x + 4y  24 = 0 at the point (4, 3).

12306007

3. A Circle Passing Through Two Points And Touching A Given Line.

Example:

Find an equation of the circle passing through the points A(1, 2) and B(1, 2) and touching the line x + 2y + 5 = 0. 12306008

EXERCISE 6.1

Q.1 In each of the following, find an equation of the circle with 12306009

(a) Centre at (5, 2) and radius 4 12306010

(Board 2007, 08, 09)

(b) Centre at 2  33 and radius 22

(Board 2009) 12306011

(5, 6). (Board 2008, 09, 10) 12306012

Q.2 Find the centre and radius of the circle with the given equation 12306013

(Board 2009, 10)

(a) x2 + y2 + 12x  10y = 0 12306014

(b) 5x2 + 5y2 + 14x + 12y  10 = 0 12306015

(Board 2009, 12)

x2 + y2  6x + 4y + 13 = 0 ….. (i) (Board 2009)

(d) 4x2 + 4y2  8x + 12y  25 = 0 12306017

(Board 2006, 08, 09)

Q.3 Write an equation of circle passing through the given points 12306018

(a) A(4, 5), B(4, 3) and C(8, 3) 12306019

(Board 2008, 09, 11)

(b) A(7, 7) B(5, 1), C(10, 0) 12306020

(d) A(5, 6), B(3, 2), C(3, 4) 12306022

Q.4 In each of following, find an equation of the circle passing through 12306023

(a) A(3, 1), B(0, 1) and having centre at

4x  3y  3 = 0 (Board 2008) 12306024

(b) A(3, 1) with radius 2 and centre on

2x  3y + 3 = 0 12306025

2x  y  10 = 0 at B(3,  4) 12306026

(d) A(1, 4), B(1, 8) and tangent to the line x + 3y  3 = 0 12306027

Q.5 Find an equation of a circle of radius a and lying in the second quadrant such that it is tangent to both the axes. 12306028

Q.6 Show that the lines 3x  2y = 0 and

2x + 3y  13 = 0 are tangents to circle

x2 + y2 + 6x  4y = 0 12306029

Q.7 Show that the circles x2 + y2 +2x  2y  7= 0 and x2 + y2  6x + 4y + 9 = 0 touch externally.

(Board 2009) 12306030

Q.8 Show that the circles x2+y2 + 2x  8 = 0 and x2 + y2  6x + 6y  46 = 0 touch internally. 12306031

Q.9 Find equations of the circles of radius 2 and tangent to the line x  y  4 = 0 at A(1, 3).

12306032

Tangents And Normals:

Rule For Writing The Equation Of A Tangent To A Circle: (Board 2009) 12306034

Theorem:

The point P(x1, y1) lies outside, on or inside the circle

x2 + y2 + 2gx + 2f + c = 0

according as

x12 + y12 + 2gx1 + 2fy1 + c 0

12306035

Example: (Board 2008)

Determine whether the point P(5, 6) lies outside, on or inside the circle:

x2 + y2 + 4x  6y  12 = 0 12306036

Intersection Of A Line And A Circle:

Theorem:

The line y = mx + c intersects the circle x2 + y2 = a2 at the most at two points.

(Board 2009) 12306037

Theorem:

Two tangents can be drawn to a circle from any point P (x1 , y1). The tangents are real and distinct, coincident or imaginary according as the point lies outside, on or inside the circle. 12306038

Example 1: 12306039

Write equations of two tangents from (2, 3) to the circle x2 + y2 = 9.

Example 2:

Write equations of two tangents to the circle x2 + y2  4x + 6y + 9 = 0

at the points on the circle whose ordinate is 2.

12306040

Example 3:

Find a joint equation to the pair of tangents drawn from (5, 0) to the circle: 12306041

x2 + y2 = 9

Theorem:

Find the length of the tangent drawn from the point (x1, y1) to the circle

x2 + y2 + 2gx + 2fy + c = 0. 12306042

Example 1:

Find the length of the tangent from point

P(5, 10) to the circle 5x2 +5y2 +14x+12y10 = 0.

(Board 2009) 12306043

Example 2:

Write equations of the tangent lines to the circle x2 + y2 + 4x + 2y = 0 12306044

drawn from P(1, 2). Also find the tangential distance.

Example 3:

Tangents are drawn from (3, 4) to the circle x2 + y2 = 21. Find an equation of the line joining the points of contact (The line is called the chord of contact). 12306045

Length of a Chord of a Circle:

Find the length of the chord of the circle

x2 + y2 = a2 on the line y = mx + c. 12306046

EXERCISE 6.2

Q.1 Write down equations of the tangent and normal to the circle 12306047

(Board 2009, 11)

(i) x2 + y2 = 25 at (4,3) and at (5cos, 5sin )

x2 + y2 = 25 at (5 cos , 5 sin )

(Board 2009, 11) 12306049

(ii) 3x2 + 3y2 + 5x  13y + 2 = 0 at 1 103

(Board 2010) 12306050

Q.2 Write down equations of the tangent and normal to the circle. 12306051

4x2 + 4y2 – 16x + 24y – 117 = 0 at the points whose abscissa is – 4. (Board 2008)

a. Q.3 Check the position of the point (5, 6) with respect to circle 12306052

(Board 2007, 09, 10, 11)

(i) x2 + y2 = 81 12306053

(ii) 2x2 + 2y2 + 12x  8y + 1 = 0 12306054

(Board 2011, 12)

Q.4 Find the length of the tangent drawn from the point (5, 4) to the circle

5x2 + 5y2 – 10x + 15y – 131 = 0 12306055

(Board 2007, 08, 12)

Q.5 Find the length of the chord cut off from the line 2x + 3y = 13 by the circle

x2 + y2 = 26. 12306056

Q.6 Find the coordinates of the points of intersection of the line x + 2y = 6 with the circle x2 + y2 – 2x – 2y – 39 = 0 12306057

(Board 2009)

Q.7 Find equation of the tangents to the circle x2 + y2 = 2 12306058

(i) parallel to the line x – 2y = 1 12306059

(ii) perpendicular to the line 3x+2y=6 12306060

(ii) Find the equations of tangents to the circle x2 + y2 = 2 (Board 2009) 12306061

Q.8 Find equations of the tangents drawn (i) from (0, 5) to the circle. Also find the points of contact. 12306062

x2 + y2 = 16

(ii) Find equations of the tangents drawn from (1, 2) to x2 + y2 + 4x + 2y = 0.

(Board 2010) 12306063

(iii) Find equations of tangents drawn from (7, 2) to (x + 1)2 + (y  2)2 = 26 12306064

Q.9 Find an equation of the chord of contact of the tangents drawn from (4, 5) to the circle 2x2+ 2y2 – 8x + 12y + 21 = 0 12306065

Theorem 1:

Length of a diameter of the circle

x2 + y2 = a2 is 2a. (Board 2010) 12306066

Theorem 2:

Perpendicular dropped from the center of a circle on a chord bisects the chord. 12306067

Theorem 3:

The perpendicular bisector of any chord of a circle passes through the center of the circle. 12306068

Theorem 4:

The line joining the center of a circle to the midpoint of a chord is perpendicular to the chord. (Board 2010) 12306069

Theorem 5:

Congruent chords of a circle are equidistant from its center. 12306070

Theorem 6:

Show that measure of the central angle of a minor arc is double the measure of the angle subtended in the corresponding major arc. 12306071

Theorem 7:

An angle in a semi-circle is a right angle.

12306072

Theorem 8:

The tangent to a circle at any point of the circle is perpendicular to the radial segment at that point. 12306073

Theorem 9:

The perpendicular at the outer end of a radial segment is tangent to the circle.

12306074

EXERCISE 6.3

Q.1 Prove that normal lines of a circle pass through the centre of circle.

(Board 2009, 12) 12306075

Q.2 Prove that the straight line drawn from centre of a circle perpendicular to a tangent passes through the point of tangency. 12306076

Q.3. Prove that the mid point of the hypotenuse of a right angled triangle is the circumcentre of the triangle. (Board 2011)

Q.4 Prove that the perpendicular dropped from a point of a circle on a diameter is a mean proportional between the segments into which it divides the diameters. 12306078

(Board 2008)

Analyze the parabola x2 = 16y and draw its graph. 12306079

Find an equation of the parabola whose focus is F(3, 4) and directrix is 3x4y+5=0.

(Board 2009, 11)

Analyze the parabola

x2  4x  3y + 13 = 0

and sketch its graph. 12306081

(Board 2007)

The point of a parabola which is closest to the focus is the vertex of the parabola.

12306082

A comet has a parabolic orbit with the sun at the focus. When the comet is 100 million km from the sun, the line joining the sun and the comet makes an angle of 60o with the axis of the parabola. How close will the comet get to the sun?

A suspension bridge with weight uniformly distributed along the length has two towers of 100 m height above the road surface and are 400 m apart. The cables are parabolic in shape and are tangent to road surface at the center of the bridge. Find the height of the cables at a point 100 m from the center.

12306084

Q.1 Discuss and sketch the graph of the following parabolas: 12306085

(i) y2 = 8 x (Board 2008, 12) 12306086

(ii) x2 = 16y (Board 2008, 11) 12306087

(iii) x2 = 5y (Board 2012) 12306088

(iv) y2 = –12 x 12306089

(v) x2 = 4(y – 1) (Board 2009, 11) 12306090

(vi) y2 = –8(x – 3) 12306091

(vii) (x – 1)2 = 8 (y + 2) (Board 2009) 12306092

(viii) y = 6x2 – 1 12306093

(ix) x + 8 – y2 + 2y = 0 (Board 2011) 12306094

(x) x2 – 4x – 8y + 4 = 0 12306095

(Board 2006, 08, 09, 11)

Q.2 Write an equation of the parabola with given elements. (Board 2010) 12306096

(i) Focus (–3, 1); directrix x = 3 12306097

(ii) Focus (2, 5) ; directrix y = 1 12306098

(iii) Focus (–3, 1); directrix x – 2 y – 3 = 0

(Board 2007, 08) 12306099

(iv) Focus (1, 2), vertex (3, 2) 12306100

(Board 2008, 12)

(v) Focus (1, 0), vertex (1, 2) (Board 2005)

12306101

(vi) Directrix x = 2, focus (2, 2)

(Board 2010) 12306102

(vii) Directrix y = 3, vertex (2, 2) 12306103

(viii) Directrix y = 1, length of latus-rectum is 8. Opens downward. 12306104

(ix) Axis y = 0, through (2, 1) and (11, 2).

12306105

(x) Axis parallel to y-axis, the points (0, 3), (3, 4) and (4, 11) lie on the graph.

(Board 2009) 12306106

Q.3 Find the equation of the parabola having its focus at the origin and directrix is parallel to 12306107

(i) x-axis (ii) y-axis. (Board 2010)

Sol:

(i) Directrix parallel to x-axis 12306108

(ii) Directrix parallel to y-axis 12306109

Q.4 Show that the an equation of parabola with focus at (a cos , a sin ) and directrix

x cos  + y sin  + a = 0 is 12306110

x sin  – y cos 2 = 4a x cos  + y sin 

(Board 2011)

Q.5 Show that the ordinate at any point P of the parabola is the mean proportional between the length of the latus rectum and the abscissa of P. 12306111

Q.6 An comet has a parabolic orbit with the earth at the focus. When the comet is 150,000 km from the earth, the line joining the comet and the earth makes an angle of 30 with the axis of the parabola. How close will the comet come to the earth? 12306112

Q.7 Find an equation of the parabola formed by the cables of a suspension bridge whose span is a m and the vertical height of the supporting towers is b m. 12306113

Q.8 A parabolic arch has 100 m base and height 25 m. Find the height of the arch at a point 30 m from the centre of the base. 12306114

Q.9 Show that tangent at any point P of a parabola makes equal angles with the line PS and the line through P parallel to the axis of the parabola, S being focus. (These angles are called respectively angle of incidence and angle of reflection). 12306115

Show that the equation

9x2 18x + 4y2 + 8y  23 = 0 represents an ellipse. Find its elements and sketch its graph. 12306116

An arch in the form of half an ellipse is 40 m wide and 15 m high at the center. Find the height of the arch at a distance of 10 m from its center.

Q.1 Find an equation for the ellipse with given data and sketch its graph: 12306118

(Board 2009)

(i) Foci (3, 0) and minor axis of length 10 .

12306119

(ii) Foci (0, –1) and (0, –5) and major axis of length 6. 12306120

(iii) Foci (33 , 0) and vertices (6, 0)

(Board 2009, 10, 12) 12306121

(iv) Vertices (5, 1) , (1, 1) , foci (4, 1) and (0, 1)

12306122

(v) Foci (5, 0) and passing through the point 32  3. 12306123

(vi) Vertices (0 , 5) , eccentricity = 45

(Board 2008, 12) 12306124

(vii) Centre (0, 0), focus (0, –3), vertex (0, 4)

(Board 2011) 12306125

(viii) Centre (2, 2), major axis parallel to y-axis and of length 8 units, minor axis parallel to x-axis and of length 6 units.

12306126

(ix) Centre (0, 0), symmetric with respect to both the axis and passing through the points (2, 3) and (6, 1). 12306127

(x) Center (0, 0) , major axis horizontal, the points (3, 1), (4, 0) lie on the graph. 12306128

Q.2 Find the centre, foci, eccentricity, vertices and directrices of the ellipse whose equation is given. (Board 2008, 09, 11) 12306129

(i) x2 + 4y2 = 16 12306130

(ii) 9x2 + y2 = 18 (Board 2011) 12306131

(iii) 25 x2 + 9 y2 = 225

(Board 2008, 10, 11) 12306133

(iv) (2x – 1)2 4 + (y + 2)2 16 = 1

(Board 2006, 11) 12306134

(v) x2 + 16x + 4y2 – 16y + 76 = 0 (Board 2008)

12306135

(vi) 25x2 + 4y2 – 250 x – 16y + 541 = 0

(Board 2007) 12306136

Q.3 Let a be a positive number and

0 < c < a. Let F ( c , 0) and F(c , 0) be two given points. Prove that the locus of points P (x , y) such that |PF| + |PF| = 2a , is an ellipse. 12306137

Q.4 Use problem 3 to find equation of the ellipse as locus of points P(x, y) such that the sum of the distances from P to the points (0, 0) and (1, 1) is 2. 12306138

Q.5 Prove that the latus rectum of the ellipse x2a2 + y2b2 = 1, is . (Board 2008) 12306139

Q.6 The major axis of an ellipse in standard form lies along the x-axis and has length 42. The distance between the foci equals the length of the minor axis. Write an equation of the ellipse. 12306140

Q.7 An astroid has an elliptic orbit with the sun at one focus. Its distance from the sun ranges from 17 million to 183 million miles. Write an equation of the orbit of the astroid. 12306141

Q.8 An arch in the shape of a semi ellipse in 90 m wide at the base and 30 m high at the centre. At what distance from the center is the arch 202 m high? 12306142

Q.9 The moon orbits the earth in an elliptic path with earth at one focus. The major and minor axes of the orbit are 768,806 km and 767,746km respectively. Find the greatest and least distances (in Astronomy called the apogee and perigee) of the moon from the earth. 12306143

Discuss and sketch the graph of the equation

4x2  8x  y2  2y  1 = 0 … (1) 12306144

Q.1 Find an equation of the hyperbola with the given data. Sketch the graph of each 12306145

(i) Centre (0, 0), focus (6, 0), vertex (4, 0)

(Board 2007, 08, 11) 12306146

(ii) Foci ( 5, 0) , Vertex (3, 0)

(Board 2011, 12) 12306147

(iii) Foci (2  52, –7), length of the transverse axis 10. 12306148

(iv) Foci (0, 6), e = 2 12306149

(Board 2008, 11)

(v) Foci (0, 9) , Directrices y = 4

12306150

(vi) Centre (2, 2), horizontal transverse axis of length 6 and eccentricity e = 2

(Board 2008, 11) 12306151

(vii) Vertices (2, 3), (0, 5) lie on the curve.

12306152

(viii) Foci (5, –2), (5, 4) and one vertex

(5, 3) 12306153

Q.2 Find the centre, foci, eccentricity, vertices and directrices of the hyperbolas whose equations are given: 12306154

(i) x2 – y2 = 9 (Board 2006, 11) 12306155

(ii) x2 4 – y2 9 = 1 (Board 2006, 11) 12306156

(iii) y2 16 – x2 9 = 1 (Board 2007, 09, 10, 11) 12306157

(iv) y2 4 – x2 = 1 (Board 2009) 12306158

(v) (x – 1)2 2 – (y – 1)2 9 = 1 (Board 2006)

12306159

(vi) (y + 2)2 9 – (x – 2)2 16 = 1 12306160

(vii) 9x2 – 12x – y2 – 2y + 2 = 0 12306161

(viii) 4y2 + 12y – x2 + 4x + 1 = 0 12306162

(ix) x2  y2 + 8x  2y  10 = 0 (Board 2006)

12306163

(x) 9x2  y2  36x  6y + 18 = 0 12306164

Q.3 Let 0 < a < c and F (–c, 0) , F(c, 0) be two fixed points. Show that the set of points P(x, y) such that PF – =  2a, is the hyperbola x2a2 – y2c2 – a2 = 1 (F, F are foci of the hyperbola). 12306165

Q.4 Using Problem 3, find an equation of the hyperbola with foci (5, 5) and (5, 5), vertices ( 32,  32) and (32, 32).

12306166

Q.5 For any point on the hyperbola the difference of its distances from the points

(2, 2) and (10, 2) in 6. Find an equation of the hyperbola. 12306167

Q.6 Two listening posts hear the sound of an enemy gun. The difference in time is one second. If the listening posts are 1400 feet apart, write an equation of the hyperbola passing through the position of the enemy gun. (Sound travels at 1080 ft/sec). 12306168

Find equation of the tangent and normal to

(i) y2 = 4ax

(ii) x2 a2 + y2 b2 = 1

(iii) x2 a2  y2 b2 = 1

at the point (x1, y1).

i. Find an equation of the tangent to the parabola y2 = 6x which is parallel to the line 2x + y + 1 = 0. Also find the point of tangency.

ii. Find equations of the tangent to the ellipse

iii. x2 128 + y2 18 = 1

iv. which are parallel to the line 3x + 8y + 1 = 0. Also find the points of contact.

v. Show that the product of the distances from the foci to any tangent to the hyperbola.

vi. x2 a2  y2 b2 = 1 is constant.

vii. Find the points of intersection of the ellipse x2 43/3 + y2 43/4 = 1 and the hyperbola

x2 7 – y2 14 = 1 12306173

Also sketch the graph of the two conics.

Example 8: (Board 2010)

viii. Find the points of intersection of the conics.

ix. y = 1 + x2 and y = 1 + 4x  x2

Also draw the graph of the conics. 12306174

x. Find equations of the common tangent to the two conics and .

12306175

Q.1 Find equations of the tangent and normal to each of the following at the indicated point 12306176

(i) y2 = 4ax at (at2, 2at) 12306177

(ii) x2a2 + y2b2 = 1 at (a cos , b sin ) 12306178

(iii) x2a2  y2b2 = 1 at (a sec , b tan )

(Board 2007) 12306179

Q.2 Write equation of the tangent to the given conic at the indicated point. 12306180

(i) 3x2 = –16y at the points whose ordinate is –3

12306181

(ii) 3x2  7y2 =20 at the points where y= 1.

12306182

(iii) 3x2 – 7y2 + 2x – y – 48 = 0 at the point where x = 4 12306183

Q.3 Find equations of tangents which passes through the given points to the given conics.

(i) x2 + y2 = 25 through (7, –1) 12306184

(ii) y2 = 12x through (1, 4) 12306185

(iii) x2 – 2y2 = 2 through (1, –2) 12306186

Q.4 Find equations of the normal to the parabola y2 = 8x which are parallel to the line 2x + 3y = 10. 12306187

Q.5 Find equations of the tangents to the ellipse x2 4 + y2 = 1 which are parallel to the line 2x – 4y + 5 = 0. 12306188

Q.6 Find equations of the tangents to the conic 9x2  4y2 = 36 parallel to 5x  2y + 7 = 0

(Board 2009) 12306189

Q.7 Find equations of the common tangents to the given conics

(i) x2 = 80y and x2 + y2 = 81 12306190

(ii) y2 = 16 x and x2 = 2y 12306191

Q.8 Find the points of intersection of the given conics. 12306192

(i) x2 18 + y2 8 = 1 and x2 3 – y2 3 = 1 12306193

(ii) x2 + y2 = 8 and x2 – y2 = 1 12306194

(iii) 3x2 – 4y2 = 12 and 3y2 – 2x2 = 7 12306195

(iv) 3x2 + 5y2 = 60 and 9x2 + y2 = 124

12306196

(v) 4x2 + y2 = 16 and x2 + y2 + 2y  8 = 0

12306197

EXERCISE 6.8

Q.1 Find an equation of each of the following curves with respect to new parallel axes obtained by shifting the origin to the indicated point. 12306198

(i) x2 + 16y – 16 = 0, O(0, 1) 12306199

(ii) 4x2 + y2 + 16x – 10y + 37 = 0, O(–2, 5)

12306200

(iii) 9x2 + 4y2 + 18x – 16y – 11 = 0, O(–1, 2)

12306201

(iv) x2 – y2 + 4x + 8y – 11 = 0, O(–2, 4)

12306202

(v) 9x2 – 4y2 + 36x + 8y – 4 = 0, O(2, 1)

(Board 2008) 12306203

Q.2 Find coordinates of the new origin (axes remaining parallel) so that first degree terms are removed from the transformed equation of each of the following. Also find the transformed equation.

(i) 3x2 – 2y2 + 24x + 12y + 24 = 0 12306204

(ii) 25x2 + 9y2 + 50x – 36y – 164 = 0 12306205

(iii) x2 – y2 – 6x + 2y + 7 = 0 12306206

Q.3 In each of the following, find an equation of the curve referred to the new axes obtained by rotation of axes about the origin through the given angle. 12306207

(i) x y = 1 ,  = 45o 12306208

(ii) 7x2 – 8xy + y2 – 9 = 0,  = arctan 206(181)

(iii) 9x2 + 12xy + 4y2 – x – y = 0,  = arc tan 23

12306209

(iv) x2 – 2xy + y2 – 22 x – 22 y + 2 = 0,

 = 45o 12306210

Q.4 Find measure of the angle through which the axes be rotated so that the product term XY is removed from the transformed equation. Also find the transformed equation. 12306211

(i) 2x2 + 6xy + 10y2 – 11 = 0 12306212

(ii) xy + 4x – 3y – 10 = 0 12306213

(iii) 5x2 – 6xy + 5y2 – 8 = 0 12306214

Q.1 By a rotation of axes, eliminate the

xy-term in each of the following equations. Identify the conic and find its elements: 12306215

(i) 4x2 – 4xy + y2 – 6 = 0 12306216

(ii) x2 – 2xy + y2 – 8x – 8y = 0 12306217

(iii) x2 + 2xy + y2 + 22 x – 22 y + 2 = 0

12306218

(iv) x2 + xy + y2 – 4 = 0 12306219

(v) 7x2 – 63 xy + 13y2 – 16 = 0 12306220

(vi) 4x2 – 4xy + 7y2 + 12x + 6y – 9 = 0

12306221

(vii) xy – 4x – 2y = 0 12306222

(viii) x2 + 4xy – 2y2 – 6 = 0 12306223

(ix) x2  4xy  2y2 + 10 x + 4y = 0 12306224

Q.2 Show that (i) 10 xy + 8x  15y  12 = 0 and (ii) 6x2 + xy  y2  21x  8y + 9 = 0 each represents a pair of straight lines and find an equation of each line. 12306225

(i) 10 xy + 8x  15y  12 = 0 12306226

(ii) 6x2 + xy  y2  21x  8y + 9 = 0 12306227

Q.3 Find an equation of the tangent to each of the given conics at the indicated point. 12306228

(i) 3x2  7y2 + 2x  y  48 = 0 at (4, 1)

12306229

(ii) x2 + 5xy  4y2 + 4 = 0, at y =  1 12306230

(iii) x2 + 4xy  3y2  5x  9y + 6 = 0 at x = 3

Unit Vectors

Subtraction of Vectors:

12307001

Position Vector:

(Board 2009, 10) 12307002

17. Example: ( (Board 2006)

Find the unit vector in the direction as the vector . 12307003

Theorem:

The Ratio Formula. Let A and B be two points whose position vectors (p.v’s) are a and b respectively. If a point P divides AB in the ratio p:q, then the position vector of P is given by r = . 12307004

18. Example 1:

If a and b be the P.Vs of A and B respectively w.r.t. origin O and C be a point on such that = , then show that C is the midpoint of AB. 12307005

19. Example 2: (Board 2007, 08, 10)

Use vectors to prove that the diagonals of a parallelogram bisect each other. 12307006

EXERCISE 7.1

Q.1 Write the vector in the form

xi + yj. 12307007

(i) P = , Q =

(ii) P = Q =

Q.2 Find the magnitude of the vector u.

(i) u = 2i – 7j 12307008

(ii) u = i + j 12307009

(iii) u = 12307010

Q.3 If u = 2i – 7j , v = i – 6j and w = –i + j. Find the following vectors. (Board 2010) 12307011

Q.4 Find the sum of the vectors and , given the four points , and . 12307015

Q.5 Find the vector from the point A to the origin where = and B is the point . (Board 2010) 12307016

Q.6 Find a unit vector in the direction of the vector given below: 12307017

(Board 2008, 09, 10)

(iii) v = 12307020

Q.7 If A, B and C are respectively the points (2, –4), (4, 0) and (1, 6). Use vector method to find the coordinates of the point D if 12307021

(i) ABCD is a parallelogram. 12307022

(ii) ADBC is a parallelogram. 12307023

Q.8 If B, C and D are respectively (4, 1),

(–2, 3) and (–8, 0). Use vector method to find the coordinates of the point: 12307024

(i) A if ABCD is a parallelogram. 12307025

(ii) E if AEBD is a parallelogram. 12307026

Q.9 If O is the origin and , find the point P when A and B are (–3, 7) and

(1, 0) respectively. 12307027

Q.10 Use vectors, to show that ABCD is a parallelogram, when points A, B, C and D are respectively (0, 0), (a, 0), (b, c) and (b – a, c).

12307028

Q.11 If = , find the coordinates of the point A when points B, C, D are

(1, 2), (–2, 5), (4, 11) respectively. 12307029

Q.12 Find the position vectors of the point of division of the line segments joining the following pair of points, in the given ratio:

12307030

(i) Point C with position vector 2 – 3 and point D with position vector 3i + 2j in the ratio 4 : 3. 12307031

(ii) Point E with position vector 5i and point F with position vector 4i + j in the ratio 2:5.

12307032

Q.13 Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. 12307033

(Board 2009, 11, 12)

Q.14 Prove that the line segments joining the midpoints of the sides of a quadrilateral taken in order form a parallelogram. 12307034

(Board 2009)

Set of vectors in R2:

Let v be a vector in the plane or in space and let c be a real. Then

(i) |v| > 0 and |v| = 0 iff v = 0 12307036

(ii) |cv| = |c||v| 12307037

Theorem: (Board 2008)

Show that If , ,  are direction angles then cos2  + cos2  + cos2  = 1 12307038

Example 1:

For the vectors, v = and

w = , we have the following:

Example 2:

If u = , = and

w = then 12307040

(a) Find (i) (ii)

(Board 2009) 12307041

(b) Show that and are parallel to each other.

Q.1 Let A(2, 5), B(–1, 1) and C(2, –6). Find

12307042(a)

(i) 12307043

(ii) 12307044

(iii) 12307045

Q.2 Let = = , = . Find the indicated vector or number. 12307046

(i) 12307047

(ii) 12307048

(iii) (Board 2011) 12307049

Q.3 Find the magnitude of the vector and write the direction cosines of . 12307050

(Board 2008, 09)

(i) 12307051

(ii) 12307052

(iii) 12307053

Q.4 Find α, so that = 3.

(Board 2007, 08, 09) 12307054

Q.5 Find a unit vector in the direction of

= . (Board 2009) 12307055

Q.6 If = , = and = .

Find a unit vector parallel to .

(Board 2009, 11, 12) 12307056

Q.7 Find a vector whose

(i) Magnitude is 4 and is parallel to (Board 2008, 09, 12) 12307057

(ii) Magnitude is 2 and is parallel to (Board 2012) 12307058

Q.8 If u = v = and

w = represent the sides of a triangle. Find the value of z. (Board 2010) 12307059

Q.9 The position vectors of the points A, B, C and D are , and respectively. Show that is parallel to . 12307060

Q.10 We say that two vectors and in space are parallel if there is a scalar c such that = . The vector point in the same direction if c > 0, and the vector point in the opposite direction if c < 0. 12307061

(a) Find two vectors of length 2 parallel to the vector = 12307062

(b) Find the constant a so that the vectors

= and = are parallel.

(d) Find a and b so that the vectors and are parallel.

12307064

Q.11 Find the direction cosines for the given vector. 12307065

(i) (Board 2007) 12307066

(ii) (Board 2008, 11) 12307067

Q.12 Which of the following triples can be the direction angles of a single vector.

12307068

(i) 12307069

(ii) 12307070

(iii) 12307071

(i) If = and = are two vectors in the plane, then

= 12307072

(ii) and are two non-zero vectors in the plane, then

= 12307073

where  is the angle between and and .

\Example 2: (Board 2012)

Find the angle between the vectors

= and = . 12307074

Find a scalar α so that the vectors and are perpendicular. 12307075

Show that the vectors , and form the sides of a right triangle. 12307076

Show that the components of a vector are projections of that vector along and respectively.

Prove that in any triangle ABC. 12307078

(i) = (Cosine Law)

(Board 2009) 12307079

(ii) a = (Projection Law)

20. Prove that

= .

Q.1 Find cosine of the angle  between

and . (Board 2011) 12307082

(i) = , = 12307083

(ii) = , =

(Board 2011) 12307084

(iii) = , = 12307085

(iv) = , = 12307086

Q.2 Calculate the projection of along and projection of along when: 12307087

(i) = = (Board 2009) 12307088

(ii) = , = 12307089

(Board 2008)

Q.3 Find a real number α so that the vectors and are perpendicular. 12307090

(Board 2007, 09, 10, 11 ,12)

(Board 2007, 09) 12307092

Q.4 Find the number z so that the triangle with vertices and is a right triangle with right angle at C. (Board 2011) 12307093

Q.5 If is a vector for which = 0,

= 0, = 0, find . (Board 2009) 12307094

Q.6 (i)

Show that the vectors and form a right triangle. 12307095

(ii) Show that the set of points and form a right triangle. 12307096

Q.7 Show that midpoint of hypotenuse of a right triangle is equidistant from its vertices. 12307097

Q.8 Prove that perpendicular bisectors of a triangle are concurrent. 12307098

Q.9 Prove that the altitudes of a triangle are concurrent. 12307099

Q.10 Prove that the angle in a semi-circle is a right angle. (Board 2010) 12307100

Q.11 Prove that (Board 2011) 12307101

Q.12 Prove that in any triangle ABC. 12307102

(i) b = 12307103

(ii) c = a 12307104

(iii) = 12307105

(iv) = 12307106

Find a vector perpendicular to each of the vectors (Board 2008, 11)

= and = . 12307107

If = and =

Find a unit vector perpendicular to both and . Also find the sine of the angle between the vectors and . 12307108

Prove that:

In any triangle ABC, prove that (Law of Sines) 12307110

Find the area of the triangle with vertices and . Also find a unit vector perpendicular to the plane ABC.

Find area of the parallelogram whose vertices are and . 12307112

Example 7:

If = and = , find by determinant formula

(i) 12307113

(ii) 12307114

(iii) 12307115

e. 4

Q.1 Compute the cross product and . Check your answer by showing that each and is perpendicular to and . 12307116

(i) = and = 12307117

(ii) 12307118

(iii) = and = 12307119

Q.2 Find a unit vector perpendicular to the plane containing and . Also find sine of angle between them. 12307120

(i) = and =

(Board 2009, 11) 12307121

(ii) = and =

(Board 2011) 12307122

(iii) = and =

(Board 2011) 12307123

Q.3 Find the area of the triangle, determined by the points P, Q and R. 12307125

(i) and 07(075)

(ii) and

(Board 2008, 09) 12307126

Q.4 Find the area of parallelogram, whose vertices are: (Board 2009) 12307127

(i)

12307128

(ii)

(Board 2008) 12307129

(iii)

(Board 2008) 12307130

Q.5 Which vectors, if any, are perpendicular or parallel. 12307131

Sol:

(i) = , = , 12307132

(ii) = , = , 12307133

Q.6 Prove that: (Board 2005, 08, 09, 11)

= 0

12307134

Q.7 If = 0 , then prove that

. 12307135

(Board 2008, 09, 10, 11, 12)

Q.8 Prove that: (Board 2010)

12307136

Q.9 If = 0 and = 0, what conclusion can be drawn about or ?

(Board 2007, 08) 12307137

Let = a1i + a2j + a3k

= b1i + b2j + b3k

= c1i + c2j + c3k

then we have to prove that

[ ] = 12307138

Prove that every scalar triple product is independent of the position of the dot or cross

i.e., ·  = ·  = · 

Q.1 Find the volume of the parallelepiped for which the given vectors are three edges.

(Board 2008, 09) 12307140

(i) = , = , =

12307141

(ii) = , = ,

12307142

(iii) = , = ,

12307143

Q.2 Verify that = =

If = , = and

= 12307144

Q.3 Prove that the vectors , and are coplaner. (Board 2008)

Q.4 Find the constant α such that the vectors are coplaner. (Board 2008, 10) 12307146

(i) Let = , = and

(ii) Let = , = and

= (Board 2010) 12307147

Q.5 (a) Find the value of 12307148

(i)

(ii)

(iii)

(iv)

(b) Prove that: (Board 2011)

12307149

Q.6 Find volume of the tetrahedron with the vertices. 12307150

Sol:

(i) Let

and

(ii) Let

and (Board 2010)

Q.7 Find the workdone, if the point at which the constant force = is applied to an object, moves from and .

(Board 2010) 12307151

Q.8 A particle, acted by constant forces and , is displaced from to . Find the workdone. (Board 2012) 12307152

Q.9 A particle is displaced from the point to the point under the action of constant forces defined by , and . Show that the total workdone by the forces is 67 units. 12307153

Q.10 A force of magnitude 6 units acting parallel to displaces, the point of application from to .Find the workdone. (Board 2009) 12307154

Q.11 A force = is applied at the point . Find the moment of the force about the point . 12307155

Q.12 A force = , passes through the point . Find the moment of about the point . 12307156

(Board 2009, 12)

Q.13 Given a force = acting at a point . Find the moment of about the point . 12307157

Q.14 Find the moment about of each of the concurrent forces where is their point of concurrency. 12307158

Q.15 A force = is applied at . Find its moment about the point .

Causative

12th Math Subjective + Exercises

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