KENDRIYA VIDYALAYA DANAPUR CANTT

MINIMUM PROGRAMME FOR

AISSCE – 2018

SUBJECT : MATHEMATICS

(CLASS – XII)

Prepared By :

K. N. P. SinghVice-Principal

K.V. Danapur Cantt

(2) XII/Maths.

MATRIX

1. Find X and Y if

211

123y2x&

437

744yx2

2. If ,

363

121

484

X

3

1

4

, find X (Ans. X = [–1 2 1]

3. Find x such that 0

x

2

1

2315

152

231

1x1

(Ans. x = –14 or –2)

4. If

11

43A then Prove that NnV

n21n

n4n21An

5. If 213B&

7

5

1

A

verify that (AB)' = B'A'

6. Express the Matrix

564

386

531

as the sum of a symmetric and skew symmetric

Matrix.

7. Using elementary row transformation find the inverse of Matrix

102

114

123

.Ans

542

752

321

8. Show that

42

58A satisfies the equation x2 + 4x – 42 = 0. Hence find A–1

Ans.

82

54

421

9. If

321

213

812

A show that A (adjA) = (adj A) A = |A| I3

10. Using Matrix method, solve the following system of linear equations :

11z4y3x3

2z2y3x2

4z3y2x

(Ans. x = 3, y = –2 & z = 1

(3) XII/Maths.

DETERMINANT

1. Evaluatecbbaac

baaccb

accbba

2. Using properties of determinant prove that :

(i) xzzyyxxyz

zyx

zyx

zyx

333

222

(ii)

222

(iii)

zyx

rqp

cba

2

yxxzzy

qpprrq

baaccb

(iv) 3cba

bacc2c2

b2acbb2

a2a2cba

(v)

c1

b1

a1

1abc

c111

1b11

11a1

(vi) 322

22

22

22

ba1

ba1a2b2

a2ba1ab2

b2ab2ba1

(vii) cbaaccbba

cba

cba

111

333

(viii)

333

222

111

(ix) If x, y, z are different and

0

z1zz

y1yy

x1xx

A32

32

32

, then show that 1+ xyz = 0

(4) XII/Maths.

RELATION & FUNCTIONS

1. Prove that the relation R on the set N × N defined by (a, b) R (c, d) a + d = b + c

V (a, b), (c, d) N × N is an equivalance relation.

2. Consider the set N × N, the set of all ordired pairs of natural numbers. Let R be

the relation in N × N which is defined by (a, b) R (c, d) if and only if ad = bc. Prove

that R is an equivalence relation.

3. Prove that the relation R on set Z of all integers defined by (a, b) R (a – b) is

divisible by 5 is an equivalence relation on Z.

4. Show that the function f : R R given by f (x) = ax + b, a, b r, a o is a

bijection.

5. Show that the function f : R – {–1} R – {1} given that 1x3x

xf

is a bi jective

function.

6. Let f : R+ [-5, ] given by f (x) = 9x2 + 6x – 5. Show that f is invertible and

3

16yyf 1

7. Let A be the set of all real numbers except – 1

Let an operation * be defined on A as a * b = a + b + ab V a, b A

Prove that (i) A is closed under the given operation

(ii) * is commutative as well as associative

(iii) the number O is the identity element

(iv) every element a A hasa1

a

as inverse

8. Let

53

R57

R:f be defined as 7x54x3

xf

and

57

R53

R:g

be defined as 3x54x7

xg

Show that fog = IA and gof = IB where IA and IB are idnetity functions on A and B

respectively.

(5) XII/Maths.

INVERSE TRIGONOMETRIC FUNCTIONS

1. Find the value of

21

sin21

cos1tan 111

1213

632

4.Ans

2. Find the value of 3coteccos2tansec 1212 (Ans. 15)

3. Prove that

1731

tan71

tan21

tan2 111

4. Prove that tan–1 1 + tan–1 2 + tan–1 3 =

5. Prove that47

1tan

31

tan2 11

6. Prove that48

1tan

71

tan51

tan31

tan 1111

7. Prove thatab2

ba

cos21

4tan

ba

cos21

4tan 11

8. Prove that [4

,0]x,2x

xsin1xsin1

xsin1xsin1cot 1

9. Prove that 21

22

221 xcos

21

4x1x1

x1x1tan

10. If cos–1x + cos–1y + cos–1z = Prove that x2 + y2 + z2 + 2xyz = 1

11. If tan–1x + tan–1y + tan–1z = Prove that x + y + z = xyz

12. (a) Solve : 4

x3tanx2tan 11

61

x.Ans

(b) Solve : 2 tan–1 (cos x) = tan–1 (2 cosec x)

4

x.Ans

13. Solve :42x

1xtan

2x1x

tan 11

2

1.Ans

14. Write

x1x1

tan2

1 in the simplest form.

15. Write

xsin1xcos

tan 1 in the simplest form.

(6) XII/Maths.

CONTINUITY & DIFFERENTIABILITY

1. A function f is defined by

3x23x2

2x1,1xxf

Find whether f is continuous at x = 2 (Ans. continuous)

2. A function f is defined by

0x5x

1xxxf

2

Show that the function is discontinuous at x = 1.

3. Discuss the continuity of the function

0x,

21

0x,x

xcos1

xf2

at x = 0

4. For what value of k is the function

2xif1x3

2xifk

2xif1x2

xf continuous at x = 2 ?

(Ans. k = 5)

5. Find the value of k if f (x) is continuous at2

x , where

2x,3

2x,

x2xcosk

xf

(Ans. k = 6)

6. If the function

1xifb2ax5

1xif11

1xifbax3

xf is continuous at x = 1, find the values

of a & b. (Ans. a = 3 & b = 2)

7. For what value of k,

0x,k

0x,x8

x4cos1xf 2 is continuous at x = 0 ? (Ans. k = 1)

8. Prove that f (x) = |x| is continuous at x = 0 but not differentiable at x = 0

(7) XII/Maths.

DIFFERENTIATION

Find the derivative w.r. to x of the following :

1.

xsinxcosxsinxcos

tan 1 (Ans. 1)

2.

xsinaxcosbxsinbxcosa

tan 1 (Ans. –1)

3.

x1x1

tan2

1

2x121

.Ans

4.

22

221

x1x1

x1x1tan

4x1

x.Ans

5.

13x112x5

sin2

1

2x1

1.Ans

6. If

xsin1xsin1

xsin1xsin1coty 1 Prove that

dxdy

is independent of x.

7. If sin y = x sin (a + y) Prove that asin

yasindxdy 2

8. If t1sinax

and t1cosay

Prove that 0xy

dxdy

9. If y = tan–1 x show that 0dxdy

x2dx

ydx1 2

22

10. If 22 axxlogy Prove that 0dxdy

xdx

ydax 2

222

11. If n2 1xxy Prove that (x2 + 1) y2 + xy1 – n2 y = 0

12. If y = A cos n x + B sin n x Prove that 0yndx

yd 22

2

13. If y = a cos (log x) + b sin (log x) prove that 0ydxdy

xdx

ydx 2

22

14. If y = sin (m sin–1x) show that (1 – x2) y2 – xy1 + m2y = 0

15. If y = xsinx + (sinx)x finddxdy

16. If

...xxxxy, then prove that xlogy1x

ydxdy 2

(8) XII/Maths.

APPLICATION OF DERIVATIVES

1. Verify Lagrange's Mean Value Theorem for the function f (x) = (x – 1) (x – 2) (x – 3)

in the interval [1, 4] (Ans. 3)

2. Verify Rolle's Theorem for the function

2,0x,xcosxsinxf

4c.Ans

3. A particle moves along the curve 6y = x3 + 2. Find the points on the curve at

which y co-ordinate is changing 8 times as fast as the x-co-ordinate.

Ans. (4, 11) &

331

,4

4. Find the intervals in which the function f (x) = x3 – 12x2 + 36x + 17 is (a) increasing

(b) decreasing Ans. (Increasing in (–, 2) U (6, ) decreasing in (2, 6)

5. Find the intervals in which the function f given by f (x) = sin x – cos x

2x0 (i) is increasing (ii) is decreasing

(Ans. Increasing when43

x0

or

2x47

Decreasing when47

x43

)

6. Find the intervals in which the function f (x) = 7 + 12x – 3x2 – 2x3 is increasing or

decreasing (Ans. (i) –2 x 1 & x < –2 or x > –1)

7. At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, is the tangent parallel to

y-axis ? (Ans. (3, 2) & (–1, 2)

8. Prove that the curves x = y2 & xy = k cut at right angles if 8k2 = 1.

9. Find the Co-ordinates of the points on the curve y = 2x3 – 6x + 3 where the

tangents are || to x-axis. (Ans. (1, –1) & (–1, 7))

10. Find the equation of the tangent & normal to the curve.

x = 1 – cos , y = – sin at4

04

yx12normalofEquation

1224

yx12genttanofEquation.Ans

(9) XII/Maths.

11. Using differentials, find the approximate value of

(i) 037.0 (Ans. 0.1925)

(ii) 48.0 (Ans. 0.693)

12. Find the approximate value of f (2.01), when f (x) = 4x2 + 5x + 2 (Ans. 28.21)

13. If y = x4 – 10 and if x changes from 2 to 1.97, using differential, find the approxi-

mate change in y. (Ans. – 0.96)

14. Find the points of local maxima or local minima for the function f (x) = sin x – cos x,

0 < x < 2. (Ans. max value =43

xat2

, min value =47

xat2

15. Show that the rectangle of max. area that can be inscribed in a circle of radius r is

a square of side 2 r..

16. Find the point on the curve y2 = 4x which is nearest to the point (2, –8). (Ans. (4, –y)

17. Two sides of a triangle have lengths 'a' & 'b' and the angle between them is .What value of will maximize the area of the triangle ? Find the max. area of the

triangle also. (Ans. = /2, max area = ½ ab

18. A wire of length 25 m is to be cut into two pieces. One of the two pieces is to be

made into a square and the other into a circle. What should be the lengths of the

two pieces so that the combined area of the square & the circle is minimum ?

m425

&m4100

.Ans(

)

19. An open 60x, with a square base is to be made out of a given quantity of metal

sheet of area c2. Show that the max volume is36

c3

.

20. A window is in the form of rectangle surmounted by a Semi circle. If the total

perimeter of the window is 30 m, find the dimensions of the window so that max.

light in admitted.

m

430

.Ans

21. Show that the height of the cylinder of max. volume that can be inscribed in a

sphere of radius R is3

R2.

(10) XII/Maths.

INTEGRATION

Evaluate :

1. dxx3sinx2sinxsin

2. dxx3cosx2cos.xcos

3. dxxsin 4

4. dxcos 4

5. dxxtan4

6.

dxcosxcos

2cosx2cos

7. dx

xsinxsindx

8. xtan1dx

9. dxxlogx

10. dxxsec3

11. dxx1

xe2

x

12.

dxex2cos1x2sin2 x

13. dxxsinx 1

14. dxxtanx 1

15. dxxtanx 12

16.

dxx11x4

2

17.

dxx11x4

2

18. 4x1dx

19. dxbxsineax

20. dxbxcoseax

21. dxxtan

22. dxxcot

23. dxxcotxtan

24. xsin45dx

25. 1xxxxdx

23

26. 1xxdxn

27. xsin2xsin1xdxcos

28. 1xdx3

29. 2xx1

dx

30. dx1x4x

1x22

31. 2x1xdx2

32. dxxx431x2 2

(11) XII/Maths.

INTEGRATION AS A LIMIT OF SUM

Evaluate following integrals as the limit of sums :

1. 5

0

dx1x

2. 3

2

2dxx

3. 2

0

2 dx3x

4. 4

1

dx2x3

5. 3

1

2dxx

6. 3

0

2 dx5x2

7. 2

0

x dxe

8. 3

1

2 dxx5x2

9. 3

1

3 dxx

10. 2

0

2 dx2xx

(12) XII/Maths.

DEFINITE INTEGRALS

1.

4

0

dxx3sin.x2sin

25

3.Ans

2. 2.Ansdxxcotxtan2

0

3.

31

tan32

.Ansxsin45

dx 12

0

4.

4.Ansdx

xcosxsinxsin2

05

5

5.

2

19.Ansdx|3x||2x||1x|

4

1

6.

2log8

.Ansdtan1log4

1

7.

12.Ans

xtan1

dx3

6

8.

22

0

log2

.Ansdxxsinlog

9.

22

0

log2

.Ansdxx2sinlogxsinlog2

10. 0.Ansdxxcosxsin1xcosxsin2

0

11.

ab2.Ans

xsinbxcosaxdx 2

02222

12.

4.Ansdx

xcos1xsinx 2

02

(13) XII/Maths.

AREA OF THE BOUNDED REGION

1. Find the area of the region bounded by the ellipse 19y

4x 22

. (Ans. 6 sq unit)

2. Find the area of the region included between the Parabolas y2 = 4ax & x2 = 4ay.

unitsq

3a16

.Ans2

3. Find the area of the region {(x, y) x2 y x}

unitsq

61

.Ans

4. Find the area of the region {(x, y) x2 + y2 1 x + y}

unitsq21

4.Ans

5. Find the area of the region enclosed between the two circles x2 + y2 = 1 and

(x – 1)2 + y2 = 1

unitsq23

32

.Ans

6. Find the area of the region {(x, y) : x2 y |x|}

unitsq

31

.Ans

7. Find the area enclosed by the Parabola x2 = 4y and the straight line x = 4y – 2

unitsq

89

.Ans

8. Find the area of the region enclosed between the circles x2 + y2 = 4 and

(x – 2)2 + y2 = 4

unitsq3238

.Ans

9. Find the area of the smaller region bounded by the ellipse 1by

ax

2

2

2

2

& the

line 1by

ax

unitsq122

ab.Ans

10. Using integration, find the area of the triangle whose vertices are (1, 0), (2, 2)

& (3, 1)

unitsq

23

.Ans

11. Using integration, find the area of ABC when A is (2, 3), B is (4, 7) & C is (6, 2)

(Ans. 9 sq unit)

12. Find the area of the region bounded by the lines x + 2y = 2, y – x = 1 & 2x + y – 7 = 0

(Ans. 6 sq unit)

(14) XII/Maths.

DIFFERENTIAL EQUATION

1. Verfiy that x1sinmey

is a solution of the differential equation

0ymdxdy

xdx

ydx1 2

2

22

2. F o r m t h e d i f f e r e n t i a l e q u a l c o r r e s p o n d i n g t o ( x – a )

2 + (y – b)2 = r2 by eliminating

a & b. (Ans. (1+y12)3 = r2 (y2)

2)

3. Solve the differential equation

dxdy

yadxdy

xy 2

4. Solve the differential equation (1 + e2x)dy + ex (1 + y2) dx = 0 Given y = 1, when x = 0

5. Solve the differential equation 0y1dxdy

x1 22 Given y = 1 when x = 0

6. Solve the differential equationdxdy

yxdxdy

xy

7. Solve the differential equation 0dyyx

1edxe1 yx

yx

8. Solve the differential equation dxyxdxydyx 22

9. Solve the differential equation dxyxdxydyx 22

10. Solve the differential equation x2 dy + (xy + y2) dx = 0 given that y = 1 when x = 1

11. Solve the differential equation (1 + y2)dx = (tan–1y – x)dy

12. Solve the differential equation ydxdy

y2x 2 given y = 1 when x = 2

13. Solve the differential equation xtanydxdy

x1 12

14. Solve the differential equation 22 yxy2dxdy

x

15. Solve the differential equation xlogxy2dxdy

x 2

16. Solve the differential equation xsinydxdy

xcos given that y = 2 when x = 0

(15) XII/Maths.

VECTORS & 3–DIMENSIONAL GEOMETRY

1. If ji3c&kib,k3j2ia

find the value of such that ca is to b.

34

.Ans

2. Find a unit vector in the direction of the sum of the vectors

k3ji2b&kjia

k2j2i

31

.Ans

3. Find the value of for which the vectors k8j4ib&k4j2i3a

are

parallel. (Ans. = 6)

4. If kjb&kjia

, find the vector c such that 3c.a&bca

k

32

j32

i35

c.Ans

5. If |b.a|find,25|ba|&13|b|,5|a|

(Ans. 60)

6. Find the area of the ||gm whose adjacent sides are determined by the vectors

kj7i2b&k3jia (Ans. 15 2 sq unit)

7. If |ba|find,12b.a&2|b|,10|a| (Ans. 16)

8. Show that the points (2, –1, 2), (4, 5, –6) & (3, 2, –2) are collinear.

9. If 0cba

Prove that accbba

10. Prove that cba2accbba

11. Find the foot of the r from the point (1, 2, 3) on the line23z

31y

12x

4

21

,23

.Ans

12. Find the shortest distance between the lines

k2ji2kji2r&kjikj2ir

unit

2

3.Ans

13. Find the shortest distance between the following lines

11z

61Y

71x

&1

7z25Y

13x

unit292.Ans

(16) XII/Maths.

14. Find the equation of the plane Passing through the points (1, 1, 0), (1, 2, 1) & (–2, 2, –1)

(Ans. –2x – 3y + 3z + 5 = 0)

15. Find the equation of the plane passing through the points (2, 1, –1) & (–1, 3, 4)

and r to the plane x – 2y + 2 = 0 (Ans. 10x + 5y + 4z – 21 = 0)

16. Find the equation of the plane passing through the points A (2, 3, –1), B (–2, 4, 5)

& C (0, 1, 2) in vector form. Also express it in Cartesian form.

17. The Cartesian equations of a line are 3x + 1 = 6y – 2 = 1 – z. Find the fixed point

through which it passes, its direction rations and also its vector equation.

6,1,2;1,31

,31

.Ans

k6ji2kj31

i31

r

18. Find the value of so that lines5

z61

5y3

x77&

23z

214y7

3x1

are at right angle.

1170

.Ans

19. Find the image of the point (1, 6, 3) in the line3

2z2

1y1x

. Also, write the

equation of the line Joining the given point and its image and find the length of

the segment joining the given point and its image.

132,y

3z66y

01x

,7,0,1.Ans

20. Find the angle between the planes 5k2ji.r&6kji2.r 3.Ans

21. Find the image of the point (3, –2, 1) in the plane 3x – y + 4z = 2 (Ans. 0, –1, –3)

22. Prove that the lines7

6z4

4y1

2x&

75z

53y

31x

are coplanar. Also

find the plane containing these two lines.

23. Find the equation of the plane passing through the intersection of the planes

5k4j3i2.r&6kji.r

and the point (1, 1, 1)

69k26j23i20.r.Ans

(17) XII/Maths.

LINEAR PROGRAMMING

1. Solve the following Linear Programming Problem graphically

Maximize z = 50 x + 15 y

Subject to 5x + y 100

x+ y 60

x, y 0 (Ans. 1250)

2. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two

foods F1 & F2 are available. Food F1 costs Rs. 4 per unit and F2 costs Rs. 6 per

unit, one unit of food F1 contains 3 units of vitamin A and 4 units of minerals.

One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formu-

late this as a linear programming problem and find graphically the minimum

cost for diet that consists of mixture of these foods & also meets the mineral

nutritional requirements. (Ans. Rs. 104)

3. One kind of cake requires 300 gm of flour and 15 gm of fat, another kind of cake

requires 150 gm of flour & 30 gm of fat. Find the max. number of cakes which

can be made from 7.5 kg of flour and 600 gm of fat, assuming that there is no

shortage of the other ingradients used in making the cakes. Make it as an LPP

and solve it graphically. (Ans. 20, 10)

4. There are two types of fertilisers 'A' and 'B' A' consists of 12% Nitrogen & 5%

phosphoric acid where as 'B' consists of 4% nitrogen and 5% phospheric acid.

After testing the soil conditions, farmers finds that he needs at least 12 kg of

nitrogen & 12 kg of phosphoric acid for his crops. It 'A' costs Rs. 10 per kg and 'B'

cost Rs. 8 per kg, then graphically determine how much of each type of fertiliser

should be used so that nutrient requirements are met at a minimum cost.

(Ans. A 0.3 kg, B : 0.21 kg)

5. Solved the following linear programming problem graphically

Maximize z = 60x + 15y Subject to constraints

x + y 50

3x + y 90

x, y 0

(Ans. x = 20, y = 30, z = 1650)

(18) XII/Maths.

6. A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per

bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units

of element C whereas brand Q costing Rs. 200 per bag contains 1.5 units of

nutritional element A, 11.25 units of element B and 3 units of element C. The

minimum requirements of nutrients A, B and C are 18 units, 45 units and 24

units respectively. Determine the number of bags of each brand which should be

mixed in order to produce a mixture having a minimum cost per bag ? What is

the minimum cost of the mixture per bag ? (Ans. Rs. 1950 at (3, 6)

7. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is

made on each exclusive class ticket and a profit of Rs. 600 is made on each

economy class ticket. The airline reserves at least 20 seats for executive class.

However at least 4 times as many passengers prefer to travel by economy class

than by the executive class. Determine how many tickets of each type must be

sold in order to maximise the profit for the airline. What is the maximum profit ?

(Ans. Max profit Rs. 136000 at (40,160)

8. Two godowns A and B have grain capacity of 100 quintals and 50 qunitals re-

spectively. They supply to 3 ration shops D, E and F, whose requirements are 60,

50 and 40 quintals respectively. The cost of transportation per quintal from the

godowns to the shops are given in the following table :

How should the supplies be transported in order that the transportation cost in

minimum. What is the minimum cost ?

Transportation cost per quintal (in Rs.)

From/To A B

D 6 4

E 3 2

F 2.50 3

(Ans. : Minimum cost Rs. 510 at (10, 50))

(19) XII/Maths.

PROBABILITY

1. If A & B are two events such that 41

BAP&31

BP,21

AP Find

(i) P (A/B) (ii) P (B/A) (iii) B/AP (iv) B/AP

85

,41

,21

,43

.Ans

2. A problem in Mathematics is given to 3 students whose chances of solving it are

41

,31

,21

. What is the probability that te problem is solved ?

43

.Ans

3. Two persons A & B throw a die alternately till one of them gets a 'three' and wins

the game. Find their respectively probabilities of winning, if A begins.

115

,116

.Ans

4. Three persons A, B, C throw a die in succession till one gets a 'six'and wins the

game. Find their respective probabilities of winning, if A begins.

9125

&9130

,9136

.Ans

5. A husband and wife appear in an interview for two vacancies for the same post.

The probability of husband's selection is71

and that of wife's selection is51

.

What is the probability that

(i) both of them will be selected ?

351

.Ans

(ii) only one of them will be selected ?

72

.Ans

(iii) none of them will be selected ?

3524

.Ans

6. In a bolt factory, machines A, B and C manufacture respectively 25%, 35% and

40% of the total bolts of their output 5, 4 and 2 percent are respectively defective

bolts. A bold is drawn at random from the product. If the bolt drawn is found to

be defective, what is the probability that it is manufactured by the machine B ?

6928

.Ans

7. A Insurance company insured 2000 scooter drivers, 4000 car drivers and 6000

truck drivers. The probabilities of an accident involving a scooter driver, car driver

& a truck driver are 0.01, 0.03 and 0.15 respectively. One of the insured person

meet with an accident. What is the probability that he is a scooter driver ?

521

.Ans

(20) XII/Maths.

8. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two

cards are drawn and are found to be hearts. Find the probability of missing card

to be a heart.

5011

.Ans

9. A man is known to speak truth 3 out of 4 times. He throws a die and reports that

it is a six. Find the probability that it is actually a six.

83

.Ans

10. There are three coins. One is two headed coin, another is a biased coin that

comes up heads 75% of the time and third is an unbiased coin. One of the three

coins is choosen at random and tossed, it shows heads, what is the probability

that it was the two headed coin ?

94

.Ans

11. Three urns A, B and C contain 6 red an 4 white ; 2 red and 6 white; and 1 red and

5 white balls respectively. An urn is chosen at random and a ball is drawn. If the

ball drawn is found to be red, find the probability that the ball was drawn from

urn A.

6136

.Ans

12. A random variable X has the following probability distribution :

X : 0 1 2 3 4 5 6 7

P (x) : 0 k 2k 2k 3k k2 2k2 7k2+k

Find each of the following :

(i) k (ii) P (x < 6) (iii) P (x 6) (iv) P (0< × <5)

54

,10019

,10081

,101

.Ans

13. Find the mean & variance of the number of tails in three tosses of a coin.

43var,5.1Mean