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HOLIDAY HOMEWORK (AUTUMN BREAK) CLASS:- XII (2015-16) (RELATIONS AND FUNCTIONS) (1). Show that the relation R defined by N b a by divisible is b a b a R , ; 3 , , , , : ) , ( is an equivalence relation. (2). Let N N f : be defined by ) (n f even is n if n odd is n if n , , , , 2 , , , , 2 1 , Find whether the function f is bijective. (3). ). Let f: R R be defined by 0 1 0 0 0 1 ) ( x if x if x if x f Is f(x) one-one and onto. (4) Show that the relation R defined by c b d a d c R b a ) , ( ) , ( on the set NxN is an equivalence relation. (5). Consider f : R+→(-5, ) given by f(x) = 9 x 2 + 6x -5. Show that f is invertible with f -1 (y) = 3 1 6 y (INVERSE TRIGONOMETRIC FUNCTIONS) (6). Prove that:- 4 , 0 , 2 sin 1 sin 1 sin 1 sin 1 cot 1 x x x x x (7). Prove that:- 5 3 cos 2 1 9 2 tan 4 1 tan 1 1 1 . (8). Prove that:- x b a x a b x b a b a cos cos cos 2 tan tan 2 1 1 (9). Prove that . cos 2 1 4 1 1 1 1 tan 2 1 2 2 2 2 1 x x x x x (10). Prove that 2 1 tan 1 x x + 2 1 tan 1 x x = (11). + = (12). Find the value of 1 , 0 , 1 , 1 1 cos 1 2 sin 2 1 tan 2 2 1 2 1 xy y x y y x x (MATRICES) (13). By using elementary operations, find the inverse, if exists of the following: (i) 2 1 1 3 2 3 1 2 1 A (ii) 1 2 1 3 1 2 2 1 3 A (iii) 1 2 0 0 3 1 2 2 1 A (14). If A = 0 2 tan 2 tan 0 and I = 1 0 0 1 show that I + A = cos sin sin cos A I

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Page 1: HOLIDAY HOMEWORK (AUTUMN BREAK) CLASS:- XII … · HOLIDAY HOMEWORK (AUTUMN BREAK) CLASS:- XII (2015-16) (RELATIONS AND FUNCTIONS) (1). ... Show that the right circular cylinder …

HOLIDAY HOMEWORK (AUTUMN BREAK)

CLASS:- XII (2015-16)

(RELATIONS AND FUNCTIONS)

(1). Show that the relation R defined by NbabydivisibleisbabaR ,;3,,,,:),( is an

equivalence relation.

(2). Let NNf : be defined by )(nf

evenisnifn

oddisnifn

,,,,2

,,,,2

1

, Find whether the function

f is bijective.

(3). ). Let f: RR be defined by

01

00

01

)(

xif

xif

xif

xf Is f(x) one-one and onto.

(4) Show that the relation R defined by cbdadcRba ),(),( on the set NxN is an

equivalence relation.

(5). Consider f : R+→(-5, ) given by f(x) = 9 x2 + 6x -5. Show that f is invertible with

f-1

(y) =3

16 y

(INVERSE TRIGONOMETRIC FUNCTIONS)

(6). Prove that:-

4,0,

2sin1sin1

sin1sin1cot 1

xxx

xx

(7). Prove that:-

5

3cos

2

1

9

2tan

4

1tan 111 .

(8). Prove that:-

xba

xabx

ba

ba

cos

coscos

2tantan2 11

(9). Prove that .cos2

1

411

11tan 21

22

221 x

xx

xx

(10). Prove that

2

1tan 1

x

x+

2

1tan 1

x

x=

(11).

+

=

(12). Find the value of 1,0,1,1

1cos

1

2sin

2

1tan

2

21

2

1

xyyxy

y

x

x

(MATRICES)

(13). By using elementary operations, find the inverse, if exists of the following:

(i)

211

323

121

A (ii)

121

312

213

A (iii)

120

031

221

A

(14). If A =

02

tan

2tan0

and I =

10

01 show that I + A =

cossin

sincosAI

Page 2: HOLIDAY HOMEWORK (AUTUMN BREAK) CLASS:- XII … · HOLIDAY HOMEWORK (AUTUMN BREAK) CLASS:- XII (2015-16) (RELATIONS AND FUNCTIONS) (1). ... Show that the right circular cylinder …

(DETERMINANTS) (15). Use the matrix method to solve the system linear equations.

22

632

3223

zyx

zyx

zyx

(16). Given that

312

221

111

,,

135

317

444

BandA , find AB. Use this result to

solve the following system of linear equations:

132

922

4

zyx

zyx

zyx

(17).Using three types of materials(plastic) P1, P2 and P3, a factory produces three types of

monkey toys T1,T2 and T3, one with eye closed, second with mouth closed and third with ears

closed. Plastic requirement for each type of toy and total available plastic of all three types is

summarized in the following table:

Plastic

T1 T2 T3 Total available Plastic

P1 1 1 1 6

P2 2 5 5 27

P3 2 5 11 45

(a). Represent the above data by linear equations and write its matrix form.

(b). Is it possible to solve the system of linear equations so obtained using matrices?

(c). Write the importance of each type of toy.

(18). Two schools A and B decided to award prizes to their students for three values

honesty (x), punctuality (y) and obedience (z). School A decided to award a total

of Rs. 11000 for the three values to 5, 4 and 3 students respectively while school B

decided to award Rs. 10700 for the three values to 4, 3 and 5 students respectively.

If all the three prizes together amount to Rs. 2700, then.

i). Represent the above situation by a matrix equation and form Linear equations using matrix

multiplication.

ii). Is it possible to solve the system of equations so obtained using matrices?

iii).Which value you prefer to be rewarded most and why?

COUNTINUITY AND DIFFERENTIABILITY

(19). If 21

1,011

xdx

dythatprovexyyx

(20). If dx

dyfindxxy xx ,)(sin)( cossin

(21). If 2121,tan 1

2

2

2221 yxxyxthatshowxy

(22). If 4

),cos(sin),sin(cos2

2

at

d

yfindayax

xd

(23).If xf =

1,25

1,11

1,3

xifbax

xif

xifbax

is continuous at 1x , find the value of a and .b

Page 3: HOLIDAY HOMEWORK (AUTUMN BREAK) CLASS:- XII … · HOLIDAY HOMEWORK (AUTUMN BREAK) CLASS:- XII (2015-16) (RELATIONS AND FUNCTIONS) (1). ... Show that the right circular cylinder …

APPLICATION OF DERIVATIVE

(24). The section of corner window is a rectangle surmounted by an equilateral triangle.

Given the perimeter is 16 m. find the width of the window in order that the maximum light

may be admitted.

(25). Show that the right circular cylinder of given surface and maximum volume is such

that its height is equal to the diameter of the base.

(26). A wire of length 28 meter is to be cut into two pieces, one of the pieces is to be made

into a square and the other into a circle. What should be the length of the two pieces so that

the combined area of the square and circle is maximum?

(27). Prove that the volume of the largest cone that can be inscribed in a sphere of radius R

is

of the volume of the sphere.

(28). An open box, with a square base is to be made out of a given quantity of a metal sheet

of area c2.Show that the maximum volume is 36

3c

INTEGRALS

(29). Evaluate: ∫ | |

(30). Evaluate: ∫

, using integral as a limit of sum.

(31). Evaluate: ∫

, using the property of definite integral.

(32) Evaluate: ∫

, using the property of definite integral.

(33). Evaluate: ∫

, using the property of definite integral.

(34) Evaluate: ∫

, using the property of definite integral.

APPLICATION OF INTEGRALS

(35).Draw a rough sketch of the region enclosed between the circles x2 + y

2 = 9 and

( x – 3 )2 + y

2 = 9. Using integration find the area of the enclosed region.

(36). Find the area of the circle 4x2 + 4y

2 = 9 which is interior to the parabola x

2 = 4y.

(37).Using integration find the area of region bounded by the triangle whose vertices are

(– 1, 0), (1, 3) and (3, 2).

(38). Find the area in the first quadrant enclosed by x-axis, the line yx 3 and the circle

422 yx

(39). Find the area lying above x-axis and included between the circles x2 +y

2=8x and the

parabola y2=4x.

DIFFERENTIAL EQUATIONS

(40). Solve the following differential equation:

0,log xxxydx

dyx

(41). Solve the following differential equation:

,tancos2 xydx

dyx

Page 4: HOLIDAY HOMEWORK (AUTUMN BREAK) CLASS:- XII … · HOLIDAY HOMEWORK (AUTUMN BREAK) CLASS:- XII (2015-16) (RELATIONS AND FUNCTIONS) (1). ... Show that the right circular cylinder …

(42). Solve the following differential equation:

02)( 22 dyxydxyx if y=1 when x=1

(43). Solve the following differential equation:

30,sintan

xatythatgivenxxy

dx

dy

(44). Solve the following differential equation:

0)()3( 22 dyxyxdxyxy

VECTOR ALGEBRA

(45). Find the value of for which the vector

(46). If p is a unit vector and | |

(47) If , find a unit vector in the direction of ( )

(48). Find the projection of

(49). If dcba

and dbca

.Show that da

is parallel to cb

where

cbda

&

(50). Using vector find the area of the triangle with vertices A(2, 3, 5), B(3, 5, 8)and C(2, 7,

8).

(51). Let

a ,

b and

c be the three vectors such that 543

candba and each of them

is perpendicular to the of other two, find

cba .

(HAPPY DURGA PUJA & AUTUMN BREAK)