M_2Y_Function_Revision Test-4.docx

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M_2Y_Function_Assignment 1 VMC/JEE

Aggarwal Corporate Heights, 3rd Floor, Plot No. A - 7, Netaji Subhash Place,

Pitam Pura, Delhi - 110034 Phone: 011-45221190 - 93. Fax : 25222953 Revision Test-4 FUNCTION Maths

Time: M.M. :

1. A natural number ' ' x is chosen at random from the first 1000 natural numbers. If [.] denotes the

greatest integer function and the probability that 312 3 5 30 x x x x

A)31

1000 B)

33999

C)33

1000 D)

67

1000

*2. If 1 2 , 0 4 f x x x and 2 , 1 3 g x x x then fog x is

A) Discontinuous at 0 x B) Continuous at 0 x C) Not differentiable at 0 x D) Differentiable at 0 x

P) Two curves 1 32 31 0:c f y f x and 2 32 3

2 12:c f y f x

satisfying 2 24 x y f x y x y f x y x y x y

3. The area bounded by 1c and 2c (in square units) is

A) 2 3 B) 2 3 C) 3 D) 3

4. The area bounded by 2c and 12 x y (in square units) is

A) 12 24 B) 6 12 C) 6 12 D) 12 24

5. The area bounded by 1c and 2 0 x y (in square units) is

A)7

2 B)

11

2 C)

9

2 D)

5

2

6. The number of solutions of the equation 28 4 13 12 sin x x x x is (where [.] represents

greatest integer function)A) 0 B) 2 C) 4 D) 6

7. If 02

and 2sec 2 cosec 22

.cos .sin f , then

A) 2 f B) 2 f C) f D) 2 f



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8. If the graphs of the functions & y ln x y ax intersect at exactly two points, then

A) 0,a e B)1

0,ae

C) ,1a e D) 1,a e

*9. Let tan cot ,2

g x f x f x x

. If " 0 ,2

f x x

,then

A) g x is increasing in3

,2 4

B) g x is increasing in

3,4

C) g x is decreasing in3

,4

D) g x has local maximum at34

x

10. Let : f R R be a function defined by

51

3

f x f x x R

f x . Then which of the following

statement(s) is/are trueA) 2 f x f x B) 4 f x f x

C) 6 f x f x D) 8 f x f x

11. If ' ' x is positive and , x x x and ' ' x are in G.P., (where [.] denotes greatest integer function), then

22 2 1 x x is equal to

12. Let , 100S a N a . If the equation 2tan tan 0 x x a has real roots, then number of elements

in ' 'S is (where [.] represents greatest integer

13. If 2 25 3 1 2 y y x x x x R , then the number of integers lying in the range of ' ' y are

14. If 2

1 , 0

1 , 1

x x f x

x x

, then the number of solutions of the equation

1 0 f x f x is/are

15. f x is a polynomial of 6 th degree and 2 . f x f x x R If 0 f x has 4 distinct real roots

and two real and equal roots then sum of roots of 0 f x

*16. If 1 11 sin x cos x then ([x] denotes greatest integer function)

(A) x cos1, sin1 (B) x sin1, 1 (C) x (cos2, 0) (D) x (cos1, 1] .

17. The total number of function f' from the set {1, 2, 3} into the set {1, 2, 3, 4, 5} such that f(i) f(j) i <

j is equal to 40 – k then k is

P) Let f : R R be a continuous function such that f(x) – 2f (x/2) + f(x/4) = x 2

18. f(3) is equal to(A) f(0) (B) 4 + f(0) (C) 9 + f(0) (D) 16 + f(0)

19. The equation f(x) – x – f(0) = 0 has(A) no solution (B) one solution (C) two solution (D)infinitely many solution.

20. f (0) is equal to(A) 0 (B) 1 (C) f(0) (D) – f(0)



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21. Let f x be continuous function on 0,1 and if 1 1

1 2

00 0

1, 2 3 f x dx xf x dx and x f x dx .

Then the number of roots of 0 f x in 0,1 is _____

a) exactly one b) atleast one c) atmost one d) zero

*22. Which of the following statements is / are true ?

A) 1 1cos cos sin 2 0 x x

B) 1 11 1sin sin cos 1 2 6

2 3

C) minimum value of1 2 1 2cos cos tan cot x x

e

is 2e

D) 1 1 3tan 2 tan 3

4

PASSAGE

Let : f R R be continous function and bijective, defined such that 0 0 f . The area bounded by , , y f x x x t is equal to area bounded by , , y f x x x t , t R then

23. y f x is symmetrical about the point

a) 0,0 b) 0, c) ,0 d) ,

24. 2 f equal to _______

a) f b) f c) 0 f d) 0 f

25. Value of 1 ____ f t dt

a) 0 b) 2 c) d) none

26. Let 2009 K

f K and

4

4 41

f K g K

f K f K and

2009

0 K

S g K then sum of the digits in _____ S

27. If : f D R such that 2sin cos log 2cos 3cos 1e f k x x x then2

1

1cos

2

x

x

x dx

is equal to,

where 1 2, x x D and . denotes the greatest integer function

28. A continuous function y f x is defined in a closed interval 7,5 .

7, 4 , 2,6 , 0,0 , 1,6 , 5, 6 A B C D E are consecutive points on the graph of ' ' f and

, , , AB BC CD DE are line segments. The minimum number of real roots of the equation 6 f f x is

A) 6 B) 4 C) 2 D) 0

*29. Let ,: f R R such that 2 2" ' x f x f x f x e and 0, ,' f x x R then which of the

following can be correctA) , f x f x x R B) , f x f x x R

C) 3 5 f D) 3 7 f



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30. If x x x f cossinsincos , then the range of f ( x) is

(A) cos1, sin1 (B) cos1, 1 sin1

(C) 1 cos1, sin1 (D) cos1,1

P) f(x) is a polynomial function f : R R such that f 2x f ' x .f '' x .

31. The value of f(3) is(A) 4 (B) 12 (C) 15 (D) 18

32. f(x) is(A) one-one and onto (B) one-one and into(C) many-one and onto (D) many-one and into

33. Equation f(x) = x has(A) Three real and distinct roots (B) one real root(C) Four real and distinct roots (D) Two real and distinct roots

34. Suppose a cubic polynomial f(x) = x 3 + px 2 + qx + 72 is divisible by both x 2 + ax + b and x 2 + bx + a(where a, b, p, q are constants and a b ), then the value of p is (0)

35. The greatest value of 2 3

2 2f x 3 4 x 1 4 x is A, then A/7 =

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