Revision Plan-II (Dpp # 2)_mathematics_english

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DATE : 13.04.2016 PART TEST-01 (PT-01)

Syllabus : Limits, Continuity & Derivability, Quadratic Equation

TARGET : JEE (Advanced) 2016TEST INFORMATION

Course: VIJETA(ADP) & VIJAY(ADR) Date : 10-04-2016

DPP

NO.

02

MMAATTHHEEMMAATTIICCSS

DDPPPPDDAAIILLYYPPRRAACCTTIICCEEPPRROOBBLLEEMMSS

TTEESSTTIINNFFOORRMMAATTIIOONN

REVISION DPPOFLimits, Continuity & Derivability, Quadratic Equation

Total Marks : 137 Max. Time : 120 min.Single choice Objective ('1' negative marking) Q.1 to Q.16 (3 marks 3 min.) [48, 48]Multiple choice objective ('1' negative marking) Q.17 to Q.33 (4 marks 3 min.) [68, 51]Comprehension ('1' negative marking) Q.34 to Q.40 (3 marks 3 min.) [21, 21]

1. Let f be a continuous real function such that f(11) = 10 and for all x, f(x) f(f(x)) = 1 then f(9) =

(A) 9 (B)91 (C)

910 (D)

109

2. The value of2x

x 0

e cosx n(1 2x)lim

xtanx sinx

is

(A) 0 (B) 1 (C) 2 (D) 3

3. The value ofx 0

1lim x

x

, where [.] denotes GIF, is

(A) 0 (B) 1 (C) 2 (D) infinite

4.n

rnr 1

1lim

2

, where [.] denotes GIF is equal to

(A) 1 (B) 0 (C) non-existent (D) 2

5. The function f(x) =

2[x ] sgn(x 1) , x 2

x 2sin{2x 3},

x 2

, where [.] and {.} denotes GIF and FPF respectively is

(A) continuous at x = 2 (B) differentiable at x = 2(C) continuous and differentiable at x = 2 (D) discontinuous at x = 2

6. Let f : R+R+be a monotonic continuous function so that there exists at least one c [a, b], a, b > 0

such that f(c) =f(a) f(b)

1

, > 0 then possible value(s) of 'c' is given by

(A) c=f12f(a) f (b)

f(a) f(b)

(B) c=f1

2 2f (a) f (b)

f(a) f(b)

(C) c=f1

2f (a) f (b)

f(a) f(b)

(D) c=f1

3 3f (a) f (b)

f(a) f(b)

7. If f(x)=1 x , 0 x 2

3 x , 2 x 3

then the number of values of x at which the function fof is not differentiable is

(A) 2 (B) 1 (C) 3 (D) 0

8. If a2x2+ bx + c = 0 has roots and (< ) and p2x2+ qx + r = 0 has roots + 5 and 5 then(a2+ p2)x2+ (b q)x + (c r) = 0 has a root in interval (given + 5 < 5)(A) (, + 5) (B) (+ 5, 5) (C) ( , ) (D) (, )

9. Let f(x) be a function which is differentiable everywhere any number of times and f(2x21)=2x

3f(x) xR

then f2010

(0) is equal to (fn(x) is n

thderivative of f(x))

(A) 1 (B) 1

(C) 0 (D) Data provided is insufficient


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10. If y = x + exthen

2nx

2

2

dy

xd

is

(A) 9

1 (B)

27

2 (C)

27

2 (D)

9

1

11_. If f(x) =2

2

Minimum {x, x } ; x 0

Minimum {2x, x 1} ; x 0

, then number of points in the interval [2, 2] where f(x) is

non-differentiable is :(A) 1 (B) 2 (C) 3 (D) 4

12_. Let f"(x) be continues at x = 0, f(0) 0, f(0) 0 and

2x 0

2f x 3af 2x bf 8xlim

sin x

exists, then the

value of a + b is :(A) 0 (B) 2 (C) 1 (D) 3

13. If f(x) =

xn

e 1 | x sinx |, x 0

x

0 , x 0

is derivable at x = 0 then interval of possible values of 'n' is

(A) [1, ) (B) (, 1] (C) (, 2] (D) [1, 2]

14. Find 'a' if equations x3

+ ax + 1 = 0 and x4

+ ax2

+ 1 = 0 have a common root(A) 0 (B) 1 (C) 2 (D) 2

15. If f(x) = ax2+ bx + c such that f(p) + f(q) = 0 where a 0; a, b, c, p, q R then number of real roots ofequation f(x) = 0 in interval [p, q] is(A) Exactly one (B) at least one(C) at most one (D) data provided is insufficient

16. If a =16

x25 2and loga

224 2x x14

> 1, then x

(A) (0, 1) (B) (3, 1) (3, 4) (C) ( 17) (1, ) (D) (4, 5)

17_. If f(x) =2

x 3

5x x 1

& g(x) =

2

2

2x 3x

20 2x x

such f(x) and g(x) are differentiable functions in their domains, then which of the following is/are true(A) 2f'(2) + g'(1) = 0 (B) 2f'(2) g'(1) = 0 (C) f'(1) + 2g'(2) = 0 (D) f'(1) 2g'(2) = 0

18. If f(x) is a polynomial of degree five with leading coefficient one such that f(1) = 2, f(2) = 8, f(3) = 18,f(4) = 32, f(5)= 50 then

(A) f(6) = 192 (B)

5x x

xfLim

= 1 (C) f(0) < 0 (D) f(0) > 0

19_. If f(x) =3 2x 6x ; if x is rational

11x 6 ; if x is irrational

then

(A) f(x) is continuous for all real x. (B) f(x) is continuous at x = 1, 2, 3.(C) f(x) is discontinuous at infinite points (D) f(x) discontinuous for all real x.

20. Ifx=a satisfies equation tan1

(x+2)+cot1

20x4 =

2x

Lim

xsinxcos

xcosxsinand

0xLim

32

x

1

a

xkxx5

bxe

x1x1

= 0

then

(A) a = 2 (B) ab = 3 (C) a = 1, b = 3 (D) kR

21. If f(x) =

0x,0

0x,x

xtan

, where [.] = GIF, then

(A)0x

Lim f(x) = 1 (B)0x

Lim f(x) = 1 (C)0x

Lim f(x) = 0 (D) f(0) = 0


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22. If f(x) =

0x,bxsin

0x,x

1

x

1

1

22 then at x = 0, f(x) is

(A) continuous if b = 0 (B) discontinuous for any real b(C) differentiable for b = 1 (D) non-differentiable for any real b

23. Consider a continuous function f:[0, ) [0, ). If f(ab) = f(a) f(b) for all a, b in the domain of ' f ' and

xLim f(x) is a non-zero finite number then

(A) f(2) = 2 (B)

10

1r

rf = 55 (C)

10

0r

rf = 11 (D) f'(2) = 0

24. If f(x) = sin x 1 2 then

(A) f(x) is continuous at x = 2 (B) f(x) is differentiable at x = 2(C) f'(2) = cos1 (D) f(x) is non-differentiable at x = 0

25. A quadratic equation f(x) = ax2+ bx + c = 0 has positive distinct roots reciprocal of each other. Which

one is correct?

(A) f'(1) = 0 (B) af'(1) < 0 (C) c 0 (D) abc 0

26. If the two roots of the equation ( 2)(x2 + x + 1)2 (+ 2) (x4+ x2+ 1) = 0 are real and equal for= 1, 2then

(A) 1+ 2= 0 (B) |1 2| = 6 (C) 12+ 2

2= 32 (D) all of these

27. Which of the following is a subset of solution set of inequality |x2 2x| + |x 4| > |x2 3x + 4|

(A) (0, 2) (B) (4, ) (C) (0, 3) (D) [4, 5)

28. If p =2n

nlim n

{(n + 20)(n + 21)(n + 22) . . . . (n + 2n+1)}nthen which of the following is true?

(A) p is irrational (B) integer nearest to p is 7

(C) p >

xxn

x x

x1Lim

(D) None of these

29. If f(x) = [4x] + {3x} where [.] denotes GIF and {.} denotes FPF then for x[0, 5](A) number of points of discontinuity of f(x) are 25 (B) f'(0) = 0

(C) f'(x) = 3 wherever defined (D) f(x) < 2030. The equation 2log(x + 3) = log (x) has only one solution if

(A) = 12 (B) (, 0) (C) (0, ) (D) (0,12) {24}

31. If x2+ 2ax + a < 0 x [1, 2] then(A) a(0, 1) (B) a > 1

(C) a < 5

4 (D) x

2+ 2ax + a = 0 for some x > 2

32.

bx

x

ax 3lim

bx 2

is equal to (a and b are positive)

(A) 0 if a < b (B) e if a = b (C) if a > b (D) 1 if a > b

33_.x 0lim

sinxsinx

x sinx

is&

(A) 0 (B) 1 (C) ne (D) e1

Comprehension # 1 (Q. 34 to 36)

If f(x) is continuous at x = 0 then xf(x) is differentiable at x = 0. By changing origin we can say that iff(x a) is continuous at x = a then (x a) f(x a) is differentiable at x = a

34. The largest set over which2|x|1

|x|sinx

is differentiable is

(A) R {0, 1, 1} (B) R (C) R {1, 1} (D) None of these


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35. The number of points where f(x) = (x 3) |x2 7x + 12| + cos|x 3| is not differentiable is

(A) 1 (B) 2 (C) 3 (D) infinite

36. Let f(x) = |x|, g(x) = sinx and h(x) = g(x) f(g(x)) then(A) h(x) is continuous but not differentiable at x = 0(B) h(x) is continuous and differentiable everywhere(C) h(x) is continuous everywhere and differentiable only at x = 0(D) all of these

Comprehension # 2 (Q. No. 37 to 38)

If left hand derivative and right hand derivative of a function is same and finite then the function iscontinuous as well as differentiable. If left hand and right hand derivatives are different but finite thenthe function is continuous but not differentiable. If one of the derivatives is infinite then the function maybe continuous but not differentiable.

37. If f(x) =

x

0

t dt then

(A) f(x) is continuous and differentiable at x N (B) f(x) is continuous but not differentiable at xN(C) f(x) is discontinuous at x N (D) f(x) is continuous and differentiable at xQ

38. f(x) =1/ x

x

1 2

is

(A) continuous at all points (B) differentiable at all points

(C) continuous for x R {0} (D) non-differentiable at x = 0, 1

Comprehension # 3 (Q. No. 39 to 40)

f(x) =2n

2n 1

| x | , x [2n , 2n 1)

| x | , x [2n 1 , 2n 2)

, where n Z then

39. The number of integral points where f(x) is continuous is(A) 0 (B) 1 (C) 2 (D) all integers

40. The number of integral points where f(x) is derivable, is(A) 0 (B) 1 (C) 2 (D) all integers

DPP#01

ANSWERKEY

FUNCTIONAND INVERSETRIGONOMETRIC FUNCTION

1. (D) 2. (A) 3. (A) 4. (D) 5. (A)

6. (B) 7. (B) 8. (D) 9. (C) 10. (A,D)

11. (A,B,C) 12. (A,C,D) 13. (A,B,C,D) 14. (A,B,C,D)

15. (A,B,C,D) 16. (A,B,C) 17. (A,B) 18. (B,C) 19. (B, D)

20. (A,B,C) 21. (A,B) 22. (B,C,D) 23. (A,B,C) 24. (A,C)

25. (B,C) 26. (A,C,D) 27. (B,C,D) 28. (A,B) 29. (B,D)

30. (A,B) 31. (B) 32. (A) 33. (D)

34. (A) (p), (B) (p), (C) (p, q, s), (D) (p,q,r,s,t) 35. 5 36. 1

37. 1 38. 4 39. 3 40. 3

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