Revision Plan-II (Dpp # 3)_mathematics

8/16/2019 Revision Plan-II (Dpp # 3)_mathematics

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DATE : 17.04.2016 CUMULATIVE TEST-02 (CT-02)

Syllabus : Function & Inverse Trigonometric Function, Limits, Continuity & Derivability, Quadratic Equation,

A lication of Derivatives

TARGET : JEE (Advanced) 2016TEST INFORMATION

Course : VIJETA(ADP) & VIJAY(ADR) Date : 13-04-2016

DPP

NO.

03

MMAATTHHEEMMAATTIICCSS

DDPPPP DDAAIILLYY PPRRAACCTTIICCEE PPRROOBBLLEEMMSS

TTEESSTT IINNFFOORRMMAATTIIOONN

REVISION DPP OFAPPLICATION OF DERIVATIVES

Total Marks : 139 Max. Time : 120 min.Single choice Objective (–1 negative marking) Q. 1 to 17 (3 marks 3 min.) [51, 51]Multiple choice objective (–2 negative marking) Q. 18 to 36 (4 marks, 3 min.) [76, 57]Comprehension (–1 negative marking) Q.37 to Q.40 (3 marks 3 min.) [12, 12]

1. There are 50 machines in a factory each producing 1000 bolts daily. For each additional machineinstalled, the output per machine drops by 10 bolts. How many additional machines should be installed

to maximize the total output per day?(A) 20 (B) 30 (C) 50 (D) 25

2. If a2 x4 + b2y4 = c6 then the maximum value of xy is (a,b,c > 0)

(A)2c

ab (B)

3c

ab (C)

3c

2ab (D)

3c

2ab

3. The point on the curve xy2 = 1 nearest to origin is

(A) 1/ 3 1/ 62 , 2 (B) 1/ 3 –1/ 62 ,2 (C) 1/ 3 1/ 62 , 2 (D) (1, 1) 4. The fraction exceeding its own nth power (n N) by the maximum possible value is

(A)

1

n–11

n

(B)

n–11

n

(C)

n1

n

(D)

1

n–111

n

5. A person is standing at the edge of a slow moving river which is 1 km wide. He wishes to return to the

camp ground on the opposite side of the river for which he may swim to any point on the opposite bankand then walk for the rest of the distance. The campground is 1 km away from the point on the oppositebank directly across from where he starts to swim. If he swims at the rate of 2 km/hr and walks at therate of 3 km/hr, the minimum time taken by him is approximately(A) 0.6 hr (B) 0.7 hr (C) 0.8 hr (D) 0.9 hr

6. The greatest value of the f(x) = 2 3

2 23 – 4 – x 1 4 – x is

(A) 25 (B) 26 (C) 28 (D) 24

7. Let f(x) be a non-negative continuous function satisfying f (x)cos x f(x)sin xx 0. Then f 5

3

=

(A) e –1/2 (B)1

2 (C) 0 (D)

1

2

8. If f(x) =

2

2

x –cos , 0 x 1

2

1 x , 1 x 2

then number of values of 'c' obtained by applying LMVT on f(x) in interval

[0, 2] is(A) 1 (B) 2 (C) 3 (D) LMVT is not applicable

9_. Tangent lines are drawn at the points of inflexion for the function f(x) = cosx on [0, 2]. The linesintersect with the x-axis so as to form a triangle. The area of this triangle is

(A)

2

2 (B)

4

2 (C)

2

8

(D)

16

2


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10. Let f(x) be a twice differentiable positive function on an interval (a, b). Define a function g(x) such that

f(x) = eg(x) x (a, b). If f(x) is such that the roots of the equation f(x)t2 – [f (x)]t + f (x) = 0 are alwaysreal and distinct, then(A) g(x) is always increasing on (a, b) (B) g(x) is always decreasing on (a, b)(C) g(x) is always concave up on (a, b) (D) g(x) is always concave downward on (a, b)

11. A lamp of negligible height is placed on the ground at a distance of 10m from a wall. A man 2m tall is

walking at a speed of 1 m/s from the lamp towards the wall. When he is 5m away from the wall, the rateat which his shadow shortens on the wall is

(A) 32

m/s (B) 35

m/s (C) 45

m/s (D) 1 m/s

12. The point of intersection of tangent to the curve y = x4 at (1, 1) with the normal to the same curve at(–1, 1) is

(A)17 13

,15 15

(B)7 3

,5 5

(C)17 23

,15 15

(D) does not exist

13_. The tangent and normal at the point P(at2, 2at) to the parabola y

2 = 4ax meet the x-axis in T and G

respectively, then the angle at which the tangent at P to the parabola is inclined to the tangent at P tothe circle through P, T, G is(A) tan

–1(t

2) (B) cot

–1(t

2) (C) tan

–1(t) (D) cot

–1(t)

14_. If graphs of y = logax and y = ax (a > 1) intersect at exactly one point then a =

(A) e (B) e (C) ee (D) e1/e

15_. Minimum value of (x1 – x2)2 +

221

2 2

x17 x x 13

20

, where x1 R

+, x2 (13, 17) is

(A) 2

5 2 2 (B) 5 2 + 2 (C) 2

5 2 2 (D) (5 2 – 2)

16. Let f(x) be a differentiable real valued function satisfying f x) – 6f (x) > 6 x 0. If f (0) = –1 andg(x) = f(x) + x then g(x) is

(A) increasing x 0 (B) decreasing x 0(C) a constant function x 0 (D) None of these

17. Let f(x) be a differentiable function on [0, 8]. Such that f(1) = 3, f(2) = 1/2, f(3) = 4, f(4) = –2, f(5) = 6,

f(6) = 1/3, f(7) = –1/4. Then the minimum number of points of intersection of the curves y = f (x) andy = f (x) [f(x)]2 is(A) 10 (B) 6 (C) 11 (D) 25

18_. Let x2/3

+ y2/3

= a2/3

be a equation of curve, then which of the following is correct ?

(A) Equation of tangent at P(x1, y1) to the curve is 1/ 3 1/ 31 1

x y

x y = a2/3

(B) Length of portion of tangent intercepted between coordinate axes is constant(C) Equation of normal at P(x1, y1) to the curve is xx1

1/3 + yy1

1/3 = x1

4/3+ y1

4/3

(D) If tangent at (x1, y1) meet the coordinate axes in A and B, then locus of mid point of AB is

x2 + y

2 =

2a

4

19_. Equation of tangents drawn to the curve y2 – 2x

3 – 4y + 8 = 0 from the point(1, 2) is/are

(A) 32 (x – 2) – y + 2 31 = 0 (B) 32 (x – 2) – y – 2 31 = 0

(C) 32 (x – 2) + y + 2 13 = 0 (D) 32 (x – 2) + y – 2 13 = 020_. Which of the following pairs of curves is orthogonal

(A) y2 = 4ax ; y =

x

2ae

(B) y2 = 4ax ; x

2 = 4ay at (0, 0)

(C) xy = a2 ; x

2 – y

2 = b

2(D) y = ax ; x

2 + y

2 = c

2

21_. If parametric equation of any curve at is given by x = acos3, y = asin3 then a > 0,

2,0

(A) length of tangent asin2 (B) length of subtangent asin2 cos

(C) length of normal atan sin2 (D) length of subnormal asin3 tan


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22_. Tangent at point P1(other than origin) on the curve y = x3 meets the curve again at P2. The tangent at P2

meets the curve at P3 and so on. Then(A) abscissae of P1, P 2, P 3 ...... Pn form a G.P. (B) ordinates of P 1, P 2, P 3 ... P n form a G.P

(C) ratio of area of P1P2P3 and P2P3P4 is1

16 (D) ratio of area of P1P2P3 and P2P3P4 is 16

23. f(x) = 2ex + (a

2 – 5a + 6)e

–x + (10a – 2a

2 – 11)x – 3 is increasing for all real values of x if a

(A) {2} (B) [2, 3] (C) (2, 3) (D) (3, )

24. Let f(x) be a function satisfying f (x) = n 2

x x 1 and f(0) = 0, then(A) f(x) 0 x R (B) f(x) 0 x R(C) f'(x) is increasing xR (D) f(x) is even function

25. f(x) = (4sin2x – 1)n (x2 + 6x + 11) where n N, then x =6

is a point of

(A) local maximum if n is even (B) decreasing if n is odd(C) local minimum if n is even (D) increasing if n is odd

26. If f(x) = |ax – b| + c|x| x R has exactly one minimum, (a, b, c > 0), then which of the following maybe possible(A) a > c (B) a < c (C) a = c (D) a, c can be anything

27. Let f(x) = 3sin x – 4cos x + ax + b, then

(A) f(x) = 0 has only one real root which is positive if a > 5 and b < 0(B) f(x) = 0 has only one real root which is negative if a > 5 and b > 0(C) f(x) = 0 has only one real root which is negative if a < –5 and b < 0(D) f(x) = 0 has only one real root which is positive if a < –5 and b > 0

28_. If f(x) = x3 – x

2 + 100x + 2002 then

(A) f(1000) > f(1001) (B) f1

2000

>f 1

2001

(C) f(x – 1) > f(x – 2) (D) f(sin(cos1)) f(cos(sin1))

29_. If f(x) =x

1 x – n(1 + x) for x > 0 then

(A) f(x) is increasing function (B) f(x) is decreasing function

(C)e 1

n(1 e)(1 ) 1 e

(D)sin1 cos1 1 sin1

n(1 sin1)(1 cos1) 1 cos1

30. A movie screen on a wall is 20 feet high and 10 feet above the floor. If a man has to position himself at

distance x from the screen to have a maximum angle of view , then

(A) x =10

3feet (B) = 60° (C) x = 10 3 feet (D) = 30°

31. Let f(x) be a polynomial with real coefficients. Let a, b R be two consecutive roots of f(x). Then forsome c [a, b](A) f (c) + f(c) = 0 ( is constant) (B) f (c) 0 for any c(C) f (c) = 0 for atleast one c (a, b) (D) None of these

32. Consider function f(x) = |xnx|. Then

(A) maximum value of f(x) in x(0, 1) is

e

1 (B) f'(x) has local minima at x = 1

(C) Rolle's theorem can be applied to f(x) for an interval of maximum length 1 unit(D) f'(x + 2) – f'(x) < 2 for all x > 1

33. Let f(x) be a differentiable function with f(1) f(–1) 0. Define a function g(x) =2x – 1

f(x) . If g(x) does not

follow Rolle's theorem in [–1, 1], then which of the following options is/are FALSE?(A) f(x) = 0 cannot have any root in [–1, 1] (B) f(x) = 0 has at least one root in [–1, 1]

(C) f (x) is zero at at least one point in [–1, 1] (D) f(x) cannot satisfy Rolle's theorem in [–1,1]

34. Let f : (a, b) R is a differentiable function such thatax

Lim f 2(x) = 0,

bxLim f

2(x) = e – 1 and

2f(x) f'(x) – f 2(x) 1 for all x (a, b) then value of (b – a) can be

(A) 0 (B)

2

1 (C) 1 (D) 2


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35. Which of the following are incorrect given x y?

(A)1 –1cos x – cos y

y – x 1 x, y [–1, 1] (B)

1 –1cot x –cot y

y – x 1 x, y R

(C)1 –1tan x – tan y

x – y 1 x, y R (D)

1 1sin x sin y

x y

1 x, y [–1, 1]

36. if y = p(x) be a polynomial of least degree passing through A(–1, 1) and whose graph has two points of

inflexion B(1, 1) and C with abscissa 0 at which the curve is inclined /4 to the positive direction ofX-axis then

(A) P(x) =4 3x x 1

6 x12 6 2

(B) P(x) =4 3x x 1

6 x12 6 2

(C) P(x) has local minima at x = –1

2

(D) there exists two real number a and b (b > a) such that 2P'(b) + 3b 2P'(a) + 3a

Comprehension # (Q. No. 37 to 38)f(x), g(x) and h(x) are all continuous and differentiable increasing functions in [a, b]. Also a < c < b &f(a) = g(a) = h(a). Further the point of intersection of tangent at x = c with the chord joining x = a & x = bis on the left of c in y = f(x) and on the right of c in y = h(x). In case of g(x), tangent is parallel to thechord. Given f(x), g(x), h(x) are concave up.

37. If f (x) > g(x) > h(x), then(A) f(b) < g(b) < h(b) (B) f(b) > g(b) > h(b) (C) f(b) = g(b) = h(b) (D) f(b) g(b) h(b)

38. If f(b) = g(b) = h(b), then

(A) f (c) = g(c) = h(c) (B) f (c) > g(c) > h(c) (C) f (c) < g(c) < h(c) (D) None of these

Comprehension # (39 to 40)

Let f(x) = x3 + ax

2 + bx + c be a given cubic polynomial and f(x) = 0 be the corresponding cubic equation

where a, b, c R

39_. Which of the following is/are correct.(A) If a

2 – 3b < 0 and c > 0 then f(x) = 0 has only one negative real root

(B) If a2 – 3b < 0 and c < 0 then f(x) = 0 has only one positive real root

(C) If a2 – 3b = 0 then f(x) = 0 may have three real and equal roots

(D) If a2 – 3b = 0 then f(x) = 0 may have three real roots in which two roots are equal

40_. If a2 – 3b > 0 and x1 and x2 are real roots of f (x) = 0 such that x2 > x1, then which of the following is/are

correct.(A) If f(x1).f(x2) > 0 then f(x) = 0 would have just one real root(B) If f(x1).f(x2) < 0 then f(x) = 0 would have three real and distinct roots(C) If (x1).f(x2) = 0 then f(x) = 0 would have three real roots but one of which repeated.

(D) f (x) < 0 x (x1, x2) and f (x) > 0 x (–, x1) (x2, ) and x = x1 is point of local minima andx = x2 is point of local max.

DPP # 02 ANSWER KEY

Limits, Continuity & Derivability, Quadratic Equation

1. (B) 2. (A) 3. (B) 4. (B) 5. (D) 6. (A) 7. (A)

8. (A) 9. (C) 10. (B) 11. (C) 12. (B) 13. (C) 14. (D)

15. (B) 16. (B) 17. (A, C) 18. (A,B,C) 19. (B,C) 20. (B, D)

21. (A,C,D) 22. (A, D) 23. (C, D) 24. (A,B,D) 25. (B,C,D) 26. (A,C) 27. (A,B)

28. (A,B,C) 29. (A, C) 30. (A, B) 31. (C, D) 32. (A,B,C) 33. (B,C) 34. (C)

35. (A) 36. (B) 37. (B) 38. (C) 39. (C) 40. (A)

Revision Plan-II (Dpp # 3)_mathematics

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