Maths quest.pdf

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SPEED - IIQUESTION BANK

FOR IITJEE

MATHEMATICS


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East Delhi : North Delhi :No. 1 Vigyan Vihar, New Delhi. Ph. 65270275 : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439


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APPLICATION OF DERIVATIVE

MATHEMATICS

Time Limit : 5 Sitting Each of 80 Minutes duration approx.

TARGET IIT JEE


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[2] Quest TutorialsNorth Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest

Question bank on Application of Derivative

Select the correct alternative : (Only one is correct)

Q.1 Suppose x1 & x

2are the point of maximum and the point of minimum respectively of the function

f(x) = 2x3 − 9 ax2 + 12 a2x + 1 respectively, then for the equality 21

x = x2 to be true the value of 'a' must be

(A) 0 (B) 2 (C) 1 (D) 1/4

Q.2 Point 'A' lies on the curve2

xey −= and has the coordinate (x,2

xe− ) where x > 0. Point B has thecoordinates (x, 0). If 'O' is the origin then the maximum area of the triangle AOB is

(A)e2

1(B)

e4

1(C)

e

1(D)

e8

1

Q.3 The angle at which the curve y = KeKx intersects the y-axis is :

(A) tan−1 k 2 (B) cot−1 (k 2) (C) sec−1 1 4+ k (D) none

Q.4 {a1, a

2, ....., a

4, ......} is a progression where a

n =

n

n

2

3 200+ . The largest term of this progression is :

(A) a6 (B) a7 (C) a8 (D) none

Q.5 The angle between the tangent lines to the graph of the function f (x) = ∫ −x

2

dt)5t2( at the points where

the graph cuts the x-axis is

(A) π /6 (B) π /4 (C) π /3 (D) π /2

Q.6 The minimum value of the polynomial x(x + 1) (x + 2) (x + 3) is :

(A) 0 (B) 9/16 (C) − 1 (D) − 3/2

Q.7 The minimum value of( )tan

tan

x

x

+ π6

is :

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

Q.8 The difference between the greatest and the least values of the function, f (x) = sin2x – x on

ππ−2

,2

(A) π (B) 0 (C)32

3 π+ (D)

3

2

2

3 π+−

Q.9 The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the

rate of 0.2 cm/min. The rate of change of the volume of the cylinder, in cm3 /min, when the radius is 2 cm

and the height is 3 cm is

(A) – 2π (B) – 58π

(C) – 5

3π(D) 5

2π

Q.10 If a variable tangent to the curve x2y = c3 makes intercepts a, b on x and y axis respectively, then the

value of a2b is

(A) 27 c3 (B)3c

27

4(C)

3c4

27(D)

3c9

4

Q.11 Difference between the greatest and the least values of the function

f (x) = x(ln x – 2) on [1, e2] is

(A) 2 (B) e (C) e2 (D) 1


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Quest

Q.12 Let f (x) =

∑=

n2

0r

r

n

xtan

xtan, n ∈ N, where x ∈

π2

,0

(A) f (x) is bounded and it takes both of it's bounds and the range of f (x) contains exactly one integral point.

(B) f (x) is bounded and it takes both of it's bounds and the range of f (x) contains more than one integral point.

(C) f (x) is bounded but minimum and maximum does not exists.(D) f (x) is not bounded as the upper bound does not exist.

Q.13 If f (x) = x3 + 7x – 1 then f (x) has a zero between x = 0 and x = 1. The theorem which best describes

this, is

(A) Squeeze play theorem (B) Mean value theorem

(C) Maximum-Minimum value theorem (D) Intermediate value theorem

Q.14 Consider the function f (x) =

=

>π

0xfor0

0xforx

sinx

then the number of points in (0, 1) where the

derivative f ′(x) vanishes , is(A) 0 (B) 1 (C) 2 (D) infinite

Q.15 The sum of lengths of the hypotenuse and another side of a right angled triangle is given. The area of the

triangle will be maximum if the angle between them is :

(A)π6

(B)π4

(C)π3

(D)5

12

π

Q.16 In which of the following functions Rolle’s theorem is applicable?

(A) f(x) =

=


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Quest

Q.20 The vertices of a triangle are (0, 0), (x, cos x) and (sin3x, 0) where 0 < x <2

π. The maximum area for

such a triangle in sq. units, is

(A)32

33(B)

32

3(C)

32

4(D)

32

36

Q.21 The subnormal at any point on the curve xyn = an + 1 is constant for :

(A) n = 0 (B) n = 1 (C) n = − 2 (D) no value of n

Q.22 Equation of the line through the point (1/2, 2) and tangent to the parabola y =− x2

2 + 2 and secant to

the curve y = 4 2− x is :

(A) 2x + 2y − 5 = 0 (B) 2x + 2y − 3 = 0 (C) y − 2 = 0 (D) none

Q.23 The lines y = − 3

2x and y = −

2

5x intersect the curve 3x2

+ 4xy + 5y2 − 4 = 0 at the points P and Q respectively.The tangents drawn to the curve at P and Q

(A) intersect each other at angle of 45º

(B) are parallel to each other

(C) are perpendicular to each other

(D) none of these

Q.24 The least value of 'a' for which the equation,

4 1

1sin sinx x+

− = a has atleast one solution on the interval (0, π /2) is :

(A) 3 (B) 5 (C) 7 (D) 9

Q.25 If f(x) = 4x3 − x2 − 2x + 1 and g(x) = [ { }Min f t t x xx x

( ) : ;

;

0 0 1

3 1 2

≤ ≤ ≤ ≤− < ≤

then

g1

4

+ g

3

4

+ g

5

4

has the value equal to :

(A)7

4(B)

9

4(C)

13

4(D)

5

2

Q.26 Given : f (x) =

3 / 2

x2

14

−− g (x) =

=

≠

0x,1

0x,x

]x[nta

h (x) = {x} k (x) = )3x(log25 +

then in [0, 1] Lagranges Mean Value Theorem is NOT applicable to(A) f, g, h (B) h, k (C) f, g (D) g, h, k

Q.27 Two curves C1 : y = x2 – 3 and C

2 : y = kx2 , Rk ∈ intersect each other at two different points. The

tangent drawn to C2 at one of the points of intersection A ≡ (a,y1) , (a > 0) meets C1 again at B(1,y2)

( )21 yy ≠ . The value of ‘a’ is(A) 4 (B) 3 (C) 2 (D) 1

Q.28 f (x) = dxx1

1

x1

12

22∫

+−

+− then f is

(A) increasing in (0, ∞) and decreasing in (– ∞, 0) (B) increasing in (– ∞, 0) and decreasing in (0, ∞)(C) increasing in (– ∞ , ∞) (D) decreasing in (– ∞ , ∞)


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Quest

Q.29 The lower corner of a leaf in a book is folded over so as to just reach the inner edge of the page. The

fraction of width folded over if the area of the folded part is minimum is :

(A) 5/8 (B) 2/3 (C) 3/4 (D) 4/5

Q.30 A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane

bounded by the lines y = 0, y = 3x, and y = 30 – 2x. The largest area of such a rectangle is

(A) 8

135

(B) 45 (C) 2

135

(D) 90

Q.31 Which of the following statement is true for the function

0

Q.32 A closed vessel tapers to a point both at its top E and its bottom F and is fixed with EF vertical when the

depth of the liquid in it is x cm, the volume of the liquid in it is, x2 (15 − x) cu. cm. The length EF is:(A) 7.5 cm (B) 8 cm (C) 10 cm (D) 12 cm

Q.33 Coffee is draining from a conical filter, height and diameter both 15 cms into a cylinderical coffee pot

diameter 15 cm. The rate at which coffee drains from the filter into the pot is 100 cu cm /min.

The rate in cms/min at which the level in the pot is rising at the instant when the coffee in the pot is 10 cm, is

(A)π16

9(B)

π925

(C)π35

(D)π9

16

Q.34 Let f (x) and g (x) be two differentiable function in R and f (2) = 8, g (2) = 0, f (4) = 10 and g (4) = 8

then

(A) g ' (x) > 4 f ' (x) ∀ x ∈ (2, 4) (B) 3g ' (x) = 4 f ' (x) for at least one x ∈ (2, 4)(C) g (x) > f (x) ∀ x ∈ (2, 4) (D) g ' (x) = 4 f ' (x) for at least one x ∈ (2, 4)

Q.35 Let m and n be odd integers such that o < m < n. If f(x) = x

m

n for x ∈ R, then(A) f(x) is differentiable every where (B) f ′ (0) exists(C) f increases on (0, ∞) and decreases on (–∞, 0) (D) f increases on R

Q.36 A horse runs along a circle with a speed of 20 km/hr . A lantern is at the centre of the circle . A fence is

along the tangent to the circle at the point at which the horse starts . The speed with which the shadow of

the horse move along the fence at the moment when it covers 1/8 of the circle in km/hr is

(A) 20 (B)40 (C) 30 (D) 60

Q.37 Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is

false.

Statement-1: If f : R → R and c ∈ R is such that f is increasing in (c – δ, c) and f is decreasing in(c, c + δ) then f has a local maximum at c. Where δ is a sufficiently small positive quantity.Statement-2 : Let f : (a, b) → R, c ∈ (a, b). Then f can not have both a local maximum and a point of inflection at x = c.

Statement-3 : The function f (x) = x2 | x | is twice differentiable at x = 0.

Statement-4 : Let f : [c – 1, c + 1] → [a, b] be bijective map such that f is differentiable at c then f –1is also differentiable at f (c).

(A) FFTF (B) TTFT (C) FTTF (D) TTTF


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Quest

Q.38 Let f : [–1, 2] → R be differentiable such that 0 ≤ f ' (t) ≤ 1 for t ∈ [–1, 0] and – 1 ≤ f ' (t) ≤ 0 fort ∈ [0, 2]. Then(A) – 2 ≤ f (2) – f (–1) ≤ 1 (B) 1 ≤ f (2) – f (–1) ≤ 2(C) – 3 ≤ f (2) – f (–1) ≤ 0 (D) – 2 ≤ f (2) – f (–1) ≤ 0

Q.39 A curve is represented by the equations, x = sec2 t and y = cot t where t is a parameter. If the tangent

at the point P on the curve where t = π /4 meets the curve again at the point Q then PQ is equal to:

(A)5 3

2(B)

5 5

2(C)

2 5

3(D)

3 5

2

Q.40 For all a, b ∈ R the function f (x) = 3x4 − 4x3 + 6x2 + ax + b has :(A) no extremum (B) exactly one extremum

(C) exactly two extremum (D) three extremum .

Q.41 The set of values of p for which the equation ln x − px = 0 possess three distinct roots is

(A)

e

1,0 (B) (0, 1) (C) (1,e) (D) (0,e)

Q.42 The sum of the terms of an infinitely decreasing geometric progression is equal to the greatest value of the function f (x) = x3 + 3x – 9 on the interval [– 2, 3]. If the difference between the first and the second

term of the progression is equal to f ' (0) then the common ratio of the G.P. is

(A) 1/3 (B) 1/2 (C) 2/3 (D) 3/4

Q.43 The lateral edge of a regular hexagonal pyramid is 1 cm. If the volume is maximum, then its height must

be equal to :

(A)1

3(B)

2

3(C)

1

3(D) 1

Q.44 The lateral edge of a regular rectangular pyramid is 'a' cm long . The lateral edge makes an angleα with

the plane of the base. The value of α for which the volume of the pyramid is greatest, is :(A)

π4

(B) sin−12

3(C) cot−1 2 (D)

3

Q.45 In a regular triangular prism the distance from the centre of one base to one of the vertices of the other

base is l. The altitude of the prism for which the volume is greatest :

(A)

2(B)

3(C)

3(D)

4

Q.46 Let f (x) =

1xif )2x(

1xif x

3

53

>−−

≤

then the number of critical points on the graph of the function is

(A) 1 (B) 2 (C) 3 (D) 4

Q.47 The curve y − exy + x = 0 has a vertical tangent at :(A) (1, 1) (B) (0, 1) (C) (1, 0) (D) no point

Q.48 Number of roots of the equation x2 . e2 − x = 1 is :

(A) 2 (B) 4 (C) 6 (D) zero

Q.49 The point(s) at each of which the tangents to the curve y = x3 − 3x2 − 7x + 6 cut off on the positivesemi axis OX a line segment half that on the negative semi axis OY then the co-ordinates the point(s) is/

are given by :(A) (− 1, 9) (B) (3, − 15) (C) (1, − 3) (D) none


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Quest

Q.50 A curve with equation of the form y = ax4 + bx3 + cx + d has zero gradient at the point (0, 1) and also

touches the x-axis at the point (− 1, 0) then the values of x for which the curve has a negative gradient are(A) x > − 1 (B) x < 1 (C) x < − 1 (D) − 1 ≤ x ≤ 1

Q.51 Number of solution(s) satisfying the equation, 3x2 − 2x3 = log2 (x2 + 1) − log2 x is :

(A) 1 (B) 2 (C) 3 (D) none

Q.52 Consider the function

f (x) = x cos x – sin x, then identify the statement which is correct .(A) f is neither odd nor even (B) f is monotonic decreasing at x = 0

(C) f has a maxima at x = π (D) f has a minima at x = – π

Q.53 Consider the two graphs y = 2x and x2 – xy + 2y2 = 28. The absolute value of the tangent of the angle

between the two curves at the points where they meet, is

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

Q.54 The x-intercept of the tangent at any arbitrary point of the curve22 y

b

x

a+ = 1 is proportional to:

(A) square of the abscissa of the point of tangency

(B) square root of the abscissa of the point of tangency(C) cube of the abscissa of the point of tangency

(D) cube root of the abscissa of the point of tangency.

Q.55 For the cubic, f (x) = 2x3 + 9x2 + 12x + 1 which one of the following statement, does not hold good?

(A) f (x) is non monotonic

(B) increasing in (– ∞, – 2) ∪ (–1, ∞) and decreasing is (–2, –1)(C) f : R → R is bijective(D) Inflection point occurs at x = – 3/2

Q.56 The function 'f' is defined by f(x) = xp (1 − x)q for all x ∈ R, where p,q are positive integers, has amaximum value, for x equal to :

(A)pq

p q+(B) 1 (C) 0 (D)

p

p q+

Q.57 Let h be a twice continuously differentiable positive function on an open interval J. Let

g(x) = ln( ))x(h for each x ∈ J

Suppose ( )2)x('h > h''(x) h(x) for each x∈ J. Then

(A) g is increasing on J (B) g is decreasing on J

(C) g is concave up on J (D) g is concave down on J

Q.58 Let f (x) =

2

1xif 0

21xif

1x2)1x6)(1x(

=

≠− −− then at x =

2

1

(A) f has a local maxima (B) f has a local minima

(C) f has an inflection point (D) f has a removable discontinuity

Q.59 Let f (x) and g (x) be two continuous functions defined from R → R, such that f (x1) > f (x2) and

g (x1) < g (x

2), ∀ x

1 > x

2 , then solution set of )2(gf 2 α−α > ( ))43(gf −α is

(A) R (B) φ (C) (1, 4) (D) R – [1, 4]


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Quest

Q.60 If f(x) =x

x2

∫ (t − 1) dt , 1 ≤ x≤ 2, then global maximum value of f(x) is :

(A) 1 (B) 2 (C) 4 (D) 5

Q.61 A right triangle is drawn in a semicircle of radius 1/2 with one of its legs along the diameter. The maximum

area of the triangle is

(A)4

1(B)

32

33(C)

16

33(D)

8

1

Q.62 At any two points of the curve represented parametrically by x = a (2 cos t − cos 2t) ;y = a (2 sin t − sin 2t) the tangents are parallel to the axis of x corresponding to the values of theparameter t differing from each other by :

(A) 2π /3 (B) 3π /4 (C) π /2 (D) π /3

Q.63 Let x be the length of one of the equal sides of an isosceles triangle, and let θ be the angle between them.If x is increasing at the rate (1/12) m/hr, and θ is increasing at the rate of π /180 radians/hr then the ratein m2 /hr at which the area of the triangle is increasing when x = 12 m and θ = π /4

(A) 21/2

π+5

21 (B)

2

73 · 21/2 (C)

52

3 21 π+ (D) 21/2

π+52

1

Q.64 If the function f (x) =4x

xx3t 2

−−+

, where 't ' is a parameter has a minimum and a maximum then the

range of values of 't' is

(A) (0, 4) (B) (0, ∞) (C) (– ∞, 4) (D) (4, ∞)

Q.65 The least area of a circle circumscribing any right triangle of area S is :

(A) π S (B) 2 π S (C) 2 π S (D) 4 π S

Q.66 A point is moving along the curve y3 = 27x. The interval in which the abscissa changes at slower rate thanordinate, is

(A) (–3 , 3) (B) (– ∞ , ∞ ) (C) (–1, 1) (D) (–∞ , –3) ∪ (3,∞ )

Q.67 Let f (x) and g (x) are two function which are defined and differentiable for all x≥ x0. If f (x

0) = g (x

0) and

f ' (x) > g ' (x) for all x > x0 then

(A) f (x) < g (x) for some x > x0

(B) f (x) = g (x) for some x > x0

(C) f (x) > g (x) only for some x > x0

(D) f (x) > g (x) for all x > x0

Q.68 P and Q are two points on a circle of centre C and radius α, the angle PCQ being 2θ then the radius of the circle inscribed in the triangle CPQ is maximum when

(A)22

13sin −=θ (B)2

15sin −=θ (C)2

15sin +=θ (D)4

15sin −=θ

Q.69 The line which is parallel to x-axis and crosses the curve y = x at an angle ofπ4

is

(A) y = − 1/2 (B) x = 1/2 (C) y = 1/4 (D) y = 1/2

Q.70 The function S(x) = ∫

πx

0

2

dt2

tsin has two critical points in the interval [1, 2.4]. One of the critical

points is a local minimum and the other is a local maximum. The local minimum occurs at x =

(A) 1 (B)2

(C) 2 (D)2

π


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Quest

Q.71 For a steamer the consumption of petrol (per hour) varies as the cube of its speed (in km). If the speed

of the current is steady at C km/hr then the most economical speed of the steamer going against the

current will be

(A) 1.25 C (B) 1.5 C (C) 1.75C (D) 2 C

Q.72 Let f and g be increasing and decreasing functions, respectively from [0 , ∞) to [0 , ∞). Leth (x) = f [g (x)] . If h (0) = 0, then h (x) − h (1) is :

(A) always zero (B) strictly increasing (C) always negative (D) always positive

Q.73 The set of value(s) of 'a' for which the function f (x) =a x3

3 + (a + 2) x2 + (a − 1) x + 2 possess a

negative point of inflection .

(A) (− ∞, − 2) ∪ (0, ∞) (B) {− 4/5 }(C) (− 2, 0) (D) empty set

Q.74 A function y = f (x) is given by x =

1

1 2+ t & y =

1

1 2t t( )+ for all t > 0 then f is :

(A) increasing in (0, 3/2) & decreasing in (3/2, ∞)(B) increasing in (0, 1)(C) increasing in (0, ∞)(D) decreasing in (0, 1)

Q.75 The set of all values of ' a

' for which the function

,

f (x) = (a2 − 3 a + 2) cos sin2 2

4 4

x x−

+ (a − 1) x + sin 1 does not possess critical points is:

(A) [1, ∞) (B) (0, 1) ∪ (1, 4) (C) (− 2, 4) (D) (1, 3) ∪ (3, 5)

Q.76 Read the following mathematical statements carefully:

I. Adifferentiable function ' f ' with maximum at x = c ⇒ f ''(c) < 0.

II. Antiderivative of a periodic function is also a periodic function.

III. If f has a period T then for any a ∈ R. ∫T

0

dx)x(f = ∫ +T

0

dx)ax(f

IV. If f (x) has a maxima at x = c

, then ' f

' is increasing in (c – h, c) and decreasing in (c, c + h)

as h → 0 for h > 0.Now indicate the correct alternative.

(A) exactly one statement is correct. (B) exactly two statements are correct.

(C) exactly three statements are correct. (D) All the four statements are correct.

Q.77 If the point of minima of the function, f(x) = 1 + a2x – x3 satisfy the inequality

x x

x x

2

2

2

5 6

+ ++ +

< 0, then 'a' must lie in the interval:

(A) ( )−3 3 3 3, (B) ( )− −2 3 3 3,

(C) ( )2 3 3 3, (D) ( ) ( )− −3 3 2 3 2 3 3 3, ,∪

Q.78 The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the

radius and increases three times as fast as radius. When the radius is 1cm the altitude is 6 cm. When the

radius is 6cm, the volume is increasing at the rate of 1Cu cm/sec. When the radius is 36cm, the volume

is increasing at a rate of n cu. cm/sec. The value of 'n' is equal to:(A) 12 (B) 22 (C) 30 (D) 33


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Quest

Q.79 Two sides of a triangle are to have lengths 'a' cm & 'b' cm. If the triangle is to have the maximum area,

then the length of the median from the vertex containing the sides 'a' and 'b' is

(A)1

2

2 2a b+ (B)

2

3

a b+(C)

a b2 2

2

+(D)

a b+ 23

Q.80 Let a > 0 and f be continuous in [– a, a]. Suppose that f ' (x) exists and f ' (x) ≤ 1 for all x ∈ (– a, a). If

f (a) = a and f (– a) = – a then f (0)(A) equals 0 (B) equals 1/2 (C) equals 1 (D) is not possible to determine

Q.81 The lines tangent to the curves y3 – x2y + 5y – 2x = 0 and x4 – x3y2 + 5x + 2y = 0 at the origin intersect

at an angle θ equal to(A) π /6 (B) π /4 (C) π /3 (D) π /2

Q.82 The cost function at American Gadget is C(x) = x3 – 6x2 + 15x (x in thousands of units and x > 0)

The production level at which average cost is minimum is

(A) 2 (B) 3 (C) 5 (D) none

Q.83 A rectangle has one side on the positive y-axis and one side on the positive x - axis. The upper right hand

vertex on the curve y =nx

x2 . The maximum area of the rectangle is

(A) e–1 (B) e – ½ (C) 1 (D) e½

Q.84 A particle moves along the curve y = x3/2 in the first quadrant in such a way that its distance from the

origin increases at the rate of 11 units per second. The value of dx/dt when x = 3 is

(A) 4 (B)9

2(C)

3 3

2(D) none

Q.85 Number of solution of the equation 3tanx + x3 = 2 in

π

4

,0 is

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

Q.86 Let f (x) = ax2 – b | x |, where a and b are constants. Then at x = 0, f (x) has

(A) a maxima whenever a > 0, b > 0 (B) a maxima whenever a > 0, b < 0

(C) minima whenever a > 0, b > 0 (D) neither a maxima nor minima whenever a > 0, b < 0

Q.87 Let f (x) = ∫

−x

1

dtt

tn)t(nt

ll (x > 1) then

(A) f (x) has one point of maxima and no point of minima.

(B) f ' (x) has two distinct roots(C) f (x) has one point of minima and no point of maxima

(D) f (x) is monotonic

Q.88 Consider f (x) = | 1 – x | 1 ≤ x ≤ 2 and

g (x) = f (x) + b sin2

πx, 1 < x < 2

then which of the following is correct?

(A) Rolles theorem is applicable to both f, g and b = 3/2

(B) LMVT is not applicable to f and Rolles theorem if applicable to g with b = 1/2

(C) LMVT is applicable to f and Rolles theorem is applicable to g with b = 1

(D) Rolles theorem is not applicable to both f, g for any real b.


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[11] Quest TutorialsN o r t h D e l h i : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest

Q.89 Consider f (x) = ∫

+x

0

dtt

1t and g (x) = f ′ (x) for x

∈ 3,2

1

If P is a point on the curve y = g(x) such that the tangent to this curve at P is parallel to a chord joining

the points

2

1g,

2

1 and (3, g(3) ) of the curve, then the coordinates of the point P

(A) can't be found out (B)

28

65,

4

7(C) (1, 2) (D)

6

5,

2

3

Q.90 The co-ordinates of the point on the curve 9y2 = x3 where the normal to the curve makes equal

intercepts with the axes is

(A)

3

1,1 (B) ( )3,3 (C)

3

8,4 (D)

5

6

5

2,

5

6

Q.91 The angle made by the tangent of the curve x = a (t + sint cost) ; y = a (1 + sint)2 with the x-axis at any

point on it is

(A) ( )t24

1+π (B)

tcos

tsin1−(C) ( )π−t2

4

1(D)

t2cos

tsin1+

Q.92 If f (x) = 1 + x + ( )∫ +x

1

2 dtnt2tn ll , then f (x) increases in

(A) (0, ∞) (B) (0, e–2) ∪ (1, ∞) (C) no value (D) (1, ∞)

Q.93 The function f (x) =)xe(n

)x(n

++π

l

l is :

(A) increasing on [0, ∞) (B) decreasing on [0, ∞)(C) increasing on [0, π /e) & decreasing on [π /e, ∞) (D) decreasing on [0, π /e) & increasing on [π /e,∞)

Directions for Q.94 to Q.96

Suppose you do not know the function f (x), however some information about f (x) is listed below. Read

the following carefully before attempting the questions

(i) f (x) is continuous and defined for all real numbers

(ii) f '(–5) = 0 ; f '(2) is not defined and f '(4) = 0

(iii) (–5, 12) is a point which lies on the graph of f (x)

(iv) f ''(2) is undefined, but f ''(x) is negative everywhere else.

(v) the signs of f '(x) is given below

Q.94 On the possible graph of y = f (x) we have

(A) x = – 5 is a point of relative minima.

(B) x = 2 is a point of relative maxima.

(C) x = 4 is a point of relative minima.

(D) graph of y = f (x) must have a geometrical sharp corner.


14/226


Quest

Q.95 From the possible graph of y = f (x), we can say that

(A) There is exactly one point of inflection on the curve.

(B) f (x) increases on – 5 < x < 2 and x > 4 and decreases on – ∞ < x < – 5 and 2 < x < 4.(C) The curve is always concave down.

(D) Curve always concave up.

Q.96 Possible graph of y = f (x) is

(A) (B)

(C) (D)

Directions for Q.97 to Q.100

Consider the function f (x) =

x

x

11

+ then

Q.97 Domain of f (x) is

(A) (–1, 0) ∪ (0, ∞) (B) R – { 0 } (C) (–∞, –1) ∪ (0, ∞) (D) (0, ∞)

Q.98 Which one of the following limits tends to unity?

(A))x(Lim

x

f ∞→ (B)

)x(Lim0x

f +→ (C)

)x(Lim1x

f −−→ (D)

)x(Limx

f −∞→

Q.99 The function f (x)

(A) has a maxima but no minima (B) has a minima but no maxima

(C) has exactly one maxima and one minima (D) has neither a maxima nor a minima

Q.100 Range of the function f (x) is

(A) (0, ∞) (B) (– ∞, e) (C) (1, ∞) (D) (1, e) ∪ (e, ∞)

Q.101 A cube of ice melts without changing shape at the uniform rate of 4 cm3 /min. The rate of change of the

surface area of the cube, in cm2 /min, when the volume of the cube is 125 cm3, is

(A) – 4 (B) – 16/5 (C) – 16/6 (D) – 8/15

Q.102 Let f (1) = – 2 and f ' (x) ≥ 4.2 for 1 ≤ x ≤ 6. The smallest possible value of f (6), is(A) 9 (B) 12 (C) 15 (D) 19

Q.103 Which of the following six statements are true about the cubic polynomial

P(x) = 2x3 + x2 + 3x – 2?

(i) It has exactly one positive real root.

(ii) It has either one or three negative roots.

(iii) It has a root between 0 and 1.

(iv) It must have exactly two real roots.

(v) It has a negative root between – 2 and –1.

(vi) It has no complex roots.

(A) only (i), (iii) and (vi) (B) only (ii), (iii) and (iv)(C) only (i) and (iii) (D) only (iii), (iv) and (v)


15/226


Quest

Q.104 Given that f (x) is continuously differentiable on a ≤ x≤ b where a < b, f (a) < 0 and f (b) > 0, which of the following are always true?

(i) f (x) is bounded on a ≤ x ≤ b.(ii) The equation f (x) = 0 has at least one solution in a < x < b.

(iii) The maximum and minimum values of f (x) on a ≤ x≤ b occur at points where f ' (c) = 0.(iv) There is at least one point c with a < c < b where f ' (c) > 0.

(v) There is at least one point d with a < d < b where f

' (c) < 0.(A) only (ii) and (iv) are true (B) all but (iii) are true

(C) all but (v) are true (D) only (i), (ii) and (iv) are true

Q.105 Consider the function f (x) = 8x2 – 7x + 5 on the interval [–6, 6]. The value of c that satisfies the

conclusion of the mean value theorem, is

(A) – 7/8 (B) – 4 (C) 7/8 (D) 0

Q.106 Consider the curve represented parametrically by the equation

x = t3 – 4t2 – 3t and

y = 2t2 + 3t – 5 where t ∈ R.If H denotes the number of point on the curve where the tangent is horizontal and V the number of point

where the tangent is vertical then(A) H = 2 and V = 1 (B) H = 1 and V = 2

(C) H = 2 and V = 2 (D) H = 1 and V = 1

Q.107 At the point P(a, an) on the graph of y = xn (n ∈ N) in the first quadrant a normal is drawn. The normal

intersects the y-axis at the point (0, b). If2

1bLim

0a=

→, then n equals

(A) 1 (B) 3 (C) 2 (D) 4

Q.108 Suppose that f is a polynomial of degree 3 and that f ''(x) ≠ 0 at any of the stationary point. Then(A) f has exactly one stationary point. (B) f must have no stationary point.

(C) f must have exactly 2 stationary points. (D) f has either 0 or 2 stationary points.

Q.109 Let f (x) =

0xfor8x

0xforx

2

2

≥+


16/226


Quest

Select the correct alternatives : (More than one are correct)

Q.113 The equation of the tangent to the curvex

a

y

b

n n

+

= 2 (n ∈ N) at the point with abscissa equal to

'a' can be :

(A)x

a

y

b

+

= 2 (B)

x

a

y

b

−

= 2 (C)

x

a

y

b

−

= 0 (D)

x

a

y

b

+

= 0

Q.114 The function y =2 1

2

x

x

−−

(x ≠ 2) :

(A) is its own inverse (B) decreases for all values of x

(C) has a graph entirely above x-axis (D) is bound for all x.

Q.115 Ifx

a

y

b+ = 1 is a tangent to the curve x = Kt, y =

K

t, K > 0 then :

(A) a > 0, b > 0 (B) a > 0, b < 0 (C) a < 0, b > 0 (D) a < 0, b < 0

Q.116 The extremum values of the function f(x) = 14

1

4sin cosx x+ −− , where x ∈ R is :

(A)4

8 2−(B)

2 2

8 2−(C)

2 2

4 2 1+(D)

4 2

8 2+

Q.117 The function f(x) = 1 4

0

−∫ tx

dt is such that :

(A) it is defined on the interval [− 1, 1] (B) it is an increasing function(C) it is an odd function (D) the point (0, 0) is the point of inflection

Q.118 Let g(x) = 2 f (x/2) + f (1 − x) and f ′′ (x) < 0 in 0 ≤ x ≤ 1 then g(x) :

(A) decreases in [0, 2/3) (B) decreases in (2/3, 1]

(C) increases in [0, 2/3) (D) increases in (2/3, 1]

Q.119 The abscissa of the point on the curve xy = a + x, the tangent at which cuts off equal intersects from

the co-ordinate axes is : ( a > 0)

(A)a

2(B) −

a

2(C) a 2 (D) − a 2

Q.120 The functionsin ( )

sin ( )

x a

x b

+

+

has no maxima or minima if :

(A) b − a = n π , n ∈ I (B) b − a = (2n + 1) π , n ∈ I(C) b − a = 2n π , n ∈ I (D) none of these .

Q.121 The co-ordinates of the point P on the graph of the function y = e–|x| where the portion of the tangent

intercepted between the co-ordinate axes has the greatest area, is

(A) 11

,e

(B) −

1

1,

e(C) (e, e–e) (D) none

Q.122 Let f(x) = (x2 − 1)n (x2 + x + 1) then f(x) has local extremum at x = 1 when :(A) n = 2 (B) n = 3 (C) 4 (D) n = 6


17/226


Quest

Q.123 For the function f(x) = x4 (12 ln x − 7)(A) the point (1, − 7) is the point of inflection (B) x = e1/3 is the point of minima(C) the graph is concave downwards in (0, 1) (D) the graph is concave upwards in (1, ∞)

Q.124 The parabola y = x2 + px + q cuts the straight line y = 2x − 3 at a point with abscissa 1. If the distancebetween the vertex of the parabola and the x − axis is least then :

(A) p = 0 & q = −

2(B) p = − 2 & q = 0(C) least distance between the parabola and x − axis is 2(D) least distance between the parabola and x − axis is 1

Q.125 The co-ordinates of the point(s) on the graph of the function, f(x) =x x3 2

3

5

2− + 7x - 4 where the

tangent drawn cut off intercepts from the co-ordinate axes which are equal in magnitude but opposite in

sign, is

(A) (2, 8/3) (B) (3, 7/2) (C) (1, 5/6) (D) none

Q.126 On which of the following intervals, the function x100 + sin x − 1 is strictly increasing.(A) (− 1, 1) (B) (0, 1) (C) (π /2, π) (D) (0, π /2)

Q.127 Let f(x) = 8x3 – 6x2 – 2x + 1, then

(A) f(x) = 0 has no root in (0,1) (B) f(x) = 0 has at least one root in (0,1)

(C) f ′(c) vanishes for some )1,0(c∈ (D) none

Q.128 Equation of a tangent to the curve y cot x = y3 tan x at the point where the abscissa isπ4

is :

(A) 4x + 2y = π + 2 (B) 4x − 2y = π + 2 (C) x = 0 (D) y = 0

Q.129 Let h (x) = f

(x) − {f (x)}2 + {f

(x)}3 for every real number '

x

' , then :

(A) ' h ' is increasing whenever ' f ' is increasing(B) '

h

' is increasing whenever '

f ' is decreasing

(C) ' h

' is decreasing whenever '

f ' is decreasing

(D) nothing can be said in general.

Q.130 If the side of a triangle vary slightly in such a way that its circum radius remains constant, then,

da

A

d b

B

d c

Ccos cos cos+ + is equal to:

(A) 6 R (B) 2

R (C) 0 (D) 2R(dA + dB + dC)

Q.131 In which of the following graphs x = c is the point of inflection .

(A) (B) (C) (D)

Q.132 An extremum value of y =0

x

∫ (t − 1) (t − 2) dt is :

(A) 5/6 (B) 2/3 (C) 1 (D) 2


18/226

Q . 1 B Q . 2 D Q . 3 B Q . 4 B Q . 5 D Q . 6 C Q . 7 D

Q . 8 A Q . 9 D Q . 1 0 C Q . 1 1 B Q . 1 2 A Q . 1 3 D Q . 1 4 D

Q . 1 5 C Q . 1 6 D Q . 1 7 A Q . 1 8 D Q . 1 9 A Q . 2 0 A Q . 2 1 C

Q . 2 2 A Q . 2 3 C Q . 2 4 D Q . 2 5 D Q . 2 6 A Q . 2 7 B Q . 2 8 C

Q . 2 9 B Q . 3 0 C Q . 3 1 C Q . 3 2 C Q . 3 3 D Q . 3 4 D Q . 3 5 D

Q . 3 6 B Q . 3 7 A Q . 3 8 A Q . 3 9 D Q . 4 0 B Q . 4 1 A Q . 4 2 C

Q . 4 3 C Q . 4 4 C Q . 4 5 B Q . 4 6 C Q . 4 7 C Q . 4 8 B Q . 4 9 B

Q . 5 0 C Q . 5 1 A Q . 5 2 B Q . 5 3 C Q . 5 4 C Q . 5 5 C Q . 5 6 D

Q . 5 7 D Q . 5 8 C Q . 5 9 C Q . 6 0 C Q . 6 1 B Q . 6 2 A Q . 6 3 D

Q . 6 4 C Q . 6 5 A Q . 6 6 C Q . 6 7 D Q . 6 8 B Q . 6 9 D Q . 7 0 C

Q . 7 1 B Q . 7 2 A Q . 7 3 A Q . 7 4 B Q . 7 5 B Q . 7 6 A Q . 7 7 D

Q . 7 8 D Q . 7 9 A Q . 8 0 A Q . 8 1 D Q . 8 2 B Q . 8 3 A Q . 8 4 A

Q . 8 5 B Q . 8 6 A Q . 8 7 D Q . 8 8 C Q . 8 9 D Q . 9 0 C Q . 9 1 A

Q . 9 2 A Q . 9 3 B Q . 9 4 D Q . 9 5 C Q . 9 6 C Q . 9 7 C Q . 9 8 B

Q . 9 9 D Q . 1 0 0 D Q . 1 0 1 B Q . 1 0 2 D Q . 1 0 3 C Q . 1 0 4 D Q . 1 0 5 D

Q . 1 0 6 B Q . 1 0 7 C Q . 1 0 8 D Q . 1 0 9 B Q . 1 1 0 B Q . 1 1 1 C Q . 1 1 2 D

Q . 1 1 3 A , B Q . 1 1 4 A , B Q . 1 1 5 A , D Q . 1 1 6 A , C Q . 1 1 7 A , B , C , D Q . 1 1 8 B , C

Q . 1 1 9 A , B Q . 1 2 0 A , B , C Q . 1 2 1 A , B Q . 1 2 2 A , C , D Q . 1 2 3 A , B , C , D Q . 1 2 4 B , D

Q . 1 2 5 A , B Q . 1 2 6 B , C , D Q . 1 2 7 B , C Q . 1 2 8 A , B , D Q . 1 2 9 A , C Q . 1 3 0 C , D

Q . 1 3 1 A , B , D Q . 1 3 2 A , B

ANSWER KEY


19/226

DEFINITE & INDEFINITE

INTEGRATION

MATHEMATICS

TARGET IIT JEE


20/226


Quest

Question bank on Definite & Indefinite Integration

There are 168 questions in this question bank.

Select the correct alternative : (Only one is correct)

Q.1 The value of the definite integral, ∫∞

−−+ +1

1x31x dx)ee( is

(A) 2e4π (B)

e4π (C)

−π −

e1tan

2e1 12 (D) 2e2

π

Q.2 The value of the definite integral, dxex2·ecos22 x

2n

0

x∫

π

l

is

(A) 1 (B) 1 + (sin 1) (C) 1 – (sin 1) (D) (sin 1) – 1

Q.3 Value of the definite integral ∫−−−

−−−

21

21

3131

dx))x3x4(cos)x4x3(sin(

(A) 0 (B)2

π− (C)

2

7π(D)

2

π

Q.4 Let f (x) = ∫+

x

24t1

dt and g be the inverse of f. Then the value of g'(0) is

(A) 1 (B) 17 (C) 17 (D) none of these

Q.5 ∫−

x

x1

e

)e(cot dx is equal to :

(A)2

1 ln (e2x + 1) − x

x1

e

)e(cot−

+ x + c (B)2

1 ln (e2x + 1) + x

x1

e

)e(cot−

+ x + c

(C)2

1 ln (e2x + 1) − x

x1

e

)e(cot−

− x + c (D)2

1 ln (e2x + 1) + x

x1

e

)e(cot−

− x + c

Q.6 ∫ +→k

0

x

1

0k dx)x2sin1(

k 1Lim

(A) 2 (B) 1 (C) e2 (D) non existent

Q.7 ∫ +−5n

0x

xx

3e

1eel

dx =

(A) 4 − π (B) 6 − π (C) 5 − π (D) None


21/226


Quest

Q.8 If x satisfies the equation2

1

02

x1cost2t

dt

+α+∫ – xdt

1t

t2sint3

32

2

+∫− – 2 = 0 (0 < α < π), then the

value x is

(A) ±α

αsin2

(B) ±α

αsin2(C) ±

αα

sin(D) ±

ααsin

2

Q.9 If f (x) = eg(x) and g(x) =2

x

∫t d t

t1 4+ then f ′ (2) has the value equal to :

(A) 2/17 (B) 0 (C) 1 (D) cannot be determined

Q.10 ∫ etan θ (sec θ – sin θ) dθ equals :(A) − etan θ sin θ + c (B) etan θ sin θ + c (C) etan θ sec θ + c (D) etan θ cos θ + c

Q.11 ∫π

0

(x · sin2x · cos x) dx =

(A) 0 (B) 2/9 (C) − 2/9 (D) − 4/9

Q.12 The value of( )

∑=

=∞→ +

n4r

1r2n n4r3r

nLim is equal to

(A)35

1(B)

14

1(C)

10

1(D)

5

1

Q.13 ∫−

−

+cb

ca

dx)cx( f =

(A) ∫b

a

dx)x(f (B) ∫ +b

a

dx)cx(f (C) ∫−

−

c2b

c2a

dx)x(f (D) ∫ +b

a

dx)c2x(f

Q.14 Let I1 = ∫

π

+−2 /

0

dxxcos.xsin1

xcosxsin; I

2 = ∫

π2

0

6 dx)xcos( ; I3 = ∫

π

π−

2 /

2 /

3 dx)x(sin & I4 = ∫

−1

0

dx1x

1nl then

(A) I1 = I

2 = I

3 = I

4 = 0 (B) I

1 = I

2 = I

3 = 0 but I

4 ≠ 0

(C) I1 = I

3 = I

4 = 0 but I

2 ≠ 0 (D) I

1 = I

2 = I

4 = 0 but I

3 ≠ 0

Q.15 ∫ +− )x1(xx1

7

7

dx equals :

(A) ln x +7

2 ln (1 + x7) + c (B) ln x −

7

2 ln (1 − x7) + c

(C) ln x − 7

2 ln (1 + x7) + c (D) ln x +

7

2 ln (1 − x7) + c

Q.16 ∫π

+

n2 /

0n nxtan1

dx =

(A) 0 ( B )n4π (C) 4

nπ (D) n2π


22/226


Quest

Q.17 f (x) = ∫ −−x

0

dt)2t()1t(t takes on its minimum value when:

(A) x = 0 , 1 (B) x = 1 , 2 (C) x = 0 , 2 (D) x =3 3

3

+

Q.18 ∫−

a

adx)x(f =

(A) [ ]∫ −+a

0

dx)x(f )x(f (B) [ ]∫ −−a

0

dx)x(f )x(f (C) 2 ∫a

0

dx)x(f (D) Zero

Q.19 Let f (x) be a function satisfying f ' (x) = f (x) with f (0) = 1 and g be the function satisfying f (x) + g (x) = x2.

The value of the integral ∫1

0

dx)x(g)x(f is

(A) e –21 e2 –

25 (B) e – e2 – 3 (C)

21 (e – 3) (D) e –

21 e2 –

23

Q.20 ∫ + |x|n1x|x|n

l

l dx equals :

(A) |x|n13

2l+ (lnx − 2) + c (B) |x|n1

3

2l+ (lnx + 2) + c

(C) |x|n13

1l+ (lnx − 2) + c (D) |x|n12 l+ (3 lnx − 2) + c

Q.21 ( )∫

−−+−

213

2

1

dx4|x1||3x|2

1equals:

(A)2

3− (B)

8

9(C)

4

1(D)

2

3

Where {*} denotes the fractional part function.

Q.22 ∫π / 4

0

−

x

1cos.x

x

1sin.x3 2 dx has the value :

(A)8 2

3π(B)

24 23π

(C)32 2

3π(D) None

Q.23

+

π−++

π+

ππ∞→ 3

4

n6)1n(sec.....

n6·2sec

n6sec

n6Lim 222

n has the value equal to

(A)3

3(B) 3 (C) 2 (D) 3

2


23/226


Quest

Q.24 Suppose that F (x) is an antiderivative of f (x) =x

xsin, x > 0 then ∫

3

1

dxx

x2sin can be expressed as

(A) F (6) – F (2) (B)2

1( F (6) – F (2) ) (C)

2

1( F (3) – F (1) ) (D) 2( F (6) – F (2) )

Q.25 Primitive of 24

4

)1xx(1x3

++− w.r.t. x is :

(A)1xx

x4 ++

+ c (B) − 1xx

x4 ++

+ c (C)1xx

1x4 ++

+ + c (D) −

1xx

1x4 ++

+ + c

Q.26∞→n

Lim π π π π2

12

2

2

1

2n n n

n

n+ + + +

−

cos cos ..... cos

( ) equal to

(A) 1 (B)1

2(C) 2 (D) none

Q.27( )

loglog

x

x

n2

2

2

2

2

4

−

∫

dx =

(A) 0 (B) 1 (C) 2 (D) 4

Q.28 If m & n are integers such that (m − n) is an odd integer then the value of the definite integral

∫π

0

dxnx·sinmxcos =

(A) 0 (B) 22 mnn2

−(C)

22 mnm2−

(D) none

Q.29 Let y={x}[x] where {x}denotes the fractional part of x & [x] denotes greatest integer ≤ x, then ∫3

0

dxy =

(A) 5/6 (B) 2/3 (C) 1 (D) 11/6

Q.30 If

( )

∫+

+22

4

1xx

1x dx = A ln x +

2x1

B

+ + c , where c is the constant of integration then :

(A) A = 1 ; B = − 1 (B) A = − 1 ; B = 1 (C) A = 1 ; B = 1 (D) A = − 1 ; B = − 1

Q.31 ∫π

π −−

2 / xcos1

xsin1 dx =

(A) 1 − ln 2 (B) ln 2 (C) 1 + ln 2 (D) none

Q.32 Let f : R → R be a differentiable function & f (1) = 4 , then the value of ;1x

Lim→ ∫ −

)x(f

41x

dtt2 is :

(A) f ′ (1) (B) 4 f ′ (1) (C) 2 f ′ (1) (D) 8 f ′ (1)


24/226


Quest

Q.33 If ∫)x(f

0

2 dtt = x cos πx , then f ' (9)

(A) is equal to –9

1(B) is equal to –

3

1(C) is equal to

3

1(D) is non existent

Q.34 ∫π 3 / 1)2 / (

0

35 dxx·sinx =

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

Q.35 Integral of )xeccosx(cotxcot21 ++ w.r.t. x is :

(A) 2 ln cos2

x + c (B) 2 ln sin

2

x + c

(C) 2

1 ln cos 2

x + c (D) ln sin x − ln(cosec x − cot x) + c

Q.36 If f (x) = x + x − 1 + x − 2 , x ∈ R then ∫3

0

dx)x( f =

(A) 9/2 (B) 15/2 (C) 19/2 (D) none

Q.37 Number of values of x satisfying the equation ∫−

++x

1

2 dt4t3

28t8 =

( )1xlog

1x

)1x(

23

+

+

+, is

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

Q.38 ∫−1

0

1

dxx

xtan=

(A) ∫π 4 /

0

dxx

xsin(B) ∫

π 2 /

0

dxxsin

x(C) ∫

π 2 /

0

dxxsin

x

2

1(D) ∫

π 4 /

0

dxxsin

x

2

1

Q.39 Domain of definition of the function f (x) = ∫+

x

022 tx

dt is

(A) R (B) R+ (C) R+ ∪ {0} (D) R – {0}

Q.40 If ∫ e3x cos 4x dx = e3x (A sin 4x + B cos 4x) + c then :(A) 4A = 3B (B) 2A = 3B (C) 3A = 4B (D) 4B + 3A = 1

Q.41 If f (a + b − x) = f (x) , then ∫ −+b

a

dx)xba(f .x =

(A) 0 (B)2

1(C) ∫

+ b

a

dx)x(2

ba f (D) ∫

− b

a

dx)x(2

ba f


25/226


Quest

Q.42 The set of values of 'a' which satisfy the equation ∫ −2

0

2dt)alogt( = log2

2a

4 is

( A ) a ∈ R (B) a ∈ R+ (C) a < 2 (D) a > 2

Q.43 The value of the definite integral dx)5x4(5x2)5x4(5x2

3

2∫ −++−− =

(A)7 3 3 5

3 2

+(B) 4 2 (C) 4 3 +

4

3(D)

7 7 2 5

3 2

−

Q.44 Number of ordered pair(s) of (a, b) satisfying simultaneously the system of equation

0dxx

b

a

3 =∫ and 32

dxx

b

a

2 =∫ is

(A) 0 (B) 1 (C) 2 (D) 4

Q.45 ∫ −−−−

+−

xcotxtan

xcotxtan11

11

dx is equal to :

(A)π4

x tan−1 x +π2

ln (1 + x2) − x + c (B)π4

x tan−1 x − π2

ln (1 + x2) + x + c

(C)π4

x tan−1 x +π2

ln (1 + x2) + x + c (D)π4

x tan−1 x − π2

ln (1 + x2) − x + c

Q.46 Variable x and y are related by equation x = ∫+

y

0

2

t1

dt. The value of 2

2

dx

yd is equal to

(A) 2y1

y

+ (B) y (C) 2y1

y2

+(D) 4y

Q.47 Let f (x) = ∫+

→ ++

hx

x20h t1t

dt

h

1Lim , then )x(·xLim

x f

∞−→ is

(A) equal to 0 (B) equal to2

1(C) equal to 1 (D) non existent

Q.48 If the primitive of f (x) = π sin πx + 2x − 4, has the value 3 for x = 1, then the set of x for which theprimitive of f (x) vanishes is :

(A) {1, 2, 3} (B) (2, 3) (C) {2} (D) {1, 2, 3, 4}

Q.49 If f & g are continuous functions in [0, a] satisfying f (x) = f (a − x) & g (x) + g (a − x) = 4 then

∫a

0

dx)x(g.)x(f =

(A)

∫

a

02

1f (x)dx (B)

∫

a

0

2 f (x)dx (C)

∫

a

0

f (x)dx (D)

∫

a

0

4 f (x)dx


26/226


Quest

Q.50 ∫ x . 2

2

x1

x1xn

+

++l

dx equals :

(A) 2x1+ ln

++ 2x1x − x + c (B)

2

x . ln2

++ 2x1x −

2

x1

x

+

+ c

(C)2

x . ln2

++ 2x1x + 2x1

x

+ + c (D)2x1+ ln

++ 2x1x + x + c

Q.51 If f (x) =

2x1)6x7(

1x0x1

31 ≤


27/226


Quest

Q.58 Let f be a continuous functions satisfying f ' (ln x) =

1xx for

1x0 for1

>

≤< and f (0) = 0 then f (x) can be

defined as

(A) f (x) =

0 if xe1

0if x 1

x

>−

≤ (B) f (x) =

0 if x1e

0if x 1

x

>−

≤

(C) f (x) =

0if x e

0if x x

x >

−

≤

Q.59 Let f : R → R be a differentiable function such that f (2) = 2. Then the value of Limitx → 2

4

2

3

2

t

x

f x

−∫( )

dt is

(A) 6 f ′ (2) (B) 12 f ′ (2) (C) 32 f ′ (2) (D) none

Q.60 ∫π

+

2 /

022 xsina1

dx has the value :

(A) π

2 1 2+a(B)

π

12+ a

(C)2

1 2

π

+ a(D) none

Q.61 Let f (x) =

xe

xn

x

1l then its primitive w.r.t. x is

(A)2

1ex – ln x + C (B)

2

1ln x – ex + C (C)

2

1ln2x – x + C (D)

x2

ex + C

Q.62 ∑=∞→ +

n

1k 222n xk n

nLim , x > 0 is equal to

(A) x tan–1(x) (B) tan–1(x) (C)x

)x(tan 1−(D) 2

1

x

)x(tan−

Q.63 Let f (x) =

2 2

2 2

0

2

2

cos sin( ) sin

sin sin cos

sin cos

x x x

x x x

x x

−

−

then0

2π /

∫ [f (x) + f ′ (x)] dx =

(A) π (B) π /2 (C) 2 π (D) zero

Q.64 The absolute value of sinx

x18

10

19

+∫ is less than :

(A) 10 −10 (B) 10 −11 (C) 10 −7 (D) 10 −9

Q.65 The value of the integral−∫π

π

(cos px − sin qx)2 dx where p, q are integers, is equal to :

(A) − π (B) 0 (C) π (D) 2 π


28/226


Quest

Q.66 Primitive of f (x) =)1x(n

2

2·x +l w.r.t. x is

(A))1x(2

22

)1x(n2

+

+l

+ C (B)12n

2)1x( )1x(n22

++ +

l

l

+ C

(C) )12n(2

)1x( 12n2

++ +

l

l

+ C (D) )12n(2

)1x( 2n2

+

+l

l

+ C

Q.67 ∫

++

∞→

2

0

n

ndt

1n

t1Lim is equal to

(A) 0 (B) e2 (C) e2 – 1 (D) does not exist

Q.68 Limith → 0

n t dt n t dt

h

a

x h

a

x2 2

+

∫ ∫− =

(A) 0 (B) ln2x (C)2nx

x(D) does not exist

Q.69 Let a, b, c be non−zero real numbers such that ;

0

1

∫ (1 + cos8x) (ax2 + bx + c) dx =0

2

∫ (1 + cos8x) (ax2 + bx + c) dx , then the quadratic equation

ax2 + bx + c = 0 has :

(A) no root in (0, 2) (B) atleast one root in (0, 2)

(C) a double root in (0, 2) (D) none

Q.70 Let In =

0

4π /

∫ tann x dx , then1 1 1

2 4 3 5 4 6I I I I I I+ + +, , ,.... are in :

(A) A.P. (B) G.P. (C) H.P. (D) none

Q.71 Let g (x) be an antiderivative for f (x). Then ln ( )2)x(g1+ is an antiderivative for

(A) ( )2)x(1

)x()x(2

f

g f

+ (B) ( )2)x(1)x()x(2

g

g f

+ (C) ( )2)x(1)x(2

f

f

+ (D) none

Q.720

4π /

∫ (cos 2x)3/2. cos x dx =

(A)3

16

π(B)

3

32

π(C)

3

16 2

π(D)

3 2

16

π

Q.73 The value of the definite integral ∫−+−

21

022

2

)x11(x1

dxx is

(A) 4

π(B) 2

1

4 +π

(C) 2

1

4 −π

(D) none


29/226


Quest

Q.74 The value of the definite integral ( )∫ π+37

19

2 dx)x2(sin3}x{ where { x } denotes the fractional part function.

(A) 0 (B) 6 (C) 9 (D) can not be determined

Q.75 The value of the definite integral ∫π 2

0

dxxtan , is

(A) π2 (B)2

π(C) π22 (D)

22

π

Q.76 Evaluate the integral : ∫ dxx)x6(n 2l

(A)32 )]x6(n[

8

1l + C (B) )]x6(n[

4

1 22l + C

(C) )]x6(n[2

1 2l + C (D)42 )]x6(n[

16

1l + C

Q.77 ∫π

π

θ

θ+−θ65

6

22 d)sin1(2

1)sin3(

2

1

(A) π – 3 (B) π (C) π – 32 (D) π + 3

Q.78 Let l = ∫∞→x2

xx t

dtLim and m =

∞→xLim ∫

x

1

dttnxnx

1l

l then the correct statement is

(A) l m = l (B) l m = m (C) l = m (D) l > m

Q.79 If f (x) = e–x + 2 e–2x + 3 e– 3x +...... + ∞ , then ∫3n

2n

dx)x(f

l

l

=

(A) 1 (B)2

1(C)

3

1(D) ln 2

Q.80 If I = n x(sin ) /

0

2π

∫ dx then n x x(sin cos ) /

/

+−∫π

π

4

4

dx =

(A) I2

(B) I4

(C)I

2(D) I

Q.81 The value of ∫ ∑∏

+

+

==

1

0

n

1k

n

1r

dxk x

1)rx( equals

(A) n (B) n ! (C) (n + 1) ! (D) n · n !

Q.82 ∫ ++

xsinxsin

xcosxcos42

53

dx

(A) sin x − 6 tan−1 (sin x) + c (B) sin x − 2 sin−1 x + c

(C) sin x − 2 (sin x)−1 − 6 tan−1 (sin x) + c (D) sin x − 2 (sin x)−1 + 5 tan−1 (sin x) + c


30/226


Quest

Q.830

3

∫1

4 44 4

2

2

x xx x

+ ++ − +

dx =

(A) ln 5

2

3

2− (B) ln

5

2

3

2+ (C) ln

5

2

5

2+ (D) none

Q.84 The value of the function f (x) = 1 + x +1

x

∫ (ln2t + 2 lnt) dt where f ′ (x) vanishes is :

(A) e−1 (B) 0 (C) 2 e−1 (D) 1 + 2 e−1

Q.85 Limitn → ∞1

11 2 3 3 1n

n

n

n

n

n

n

n

n n+

+ +

+ +

+ + +

+ −

.......( )

has the value equal to

(A) 2 2 (B) 2 2 − 1 (C) 2 (D) 4

Q.86 Let a function h(x) be defined as h(x) = 0, for all x ≠ 0. Also ∫

∞

∞− dx)x(·)x( f h = f (0), for every

function f (x). Then the value of the definite integral ∫∞

∞−

dxxsin·)x('h , is

(A) equal to zero (B) equal to 1 (C) equal to – 1 (D) non existent

Q.870

4π /

∫ (tann x + tann − 2x)d(x − [x]) is : ( [. ] denotes greatest integer function)

(A)1

1n −(B)

1

2n +(C)

2

1n −(D) none of these

Q.88

λλ

→λ

+∫

11

00

dx)x1(Lim is equal to

(A) 2 ln 2 (B)e

4(C) ln

e

4(D) 4

Q.89 Which one of the following is TRUE.

(A) C|x|nxx

dx.x +=

∫l (B) Cx|x|nx

x

dx.x +=

∫l

(C) Cxtandxxcos.xcos

1+=∫ (D) Cxdxxcos.xcos

1+=∫


31/226


Quest

Q.900

∞

∫ x2n + 1· e x−2

dx is equal to (n ∈ N).

(A) n ! (B) 2 (n !) (C)n !

2(D)

2

)!1n( +

Q.91 The true set of values of 'a' for which the inequality ∫

0

a

(3 −2x − 2. 3−x) dx ≥ 0 is true is:

(A) [0 , 1] (B) (− ∞ , − 1] (C) [0, ∞) (D) (− ∞ , − 1] ∪ [0, ∞)

Q.92 If α ∈ (2 , 3) then number of solution of the equation ∫α

0

cos (x + α2) dx = sin α is :

(A) 1 (B) 2 (C) 3 (D) 4.

Q.93 If x · sin πx = ∫

2x

0dt)t(f where f is continuous functions then the value of f (4) is

(A)2

π(B) 1 (C)

2

1(D) can not be determined

Q.94 ∫ +++

dx)1x4x(

)1x2(2 / 32

(A)C

)1x4x(

x

2 / 12

3

+++ (B)

C

)1x4x(

x

2 / 12

+++

(C) C)1x4x(

x2 / 12

2

+++ (D)

C)1x4x(

12 / 12

+++

Q.95 If the value of the integral ex2

1

2

∫ dx is α , then the value of nxe

e4

∫ dx is :

(A) e4 − e − α (B) 2 e4 − e − α (C) 2 (e4 − e) − α (D) 2 e4 – 1 – α

Q.96

−−∫ 2x1x2

1tandx

d

2

13

0

equals

(A)3

π(B)

6

π− (C)

2

π(D)

4

π

Q.97 Let A =0

1

∫e d t

t

t

1 + then ∫

−

−

−−

a

1a

t

1at

dte has the value

(A) Ae−a (B) − Ae−a (C) − ae−a (D) Aea


32/226


Quest

Q.98 sin /

20

2

θπ

∫ sin θ dθ is equal to :

(A) 0 (B) π /4 (C) π /2 (D) π

Q.99 dx4x

2x

4

2

∫ ++

is equal to

(A) Cx2

2xtan

2

1 21 ++−

(B) C)2x(tan2

1 21 ++−

(C) C2x

x2tan

2

12

1 +−

−(D) C

x2

2xtan

2

1 21 +−−

Q.100 Ifβ + 2 x e x20

12−

∫ dx = ex−

∫2

0

1

dx then the value of β is

(A) e−1

(B) e (C) 1/2e (D) can not be determined

Q.101 A quadratic polynomial P(x) satisfies the conditions, P(0) = P(1) = 0 &0

1

∫ P(x) dx = 1. The leading

coefficient of the quadratic polynomial is :

(A) 6 (B) − 6 (C) 2 (D) 3

Q.102 Which one of the following functions is not continuous on (0,π)?

(A) f(x)= cotx (B) g(x) = ∫

x

0

dtt

1sint

(C) h (x) =

π


33/226


Quest

Q.104 Consider f(x) =x

x

2

31+; g(t) = f t dt( )∫ . If g(1) = 0 then g(x) equals

(A)1

31 3n x( )+ (B)

1

3

1

2

3

nx+

(C)

1

2

1

3

3

nx+

(D)

1

3

1

3

3

nx+

Q.105 The value of the definite integral, ∫100

0x

dxe

x2 is equal to

( A )

2

1(1 – e–10) (B) 2(1 – e–10) (C)

2

1(e–10 – 1) (D)

2

1

− −

410e1

Q.1060

∞

∫ [2 e−x] dx where [x] denotes the greatest integer function is

(A) 0 (B) ln 2 (C) e2 (D) 2/e

Q.107 The value of ∫−

1

1|x|

dx is

(A)2

1(B) 2 (C) 4 (D) undefined

Q.108 x nx

dxl 12

0

1

+

∫ =

(A)3

41 2

3

2−

ln (B)

3

2

7

2

3

2− ln (C)

3

4

1

2

1

54+ ln (D)

1

2

27

2

3

4ln −

Q.109 The evaluation ofp x q x

x xdx

p q q

p q p q

+ − −

+ +

−+ +z

2 1 1

2 2 2 1 is

(A) –x

xC

p

p q+ + +1

(B)x

xC

q

p q+ + +1

(C) −+

++x

xC

q

p q 1(D)

x

xC

p

p q+ + +1

Q.110x x

x x

3

21

11

2 1

+ ++ +−

∫| |

| | dx = a ln 2 + b then :

(A) a = 2 ; b = 1 (B) a = 2 ; b = 0 (C) a = 3 ; b = − 2 (D) a = 4 ; b = − 1

Q.111a

b

∫ [x] dx +a

b

∫ [ − x] dx where [. ] denotes greatest integer function is equal to :

(A) a + b (B) b − a (C) a − b (D)a b+

2

Q.112 If0

2

∫ 375 x5 (1 + x2) −4 dx = 2n then the value of n is :

(A) 4 (B) 5 (C) 6 (D) 7


34/226


Quest

Q.113 ∫ −+

−

2 / 1

02 x1

x1n

x1

1 dx is equal to :

(A)3

1n

4

1 2 (B)

2

1 ln2 3 (C) −

4

1 ln2 3 (D) cannot be evaluated.

Q.114 If ∫ +− dxe)5x2x(x323

= e3x (Ax3 + Bx2 + Cx + D) then the statement which is incorrect is

(A) C + 3D = 5 (B) A + B + 2/3 = 0

(C) C + 2B = 0 (D) A + B + C = 0

Q.115 Given ∫π

++

2 /

0xcosxsin1

dx = ln 2, then the value of the def. integral. ∫

π

++

2 /

0xcosxsin1

xsindx is equal to

(A)2

1ln 2 (B)

2

π − ln 2 (C)

4

π –

2

1ln 2 (D)

2

π + ln 2

Q.116 A function f satisfying f ′ (sin x) = cos2 x for all x and f(1) = 1 is :

(A) f(x) = x +3

1

3

x3− (B) f(x) =

3

2

3

x3+

(C) f(x) = x −3

1

3

x3+ (D) f(x) = x −

3

1

3

x3+

Q.117 For 0 < x <π2

,1 2

3 2

/

/

∫ ln (ecos x). d (sin x) is equal to :

(A)12

π(B)

6

π

(C) ( ) ( )[ ]1sin3sin134

1−+− (D) ( ) ( )[ ]1sin3sin13

4

1−−−

Q.118( )∫

π

+02

xsin1

xcosxdx is equal to :

(A) π − 2 (B) − (2 + π) (C) zero (D) 2 − π

Q.119 ∫ ( ) dxxxx

e x+

(A) 2 [ ]1xxe x +− + C (B) [ ]1x2xe x +−(C) ( ) Cxxe x ++ (D) ( ) C1xxe x +++

Q.120dx

x xcos sin

/

6 60

2

+∫π

is equal to :

(A) zero (B) π (C) π /2 (D) 2 π


35/226


Quest

Q.121 The true solution set of the inequality,

π+−− ∫

x

0

2dz

2x6x5 > ∫

π

0

2dxxsinx is :

(A) R (B) ( 1, 6) ( C ) ( − 6, 1) (D) (2, 3)

Q.122 If∫

−

1

0 2x1

xn

dx = k 0

π

∫ ln (1 + cos

x) dx then the value of k is :

(A) 2 (B) 1/2 (C) − 2 (D) − 1/2

Q.123 Let a, b and c be positive constants. The value of 'a' in terms of 'c' if the value of integral

∫ ++ +

1

0

5b331b dx)bxaacx( is independent of b equals

(A)2

c3(B)

3

c2(C)

3

c(D)

c2

3

Q.124 θθ+θθ∫ d)tan(secsec22

(A) C)]tan(sectan2[2

)tan(sec+θ+θθ+

θ+θ

(B) C)]tan(sectan42[3

)tan(sec+θ+θθ+

θ+θ

(C) C)]tan(sectan2[3

)tan(sec+θ+θθ+

θ+θ

(D) C)]tan(sectan2[2

)tan(sec3 +θ+θθ+θ+θ

Q.125 ∫ ++2

14

2

1x

1x dx is equal to:

(A)1

2 tan−1 2 (B)

1

2 cot−12 (C)

1

2 tan−1

1

2(D)

1

2 tan−1 2

Q.126 1xxLimit

→ 1xx

x

− ∫

x

x1f(t) dt is equal to :

(A)( )f x

x

1

1

(B) x1 f (x

1) (C) f (x

1) (D) does not exist

Q.127 Which of the following statements could be true if, f ′′ (x) = x1/3.

I II III IV

f (x) =

28

9 x7/3 + 9 f ′ (x) =28

9 x7/3 − 2 f ′ (x) =4

3 x4/3 + 6 f (x) =

4

3 x4/3 − 4

(A) I only (B) III only (C) II & IV only (D) I & III only


36/226


Quest

Q.128 The value of the definite integral0

2π /

∫ sin x sin 2x sin 3x dx is equal to :

(A)1

3(B) −

2

3(C) −

1

3(D)

1

6

Q.129 dxx1

x1cosx1sec

)x1(

e2

21

221

2

x1tan

∫

+−

+

+

+−−

−

(x > 0)

(A) Cxtan.e 1x1tan +−

−(B)

( )C

2

xtan.e21x1tan

+−−

(C) Cx1sec.e2

21x1tan +

+−

−(D) Cx1eccos.e

221x1tan +

+−

−

Q.130 Number of positive solution of the equation, { }( )t tx

−∫2

0

dt = 2 (x − 1) where { } denotes the fractional

part function is :

(A) one (B) two (C) three (D) more than three

Q.131 If f (x) = cos(tan–1x) then the value of the integral dx)x(''f x

1

0

∫ is

(A)2

23−(B)

2

23+(C) 1 (D)

22

31−

Q.132 If ∫ + 2x

sin1 dx = A sin

π−44

x then value of A is:

(A) 2 2 (B) 2 (C)1

2(D) 4 2

Q.133 For Un =

0

1

∫ xn (2 − x)n dx; Vn =0

1

∫ xn (1 − x)n dx n ∈ N, which of the following statement(s)

is/are ture?(A) U

n = 2n V

n(B) U

n = 2 −n V

n(C) U

n = 22n V

n(D) U

n = 2 − 2n V

n

Q.134

+++

−

−x

1xtan)1x3x(

dx)1x(

2124

2

= ln | f (x) | + C then f (x) is

(A) ln

+x

1x (B) tan–1

+x

1x (C) cot–1

+x

1x (D) ln

+−x

1xtan 1


37/226


Quest

Q.135 Let f (x) be integrable over (a, b) , b > a > 0. If I

1 =

π

π

/

/

6

3

∫ f (tan θ + cot θ). sec2 θ d θ &

I2 =

π

π

/

/

6

3

∫ f (tan θ + cot θ). cosec2 θ d θ , then the ratioI

I

1

2

:

(A) is a positive integer (B) is a negative integer

(C) is an irrational number (D) cannot be determined.

Q.136 f (x) =

cos

sin

x

x

∫ (1 − t + 2 t3) d

t has in [

0, 2

π

]

(A) a maximum atπ4

& a minimum at3

4

π(B) a maximum at

3

4

π& a minimum at

7

4

π

(C) a maximum at5

4

π & a minimum at

7

4

π(D) neither a maxima nor minima

Q.137 Let S (x) =x

x

2

3

∫ l n t d t (x > 0) and H (x) = ′S x

x

( ). Then H(x) is :

(A) continuous but not derivable in its domain

(B) derivable and continuous in its domain

(C) neither derivable nor continuous in its domain

(D) derivable but not continuous in its domain.

Q.138 Number of solution of the equationd

dx ∫

xsin

xcos

dt

t1 2− = 2 2 in [0, π] is

(A) 4 (B) 3 (C) 2 (D) 0

Q.139 Let f (x) =xcos

1xsin22 −

+xsin1

)1xsin2(xcos

++

then

( )∫ + dx)x('f )x(f ex

(where c is the constant of integeration)

(A) ex tanx + c (B) excotx + c (C) ex cosec2x + c (D) exsec2x + c

Q.140 The value of x that maximises the value of the integral t t dtx

x

( )5

3

−+

∫ is

(A) 2 (B) 0 (C) 1 (D) none

Q.141 For a sufficiently large value of n the sum of the square roots of the first n positive integers

i.e. 1 2 3+ + + +... .. .. .. ... .. .. .. .. .. n is approximately equal to

(A)1

3

3 2n / (B)2

3

3 2n / (C)1

3

1 3n / (D)2

3

1 3n /

Q.142 The value of ∫ −

2

0

2)x1(

dx is

(A) –2 (B) 0 (C) 15 (D) indeterminate


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Quest

Q.143 If ∫∫π

θθθ

=++

8 /

0

a

0

d2sin

tan2

xax

dx, then the value of 'a' is equal to (a > 0)

(A)4

3(B)

4

π(C)

4

3π(D)

16

9

Q.144 The value of the integral( )

∫ ++

dx1x

)x22(nsin l is

(A) – cos ln (2x + 2) + C (B) ln

+1x2

sin + C

(C) cos

+1x2

+ C (D) sin

+1x2

+ C

Q.145 If f(x) = A sin

π2

x + B , f ′

2

1 = 2 and ∫

1

0

f(x) dx =πA2 , Then the constants A and B are

respectively.

(A)2

&2

ππ(B)

ππ3

&2

(C)π

−4

&0 (D) 0&4

π

Q.146 Let I1 = ∫

π−

2

0

x dx)xsin(e2

; I2 = ∫

π−

2

0

x dxe2

; I3 = ∫

π− +

2

0

x dx)x1(e2

and consider the statements

I I1 < I

2 II I

2 < I

3 III I

1 = I

3

Which of the following is(are) true?

(A) I only (B) II only(C) Neither I nor II nor III (D) Both I and II

Q.147 Let f (x) =x

xsin, then ∫

π

−π2

0

dxx2

f )x(f =

(A) ∫π

π0

dx)x(f 2

(B) ∫π

0

dx)x(f (C) ∫π

π0

dx)x(f (D) ∫π

π0

dx)x(f 1

Q.148 Let u = ∫ ++1

02

dx1x

)1x(nl and v = ∫π 2

0

dx)x2(sinnl then

(A) u = 4v (B) 4u + v = 0 (C) u + 4v = 0 (D) 2u + v = 0

Q.149 If ( ) θθ+θ

= ∫π

d.cos1

·sinxsinx

2

16 / 2

x

2 f then the value of f '

π2

, is

(A) π (B) – π (C) 2π (D) 0


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Quest

Q.150 The value of the definite integral, ∫π 2

0

dxxsin

x5sin is

(A) 0 (B)2

π(C) π (D) 2π

Select the correct alternatives : (More than one are correct)

Q.151 dxxsgn

b

a

∫ = (where a, b∈ R)

(A) | b | – | a | (B) (b–a) sgn (b–a) (C) b sgnb – a sgna (D) | a | – | b |

Q.152 ∫ + xcos45dx

= λ tan−1

2

xtanm + C then :

(A) λ = 2/3 (B) m = 1/3 (C) λ = 1/3 (D) m = 2/3

Q.153 Which of the following are true ?

(A) x f xa

a

. (sin )

π −

∫ dx = π2

. f xa

a

(sin )

π −

∫ dx (B) f xa

a

( )2

−∫ dx = 2. f x

a

( )2

0

∫ dx

(C) ( )f xn

cos2

0

π

∫ dx = n. ( )f xcos20

π

∫ dx (D) f x cb c

( )+−

∫0

dx = f xc

b

( )∫ dx

Q.154 The value of

( )

2 3 3

1 2 2

2

2

0

1x x

x x x

+ +

+ + +∫ ( ) dx is :

(A)π4

+ 2 ln2 − tan−1 2 (B)π4

+ 2 ln2 − tan−11

3

(C) 2 ln2 − cot−1 3 (D) − π4

+ ln4 + cot−1 2

Q.155x x

x

2 2

21

++∫cos

cosec2 x dx is equal to :

(A) cot x − cot −1 x + c (B) c − cot x + cot −1 x

(C) − tan −1 x − cossecec x

x + c (D) − e n x tan−1 − cot x + c

where 'c' is constant of integration .

Q.156 Let f (x) = sint

t

x

0

∫ dt (x > 0) then f (x) has :

(A) Maxima if x = n π where n = 1, 3, 5,.....(B) Minima if x = n π where n = 2, 4, 6,......(C) Maxima if x = n π where n = 2, 4, 6,......(D) The function is monotonic


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Quest

Q.157 If In =

( )

dx

xn

1 20

1

+∫ ; n ∈ N, then which of the following statements hold good ?

(A) 2n In + 1

= 2 −n + (2n − 1) In

(B) I2 =

π8 4

+

(C) I2 =

π

8 4

− (D) I3 =

π

16 48

−

Q.1581

1

1

12xn

x

xdx

−−+z equals :

(A)1

2 ln2

x

x

−+

1

1 + c (B)

1

4 ln2

x

x

−+

1

1 + c (C)

1

2 ln2

x

x

+−

1

1 + c (D)

1

4 ln2

x

x

+−

1

1 + c

Q.159 If An =

0

2π /

∫ sin ( )

sin

2 1n x

x

− d

x ; B

n =

0

2π /

∫ sin

sin

n x

x

2

d x ; for n ∈ N , then :

(A) An + 1

= An

(B) Bn + 1

= Bn

(C) An + 1 − An = Bn + 1 (D) Bn + 1 − Bn = An + 1

Q.1600

∞

∫x

x x( ) ( )1 1 2+ +d

x :

(A)π4

(B)π2

(C) is same as0

∞

∫dx

x x( ) ( )1 1 2+ +(D) cannot be evaluated

Q.161 1 +∫ cscx dx equals

(A) 2 sin −1 sinx + c (B) 2 cos −1 cosx + c

(C) c − 2 sin −1 (1 − 2 sin x) (D) cos −1 (1 − 2 sin x) + c

Q.162 If f (x) =

0

2π /

∫ n x( )sin

sin

12

2

+ θθ

d θ , x ≥ 0 then :

(A) f (t) = π ( )t + −1 1 (B) f ′ (t) =π

2 1t +

(C) f (x) cannot be determined (D) none of these.

Q.163 If a, b, c ∈ R and satisfy 3 a + 5 b + 15 c = 0 , the equation ax4 + b

x2 + c = 0 has :

(A) atleast one root in (− 1, 0) (B) atleast one root in (0, 1)(C) atleast two roots in (−

1, 1) (D) no root in (−

1, 1)

Q.164 Let u = ∫∞

++0

241x7x

dx & v = ∫

∞

++0

24

2

1x7x

dxx then :

(A) v > u (B) 6 v = π (C) 3u + 2v = 5π /6 (D) u + v = π /3


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Quest

Q.165 If ∫ eu . sin 2x dx can be found in terms of known functions of x then u can be :(A) x (B) sin x (C) cos x (D) cos 2x

Q.166 If f(x) = n t

t

x

11 +∫ dt where x > 0 then the value(s) of x satisfying the equation,

f(x) + f(1/x) = 2 is :(A) 2 (B) e (C) e −2 (D) e2

Q.167 A polynomial function f(x) satisfying the conditions f(x) = [f ′ (x)]2 &0

1

∫ f(x) dx =12

19 can be:

(A)4

9x

2

3

4

x2++ (B)

4

9x

2

3

4

x2+− (C)

4

x2 − x + 1 (D)

4

x2+ x + 1

Q.168 A continuous and differentiable function '

f

' satisfies the condition ,

0

x

∫ f (t) d t = f 2 (x) − 1 for all real ' x '. Then :

(A) ' f ' is monotonic increasing ∀ x ∈ R

(B) ' f ' is monotonic decreasing ∀ x ∈ R

(C) ' f ' is non monotonic

(D) the graph of y = f (x) is a straight line.


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Quest

ANSWER KEY

Q . 1 A Q . 2 C Q . 3 B Q . 4 C Q . 5 C

Q . 6 C Q . 7 A Q . 8 D Q . 9 A Q . 1 0 D

Q . 1 1 D Q . 1 2 C Q . 1 3 A Q . 1 4 C Q . 1 5 C

Q . 1 6 B Q . 1 7 C Q . 1 8 A Q . 1 9 D Q . 2 0 A

Q . 2 1 C Q . 2 2 C Q . 2 3 A Q . 2 4 A Q . 2 5 B

Q . 2 6 A Q . 2 7 A Q . 2 8 B Q . 2 9 D Q . 3 0 C

Q . 3 1 A Q . 3 2 D Q . 3 3 A Q . 3 4 D Q . 3 5 B

Q . 3 6 C Q . 3 7 B Q . 3 8 C Q . 3 9 D Q . 4 0 C

Q . 4 �

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