Maths+quest

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SPEED - II

QUESTION BANK

FOR IITJEE

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APPLICATION OF DERIVATIVE

MATHEMATICS

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

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Quest

Question bank on Application of Derivative

Select the correct alternative : (Only one is correct)

Q.1 Suppose x₁ & x₂are the point of maximum and the point of minimum respectively of the function f(x) = 2x3 − 9_ax2_{+ 12}_a2_{x + 1}_{respectively, then for the equality} 2

1

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 curve y=e−x2 and has the coordinate (x, e−x2) where x > 0. Point B has the coordinates (x, 0). If 'O' is the origin then the maximum area of the triangle AOB is

(A) e 2 1 (B) e 4 1 (C) e 1 (D) e 8 1

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

(A) tan−1 k2 _{(B) cot}−1 _(k2₎ _{(C) sec}−1

1+ 4    _k _ (D) none Q.4 {a₁, a₂, ..., a₄, ...} 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 ) 5 t 2

( 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 _−π π_ 2 , 2 (A) π (B) 0 (C) 3 2 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) – 5 8π (C) – 5 3π (D) 5 2π

Q.10 If a variable tangent to the curve x2_{y = c}3_{makes intercepts a, b on x and y axis respectively, then the}

value of a2_{b is} (A) 27 c3 (B) c3 27 4 (C) c3 4 27 (D) c3 9 4

Q.11 Difference between the greatest and the least values of the function f (x) = x(ln x – 2) on [1, e2] is

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Quest Tutorials

North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest

Q.12 Let f (x) =

∑

= n 2 0 r r n x tan x tan , n ∈ N, where x ∈      π 2 , 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 0 x for 0 0 x for x sin x

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) π 6 (B) π 4 (C) π 3 (D) 5 12 π

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

(A) f(x) =  ₌ < ≤ 1 x , 0 1 x 0 , x on [0, 1] (B) f(x) =     = < ≤ π − 0 x , 0 0 x , x x sin on [– π, 0] (C) f(x) = 1 x 6 x x2 − − − on [–2,3] (D) f(x) =     = − − ≠ − + − − 1 x if 6 ] 3 , 2 [ on , 1 x if 1 x 6 x 5 x 2 x3 2

Q.17 Suppose that f (0) = – 3 and f ' (x) ≤ 5 for all values of x. Then the largest value which f (2) can attain is (A) 7 (B) – 7 (C) 13 (D) 8

Q.18 The tangent to the graph of the function y = f(x) at the point with abscissa x = a forms with the x-axis an angle of π/3 and at the point with abscissa x = b at an angle of π/4, then the value of the integral,

a b

∫

f′ (x) . f′′ (x) dx is equal to

(A) 1 (B) 0 (C) ₋ ₃ (D) –1 [ assume f′′ (x) to be continuous ]

Q.19 Let C be the curve y = x3_{(where x takes all real values). The tangent at A meets the curve again at B. If}

the gradient at B is K times the gradient at A then K is equal to

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Quest

Q.20 The vertices of a triangle are (0, 0), (x, cos x) and (sin3_{x, 0) where 0 < x <}

2 π

. The maximum area for such a triangle in sq. units, is

(A) 32 3 3 (B) 32 3 (C) 32 4 (D) 32 3 6

Q.21 The subnormal at any point on the curve xyn_{= a}n + 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 = − x

2 2 + 2 and secant to the curve y = 4− x2 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 3x

2

+ 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

1

sinx + −sinx = 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

}

x x x ( ) : ; ; 0 0 1 3 1 2 ≤ ≤ ≤ ≤ − < ≤ then g 1 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 x 2 1 4       − − _{g (x) =}     = ≠ 0 x , 1 0 x , x ] x [ n ta h (x) = {x} k (x) = ₅log2(x+3)

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 , k∈ intersect each other at two different points. TheR

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

(

y1≠y2

)

. The value of ‘a’ is

(A) 4 (B) 3 (C) 2 (D) 1 Q.28 f (x) = dx x 1 1 x 1 1 2 2 2

∫

       + − + − _{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 Tutorials

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 x x 4 3 x 1 x 0 x 1 x x ) x ( f 3 3

(A) It is monotonic increasing ∀ x∈R

(B) f

′

(x) fails to exist for 3 distinct real values of x (C) f

′

(x) changes its sign twice as x varies from (–∞ ,∞ )

(D) function attains its extreme values at x₁ & x₂ , such that x₁, x₂ > 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) π 16 9 (B) π9 25 (C) π3 5 (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–1

is also differentiable at f (c).

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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 for t ∈ [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_{+ 6x}2_{+ 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 angle α with the plane of the base. The value of α for which the volume of the pyramid is greatest, is :

(A) π 4 (B) sin −1 2 3 (C) cot −1 ₂ _(D)π 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) = } 1 x if ) 2 x ( 1 x if x 3 5 3 > − − ≤

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 positive

semi axis OX a line segment half that on the negative semi axis OY then the co-ordinates the point(s) is/ are given by :

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Quest Tutorials

Quest

Q.50 A curve with equation of the form y = ax4_{+ bx}3_{+ 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_{= log}

2 (x

2 + 1) − log 2 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 + 2y}2_{= 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 curve ₂ ₂

y b

xa + = 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_{+ 9x}2_{+ 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 a maximum value, for x equal to :

(A) p q

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

(

h(x)

)

for each x ∈ J

Suppose

(

h'(x)

)

2 > 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 1 x if 0 2 1 x if 1 x 2 ) 1 x 6 )( 1 x ( = ≠ − − − 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 (x₂), ∀ x₁ > x₂ , then solution set of f

(

g(α2−2α)

)

> f

(

g(3α−4)

)

is

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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 3 3 (C) 16 3 3 (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 the parameter 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 rate in m2/hr at which the area of the triangle is increasing when x = 12 m and θ = π/4

(A) 21/2       ₊ π 5 2 1 _(B) 2 73 · 21/2 _(C) 5 2 312 ₊π (D) 21/2       ₊π 5 2 1 Q.64 If the function f (x) = 4 x x x 3 t 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) πS (B) 2 πS (C) 2 πS (D) 4 πS

Q.66 A point is moving along the curve y3_{= 27x. The interval in which the abscissa changes at slower rate than}

ordinate, 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 (x₀) = g (x₀) 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) 2 2 1 3 sinθ = − _(B) 2 1 5 sinθ = − (C) 2 1 5 sinθ = + (D) 4 1 5 sinθ= −

Q.69 The line which is parallel to x-axis and crosses the curve y = x at an angle of π 4 is (A) y = − 1/2 (B) x = 1/2 (C) y = 1/4 (D) y = 1/2 Q.70 The function S(x) =

∫

_       π x 0 2 dt 2 t

sin 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 Tutorials

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,∞). Let h(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 x

3

3 + (a + 2) x

2 + (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+ t2 & y = 1 1 2

t

(

+ 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)} _cos2 _sin2 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 ) a x ( 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 + a2_{x – x}3_{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:

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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 2 a +b (B) 2 3 a+b (C) a b 2 2 2 + (D) a+ 2b 3

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_{– x}2_{y + 5y – 2x = 0 and x}4_{– x}3_y2_{+ 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_{– 6x}2_{+ 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) e}½

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 dt t t n ) t ( n tl l (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 sin 2 π x, 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 Tutorials

N 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 dt t 1 t _{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 1 g , 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_{= x}3_{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)

(

2t

)

4 1 + π (B) t cos t sin 1− (C)

(

2t−π

)

4 1 (D) t 2 cos t sin 1+ Q.92 If f (x) = 1 + x +

∫

(

+

)

x 1 2 dt nt 2 t n l l , then f (x) increases in (A) (0, ∞) (B) (0, e–2) ∪ (1, ∞) (C) no value (D) (1, ∞) Q.93 The function f(x) = ) x e ( 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.

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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 1 1       + 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) Lim (x) x→∞ f (B) ) x ( Lim 0 x f + → (C) xLim1 (x) f − − → (D) xLim→−∞ f(x) 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 cm}3_{, 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_{+ x}2_{+ 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.

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[13]

Quest Tutorials

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_{– 4t}2_{– 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 = x}n (n ∈ N) in the first quadrant a normal is drawn. The normal

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

2 1 b Lim 0 a→ = , 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) =  0 x for 8 x 0 x for x 2 2 ≥ + < −

. Then x intercept of the line that is tangent to the graph of f (x) is

(A) zero (B) – 1 (C) –2 (D) – 4

Q.110 Suppose that f is differentiable for all x and that f '(x) ≤ 2 for all x. If f (1) = 2 and f (4) = 8 then f (2) has the value equal to

(A) 3 (B) 4 (C) 6 (D) 8

Q.111 There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per additional tree drops by 10 apples. Number of trees that should be added to the existing orchard for maximising the output of the trees, is

(A) 5 (B) 10 (C) 15 (D) 20 Q.112 The ordinate of all points on the curve y =

x cos 3 x sin 2 1 2

2 ₊ where the tangent is horizontal, is

(A) always equal to 1/2 (B) always equal to 1/3

(C) 1/2 or 1/3 according as n is an even or an odd integer. (D) 1/2 or 1/3 according as n is an odd or an even integer.

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Quest

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

Q.113 The equation of the tangent to the curve x

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 If x

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) = 1

4

1 4

sinx+ − cosx− , 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 −

∫

t x 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) = 2f (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 x y = 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 function sin ( )

sin ( ) x a x b +

+ has no maxima or minima if :

(A) b − a = nπ , n ∈ I (B) b − a = (2n + 1) π , n ∈ I (C) b − a = 2nπ , 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) 1, 1 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 :}

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[15]

Quest Tutorials

Quest

Q.123 For the function f(x) = x4₍₁₂_lnx − 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 distance}

between 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 x

3 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_{– 6x}2_{– 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 c∈(0,1) (D) none

Q.128 Equation of a tangent to the curve y cot x = y3 _tan_{x at the point where the abscissa is}π

4 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,

d a A d b B d c C

cos + cos + cos is equal to:

(A) 6R (B) 2R (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

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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 00 D Q.1 01 B Q.1 02 D Q.1 03 C Q.1 04 D Q.1 05 D Q.1 06 B Q.1 07 C Q.1 08 D Q.1 09 B Q.1 10 B Q.1 11 C Q.1 12 D Q.1 13 A,B Q.1 14 A,B Q.1 15 A,D Q.1 16 A,C Q.1 17 A,B ,C ,D Q.1 18 B,C Q.1 19 A,B Q.1 20 A,B ,C Q.1 21 A,B Q.1 22 A,C ,D Q.1 23 A,B ,C ,D Q.1 24 B,D Q.1 25 A,B Q.1 26 B,C ,D Q.1 27 B,C Q.1 28 A,B ,D Q.1 29 A,C Q.1 30 C,D Q.1 31 A,B ,D Q.1 32 A,B

ANSWER KEY

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DEFINITE & INDEFINITE

INTEGRATION

MATHEMATICS

(20)

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 1 x 3 1 x dx ) e e ( is (A) ₂ e 4 π (B) e 4 π (C)       −π − e 1 tan 2 e 1 1 2 (D) 2 e 2 π

Q.2 The value of the definite integral, cos e 2 ·2xex2dx

2 n 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

∫

− − − ₋ ₋ ₋ 2 1 2 1 3 1 3 1 dx ) ) x 3 x 4 ( cos ) x 4 x 3 ( sin ( (A) 0 (B) 2 π − (C) 2 7π (D) 2 π Q.4 Let f (x) =

∫

+ x 2 1 t4 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 x 1 e ) e ( cot dx is equal to : (A) 2 1 ln (e2x + 1) − x x 1 e ) e ( cot− + x + c (B) 2 1 ln (e2x_{+ 1) +} x x 1 e ) e ( cot− + x + c (C) 2 1 ln (e2x + 1) − x x 1 e ) e ( cot− − x + c (D) 2 1 ln (e2x_{+ 1) +} x x 1 e ) e ( cot− − x + c Q.6

∫

+ → k 0 x 1 0 k k (1 sin2x) dx 1 Lim

(A) 2 (B) 1 (C) e2 _{(D) non existent}

Q.7

∫

+ − 5 n 0 x x x 3 e 1 e e l dx = (A) 4−π (B) 6−π (C) 5−π (D) None

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[3]

Quest Tutorials

Quest

Q.8 If x satisfies the equation 2

1 0 2 x 1 cos t 2 t dt         + α +

∫

– dt x 1 t t 2 sin t 3 3 2 2         +

∫

− – 2 = 0 (0 < α < π), then the value x is (A) ± α α sin 2 (B) ± α α sin 2 (C) ± α α sin (D) ± α α sin 2 Q.9 If f (x) = eg(x)_{and g(x) =} 2 x

∫

t d t t

1+ 4 then f′(2) has the value equal to :

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

Q.10

∫

etan θ (secθ – sinθ)dθ equals :

(A) − etan θ_{sin θ + c (B) e}tan θ_{sin θ + c} _{(C) e}tan θ_{sec θ + c} _{(D) e}tan θ_{cos θ + c}

Q.11

∫

π 0 (x · sin2_{x · cos}_{x) dx =} (A) 0 (B) 2/9 (C) −2/9 (D) −4/9 Q.12 The value of

(

)

∑

= = ∞ → ₊ n 4 r 1 r 2 n _r ₃ _r ₄ _n n Lim is equal to (A) 35 1 (B) 14 1 (C) 10 1 (D) 5 1 Q.13

∫

− − + c b c a dx ) c x ( f = (A)

∫

b a dx ) x ( f (B)

∫

+ b a dx ) c x ( f (C)

∫

− − c 2 b c 2 a dx ) x ( f (D)

∫

+ b a dx ) c 2 x ( f Q.14 Let I₁ =

∫

π + − 2 / 0 dx x cos . x sin 1 x cos x sin ; I₂ =

∫

π 2 0 6 dx ) x cos ( ; I₃ =

∫

π π − 2 / 2 / 3 dx ) x (sin & I₄ =

∫

      − 1 0 dx 1 x 1 n l then (A) I₁ = I₂ = I₃ = I₄ = 0 (B) I₁ = I₂ = I₃ = 0 but I₄ ≠ 0 (C) I₁ = I₃ = I₄ = 0 but I₂ ≠ 0 (D) I₁ = I₂ = I₄ = 0 but I₃ ≠ 0 Q.15

∫

+ − ) x 1 ( x x 1 7 7 dx equals : (A) lnx + 7 2 ln (1 + x7_{) + c} _{(B) ln}x − 7 2 ln (1 − x7_{) + c} (C) lnx − 7 2 ln (1 + x7_{) + c} _{(D) ln}_{x +} 7 2 ln (1 − x7_{) + c} Q.16

∫

π + n 2 / 0 n nx tan 1 dx = (A) 0 ( B ) n 4 π (C) 4 nπ (D) n 2 π

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Quest

Q.17 f(x) =

∫

− − x 0 dt ) 2 t ( ) 1 t (

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 a dx ) 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 – 2 1 e2_– 2 5 (B) e – e2_{– 3} _(C) 2 1 (e – 3) (D) e – 2 1 e2_– 2 3 Q.20

∫

₊ | x | n 1 x | x | n l l dx equals : (A) 1 n|x| 3 2 l + _{(lnx − 2) + c} (B) 1 n|x| 3 2 l + _(lnx+ 2) + c (C) 1 n|x| 3 1 l + (lnx − 2) + c (D) 2 1+ln|x| (3 lnx − 2) + c Q.21

∫

(

)

      − − + − 2 1 3 2 1 dx 4 | x 1 | | 3 x | 2 1 equals: (A) 2 3 − (B) 8 9 (C) 4 1 (D) 2 3

Where {*} denotes the fractional part function.

Q.22

∫

π / 4 0       − x 1 cos . x x 1 sin . x

3 2 _{dx has the value :}

(A) 8 2₃ π (B) 24 2 3 π (C) 32 2 3 π (D) None Q.23 _      + π − + +       π +       π π ∞ → 3 4 n 6 ) 1 n ( sec ... n 6 · 2 sec n 6 sec n 6 Lim 2 2 2

n has the value equal to

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

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[5]

Quest Tutorials

Quest

Q.24 Suppose that F (x) is an antiderivative of f (x) = x x sin , x > 0 then

∫

3 1 dx x x 2 sin 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 ₄ ₂ 4 ) 1 x x ( 1 x 3 + + − w.r.t. x is : (A) 1 x x x 4₊ ₊ + c (B) − 1 x x x 4₊ ₊ + c (C) 1 x x 1 x 4₊ ₊ + + c (D) − 1 x x 1 x 4₊ ₊ + + c Q.26 ∞ → n Lim π π π π 2 1 2 2 2 1 2 n n n n n + + + + −      

cos cos ... cos( ) equal to

(A) 1 (B) 1 2 (C) 2 (D) none Q.27 log_x

(

logx

)

n 2 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 dx nx ·sin mx cos ₌ (A) 0 (B) ₂ ₂ m n n 2 − (C) _n2 _m2 m 2 − (D) none

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

∫

3 0 dx y = (A) 5/6 (B) 2/3 (C) 1 (D) 11/6 Q.30 If

( )

∫

+ + 2 2 4 1 x x 1 x dx = A ln x+ 2 x 1 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 / 1 cosx x sin 1 dx = (A) 1−ln2 (B) ln2 (C) 1+ ln2 (D) none Q.32 Let f : R → R be a differentiable function & f(1) = 4, then the value of ;

1 x Lim →

∫

₋ ) x ( f 4 x 1 dt t 2 _{is :} (A) f′(1) (B) 4f′(1) (C) 2f′(1) (D) 8f′(1)

(24)

Quest

Q.33 If

∫

) x ( f 0 2_dt t _{= 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_/₂₎1/3 ( 0 3 5 dx x ·sin x = (A) 1 (B) 1/2 (C) 2 (D) 1/3

Q.35 Integral of 1+2cotx(cotx+cosecx) w.r.t. x is :

(A) 2 ln cos 2 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 dt 4 t 3 28 t 8 =

( )

1 x log 1 x ) 1 x ( 2 3 + + + , is (A) 0 (B) 1 (C) 2 (D) 3 Q.38

∫

− 1 0 1 dx x x tan = (A)

∫

π 4/ 0 dx x x sin (B)

∫

π 2/ 0 dx x sin x (C)

∫

π 2/ 0 dx x sin x 2 1 (D)

∫

π 4/ 0 dx x sin x 2 1

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

∫

+ x 0 x2 t2 dt is (A) R (B) R+ _{(C) R}+ ∪ {0} _{(D) R – {0}}

Q.40 If

∫

e3x_cos_4x_{dx = e}3x_{(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 ) x b a ( f . x = (A) 0 (B) 2 1 (C) +

∫

b a dx ) x ( 2 b a f (D) −

∫

b a dx ) x ( 2 b a f

(25)

[7]

Quest Tutorials

Quest

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

∫

−

2 0 2a)dt log t ( = log₂       2 a 4 is ( A ) a ∈ R (B) a ∈ R+ (C) a < 2 (D) a > 2

Q.43 The value of the definite integral 2x 5(4x 5) 2x 5(4x 5) dx

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

0 dx x b a 3 ₌

∫

and 3 2 dx x b a 2 ₌

∫

is (A) 0 (B) 1 (C) 2 (D) 4 Q.45

∫

₋ ₋ − − + − x cot x tan x cot x tan 1 1 1 1 dx is equal to : (A) _π4 x tan−1 x + _π2 ln (1 + x2) − x + c _(B) π 4 x tan−1 x − _π2 ln (1 + x2_{) + x + c} (C) _π4 x tan−1 x + _π2 ln (1 + x2_{) + x + c} _(D) π 4 x tan−1 x − _π2 ln (1 + x2) − x + c

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

∫

+ y 0 1 t2 dt . The value of ₂ 2 dx y d is equal to (A) 2 y 1 y + (B) y (C) ₁ _y2 y 2 + (D) 4y Q.47 Let f (x) =

∫

+ → ₊ ₊ h x x 2 0 h _t ₁ _t dt h 1

Lim , then Lim x· (x)

x→−∞ f

is

(A) equal to 0 (B) equal to 2 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 the primitive 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 0 2 1 f (x)dx (B)

∫

a 0 2 f (x)dx (C)

∫

a 0 f (x)dx (D)

∫

a 0 4 f (x)dx

(26)

Quest

Q.50

∫

x . 2 2 x 1 x 1 x n +       ₊ ₊ l dx equals : (A) ₁₊_x2 _ln      _x₊ ₁₊_x2 _{− x + c} (B) 2 x . ln2      _x₊ ₁₊_x2 ₋ 2 x 1 x + + c (C) 2 x . ln2      _x₊ ₁₊_x2 + 2 x 1 x + + c (D) 1+x2 ln _x+ 1+x2_ + x + c Q.51 If f (x) =     2 x 1 ) 6 x 7 ( 1 x 0 x 1 3 1 _< _≤ − ≤ ≤ − − , then

∫

2 0 dx ) x ( f is equal to (A) 6 31 (B) 21 32 (C) 42 1 (D) 42 55

Q.52 The value of the definite integral

∫

+

1 0 x e ₍₁ _x_·_e ₎_dx e x is equal to (A) ee _{(B) e}e_{– e} _{(C) e}e_{– 1} _{(D) e} Q.53

∫

      − 2 2 / 1 dx x 1 x sin x 1

has the value equal to

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

Q.54 The value of the integral

∫

∞ 0

e −2x (sin 2x + cos2x) dx =

(A) 1 (B) −2 (C) 1/2 (D) zero

Q.55 The value of definite integral

∫

∞ − − − 0 z 2 z dz e 1 e z . (A) – n2 2l π (B) n2 2l π (C) – π ln 2 (D) π ln 2

Q.56 A differentiable function satisfies

3f 2_{(x) f '(x) = 2x. Given f (2) = 1 then the value of f (3) is}

(A) 3 ₂₄ _(B)3 ₆ _{(C) 6} _{(D) 2}

Q.57 For I_n =

∫

e

1

(ln x)n_dx, n ∈ N; which of the following holds good?

(A) I_n + (n + 1) I_{n + 1} = e (B) I_{n + 1} + n I_n = e (C) I_{n + 1} + (n +1) I_n = e (D) I_{n + 1} + (n – 1) I_n = e

(27)

[9]

Quest Tutorials

Quest

Q.58 _{Let f be a continuous functions satisfying f ' (ln x) = }

1 x x for 1 x 0 for 1 > ≤ <

and f (0) = 0 then f (x) can be defined as (A) f (x) =  ₁ _e _{if x} ₀ 0 if x 1 x _> − ≤ (B) f (x) =  _e ₁_{if x} ₀ 0 if x 1 x₋ _> ≤ (C) f (x) =  _e _{if x} ₀ 0 if x x x _> < (D) f (x) =  _e ₁_{if x} ₀ 0 if x x x₋ _> ≤

Q.59 Let f : R → R be a differentiable function such that f(2) = 2. Then the value ofLimit_x_{→ 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 / 0 2 2 x sin a 1 dx

has the value :

(A) π 2 1+a2 (B) π 1+ a2 (C) 2 1 2 π + a (D) none Q.61 Let f (x) =       x e x n x 1

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

(A) 2 1 ex_{– ln x + C} _(B) 2 1 ln x – ex_{+ C} _(C) 2 1 ln2_{x – x + C} _(D) x 2 ex + C Q.62

∑

= ∞ → ₊ n 1 k 2 2 2 n _n _k _x n Lim , 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 − − then 0 2 π/

∫

[f(x) + f′(x)] dx = (A) π (B) π/2 (C) 2π (D) zero

Q.64 The absolute value of sin x

x 1 8 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 :}

(28)

Quest

Q.66 Primitive of f (x) = x·2ln(x2+1) w.r.t. x is (A) ) 1 x ( 2 2 2 ) 1 x ( n 2 + + l + C (B) 1 2 n 2 ) 1 x ( 2 n(x2 1) + + + l l + C (C) ) 1 2 n ( 2 ) 1 x ( 2 n2 1 + + + l l + C (D) ) 1 2 n ( 2 ) 1 x ( 2 n2 + + l l + C Q.67

∫

      + + ∞ → 2 0 n n n 1 dt t 1 Lim is equal to

(A) 0 (B) e2 _{(C) e}2_{– 1} _{(D) does not exist}

Q.68 Limit_h_{→ 0} _{n t dt} _{n t dt} h a x h a x 2 2 +

∫

−

∫

= (A) 0 (B) ln2 _x _(C)2n x

x (D) does not exist

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

0 1

∫

(1+ cos8_{x) (ax}2 _{+ bx}₊_{c) dx =}

0 2

∫

(1+ cos8_{x) (ax}2 _{+ 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 I_n = 0 4 π/

∫

tann _{x dx}_{, then} 1 1 1 2 4 3 5 4 6 I +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

(

1+

(

g(x)

)

2

)

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.72 0 4 π/

∫

(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

∫

− + − 2 1 0 2 2 2 ) x 1 1 ( x 1 dx x is (A) 4 π (B) 2 1 4+ π (C) 2 1 4− π (D) none

(29)

[11]

Quest Tutorials

Quest

∫

(

+ π

)

37 19 2 dx ) x 2 (sin 3 } 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 dx x tan , is (A) 2π (B) 2 π (C) 2 2π (D) 2 2 π

Q.76 Evaluate the integral :

∫

dx x ) x 6 ( n 2 l (A) [n(6x2)]3 8 1 l + C (B) [n (6x )] 4 1 2 2 l + C (C) [n(6x )] 2 1 2 l _{+ C} _(D) [n(6x2)]4 16 1 l _{+ C} Q.77

∫

π π θ       θ + − θ 6 5 6 2 2 d ) sin 1 ( 2 1 ) sin 3 ( 2 1 (A) π – 3 (B) π (C) π – ₂ ₃ (D) π + 3 Q.78 Let l = _→_∞

∫

x 2 x x _t dt Lim _{and m =} ∞ → x Lim

∫

x 1 dt t n x n x 1 l

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

∫

3 n 2 n dx ) x ( f l l = (A) 1 (B) 2 1 (C) 3 1 (D) ln 2 Q.80 If I = _n_{(sin )}_x / 0 2 π

∫

dx then _n_(sin_x _{cos )}_x

/ / + −

∫

π π 4 4 dx = (A) I 2 (B) I 4 (C) I 2 (D) I Q.81 The value of

∫

∏

∑

_       +         + = = 1 0 n 1 k n 1 r dx k x 1 ) r x ( equals (A) n (B) n ! (C) (n + 1) ! (D) n · n ! Q.82

∫

+ + x sin x sin x cos x cos 4 2 5 3 dx

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

(30)

Quest

Q.83 0 3

∫

1 4 4 4 4 2 2 x x x x + + + − +         dx = (A) ln5 2 3 2 − (B) ln5 2 3 2 + (C) ln5 2 5 2 + (D) none

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

1 x

∫

(ln2_{t + 2}_{lnt) dt where f} ′ _{(x) vanishes is :} (A) e−1 (B) 0 (C) 2e−1 (D) 1+2e−1 Q.85 Limitn→ ∞ 1 1 1 2 3 3 1 n 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

∫

∞ ∞ − dx x sin · ) x ( ' h , is

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

Q.87

0 4 π/

∫

(tann _{x + tan}n − 2 x)d(x − [x]) is : ( [. _{] denotes greatest integer function)}

(A) 1 1 n− (B) 1 2 n+ (C) 2 1 n− (D) none of these Q.88 λ λ → λ         +

∫

1 1 0 0 (1 x) dx 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) x n|x| C x dx . x

∫

= l + (B) x n|x| Cx x dx . x

∫

= l + (C) . cosx dx tanx C x cos 1 ₌ ₊

∫

(D) . cosx dx x C x cos 1 ₌ ₊

∫

(31)

[13]

Quest Tutorials

Quest

Q.90 0 ∞

∫

x2n + 1_·_e−x2_{dx is equal to (n}∈ N). (A) n ! (B) 2 (n !) (C) n ! 2 (D) 2 )! 1 n ( +

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 =

∫

2 x 0 dt ) 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 ) 1 x 4 x ( ) 1 x 2 ( 2 / 3 2 (A) C ) 1 x 4 x ( x 2 / 1 2 3 + + + (B) (x 4x 1) C x 2 / 1 2₊ ₊ + (C) C ) 1 x 4 x ( x 2 / 1 2 2 + + + (D) (x 4x 1) C 1 2 / 1 2₊ ₊ +

Q.95 If the value of the integral ex2

1 2

∫

dx is α, then the value of _{n x}

e e4

∫

dx is : (A) e4 − e − α _{(B) 2}_e4 − e − α _{(C) 2}_(e4 − e) − α _{(D) 2 e}4 – 1 – α Q.96 _ _ − −

∫

2 x 1 x 2 1 tan dx d 2 1 3 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 1 a t 1 a t dt e

has the value

(32)

Quest

Q.98 sin / 2 0 2 θ π

∫

sinθ dθ is equal to : (A) 0 (B) π/4 (C) π/2 (D) π Q.99 dx 4 x 2 x 4 2

∫

₊+ is equal to (A) C x 2 2 x tan 2 1 −₁ 2+ ₊ (B) tan (x 2) C 2 1 ₁ ₂ + + − (C) C 2 x x 2 tan 2 1 2 1 ₊ − − (D) C x 2 2 x tan 2 1 ₁ 2 + − − Q.100 If β + 2 x e2 x 0 1 2 −

∫

dx =

∫

e−x2 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 dt t 1 sin t (C) h (x) =     π < < π π ≤ < x 4 3 x 9 2 sin 2 4 3 x 0 1 (D) l (x) =     π < < π π + π π ≤ < x 2 , ) x sin( 2 2 x 0 , x sin x Q.103 If f (x) =

∫

π + 0 1 tan2x sin2t dt t sin t for 0 < x < 2 π (A) f (0+) = – π _(B) 8 4 f 2 π =      π

(C) f is continuous and differentiable in       π 2 , 0

(D) f is continuous but not differentiable in       π 2 , 0

(33)

[15]

Quest Tutorials

Quest

Q.104 Consider f(x) = x x 2 3 1+ ; g(t) = f t dt

∫

( ) . If g(1) = 0 then g(x) equals (A) 1 3 1 3 _n₍ +_x ₎ _(B)1 3 1 2 3 _n +x      _(C)1 2 1 3 3 _n +x      _(D)1 3 1 3 3 _n +x     

Q.105 The value of the definite integral,

∫

100 0 x dx e x 2 is equal to ( A ) 2 1 (1 – e–10₎ _{(B) 2(1 – e}–10₎ _(C) 2 1 (e–10_{– 1)} _(D) 2 1       − −₁₀4 e 1 Q.106 0 ∞

∫

[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 nl 1 x dx 2 0 1 +      

∫

= (A) 3 4 1 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 4 ln − Q.109 The evaluation of p x q x x x dx p q q p q p q + − − + + − + +

z

2 1 1 2 2 2 1 is (A) – x x C p p q+ + + 1 (B) x x C q p q+ + + 1 (C) − + + + x x C q p q 1 (D) x x C p p q+ ₊ + 1 Q.110 x x x x 3 2 1 1 1 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.111 a 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 If

0 2

∫

375 x5 _{(1 + x}2₎ −4_{dx = 2}n_{then the value of n is :}

(34)

Quest

Q.113

∫

− + − 2 / 1 0 2 ₁ _x x 1 n x 1 1 _{dx is equal to :} (A) 3 1 n 4 1 ₂ _(B) 2 1 ln2₃ _(C)− 4 1 ln2₃ _{(D) cannot be evaluated.}

Q.114 If

∫

(x3−2x2 +5)e3xdx = e3x_(Ax3_{+ Bx}2_{+ 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 / 0 1 sinx cosx dx

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

∫

π + + 2 / 0 1 sinx cosx x sin dx is equal to (A) 2 1 ln 2 (B) 2 π − ln 2 (C) 4 π – 2 1 ln 2 (D) 2 π + ln 2

Q.116 A function f satisfying f′(sinx) = 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 <π 2 , 1 2 3 2 / /

∫

ln (ecos x_{). d (sin}_{x) is equal to :}

(A) 12 π (B) 6 π (C)

[

( ) (

3 1 sin 3 sin1

)

]

4 1 − + − (D)

[

( ) (

3 1 sin 3 sin1

)

]

4 1 − − − Q.118

∫

₍

₎

π + 0 2 x sin 1 x cos x dx is equal to : (A) π − 2 (B) − (2 + π) (C) zero (D) 2 − π Q.119

∫

(

x x

)

dx x e x + (A) 2e x

[

x− x+1

]

+ C (B) _e x

[

_x₋₂ _x₊₁

]

(C) _e x

(

_x₊ _x

)

₊ _C _(D)

(

)

C 1 x x e x + + + Q.120 dx x x cos sin / 6 6 0 2 +

∫

π is equal to : (A) zero (B) π (C) π/2 (D) 2π

(35)

[17]

Quest Tutorials

Quest

Q.121 The true solution set of the inequality, _       π + − −

_∫

x 0 2 dz 2 x 6 x 5 >

∫

π 0 2 dx x sin x is : (A) R (B) (1, 6) ( C ) (−6, 1) (D) (2, 3) Q.122 If

∫

− 1 0 1 x2 x n dx = k 0 π

∫

ln (1 + cosx) 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 5 b 3 3 1 b dx ) bx a acx ( _{is independent of b equals} (A) 2 c 3 (B) 3 c 2 (C) 3 c (D) c 2 3

Q.124

∫

sec2θ (secθ+tanθ)2 dθ

(A) [2 tan (sec tan )] C 2

) tan

(secθ+ θ ₊ _θ _θ₊ _θ ₊

(B) [2 4tan (sec tan )] C 3 ) tan (sec + θ + θ θ + θ + θ

(C) [2 tan (sec tan )] C 3 ) tan (sec + θ + θ θ + θ + θ

(D) [2 tan (sec tan )] C 2 ) tan (sec 3 + θ + θ θ + θ + θ Q.125

∫

₊+ 2 1 4 2 1 x 1 x dx is equal to: (A) 1 2 tan −1 ₂ _(B) 1 2 cot −1 ₂ _(C)1 2 tan −1 1 2 (D) 1 2 tan −1 ₂ Q.126 1 x x Limit → 1 x x x −

∫

x x₁ f(t) dt is equal to : (A) f x

( )

x 1 1

(B) x₁ f(x₁) (C) f(x₁) (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

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Quest

0 2 π/

∫

sinx sin2x sin3x dx is equal to :

(A) 1 3 (B) − 2 3 (C) − 1 3 (D) 1 6 Q.129 dx x 1 x 1 cos x 1 sec ) x 1 ( e 2 2 1 2 2 1 2 x 1 tan

∫

                + − +       ₊ + − − − (x > 0)

(A) _etan−1x_._tan−1_x ₊ _C _(B)

(

)

_C 2 x tan . etan−1x −1 2 ₊ (C) e . sec 1 x C 2 2 1 x 1 tan _ ₊      _      ₊ − − (D) e . cosec 1 x C 2 2 1 x 1 tan _ ₊      _      ₊ − −

Q.130 Number of positive solution of the equation,

(

t

{ }

t

)

x

−

∫

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–1_{x) then the value of the integral} _x _f_'_'₍_x₎ _dx 1 0

∫

is (A) 2 2 3− (B) 2 2 3+ (C) 1 (D) 2 2 3 1− Q.132 If

∫

+ 2 x sin 1 dx = A sin       ₋π 4 4 x

then value of A is:

(A) 2 2 (B) 2 (C) 1 2 (D) 4 2 Q.133 For U_n = 0 1

∫

xn₍₂− x)n_{dx; V} n = 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 1 x tan ) 1 x 3 x ( dx ) 1 x ( 2 1 2 4 2 = ln | f (x) | + C then f (x) is (A) ln       + x 1 x _{(B) tan}–1       + x 1 x _{(C) cot}–1       + x 1 x _{(D) ln} _            + − x 1 x tan 1