Matemática - Banco de Questões p/ IIT-JEE

SPEED - IIQUESTION BANK

FOR IITJEE

MATHEMATICS

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

APPLICATION OF DERIVATIVE

MATHEMATICS

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

TARGET IIT JEE

[2]Quest Tutorials

North 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

2 are 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 curve 2xey −= and has the coordinate (x,

2xe− ) 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) 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 k2 (B) cot−1 (k2) (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) – 5

8π(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

[3]Quest Tutorials


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) =

=

<≤

1x,0

1x0,x

on [0, 1] (B) f(x) =

=

<≤π−

0x,0

0x,x

xsin

on [– π, 0]

(C) f(x) = 1x

6xx2

−−−

on [–2,3] (D) f(x) =

=−

−≠−

+−−

1xif6

]3,2[on,1xif1x

6x5x2x 23

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

(A) 4 (B) 2 (C) – 2 (D) 1/4

[4]Quest Tutorials


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 = − x

2

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 x

x 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,y

1) , (a > 0) meets C

1 again at B(1,y

2)

( )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 (– ∞ , ∞)

[5]Quest Tutorials


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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

<−

≤≤

≥

=

0xx43

x

1x0x

1xx

)x(f

3

3

(A) It is monotonic increasing Rx ∈∀(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 & x

2 , such that x

1, x

2 > 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)

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).

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

[6]Quest Tutorials


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 + 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 angle α with

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

(A) π4

(B) sin−1 2

3(C) cot−1

2 (D) π3

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(

1xifx

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 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 :

(A) (− 1, 9) (B) (3, − 15) (C) (1, − 3) (D) none

[7]Quest Tutorials


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 curve 22 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 a

maximum 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

1xif0

2

1xif

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 (x

2) 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]

[8]Quest Tutorials


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 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

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 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 (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 of π4

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 , ∞). 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 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

[10]Quest Tutorials


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+ 2

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 – 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

π4

,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.

[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

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.

[12]Quest Tutorials


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(Limx

f∞→ (B) )x(Lim

0x

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)

[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 – 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). If 2

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

≥+

<− . 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 = xcos3xsin2

122 +

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.

[14]Quest Tutorials


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

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 function sin ( )

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

[15]Quest Tutorials


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 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 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 is π4

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

Q.1BQ.2DQ.3BQ.4BQ.5DQ.6CQ.7D

Q.8AQ.9DQ.10CQ.11BQ.12AQ.13DQ.14D

Q.15CQ.16DQ.17AQ.18DQ.19AQ.20AQ.21C

Q.22AQ.23CQ.24DQ.25DQ.26AQ.27BQ.28C

Q.29BQ.30CQ.31CQ.32CQ.33DQ.34DQ.35D

Q.36BQ.37AQ.38AQ.39DQ.40BQ.41AQ.42C

Q.43CQ.44CQ.45BQ.46CQ.47CQ.48BQ.49B

Q.50CQ.51AQ.52BQ.53CQ.54CQ.55CQ.56D

Q.57DQ.58CQ.59CQ.60CQ.61BQ.62AQ.63D

Q.64CQ.65AQ.66CQ.67DQ.68BQ.69DQ.70C

Q.71BQ.72AQ.73AQ.74BQ.75BQ.76AQ.77D

Q.78DQ.79AQ.80AQ.81DQ.82BQ.83AQ.84A

Q.85BQ.86AQ.87DQ.88CQ.89DQ.90CQ.91A

Q.92AQ.93BQ.94DQ.95CQ.96CQ.97CQ.98B

Q.99DQ.100DQ.101BQ.102DQ.103CQ.104DQ.105D

Q.106BQ.107CQ.108DQ.109BQ.110BQ.111CQ.112D

Q.113A,BQ.114A,BQ.115A,DQ.116A,CQ.117A,B,C,DQ.118B,C

Q.119A,BQ.120A,B,CQ.121A,BQ.122A,C,DQ.123A,B,C,DQ.124B,D

Q.125A,BQ.126B,C,DQ.127B,CQ.128A,B,DQ.129A,CQ.130C,D

Q.131A,B,DQ.132A,B

ANSWER KEY

DEFINITE & INDEFINITE

INTEGRATION

MATHEMATICS

TARGET IIT JEE

[2]Quest Tutorials


Quest

Question bank on Definite & Indefinite Integration

There are 168 questions in this question bank.


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

−−+ +1

1x31x dx)ee( is

(A) 2e4

π(B)

e4

π(C)

−π −

e

1tan

2e

1 1

2 (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

0kdx)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

[3]Quest Tutorials


Quest

Q.8 If x satisfies the equation 2

1

02

x1cost2t

dt

+α+∫ – xdt1t

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(x

x17

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 )

n4

π(C)

4

nπ(D)

n2

π

[4]Quest Tutorials


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

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

1e2 –

2

5(B) e – e2 – 3 (C)

2

1(e – 3) (D) e –

2

1e2 –

2

3

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 ( )∫

−−+−

2

13

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

[5]Quest Tutorials


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

Limπ π π π

21

2

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 mn

n2

−(C)

22mn

m2

−(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)

[6]Quest Tutorials


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

baf (D) ∫

− b

a

dx)x(2

baf

[7]Quest Tutorials


Quest

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

0

2dt)alogt( = log

2

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 3

2dxx

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

02t1

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

xf

∞−→ 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

02

1f (x)dx (B) ∫

a

0

2 f (x)dx (C) ∫a

0

f (x)dx (D) ∫a

0

4 f (x)dx

[8]Quest Tutorials


Quest

Q.50 ∫ x . 2

2

x1

x1xn

+

++l

dx equals :

(A) 2x1+ ln

++ 2x1x − x + c (B)

2

x . ln2

++ 2x1x −

2x1

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 ≤<−

≤≤−

−, 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

xe dx)e·x1(ex

is equal to

(A) ee (B) ee – e (C) ee – 1 (D) e

Q.53 ∫

−2

2/1

dxx

1xsin

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 + cos 2x) dx =

(A) 1 (B) − 2 (C) 1/2 (D) zero

Q.55 The value of definite integral ∫∞

−

−

−

0

z2

z

dze1

ez.

(A) – 2n2

lπ

(B) 2n2

lπ

(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 In = ∫

e

1

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

(A) In + (n + 1) I

n + 1 = e (B) I

n + 1 + n I

n = e

(C) In + 1

+ (n +1) In = e (D) I

n + 1 + (n – 1) I

n = e

[9]Quest Tutorials


Quest

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

1 xx for

1x0 for 1

>

≤< 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 >

<

(D) f (x) =

0 if x1e

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

1k222n xkn

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

−

− then

0

2π /

∫ [f (x) + f ′ (x)] dx =

(A) π (B) π/2 (C) 2 π (D) zero

Q.64 The absolute value ofsinx

x1 810

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

[10]Quest Tutorials


Quest

Q.66 Primitive of f (x) = )1x(n 2

2·x +l w.r.t. x is

(A) )1x(2

22

)1x(n 2

+

+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) ln2 x (C)2�nx

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 , then 1 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

gf

+ (B) ( )2)x(1

)x()x(2

g

gf

+ (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

[11]Quest Tutorials


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 = ∞→x

Lim ∫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) I

2(B)

I

4(C)

I

2(D) I

Q.81 The value of ∫ ∑∏

+

+

==

1

0

n

1k

n

1r

dxkx

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

[12]Quest Tutorials


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 → ∞

11

1 2 3 3 1n

n

n

n

n

n

n

n

n n+

++

++

++ +

+ −

.......( )


(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( fh = 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 − 2 x)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+=∫

[13]Quest Tutorials


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 = ∫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)1x4x(

)1x2(2/32

(A) C)1x4x(

x2/12

3

+++ (B) C

)1x4x(

x2/12

+++

(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

−

−∫ 2

x1

x21tan

dx

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

[14]Quest Tutorials


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

2x4

2

∫ ++

is equal to

(A) Cx2

2xtan

2

1 21 +

+−(B) C)2x(tan

2

1 21 ++−

(C) C2x

x2tan

2

12

1 +−

−(D) C

x2

2xtan

2

1 21 +

−−

Q.100 If β + 2 x e x2

0

12−

∫ dx = e x−∫

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<<π

π≤<

x4

3x

9

2sin2

4

3x01

(D) l (x) =

π<<π

π+π

π≤<

x2

,)xsin(2

2x0,xsinx

Q.103 If f (x) = ∫π

+022 tsinxtan1

dttsint for 0 < x <

2

π

(A) f (0+) = – π (B) 84

f2π

=

π

(C) f is continuous and differentiable in

π2

,0

(D) f is continuous but not differentiable in

π2

,0

[15]Quest Tutorials


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 3

�n 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

− − 4

10e1

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 of p 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 If 0

2

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

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

[16]Quest Tutorials


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

[17]Quest Tutorials


Quest

Q.121 The true solution set of the inequality,

π+−− ∫

x

0

2 dz2

x6x5 > ∫π

0

2dxxsinx is :

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

Q.122 If ∫−

1

02x1

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(secsec 22

(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−1 2 (C)

1

2 tan−1

1

2(D)

1

2 tan−1 2

Q.1261xx

Limit→

1xx

x

− ∫x

x1

f(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

[18]Quest Tutorials


Quest

Q.128 The value of the definite integral 0

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.e2

21x1tan +

+−−

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(''fx

1

0

∫ is

(A) 2

23 −(B)

2

23 +(C) 1 (D)

22

31−

Q.132 If ∫ +2

xsin1 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) Un = 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

[19]Quest Tutorials


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 ratio

I

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 equation d

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

1xsin2 2 − +

xsin1

)1xsin2(xcos

++

then

( )∫ + dx)x('f)x(fex (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

[20]Quest Tutorials


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

+1x

2sin + C

(C) cos

+1x

2 + C (D) sin

+1x

2 + 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

2II I

2 < I

3III 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(f2

(B) ∫π

0

dx)x(f (C) ∫π

π0

dx)x(f (D) ∫π

π0

dx)x(f1

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

2f then the value of f '

π2

, is

(A) π (B) – π (C) 2π (D) 0

[21]Quest Tutorials


Quest

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

0

dxxsin

x5sin is

(A) 0 (B) 2

π(C) π (D) 2π


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 ∫ + xcos45

dx = 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

(s in )

π −

∫ dx (B) f xa

a

( )2

−∫ dx = 2. f x

a

( )2

0

∫ dx

(C) ( )f x

n

cos2

0

π

∫ dx = n. ( )f xcos2

0

π

∫ dx (D) f x c

b c

( )+−

∫0

dx = f xc

b

( )∫ dx

Q.154 The value of ( )2 3 3

1 2 2

2

20

1x x

x x x

+ +

+ + +∫( )

dx is :

(A)π4

+ 2 ln2 − tan−1 2 (B)π4

+ 2 ln2 − tan−1 1

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 − cos

sec

ec x

x + c (D) − e n x� tan−1

− cot x + c

where 'c' is constant of integration .

Q.156 Let f (x) =sin t

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

[22]Quest Tutorials


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

1

4+

(C) I2 =

π8

1

4− (D) I

3 =

π16

5

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

nx

x

2

d

x ; for n ∈ N , then :

(A) An + 1

= An

(B) Bn + 1

= Bn

(C) An + 1

− An = B

n + 1(D) B

n + 1 − B

n = A

n + 1

Q.1600

∞

∫x

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

x :

(A) π4

(B) π2

(C) is same as 0

∞

∫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

1 2

2

+ θθ

d θ , x ≥ 0 then :

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

π2 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

[23]Quest Tutorials


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 = f2 (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.

[24]Quest Tutorials


Quest

ANSWER KEY

Q.1AQ.2CQ.3BQ.4CQ.5C

Q.6CQ.7AQ.8DQ.9AQ.10D

Q.11DQ.12CQ.13AQ.14CQ.15C

Q.16BQ.17CQ.18AQ.19DQ.20A

Q.21CQ.22CQ.23AQ.24AQ.25B

Q.26AQ.27AQ.28BQ.29DQ.30C

Q.31AQ.32DQ.33AQ.34DQ.35B

Q.36CQ.37BQ.38CQ.39DQ.40C

Q.41CQ.42BQ.43DQ.44BQ.45D

Q.46BQ.47DQ.48CQ.49BQ.50A

Q.51DQ.52AQ.53AQ.54CQ.55A

Q.56BQ.57CQ.58DQ.59CQ.60A

Q.61CQ.62CQ.63AQ.64CQ.65D

Q.66CQ.67CQ.68BQ.69BQ.70A

Q.71BQ.72CQ.73CQ.74BQ.75B

Q.76BQ.77BQ.78AQ.79BQ.80A

Q.81DQ.82CQ.83CQ.84DQ.85C

Q.86CQ.87AQ.88BQ.89BQ.90C

Q.91DQ.92BQ.93AQ.94BQ.95B

Q.96AQ.97BQ.98BQ.99DQ.100A

Q.101BQ.102DQ.103CQ.104BQ.105D

Q.106BQ.107CQ.108AQ.109CQ.110B

Q.111CQ.112BQ.113AQ.114CQ.115C

Q.116CQ.117AQ.118DQ.119AQ.120B

Q.121DQ.122BQ.123AQ.124CQ.125B

Q.126BQ.127DQ.128DQ.129CQ.130B

Q.131DQ.132DQ.133CQ.134BQ.135A

Q.136BQ.137BQ.138CQ.139AQ.140C

Q.141BQ.142DQ.143DQ.144AQ.145D

Q.146DQ.147AQ.148BQ.149AQ.150B

Q.151A,CQ.152A,BQ.153A,B,C,DQ.154A,C,D

Q.155B,C,DQ.156A,BQ.157A,BQ.158B,D

Q.159A,DQ.160A,CQ.161A,DQ.162A,B

Q.163A,B,CQ.164B,C,DQ.165A,B,C,DQ.166C,D

Q.167B,D

Q.168A,D

AREA UNDER THE CURVE

&

DIFFERENTIAL EQUATION

MATHEMATICS

TARGET IIT JEE

Quest Tutorials

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

Quest


Q.1 Area common to the curve y = 9 2−x & x² + y² = 6 x is :

(A) π + 3

4(B)

π − 3

4(C) 3 π +

3

4(D) 3 π −

3 3

4

Q.2 Spherical rain drop evaporates at a rate proportional to its surface area. The differential equation

corresponding to the rate of change of the radius of the rain drop if the constant of proportionality is

K > 0, is

(A) dr

dt + K = 0 (B)

dr

dt − K = 0 (C)

dr

dt = Kr (D) none

Q.3 If y = 2 sin x + sin 2 x for 0 ≤ x ≤ 2 π , then the area enclosed by the curve and the x-axis is :

(A) 9/2 (B) 8 (C) 9 (D) 4

Q.4 Number of values of m ∈ N for which y = emx is a solution of the differential equation

D3y – 3D2y – 4Dy + 12y = 0, is

(A) 0 (B) 1 (C) 2 (D) more than 2

Q.5 The area bounded by the curve y = x2 + 4x + 5 , the axes of co-ordinates & the minimum ordinate is:

(A) 32

3(B) 4

2

3(C) 5

2

3(D) none

Q.6 The general solution of the differential equation, y ′ + y φ ′ (x) − φ (x) . φ ′ (x) = 0 where φ (x) is a known

function is :

(A) y = ce− φ (x) + φ (x) − 1 (B) y = ce+ φ (x) + φ (x) − 1

(C) y = ce− φ (x) − φ (x) + 1 (D) y = ce− φ (x) + φ (x) + 1

where c is an arbitrary constant .

Q.7 The area bounded by the curve y = x2 − 1 & the straight line x + y = 3 is :

(A) 9

2(B) 4 (C)

2

177(D)

6

1717

Q.8 Orthogonal trajectories of family of the curve 323232ayx =+ , where 'a' is any arbitrary constant, is

(A) cyx3232 =− (B) cyx

3434 =− (C) cyx3434 =+ (D) cyx

3131 =−

Q.9 The area enclosed by the curve y2 + x4 = x2 is :

(A) 2

3(B)

4

3(C)

8

3(D)

3

10

Q.10 Equation of a curve passing through the origin if the slope of the tangent drawn at any of its point (x, y)

is cos(x + y) + sin(x + y), is

(A) y = 2 tan–1(ex – 1) + x (B) y = 2 tan–1(ex – 1) – x

(C) y = 2 tan–1(ex) – x (D) y = 2 tan–1(ex) + x

[3]Quest Tutorials


Quest

Q.11 The area enclosed between the curves y = sin x , y = cos x & the x-axis if 0 ≤ x ≤ π

2 is :

(A) 2 1− (B) 2 2− (C) 2 (D) ( )122 −

Q.12 The differential equation of all parabolas having their axis of symmetry coinciding with the axis of x has

its order and degree respectively:

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

Q.13 The area bounded by the curve y = x² + 1 & the tangents to it drawn from the origin is

(A) 2

3(B)

4

3(C)

1

3(D) 1

Q.14 Which one of the following functions is not homogeneous?

(A) f (x, y) = 22yx

yx

+

−(B) f (x, y) =

y

xtany·x 13

2

3

1

−−

(C) f (x, y) = x (ln 22yx + – ln y)+yex/y (D) f(x,y)=x

+−

+)yx(n

x

yx2n

22

ll +y2tanyx3

y2x

−

+

Q.15 The area enclosed by the curve y = x & x = – y , the circle x2 + y2 = 2 above the x-axis, is

(A) π

4(B)

3

2

π(C) π (D)

π

2

Q.16 Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known

that the rate at which the water level drops is proportional to the square root of water depth y, where the

constant of proportionality k > 0 depends on the acceleration due to gravity and the geometry of the

hole. If t is measured in minutes and k = 15

1 then the time to drain the tank if the water is 4 meter deep

to start with is

(A) 30 min (B) 45 min (C) 60 min (D) 80 min

Q.17 The area bounded by x² + y² − 2 x = 0 & y = sin πx

2 in the upper half of the circle is :

(A) π

π2

4− (B)

π

π4

2− (C) π

π−

8(D)

π−

π 2

2

Q.18 The solution to the differential equation y lny + xy' = 0, where y (1) = e, is

(A) x (ln y) = 1 (B) xy (ln y) = 1 (C) (ln y)2 = 2 (D) ln y + y2

x2

= 1

Q.19 The ratio in which the x-axis divides the area of the region bounded by the curves y = x2 − 4 x &

y = 2 x − x2 is :

(A) 4

23(B)

4

27(C)

4

19(D) none

Quest Tutorials


Quest

Q.20 A curve passes through the point 14

,π

& its slope at any point is given by

y

x − cos2

y

x

. Then the

curve has the equation

(A) y=x tan–1(ln x

e) (B) y=x tan–1(ln + 2) (C) y =

x

1tan–1(ln

x

e) (D) none

Q.21 The area enclosed by the curve y = (x − 1) (x − 2) (x − 3) between the co-ordinate axes and the

ordinate at x = 3 is :

(A) 9

2(B)

11

3(C)

11

4(D)

4

9

Q.22 The x-intercept of the tangent to a curve is equal to the ordinate of the point of contact. The equation of

the curve through the point (1, 1) is

(A) eey y

x

= (B) eex y

x

= (C) eex x

y

= (D) eey x

y

=

Q.23 The line y = mx bisects the area enclosed by the curve y = 1 + 4x − x2 & the lines x = 0, x = 3

2 &

y = 0. Then the value of m is:

(A) 13

6(B)

6

13(C)

3

2(D) 4

Q.24 The differential equation of all parabolas each of which has a latus rectum '4a' & whose axes are parallel

to x-axis is :

(A) of order 1 & degree 2 (B) of order 2 & degree 3

(C) of order 2 and degree 1 (D) of order 2 and degree 2

Q.25 The area bounded by the curve y = f (x), the x-axis & the ordinates x =1 & x = b is

(b − 1)sin(3b + 4). Then f (x) is:

(A) (x − 1) cos (3x + 4) (B) sin (3x + 4)

(C) sin (3x + 4) + 3 (x − 1) . cos (3x + 4) (D) none

Q.26 The foci of the curve which satisfies the differential equation (1 + y2) dx − xy dy = 0 and passes through

the point (1 , 0) are :

(A) ( )0,2± (B) ( )2,0 ± (C) (0, ± 1) (D) (± 2, 0)

Q.27 The area of the region for which 0 < y < 3 − 2x − x2 & x > 0 is :

(A) ( )3 2 2

1

3

− −∫ x x dx (B) ( )3 2 2

0

3

− −∫ x x dx

(C) ( )3 2 2

0

1

− −∫ x x dx (D) ( )3 2 2

1

3

− −∫ x x dx

[5]Quest Tutorials


Quest

Q.28 A function y = f (x) satisfies the condition f '(x) sin x + f (x) cos x = 1, f (x) being bounded when x → 0.

If I = ∫π 2

0

dx)x(f then

(A) 2

π < I <

4

2π(B)

4

π < I <

2

2π(C) 1 < I <

2

π(D) 0 < I < 1

Q.29 The area bounded by the curve y = f(x) , the co-ordinate axes & the line x = x1 is given by x

1 . 1xe .

Therefore f (x) equals:

(A) ex (B) x ex (C) xex − ex (D) x ex + ex

Q.30 A curve is such that the area of the region bounded by the co-ordinate axes, the curve & the ordinate of

any point on it is equal to the cube of that ordinate. The curve represents

(A) a pair of straight lines (B) a circle

(C) a parabola (D) an ellipse

Q.31 The limit of the area under the curve y = e−x from x = 0 to x = h as h → ∞ is :

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

e(D) 1

Q.32 Degree of the differential equation y = a ( )axe1 −− , a being the parameter is

(A) 1 (B) 2 (C) 3 (D) not applicables

Q.33 The slope of the tangent to a curve y = f (x) at (x , f (x)) is 2x + 1 . If the curve passes through the

point (1 , 2) then the area of the region bounded by the curve , the x-axis and the line x = 1 is :

(A) 5

6(B)

6

5(C)

1

6(D) 1

Q.34 A curve satisfying the initial condition, y(1) = 0, satisfies the differential equation, xdy

dx = y – x2. The area

bounded by the curve and the x-axis is

(A) 2

1(B)

3

1(C)

4

1(D)

6

1

Q.35 The graphs of f (x) = x2 & g(x) = cx3 (c > 0) intersect at the points (0, 0) &

2

c

1,

c

1. If the region which

lies between these graphs & over the interval [0, 1/c] has the area equal to 2/3 then the value of c is

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

Q.36 Number of straight lines which satisfy the differential equation dy

dx + x

dy

dx

2

− y = 0 is:

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

Quest Tutorials


Quest

Q.37 The area bounded by the curves y = − −x and x = − −y where x, y ≤ 0

(A) cannot be determined

(B) is 1/3

(C) is 2/3

(D) is same as that of the figure bounded by the curves y = −x ; x ≤ 0 and x = −y ; y ≤ 0

Q.38 The solution of the differential equation, (x + 2y3) dy

dx = y is :

(A) x

y2 = y + c (B) x

y = y2 + c (C)

x

y

2

= y2 + c (D) y

x = x2 + c

Q.39 The area bounded by the curves y = x (1 − ln x) ; x = e−1 and positive X-axis between x = e−1 and

x = e is :

(A) e e

2 24

5

−

−

(B) e e

2 25

4

−

−

(C) 4

5

2 2e e−

−

(D) 5

4

2 2e e−

−

Q.40 The real value of m for which the substitution, y = um will transform the differential equation,

2x4ydy

dx + y4 = 4x6 into a homogeneous equation is :

(A) m = 0 (B) m = 1 (C) m = 3/2 (D) no value of m

Q.41 The area bounded by the curves y = x (x − 3)2 and y = x is (in sq. units) :

(A) 28 (B) 32 (C) 4 (D) 8

Q.42 The solution of the differential equation, x2dy

dx.cos

1

x − y sin

1

x = − 1, where y → − 1 as x → ∞ is

(A) y = sin1

x – cos

1

x(B) y =

x

x x

+ 11sin

(C) y = cos1

x + sin

1

x(D) y =

x

x x

+ 11cos

Q.43 The positive values of the parameter 'a' for which the area of the figure bounded by the curve

y = cos ax, y = 0, x = π

6a, x =

5

6

π

a is greater than 3 are :

(A) φ (B) (0, 1/3) (C) (3, ∞) (D) none of these

Q.44 The equation of a curve passing through (1, 0) for which the product of the abscissa of a point P & the

intercept made by a normal at P on the x-axis equals twice the square of the radius vector of the point P, is

(A) x2 + y2 = x4 (B) x2 + y2 = 2 x4 (C) x2 + y2 = 4 x4 (D) none

Q.45 The curvilinear trapezoid is bounded by the curve y = x2 + 1 and the straight lines x=1 and x=2. The

co-ordinates of the point ( on the given curve) with abscissa x∈ [1,2] where tangent drawn cut off from

the curvilinear trapezoid an ordinary trapezium of the greatest area, is

(A) (1,2) (B) (2,5) (C)

4

13,

2

3(D) none

[7]Quest Tutorials


Quest

Q.46 The latus rectum of the conic passing through the origin and having the property that normal at each point

(x, y) intersects the x − axis at ((x + 1), 0) is :

(A) 1 (B) 2 (C) 4 (D) none

Q.47 The value of 'a' (a>0) for which the area bounded by the curves y = 2x

1

6

x+ , y = 0, x = a and

x = 2a has the least value, is

(A) 2 (B) 2 (C) 3/12 (D) 1

Q.48 The solution of the differential equation, 2 x2y dy

dx = tan (x2y2) − 2xy2 given y(1) =

π

2 is

(A) sinx2y2 = ex–1 (B) sin(x2y2) = x (C) cosx2y2 + x = 0 (D) sin(x2y2) = e.ex

Q.49 Area of the region enclosed between the curves x = y2 – 1 and x = |y| 2y1− is

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

Q.50 Solution of the differential equation, dx

dy=

x2y1

x4y21

++

−− is

(A) 4x2 + 4xy + y2 − 2x + 2y + c = 0 (B) 4x2 – 4xy – y2 − 2x − 2y + c = 0(C) 4x2 + 4xy + y2 + 2x + 2y + c = 0 (D) 4x2 + 4xy – y2 − 2x + 2y + c = 0

Q.51 Let y = g (x) be the inverse of a bijective mapping f : R → R f (x) = 3x3 + 2x. The area bounded by the

graph of g (x), the x-axis and the ordinate at x = 5 is :

(A) 4

5(B)

4

7(C)

4

9(D)

4

13

Q.52 The solution of the differential equation, dy

dx =

y x

y x

−

− − 1, given y (− 5) = − 5 represents

(A) a pair of straight lines (B) a circle

(C) parabola (D) hyperbola

Q.53 Area enclosed by the curves y = lnx ; y = ln | x | ; y = | ln x | and y = | ln | x | | is equal to

(A) 2 (B) 4 (C) 8 (D) cannot be determined

Q.54 If y = |xc|n

x

l (where c is an arbitrary constant) is the general solution of the differential equation

dx

dy =

x

y +

φ

y

x then the function

φ

y

x is :

(A) 2

2

y

x(B) – 2

2

y

x(C) 2

2

x

y(D) – 2

2

x

y

Q.55 If the tangent to the curve y = 1 – x2 at x = α, where 0 < α < 1, meets the axes at P and Q. Also α

varies, the minimum value of the area of the triangle OPQ is k times the area bounded by the axes and the

part of the curve for which 0 < x < 1 , then k is equal to

(A) 3

2(B)

16

75(C)

18

25(D)

3

2

Quest Tutorials


Quest

Q.56 If the function y = e4x + 2e–x is a solution of the differential equation Ky

dx

dy13

dx

yd3

3

=

−

then the value of

K is

(A) 4 (B) 6 (C) 9 (D) 12

Q.57 If (a, 0); a > 0 is the point where the curve y = sin2x – 3 sinx cuts the x-axis first, A is the area

bounded by this part of the curve , the origin and the positive x-axis, then

(A) 4A + 8 cosa = 7 (B) 4A + 8 sina = 7

(C) 4A – 8 sina = 7 (D) 4A – 8 cosa = 7

Q.58 A function y = f (x) satisfies (x + 1) . f ′ (x) – 2 (x2 + x) f (x) = )1x(

e2x

+ , 1x −>∀

If f (0) = 5 , then f (x) is

(A) 2xe.

1x

5x3

+

+(B)

2xe.1x

5x6

+

+(C)

2x

2e.

)1x(

5x6

+

+(D)

2xe.1x

x65

+

−

Q.59 The curve y = ax2 + bx + c passes through the point (1, 2) and its tangent at origin is the line y = x. The

area bounded by the curve, the ordinate of the curve at minima and the tangent line is

(A) 24

1(B)

12

1(C)

8

1(D)

6

1

Q.60 The differential equation whose general solution is given by,

y = ( ) )xsinc(ec)cxcos(c5

)cx(

321)( 4 +−+

+−, where c

1, c

2, c

3, c

4, c

5 are arbitrary constants, is

(A) 0ydx

yd

dx

yd2

2

4

4

=+− (B) 0ydx

dy

dx

yd

dx

yd2

2

3

3

=+++

(C) 0ydx

yd5

5

=+ (D) 0ydx

dy

dx

yd

dx

yd2

2

3

3

=−+−

Q.61 A function y = f (x) satisfies the differential equation dx

dy – y = cos x – sin x with initial condition that y is

bounded when x → ∞. The area enclosed by y = f (x), y = cos x and the y-axis is

(A) 12 − (B) 2 (C) 1 (D) 2

1

Q.62 The curve, with the property that the projection of the ordinate on the normal is constant and has a length

equal to 'a', is

(A) cyaynax 22 =

+−+ l (B) cyax 22 =−+

(C) (y – a)2 = cx (D) ay = tan–1 (x + c)

[9]Quest Tutorials


Quest

Q.63 Area bounded by the curve y = min {sin2x, cos2x}and x-axis between the ordinates x = 0 and x =4

5π is

(A) 2

5π square units (B)

4

)2(5 −π square units

(C) 8

)2(5 −π square units (D)

−

π

2

1

8 square units

Q.64 The equation to the orthogonal trajectories of the system of parabolas y = ax2 is

(A) 2

2

y2

x+ = c (B)

2

yx

22 + = c (C)

22

y2

x− = c (D)

2

yx

22 − = c

Q.65 If ∫x

a

dt)t(yt = x2 + y (x) then y as a function of x is

(A) y = 2 – (2 + a2) 2

ax 22

e

−

(B) y = 1 – (2 + a2) 2

ax 22

e

−

(C) y = 2 – (1 + a2) 2

ax 22

e

−

(D) none

Q.66 A curve y = f (x) passing through the point

e

1,1 satisfies the differential equation

dx

dy + 2

x2

ex−

=0.

Then which of the following does not hold good?

(A) f (x) is differentiable at x = 0.

(B) f (x) is symmetric w.r.t. the origin.

(C) f (x) is increasing for x < 0 and decreasing for x > 0.

(D) f (x) has two inflection points.

Q.67 The substitution y = zα transforms the differential equation (x2y2 – 1)dy + 2xy3dx = 0 into a homogeneous

differential equation for

(A) α = – 1 (B) 0 (C) α = 1 (D) no value of α.

Q.68 A curve passing through (2, 3) and satisfying the differential equation ∫x

0

dt)t(yt = x2y (x), (x >0) is

(A) x2 + y2 = 13 (B) y2 = 2

9x (C) 1

18

y

8

x 22

=+ (D) xy = 6

Q.69 Which one of the following curves represents the solution of the initial value problem

Dy = 100 – y, where y (0) = 50

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

Q.70 Solution of the differential equation

dx

dyyee

22 yx

+ + )xxy(e 2x2

− = 0, is

(A) 2xe (y2 – 1) +

2ye = C (B) 2ye (x2 – 1) +

2xe = C

(C) 2ye (y2 – 1) +

2xe = C (D) 2xe (y – 1) +

2ye = C

Quest Tutorials


Quest

Direction for Q.71 to Q.73 (3 question together)

Consider the function

f (x) = x3 – 8x2 + 20x – 13

Q.71 Number of positive integers x for which f (x) is a prime number, is

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

Q.72 The function f (x) defined for R → R

(A) is one one onto

(B) is many one onto

(C) has 3 real roots

(D) is such that f (x1) · f(x

2) < 0 where x

1 and x

2 are the roots of f ' (x) = 0

Q.73 Area enclosed by y = f (x) and the co-ordinate axes is

(A) 12

65(B)

12

13(C)

12

71(D) none

Q.74 The area enclosed by the curves y = cos x, y = 1 + sin 2x and x = 2

3π equals

(A) 2

3π – 2 (B)

2

3π(C) 2 +

2

3π(D) 1 +

2

3π

Q.75 The area of the region under the graph of y = xe–ax as x varies from 0 to ∞, where 'a' is a positive

constant, is

(A) a

1(B) 2a

1

a

1+ (C) 2

a

1

a

1− (D) 2a

1

Q.76 The polynomial f (x) satisfies the condition f (x + 1) = x2 + 4x. The area enclosed by y = f (x – 1) and the

curve x2 + y = 0, is

(A) 3

216(B)

3

16(C)

3

28(D) none


Q.77 Family of curves whose tangent at a point with its intersection with the curve xy = c2 form an angle of π

4

is

(A) y2 − 2xy − x2 = k (B) y2 + 2xy − x2 = k

(C) y = x - 2 c tan−1 x

c

+ k (D) y = c ln

c x

c x

+

− − x + k

where k is an arbitrary constant .

Q.78 The general solution of the differential equation, x dy

dx

= y . ln

y

x

is :

(A) y = xe1 − cx (B) y = xe1 + cx (C) y = ex . ecx (D) y = xecx

where c is an arbitrary constant.

[11]Quest Tutorials


Quest

Q.79 Which of the following equation(s) is/are linear.

(A) dy

dx +

y

x = ln x (B) y

dy

dx

+ 4x = 0 (C) dx + dy = 0 (D)

d y

dx

2

2 = cos x

Q.80 The function f(x) satisfying the equation, f2(x) + 4 f ′ (x) . f(x) + [f ′ (x)]2 = 0 .

(A) f(x) = c . ( )

e2 3- x

(B) f(x) = c . ( )

e2+ 3 x

(C) f(x) = c . ( )

e3 − 2 x

(D) f(x) = c . ( )

e2+ 3− x

where c is an arbitrary constant.

Q.81 The equation of the curve passing through (3 , 4) & satisfying the differential equation,

y

2

dx

dy

+ (x − y)

dx

dy – x = 0 can be

(A) x − y + 1 = 0 (B) x2 + y2 = 25 (C) x2 + y2 − 5x − 10 = 0 (D) x + y − 7 = 0

Q.82 The area bounded by a curve, the axis of co-ordinates & the ordinate of some point of the curve is equal

to the length of the corresponding arc of the curve. If the curve passes through the point P (0, 1) then the

equation of this curve can be

(A) y = 2

1(ex − e – x + 2) (B) y =

2

1(ex + e−x)

(C) y = 1 (D) y = xx ee

2−+

Q.83 Identify the statement(s) which is/are True.

(A) f(x , y) = ey/x + tan y

x is homogeneous of degree zero

(B) x . ln y

x dx +

y

x

2

sin−1y

x dy = 0 is homogeneous of degree one

(C) f(x , y) = x2 + sin x . cos y is not homogeneous

(D) (x2 + y2) dx - (xy2 − y3) dy = 0 is a homegeneous differential equation .

Q.84 The graph of the function y = f (x) passing through the point (0 , 1) and satisfying the differential equation

dx

dy + y cos x = cos x is such that

(A) it is a constant function (B) it is periodic

(C) it is neither an even nor an odd function (D) it is continuous & differentiable for all x .

Q.85 A function y = f (x) satisfying the differential equation dx

dy·sin x – y cos x + 2

2

x

xsin= 0 is such that,

y → 0 as x → ∞ then the statement which is correct is

(A) 0x

Lim→

f(x) = 1 (B) 0

2π /

∫ f(x) dx is less than π

2

(C) 0

2π /

∫ f(x) dx is greater than unity (D) f(x) is an odd function

Quest Tutorials


QuestSelect the correct alternative : (Only one is correct)

Q.1DQ.2AQ.3BQ.4CQ.5BQ.6AQ.7D

Q.8BQ.9BQ.10BQ.11BQ.12AQ.13AQ.14D

Q.15DQ.16CQ.17AQ.18AQ.19AQ.20AQ.21C

Q.22AQ.23AQ.24CQ.25CQ.26AQ.27CQ.28A

Q.29DQ.30CQ.31DQ.32DQ.33AQ.34DQ.35C

Q.36BQ.37BQ.38BQ.39BQ.40CQ.41DQ.42A

Q.43BQ.44AQ.45CQ.46BQ.47DQ.48AQ.49D

Q.50AQ.51DQ.52CQ.53BQ.54DQ.55AQ.56D

Q.57AQ.58BQ.59AQ.60BQ.61AQ.62AQ.63C

Q.64AQ.65AQ.66BQ.67AQ.68DQ.69BQ.70A

Q.71CQ.72BQ.73AQ.74CQ.75DQ.76A


Q.77A,B,C,DQ.78A,B,CQ.79A,C,DQ.80C,D

Q.81A,BQ.82B,CQ.83A,B,CQ.84A,B,D

Q.85A,B,C

ANSWER KEY

DETERMINANT

&

MATRICES

MATHEMATICS

TARGET IIT JEE

Quest Tutorials


Quest

Question bank on Determinant & Matrices



Q.1 The value of the determinant

a a

nx n x n x

nx n x n x

2 1

1 2

1 2

cos( ) cos( ) cos( )

sin( ) sin ( ) sin ( )

+ +

+ +

is independent of :

(A) n (B) a (C) x (D) a , n and x

Q.2 A is an involutary matrix given by A =

−−

−

433434110

then the inverse of 2

A will be

(A) 2A (B) 2

A 1−

(C) 2

A(D) A2

Q.3 If a, b, c are all different from zero &

1 1 1

1 1 1

1 1 1

+

+

+

a

b

c

= 0 , then the value of a−1 + b−1 + c−1 is

(A) abc (B) a−1 b−1 c−1 (C) −a − b − c (D) − 1

Q.4 If A and B are symmetric matrices, then ABA is

(A) symmetric matrix (B) skew symmetric

(C) diagonal matrix (D) scalar matrix

Q.5 If α, β & γ are real numbers , then D =

1

1

1

cos( ) cos( )

cos( ) cos( )

cos( ) cos( )

β α γ α

α β γ β

α γ β γ

− −

− −

− −

=

(A) − 1 (B) cos α cos β cos γ(C) cos α + cos β + cos γ (D) zero

Q.6 If A = cos sin

sin cos

θ θ

θ θ

−LNM

OQP , AA

–1 is given by

(A) –A (B) AT (C) –AT (D) A

Q.7 If the system of equations ax + y + z = 0 , x + by + z = 0 & x + y + cz = 0 (a, b, c ≠ 1) has a non-trivial

solution, then the value of 1

1

1

1

1

1−+

−+

−a b c is :

(A) − 1 (B) 0 (C) 1 (D) none of these

Q.8 Consider the matrices A =

−

−

521203164

, B =

− 211042

, C =

213

. Out of the given matrix products

(i) (AB)TC (ii) CTC(AB)T (iii) CTAB and (iv) ATABBTC

(A) exactly one is defined (B) exactly two are defined

(C) exactly three are defined (D) all four are defined

Quest Tutorials

North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439 [3]

Quest

Q.9 The value of a for which the system of equations ; a3x + (a +1)3 y + (a + 2)3 z = 0 ,

ax + (a + 1) y + (a + 2) z = 0 & x + y + z = 0 has a non-zero solution is :

(A) 1 (B) 0 (C) − 1 (D) none of these

Q.10 If A = 1

0 1

aFHGIKJ , then AAn (where n ∈ N) equals

(A) 1

0 1

naFHGIKJ (B)

1

0 1

2n aFHG

IKJ (C)

1

0 0

naFHGIKJ (D)

n na

n0

FHGIKJ

Q.11 Let f (x) =

1 4 2

1 4 2

1 4 2

2 2

2 2

2 2

+

+

+

sin cos sin

sin cos sin

sin cos sin

x x x

x x x

x x x

, then the maximum value of f (x) =

(A) 2 (B) 4 (C) 6 (D) 8

Q.12 If A = 3 4

1 6−

LNMOQP and B =

−LNMOQP

2 5

6 1 then X such that A + 2X = B equals

(A) 2 3

1 0−

LNMOQP (B)

3 5

1 0−

LNMOQP (C)

5 2

1 0−

LNMOQP (D) none of these

Q.13 If px4 + qx3 + rx2 + sx + t ≡

x x x x

x x x

x x x

23 1 3

1 2 3

3 4 3

+ − +

+ − −

− +

then t =

(A) 33 (B) 0 (C) 21 (D) none

Q.14 If A and B are invertible matrices, which one of the following statements is not correct

(A) Adj. A = |A| A–1 (B) det (A–1) = |det (A)|–1

(C) (A + B)–1 = B–1 + A–1 (D) (AB)–1 = B–1A–1

Q.15 If D =

a ab ac

ba b bc

ca cb c

2

2

2

1

1

1

+

+

+

then D =

(A) 1 + a2 + b2 + c2 (B) a2 + b2 + c2 (C) (a + b + c)2 (D) none

Q.16 If A = a b

c d

FHGIKJ satisfies the equation x2 – (a + d)x + k = 0, then

(A) k = bc (B) k = ad (C) k = a2 + b2 + c2 + d2 (D) ad–bc

Quest Tutorials


Quest

Q.17 If a, b, c > 0 & x, y, z ∈ R , then the determinant

( ) ( )( ) ( )( ) ( )

a a a a

b b b b

c c c c

x x x x

y y y y

z z z z

+ −

+ −

+ −

− −

− −

− −

2 2

2 2

2 2

1

1

1

=

(A) axbycz (B) a−xb−yc−z (C) a2xb2yc2z (D) zero

Q.18 Identify the incorrect statement in respect of two square matrices A and B conformable for sum and

product.

(A) tr(A + B) = t

r(A) + t

r(B) (B) t

r(αA) = α t

r(A), α ∈ R

(C) tr(AT) = t

r(A) (D) t

r(AB) ≠ t

r(BA)

Q.19 The determinant

cos ( ) sin ( ) cos

sin cos sin

cos sin cos

θ φ θ φ φ

θ θ φ

θ θ φ

+ − +

−

2

is :

(A) 0 (B) independent of θ

(C) independent of φ (D) independent of θ & φ both

Q.20 If A and B are non singular Matrices of same order then Adj. (AB) is

(A) Adj. (A) (Adj. B) (B) (Adj. B) (Adj. A)

(C) Adj. A + Adj. B (D) none of these

Q.21 If

a a a p

a a a q

a a a r

+ + +

+ + +

+ + +

1 2

2 3

3 4

= 0 , then p, q, r are in :

(A) AP (B) GP (C) HP (D) none

Q.22 Let A =

x x x

x x x

x x x

+

+

+

L

NMMM

O

QPPP

λ

λ

λ, then AA–1 exists if

(A) x ≠ 0 (B) λ ≠ 0

(C) 3x + λ ≠ 0, λ ≠ 0 (D) x ≠ 0, λ ≠ 0

Q.23 For positive numbers x, y & z the numerical value of the determinant

1

1

1

log log

log log

log log

x x

y y

z z

y z

x z

x y

is

(A) 0 (B) 1 (C) 3 (D) none

Q.24 If K ∈ R0 then det. {adj (KI

n)} is equal to

(A) Kn – 1 (B) Kn(n – 1) (C) Kn (D) K

Quest Tutorials


Quest


b c c a a b

b c c a a b

b c c a a b

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

+ + +

+ + +

+ + +

=

(A)

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

(B) 2

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

(C) 3

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

(D) 4

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

Q.26 Which of the following is an orthogonal matrix

(A)

−

−

7/27/67/37/67/37/27/37/27/6

(B)

−−

7/27/67/37/67/37/27/37/27/6

(C)

−

−−−

7/27/67/37/67/37/27/37/27/6

(D)

−−

−

7/37/27/67/37/27/2

7/37/27/6


1

1

1

+ + + +

+ + + +

+ + + +

a x a y a z

b x b y b z

c x c y c z

=

(A) (1 + a + b + c) (1 + x + y + z) − 3 (ax + by + cz)

(B) a (x + y) + b (y + z) + c (z + x) − (xy + yz + zx)

(C) x (a + b) + y (b + c) + z (c + a) − (ab + bc + ca)

(D) none of these

Q.28 Which of the following statements is incorrect for a square matrix A. ( | A | ≠ 0)

(A) If A is a diagonal matrix, A–1 will also be a diagonal matrix

(B) If A is a symmetric matrix, A–1 will also be a symmetric matrix

(C) If A–1 = A ⇒ A is an idempotent matrix

(D) If A–1 = A ⇒ A is an involutary matrix


x x x

y y y

z z z

C C C

C C C

C C C

1 2 3

1 2 3

1 2 3

=

(A) 1

3 xyz (x + y) (y + z) (z + x) (B)

1

4 xyz (x + y − z) (y + z − x)

(C) 1

12 xyz (x − y) (y − z) (z − x) (D) none

Q.30 Which of the following is a nilpotent matrix

(A)

10

01(B)

θθ

θ−θ

cossin

sincos(C)

01

00(D)

11

11

Quest Tutorials


Quest

Q.31 If a, b, c are all different and

a a a

b b b

c c c

3 4

3 4

3 4

1

1

1

−

−

−

= 0 , then :

(A) abc (ab + bc + ca) = a + b + c (B) (a + b + c) (ab + bc + ca) = abc

(C) abc (a + b + c) = ab + bc + ca (D) none of these


false.

Statement-1 : If A is an invertible 3 × 3 matrix and B is a 3 × 4 matrix, then A–1B is defined

Statement-2 : It is never true that A + B, A – B, and AB are all defined.

Statement-3 : Every matrix none of whose entries are zero is invertible.

Statement-4 : Every invertible matrix is square and has no two rows the same.

(A) TFFF (B) TTFF (C) TFFT (D) TTTF

Q.33 If ω is one of the imaginary cube roots of unity, then the value of the determinant

1

1

1

3 2

3

2

ω ω

ω ω

ω ω

=

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

Q.34 Identify the correct statement :

(A) If system of n simultaneous linear equations has a unique solution, then coefficient matrix is singular

(B) If system of n simultaneous linear equations has a unique solution, then coefficient matrix is non

singular

(C) If A–1 exists , (adjA)–1 may or may not exist

(D) F(x) =

−

000

0xcosxsin

0xsinxcos

, then F(x) . F(y) = F(x – y)

Q.35 If the determinant

a p x u f

b q m y v g

c r n z w h

+ + +

+ + +

+ + +

1

splits into exactly K determinants of order 3, each element of

which contains only one term, then the value of K, is

(A) 6 (B) 8 (C) 9 (D) 12

Q.36 A and B are two given matrices such that the order of A is 3×4 , if A′ B and BA′ are both defined

then

(A) order of B′ is 3 × 4 (B) order of B′A is 4 × 4

(C) order of B′A is 3 × 3 (D) B′A is undefined

Q.37 If the system of equations x + 2y + 3z = 4 , x + py + 2z = 3 , x + 4y + µz = 3 has an infinite number of

solutions , then :

(A) p = 2 , µ = 3 (B) p = 2 , µ = 4 (C) 3 p = 2 µ (D) none of these

Quest Tutorials


Quest

Q.38 If A =

ααα

ααα2

2

sincossin

cossincos ; B =

βββ

βββ2

2

sincossin

cossincos

are such that, AB is a null matrix, then which of the following should necessarily be an odd integral

multiple of 2

π.

(A) α (B) β (C) α – β (D) α + β

Q.39 Let D1 =

babadcdcbaba

−++

and D2 =

cbacadbdbcaca

++++

then the value of 2

1

D

D where b ≠ 0 and

ad ≠ bc, is

(A) – 2 (B) 0 (C) – 2b (D) 2b

Q.40 For a given matrix A =

θθθ−θ

cossin

sincos which of the following statement holds good?

(A) A = A–1 R∈θ∀ (B) A is symmetric, for θ = (2n + 1) 2

π, In∈

(C) A is an orthogonal matrix for θ ∈ R (D) A is a skew symmetric, for θ = nπ ; n ∈ I

Q.41 If a2 + b2 + c2 = – 2 and f (x) =

xc1x)b1(x)a1(

x)c1(xb1x)a1(

x)c1(x)b1(xa1

222

222

222

+++

+++

+++

then f (x) is a polynomial of degree

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

Q.42 Matrix A =

z22

4y123x

, if x y z = 60 and 8x + 4y + 3z = 20 , then A (adj A) is equal to

(A)

640006400064

(B)

880008800088

(C)

680006800068

(D)

340003400034

Q.43 The values of θ, λ for which the following equations

sinθx – cosθy + (λ+1)z = 0; cosθx + sinθy – λz = 0; λx +(λ + 1)y + cosθ z = 0

have non trivial solution, is

(A) θ = nπ, λ ∈ R – {0} (B) θ = 2nπ, λ is any rational number

(C) θ = (2n + 1)π, λ ∈ R+, n ∈ I (D) θ = (2n + 1)π

2, λ ∈ R, n ∈ I

Q.44 If A is matrix such that A2 + A + 2I = O, then which of the following is INCORRECT ?

(A) A is non-singular (B) A ≠ O (C) A is symmetric (D) A–1 = –2

1(A + I)

(Where I is unit matrix of order 2 and O is null matrix of order 2 )

Quest Tutorials


Quest

Q.45 The system of equations :

2x cos2θ + y sin2θ – 2sinθ = 0

x sin2θ + 2y sin2θ = – 2 cosθ

x sinθ – y cosθ = 0 , for all values of θ, can

(A) have a unique non - trivial solution (B) not have a solution

(C) have infinite solutions (D) have a trivial solution

Q.46 The number of solution of the matrix equation X2 =

32

11 is

(A) more than 2 (B) 2 (C) 1 (D) 0

Q.47 If x, y, z are not all simultaneously equal to zero, satisfying the system of equations

(sin 3 θ) x − y + z = 0

(cos 2 θ) x + 4 y + 3 z = 0

2 x + 7 y + 7 z = 0

then the number of principal values of θ is

(A) 2 (B) 4 (C) 5 (D) 6

Q.48 Let A + 2B =

−−

135336021

and 2A – B =

−−

210612512

then Tr (A) – Tr (B) has the value equal to

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

Q.49 For a non - zero, real a, b and c

b

acbb

aa

cba

ccc

ba

22

22

22

+

+

+

= α abc, then the values of α is

(A) – 4 (B) 0 (C) 2 (D) 4

Q.50 Given A =

2231

; I =

1001

. If A – λI is a singular matrix then

(A) λ ∈ φ (B) λ2 – 3λ – 4 = 0 (C) λ2 + 3λ + 4 = 0 (D) λ2 – 3λ – 6 = 0

Q.51 If the system of equations, a2 x − ay = 1 − a & bx + (3 − 2b) y = 3 + a possess a unique solution x = 1,

y = 1 then :

(A) a = 1 ; b = − 1 (B) a = − 1 , b = 1

(C) a = 0 , b = 0 (D) none

Quest Tutorials


Quest

Q.52 Let A =

θ−−θθ−

θ

1sin1sin1sin

1sin1

, where 0 ≤ θ < 2π, then

(A) Det (A) = 0 (B) Det A ∈ (0, ∞) (C) Det (A) ∈ [2, 4] (D) Det A ∈ [2, ∞)

Q.53 Number of value of 'a' for which the system of equations,

a2 x + (2 − a) y = 4 + a2

a x + (2 a − 1) y = a5 − 2 possess no solution is


Q.54 If A =

1a3321210

, AA–1 =

−−

−

2/12/32/5c342/12/12/1

, then

(A) a = 1, c = – 1 (B) a = 2, c = – 2

1(C) a = – 1, c = 1 (D) a =

2

1, c =

2

1

Q.55 Number of triplets of a, b & c for which the system of equations,

ax − by = 2a − b and (c + 1) x + cy = 10 − a + 3 b

has infinitely many solutions and x = 1, y = 3 is one of the solutions, is :

(A) exactly one (B) exactly two

(C) exactly three (D) infinitely many

Q.56 D is a 3 x 3 diagonal matrix. Which of the following statements is not true?

(A) D′ = D (B) AD = DA for every matrix A of order 3 x 3

(C) D–1 if exists is a scalar matrix (D) none of these

Q.57 The following system of equations 3x – 7y + 5z = 3; 3x + y + 5z = 7 and 2x + 3y + 5z = 5 are

(A) consistent with trivial solution (B) consistent with unique non trivial solution

(C) consistent with infinite solution (D) inconsistent with no solution

Q.58 If A1, A

3, ..... A

2n – 1 are n skew symmetric matrices of same order then B = ∑

=

−−

−n

1r

1r21r2)A)(1r2( will

be

(A) symmetric (B) skew symmetric

(C) neither symmetric nor skew symmetric (D) data is adequate

Q.59 The number of real values of x satisfying 1x126x172x71x3x41x21x22x3x

−+−+−−+

= 0 is

(A) 3 (B) 0 (C) more than 3 (D) 1

Q.60 Number of real values of λ for which the matrix A =

+λ−λ+λ−

+λλ−λ

723312

11

has no inverse


Quest Tutorials


Quest

Q.61 If D =

1 1

1 1

2

2

2

2 2

z z

x y

zy z

x x xy y z

x z

x y z

xz

y x y

xz

−+

−+

−+ + +

−+

( )

( )

( ) ( ) then, the incorrect statement is

(A) D is independent of x (B) D is independent of y

(C) D is independent of z (D) D is dependent on x, y, z

Q.62 If every element of a square non singular matrix A is multiplied by k and the new matrix is denoted by B

then | A–1| and | B–1| are related as

(A) | A–1| = k | B–1| (B) | A–1| = k

1| B–1| (C) | A–1| = kn | B–1| (D) | A–1| = k–n | B–1|

where n is order of matrices.

Q.63 If f ′ (x) = pn2mxpn2mxn2mx

pnpnn

pmxpmxmx

−++++

−+

+−

then y = f(x) represents

(A) a straight line parallel to x- axis (B) a straight line parallel to y- axis

(C) parabola (D) a straight line with negative slope

Q.64 Let A =

−

−

111312

111

and 10B =

−α−321

05224

. If B is the inverse of matrix A, then α is

(A) – 2 (B) – 1 (C) 2 (D) 5

Q.65 If D(x) = 32

32

32

)1x()1x(x

)1x(x1x

x)1x(1x

++

+−

−−

then the coefficient of x in D(x) is

(A) 5 (B) – 2 (C) 6 (D) 0

Q.66 The set of equations

λx – y + (cosθ) z = 0

3x + y + 2z = 0

(cosθ)x + y + 2z = 0

0 < θ < 2π , has non- trivial solution(s)

(A) for no value of λ and θ (B) for all values of λ and θ(C) for all values of λ and only two values of θ(D) for only one value of λ and all values of θ

Q.67 Matrix A satisfies A2 = 2A – I where I is the identity matrix then for n ≥ 2, An is equal to (n ∈ N)

(A) nA – I (B) 2n – 1A – (n – 1)I (C) nA – (n – 1)I (D) 2n – 1A – I

Quest Tutorials


Quest

Q.68 If a, b, c are real then the value of determinant 1cbcac

bc1bab

acab1a

2

2

2

+

+

+

= 1 if

(A) a + b + c = 0 (B) a + b + c = 1 (C) a + b + c = –1 (D) a = b = c = 0


I. There can exist two triangles such that the sides of one triangle are all less than 1 cm while the

sides of the other triangle are all bigger than 10 metres, but the area of the first triangle is larger

than the area of second triangle.

II. If x, y, z are all different real numbers, then

222 )xz(

1

)zy(

1

)yx(

1

−+

−+

− =

2

xz

1

zy

1

yx

1

−+

−+

−.

III. log3x · log

4x · log

5x = (log

3x · log

4x) + (log

4x · log

5x) + (log

5x · log

3x) is true for exactly for one

real value of x.

IV. A matrix has 12 elements. Number of possible orders it can have is six.

Now indicate the correct alternatively.

(A) exactly one statement is INCORRECT.

(B) exactly two statements are INCORRECT.

(C) exactly three statements are INCORRECT.

(D) All the four statements are INCORRECT.

Q.70 The system of equations (sinθ)x + 2z = 0, (cosθ)x + (sinθ)y = 0 , (cosθ)y + 2z = a has

(A) no unique solution

(B) a unique solution which is a function of a and θ

(C) a unique solution which is independent of a and θ

(D) a unique solution which is independent of θ only

Q.71 Let A =

120502321

and b =

−

130

. Which of the following is true?

(A) Ax = b has a unique solution. (B) Ax = b has exactly three solutions.

(C) Ax = b has infinitely many solutions. (D) Ax = b is inconsistent.

Q.72 The number of positive integral solutions of the equation

1zyzxz

zy1yxy

zxyx1x

322

232

223

+

+

+

= 11 is

(A) 0 (B) 3 (C) 6 (D) 12

Q.73 If A, B and C are n × n matrices and det(A) = 2, det(B) = 3 and det(C) = 5, then the value of the

det(A2BC–1) is equal to

(A) 5

6(B)

5

12(C)

5

18(D)

5

24

Quest Tutorials


Quest

Q.74 The equation

x32x21x

x51x31x2

)x2()x1()x1( 222

−+

−+

+−−+

+

3x22x3x21

x2x3)x1(

1x1x2)x1(2

2

−−−

−

+++

= 0

(A) has no real solution (B) has 4 real solutions

(C) has two real and two non-real solutions (D) has infinite number of solutions , real or non-real

Q.75 The value of the determinant ab2aba

baab2a

b2abaa

++

++

++

is

(A) 9a2 (a + b) (B) 9b2 (a + b) (C) 3b2 (a + b) (D) 7a2 (a + b)

Q.76 Let three matrices A =

1412

; B =

3243

and C =

−

−3243

then

tr(A) + t

r

2

ABC + t

r

4

)BC(A2

+ tr

8

)BC(A3

+ ....... + ∞ =

(A) 6 (B) 9 (C) 12 (D) none

Q.77 The number of positive integral solutions

λ+−

−λ−

λ−

122

23

121

= 0 is

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

Q.78 P is an orthogonal matrix and A is a periodic matrix with period 4 and Q = PAPT then X = PTQ2005P will

be equal to

(A) A (B) A2 (C) A3 (D) A4

Q.79 If x = a + 2b satisfies the cubic (a, b∈R) f (x)=

xabb

bxab

bbxa

−

−

−

=0, then its other two roots are

(A) real and different (B) real and coincident

(C) imaginary (D) such that one is real and other imaginary

Q.80 A is a 2 × 2 matrix such that A

−11

=

−

21

and AA2

−11

=

01

. The sum of the elements of A, is

(A) –1 (B) 0 (C) 2 (D) 5

Q.81 Three digit numbers x17, 3y6 and 12z where x, y, z are integers from 0 to 9, are divisible by a fixed

constant k. Then the determinant

2y1

z67

13x

must be divisible by

(A) k (B) k2 (C) k3 (D) None

Quest Tutorials


Quest

Q.82 In a square matrix A of order 3, ai i

's are the sum of the roots of the equation x2 – (a + b)x + ab= 0;

ai , i + 1

's are the product of the roots, ai , i – 1

's are all unity and the rest of the elements are all zero. The

value of the det. (A) is equal to

(A) 0 (B) (a + b)3 (C) a3 – b3 (D) (a2 + b2)(a + b)

Q.83 Let N = 834756653842382528

, then the number of ways is which N can be resolved as a product of two

divisors which are relatively prime is

(A) 4 (B) 8 (C) 9 (D) 16

Q.84 If A, B, C are the angles of a triangle and CsinCsinBsinBsinAsinAsin

Csin1Bsin1Asin1111

222 ++++++ = 0, then

the triangle is

(A) a equilateral (B) an isosceles

(C) a right angled triangle (D) any triangle

Q.85 Let a = xnx

1

xn

xLim

1x ll−

→ ; b = 2

3

0x xx4

x16xLim

+

−

→ ; c =

x

)xsin1(nLim

0x

+

→

l and

d = ( ))1x()1xsin(3

)1x(Lim

3

1x +−+

+

−→ , then the matrix

dcba

is

(A) Idempotent (B) Involutary (C) Non singular (D) Nilpotent

Q.86 If the system of linear equations

x + 2ay + az = 0

x + 3by + bz = 0

x + 4cy + cz = 0

has a non-zero solution, then a, b, c

(A) are in G..P. (B) are in H.P.

(C) satisfy a + 2b + 3c = 0 (D) are in A.P.


false.

Statement-1 : If the graphs of two linear equations in two variables are neither parallel nor the same,

then there is a unique solution to the system.

Statement-2 : If the system of equations ax + by = 0, cx + dy = 0 has a non-zero solution, then it has

infinitely many solutions.

Statement-3 : The system x + y + z = 1, x = y, y = 1 + z is inconsistent.

Statement-4 : If two of the equations in a system of three linear equations are inconsistent, then the

whole system is inconsistent.

(A) FFTT (B) TTFT (C) TTFF (D) TTTF

Q.88 Let A =

−−+−+

+−−+−

−+−−+

222

222

222

yxz1)xyz(2)yzx(2

)xyz(2xzy1)zxy(2

)yzx(2)zxy(2zyx1

then det. A is equal to

(A) (1 + xy + yz + zx)3 (B) (1 + x2 + y2 + z2)3

(C) (xy + yz + zx)3 (D) (1 + x3 + y3 + z3)2

Quest Tutorials


Quest


Q.89 The set of equations x – y + 3z = 2 , 2x – y + z = 4 , x – 2y + αz = 3 has

(A) unique soluton only for α = 0 (B) unique solution for α ≠ 8

(C) infinite number of solutions for α = 8 (D) no solution for α = 8

Q.90 Suppose a1, a

2, ....... real numbers, with a

1 ≠ 0. If a

1, a

2, a

3, ..........are in A.P. then

(A) A =

a a a

a a a

a a a

1 2 3

4 5 6

5 6 7

L

NMMM

O

QPPP

is singular

(B) the system of equations a1x + a

2y + a

3z = 0, a

4x + a

5y + a

6z = 0, a

7x + a

8y + a

9z = 0 has infinite

number of solutions

(C) B = a ia

ia a

1 2

2 1

LNM

OQP is non singular ; where i = 1−

(D) none of these


a a b c bc

b b c a ca

c c a b ab

2 2 2

2 2 2

2 2 2

− −

− −

− −

( )

( )

( )

is divisible by :

(A) a + b + c (B) (a + b) (b + c) (c + a)

(C) a2 + b2 + c2 (D) (a − b) (b − c) (c − a)

Q.92 If A and B are 3 × 3 matrices and | A | ≠ 0, then which of the following are true?

(A) | AB | = 0 ⇒ | B | = 0 (B) | AB | = 0 ⇒ B = 0

(C) | A–1 | = | A |–1 (D) | A + A | = 2 | A |

Q.93 The value of θ lying between −π

4 &

π

2 and 0 ≤ A A ≤

π

2 and satisfying the equation

1 2 4

1 2 4

1 2 4

2 2

2 2

2 2

+

+

+

sin cos sin

sin cos sin

sin cos sin

A A

A A

A A

θ

θ

θ

= 0 are :

(A) A = π

4 , θ = −

π

8(B) A =

3

8

π = θ

(C) A = π

5 , θ = −

π

8(D) A =

π

6 , θ =

3

8

π

Q.94 If AB = A and BA = B, then

(A) A2B = A2 (B) B2A = B2 (C) ABA = A (D) BAB = B

Q.95 The solution(s) of the equation

x a b

a x a

b b x

= 0 is/are :

(A) x = − (a + b) (B) x = a (C) x = b (D) − b

Quest Tutorials


Quest

Q.96 If D1 and D

2 are two 3 x 3 diagonal matrices, then

(A) D1D

2 is a diagonal matrix (B) D

1D

2 = D

2D

1

(C) D1

2 + D22 is a diagonal matrix (D) none of these

Q.97 If

1

1

2

2

2 2

a a

x x

b ab a

= 0 , then

(A) x = a (B) x = b (C) x = 1

a(D) x =

a

b

Q.98 Which of the following determinant(s) vanish(es)?

(A)

1

1

1

b c b c b c

ca ca c a

a b a b a b

( )

( )

( )

+

+

+

(B)

1 1 1

1 1 1

1 1 1

a b a b

b cb c

ca c a

+

+

+

(C)

0

0

0

a b a c

b a b c

c a c b

− −

− −

− −

(D)

log log log

log log

log log

x x x

y y

z z

xy z y z

xy z z

xy z y

1

1

Q.99 If A = a b

c d

LNMOQP (where bc ≠ 0) satisfies the equations x2 + k = 0, then

(A) a + d = 0 (B) k = –|A| (C) k = |A| (D) none of these

Q.100 The value of θ lying between θ = 0 & θ = π/2 & satisfying the equation :

θ+θθ

θθ+θ

θθθ+

4sin41cossin

4sin4cos1sin

4sin4cossin1

22

22

22

= 0 are :

(A) 7

24

π(B)

5

24

π(C)

11

24

π(D)

π

24

Q.101 If p, q, r, s are in A.P. and f (x) =

p x q x p r x

q x r x x

r x s x s q x

+ + − +

+ + − +

+ + − +

sin sin sin

sin sin sin

sin sin sin

1 such that 0

2

∫ f (x)dx = – 4 then

the common difference of the A.P. can be :

(A) − 1 (B) 1

2(C) 1 (D) none

Q.102 Let A =

1 2 2

2 1 2

2 2 1

L

NMMM

O

QPPP

, then

(A) A2 – 4A – 5I3 = 0 (B) A–1 =

1

5(A – 4I

3)

(C) A3 is not invertible (D) A2 is invertible

Quest Tutorials


Quest

ANSWER KEY

Q.1AQ.2AQ.3DQ.4AQ.5D

Q.6BQ.7CQ.8CQ.9CQ.10A

Q.11CQ.12DQ.13CQ.14CQ.15A

Q.16DQ.17DQ.18DQ.19BQ.20B

Q.21AQ.22CQ.23AQ.24BQ.25B

Q.26AQ.27AQ.28CQ.29CQ.30C

Q.31AQ.32CQ.33CQ.34BQ.35B

Q.36BQ.37DQ.38CQ.39AQ.40C

Q.41CQ.42CQ.43DQ.44CQ.45B

Q.46AQ.47CQ.48CQ.49DQ.50B

Q.51AQ.52CQ.53CQ.54AQ.55B

Q.56BQ.57BQ.58BQ.59CQ.60D

Q.61DQ.62CQ.63AQ.64DQ.65A

Q.66AQ.67CQ.68DQ.69AQ.70B

Q.71AQ.72BQ.73BQ.74DQ.75B

Q.76AQ.77CQ.78AQ.79BQ.80D

Q.81AQ.82DQ.83BQ.84BQ.85D

Q.86BQ.87BQ.88B

Q.89B, DQ.90A,B,CQ.91A,C,DQ.92A,C

Q.93A,B,C,DQ.94A,B,C,DQ.95A,B,CQ.96A,B,C

Q.97A,DQ.98A,B,C,DQ.99A,CQ.100A,C

Q.101A,CQ.102A,B,D

DETERMINANT

&

MATRICES

MATHEMATICS

TARGET IIT JEE

Quest Tutorials


Quest

Question bank on Determinant & Matrices



Q.1 The value of the determinant

a a

nx n x n x

nx n x n x

2 1

1 2

1 2

cos( ) cos( ) cos( )

sin( ) sin ( ) sin ( )

+ +

+ +

is independent of :

(A) n (B) a (C) x (D) a , n and x

Q.2 A is an involutary matrix given by A =

−−

−

433434110

then the inverse of 2

A will be

(A) 2A (B) 2

A 1−

(C) 2

A(D) A2

Q.3 If a, b, c are all different from zero &

1 1 1

1 1 1

1 1 1

+

+

+

a

b

c

= 0 , then the value of a−1 + b−1 + c−1 is

(A) abc (B) a−1 b−1 c−1 (C) −a − b − c (D) − 1

Q.4 If A and B are symmetric matrices, then ABA is

(A) symmetric matrix (B) skew symmetric

(C) diagonal matrix (D) scalar matrix

Q.5 If α, β & γ are real numbers , then D =

1

1

1

cos( ) cos( )

cos( ) cos( )

cos( ) cos( )

β α γ α

α β γ β

α γ β γ

− −

− −

− −

=

(A) − 1 (B) cos α cos β cos γ(C) cos α + cos β + cos γ (D) zero

Q.6 If A = cos sin

sin cos

θ θ

θ θ

−LNM

OQP , AA

–1 is given by

(A) –A (B) AT (C) –AT (D) A

Q.7 If the system of equations ax + y + z = 0 , x + by + z = 0 & x + y + cz = 0 (a, b, c ≠ 1) has a non-trivial

solution, then the value of 1

1

1

1

1

1−+

−+

−a b c is :

(A) − 1 (B) 0 (C) 1 (D) none of these

Q.8 Consider the matrices A =

−

−

521203164

, B =

− 211042

, C =

213

. Out of the given matrix products

(i) (AB)TC (ii) CTC(AB)T (iii) CTAB and (iv) ATABBTC

(A) exactly one is defined (B) exactly two are defined

(C) exactly three are defined (D) all four are defined

Quest Tutorials


Quest

Q.9 The value of a for which the system of equations ; a3x + (a +1)3 y + (a + 2)3 z = 0 ,

ax + (a + 1) y + (a + 2) z = 0 & x + y + z = 0 has a non-zero solution is :

(A) 1 (B) 0 (C) − 1 (D) none of these

Q.10 If A = 1

0 1

aFHGIKJ , then AAn (where n ∈ N) equals

(A) 1

0 1

naFHGIKJ (B)

1

0 1

2n aFHG

IKJ (C)

1

0 0

naFHGIKJ (D)

n na

n0

FHGIKJ

Q.11 Let f (x) =

1 4 2

1 4 2

1 4 2

2 2

2 2

2 2

+

+

+

sin cos sin

sin cos sin

sin cos sin

x x x

x x x

x x x

, then the maximum value of f (x) =

(A) 2 (B) 4 (C) 6 (D) 8

Q.12 If A = 3 4

1 6−

LNMOQP and B =

−LNMOQP

2 5

6 1 then X such that A + 2X = B equals

(A) 2 3

1 0−

LNMOQP (B)

3 5

1 0−

LNMOQP (C)

5 2

1 0−

LNMOQP (D) none of these

Q.13 If px4 + qx3 + rx2 + sx + t ≡

x x x x

x x x

x x x

23 1 3

1 2 3

3 4 3

+ − +

+ − −

− +

then t =

(A) 33 (B) 0 (C) 21 (D) none

Q.14 If A and B are invertible matrices, which one of the following statements is not correct

(A) Adj. A = |A| A–1 (B) det (A–1) = |det (A)|–1

(C) (A + B)–1 = B–1 + A–1 (D) (AB)–1 = B–1A–1

Q.15 If D =

a ab ac

ba b bc

ca cb c

2

2

2

1

1

1

+

+

+

then D =

(A) 1 + a2 + b2 + c2 (B) a2 + b2 + c2 (C) (a + b + c)2 (D) none

Q.16 If A = a b

c d

FHGIKJ satisfies the equation x2 – (a + d)x + k = 0, then

(A) k = bc (B) k = ad (C) k = a2 + b2 + c2 + d2 (D) ad–bc

Quest Tutorials


Quest

Q.17 If a, b, c > 0 & x, y, z ∈ R , then the determinant

( ) ( )( ) ( )( ) ( )

a a a a

b b b b

c c c c

x x x x

y y y y

z z z z

+ −

+ −

+ −

− −

− −

− −

2 2

2 2

2 2

1

1

1

=

(A) axbycz (B) a−xb−yc−z (C) a2xb2yc2z (D) zero

Q.18 Identify the incorrect statement in respect of two square matrices A and B conformable for sum and

product.

(A) tr(A + B) = t

r(A) + t

r(B) (B) t

r(αA) = α t

r(A), α ∈ R

(C) tr(AT) = t

r(A) (D) t

r(AB) ≠ t

r(BA)


cos ( ) sin ( ) cos

sin cos sin

cos sin cos

θ φ θ φ φ

θ θ φ

θ θ φ

+ − +

−

2

is :

(A) 0 (B) independent of θ

(C) independent of φ (D) independent of θ & φ both

Q.20 If A and B are non singular Matrices of same order then Adj. (AB) is

(A) Adj. (A) (Adj. B) (B) (Adj. B) (Adj. A)

(C) Adj. A + Adj. B (D) none of these

Q.21 If

a a a p

a a a q

a a a r

+ + +

+ + +

+ + +

1 2

2 3

3 4

= 0 , then p, q, r are in :

(A) AP (B) GP (C) HP (D) none

Q.22 Let A =

x x x

x x x

x x x

+

+

+

L

NMMM

O

QPPP

λ

λ

λ, then AA–1 exists if

(A) x ≠ 0 (B) λ ≠ 0

(C) 3x + λ ≠ 0, λ ≠ 0 (D) x ≠ 0, λ ≠ 0

Q.23 For positive numbers x, y & z the numerical value of the determinant

1

1

1

log log

log log

log log

x x

y y

z z

y z

x z

x y

is

(A) 0 (B) 1 (C) 3 (D) none

Q.24 If K ∈ R0 then det. {adj (KI

n)} is equal to

(A) Kn – 1 (B) Kn(n – 1) (C) Kn (D) K