Question(s) from Search: IIT

Each of the following four inequalities given below define a region in the XY–plane. One of these four regions does not have the following property: For any two points (x₁, y₁) and (x₂, y₂) in the region, point  is also in the region. The inequality defining the region that does not have this property is

a) x² + 2y² ≤ 1

b) max (|x|, |y|) ≤ 1

c) x² – y² ≥ 1

d) y² – x ≤ 0

Let a hyperbola pass through the foci of the ellipse  . The transverse and conjugate axes of the hyperbola coincide with the major and minor axes of the given ellipse. Also the product of the eccentricity of the given ellipse and hyperbola is 1 then

a) Equation of the hyperbola is

b) Equation of the hyperbola is

c) Focus of the hyperbola is (5, 0)

d) Vertex of the hyperbola is

The integral $\underset{2}{\overset{4}{\int }}\frac{{\mathrm{logx}}^2}{{\mathrm{logx}}^2+\log \left(x^2-12x+36\right)}\mathrm{dx}$

is equal to

a) 2

b) 4

c) 1

d) 6

Fifteen coupons are numbered 1, 2, 3, .  .  .   ., 15 respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9 is

a)

b)

c)

d) None of these

Match the statement of column 1 and the properties of column 2

The value of the integral $\underset{\sqrt{\log 2}}{\overset{\sqrt{\log 3}}{\int }}\frac{x\sin x^2}{\sin x^2+\sin \left(\log 6-x^2\right)}\mathrm{dx}$

is equal to

a) $\frac{1}{4}\log \frac{3}{2}$

b) $\frac{1}{2}\log \frac{3}{2}$

c) $\log \frac{3}{2}$

d) $\frac{1}{6}\log \frac{3}{2}$

Let g(x) be a function of x defined on (−1, 1). If the area of the equilateral triangle with two of its vertices as (0, 0) and [x, g(x)] is , then the function g(x) is

a)

b)

c)

d) None of the above

Show that the integral of   is

A circle C of radius 1 is inscribed in an equilateral triangle PQR. The point of contacts of C with its sides PQ, QR and RP are D, E, F respectively. The line PQ is given by  and the point D is . Further, it is given that the origin and the centre of C are on the same side of the line PQ. The equation of circle C is

a)

b)

c)

d)

One or more than one correct options

Let F : ℝ → (0, 1) be a continuous function. Then which of the following function(s) has (have) the value zero at some point in the interval (0, 1)?

a) $e^x-\underset{0}{\overset{x}{\int }}f\left(t\right)\mathrm{sint}\mathrm{dt}$

b) $f\left(x\right)+\underset{0}{\overset{\frac{\pi }{2}}{\int }}f\left(t\right)\mathrm{sint}\mathrm{dt}$

c) $x-\underset{0}{\overset{\frac{\pi }{2}-x}{\int }}f\left(t\right)\mathrm{cost}\mathrm{dt}$

d) $x^9-f(x)$

Consider a branch of the hyperbola
 
with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of triangle ABC is

a)

b)

c)

d)

One or more than one correct options

The value(s) of $\underset{0}{\overset{1}{\int }}\frac{x^4{(1-x)}^4}{{1+x}^2}\mathrm{dx}$

is (are)

a) $\frac{22}{7}-\pi$

b) $\frac{2}{105}$

c) $0$

d) $\frac{71}{15}-\frac{3\pi }{2}$

 =

a) True

b) False

For non-zero vectors a, b, c,  holds if and only if

a) a . b = 0, b . c = 0

b) b . c = 0, c . a = 0

c) c . a = 0, a . b = 0

d) a . b = 0, b . c = 0, c . a = 0

$\underset{-2}{\overset{2}{\int }}\frac{{3x}^2}{1+e^x}\mathrm{dx}$

equals

a) 8

b) 2

c) 4

d) 0

The value of $\underset{0}{\overset{1}{\int }}{4x}^3\left[\frac{d^2}{{\mathrm{dx}}^2}{\left(1-x^2\right)}^5\right]\mathrm{dx}$

is

a) 4

b) 0

c) 2

d) 6

Let f be a non-negative function defined on the interval [0, 1]. If $\underset{0}{\overset{x}{\int }}\sqrt{1-{\left(f^{\prime }(t)\right)}^2}\mathrm{dt}=\underset{0}{\overset{x}{\int }}f\left(t\right)\mathrm{dt},0\le x\le 1$

and f(0) = 0, then

a) $f\left(\frac{1}{2}\right)<\frac{1}{2}\wedge f\left(\frac{1}{3}\right)>\frac{1}{3}$

b) $f\left(\frac{1}{2}\right)>\frac{1}{2}\wedge f\left(\frac{1}{3}\right)>\frac{1}{3}$

c) $f\left(\frac{1}{2}\right)<\frac{1}{2}\wedge f\left(\frac{1}{3}\right)<\frac{1}{3}$

d) $f\left(\frac{1}{2}\right)>\frac{1}{2}\wedge f\left(\frac{1}{3}\right)<\frac{1}{3}$

(One or more correct answers)
If E and F are independent events such that 0 < P (E) < 1 and 0 < P (F) < 1 then

a) E and F are mutually exclusive

b) E and  are independent

c)  are independent

d)

Match the following
Let  

Let p be the first of the n Arithmetic Means between two numbers and q be the first of n Harmonic Means between the same numbers. Then show that q does not lie between p and

a) – 1

b) 2

c) 1 + e⁻¹

d) None of these

One or more than one correct answer

Let P and Q be distinct points on the parabola y² = 2x such that the circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of triangle OPQ is $3\sqrt{2}$

then which of the following is/are the coordinates of P?

a) $\left(\mathrm{4,}2\sqrt{2}\right)$

b) $\left(\mathrm{9,}3\sqrt{2}\right)$

c) $\left(\frac{1}{4},\frac{1}{\sqrt{2}}\right)$

d) $\left(\mathrm{1,}\sqrt{2}\right)$

The area (in square units) of the region described by A = {(x, y) : x² + y² ≤ 1 and y² ≤ 1 – x} is

a) $\frac{\pi }{2}+\frac{4}{3}$

b) $\frac{\pi }{2}-\frac{4}{3}$

c) $\frac{\pi }{2}-\frac{2}{3}$

d) $\frac{\pi }{2}+\frac{2}{3}$

If the straight line x = b divides the area enclosed by y = (1 – x)² , y = 0 and x = 0 into two parts R₁ (0 ≤ x ≤ b) and R₂ (b ≤x ≤ 1) such that $R_1-R_2=\frac{1}{4}$

then b equals

a) $\frac{3}{4}$

b) $\frac{1}{2}$

c) $\frac{1}{3}$

d) $\frac{1}{4}$

Let f(x) be differentiable on the interval (0, ∞) such that f (1) = 1 and  for each x > 0. Then f(x) is

a)

b)

c)

d)