Question(s) from Search: IIT

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)

If y = y(x) satisfies the differential equation $8\sqrt{x}\sqrt{9+\sqrt{x}}\mathrm{dy}={\left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)}^{-1}\mathrm{dx},x>0$

and $y\left(0\right)=\sqrt{7}$ Then y(256) =

a) 16

b) 3

c) 9

d) 80

A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4 and the probability that the lot contains exactly three defective articles is 0.6. Articles are drawn from the lot at random one by one without replacement and tested till defective articles are found. What is the probability that the testing will end at the 12^th testing?

If the curve y = f(x) passes through the point (1, −1) and satisfies the differential equation y(1 + xy) dx = xdy then $f\left(\frac{-1}{2}\right)$

is equal to

a) $\frac{-2}{5}$

b) $\frac{-4}{5}$

c) $\frac{2}{5}$

d) $\frac{4}{5}$

One or more than one correct options

Let f : (0, ∞) → ℝ be a differentiable function such that $f^{\prime }\left(x\right)=2-\frac{f(x)}{x}$

for all x ∈ (0, ∞) and f(1) ≠ 1. Then

a) $\underset{x\to 0^{+}{f}^{\prime }\left(\frac{1}{x}\right)=1}{\lim }$

b) $\underset{x\to 0^{+}x{f}^{\prime }\left(\frac{1}{x}\right)=2}{\lim }$

c) $\underset{x\to 0^{+}{x^{2}f}^{\prime }x=0}{\lim }$

d) $\left|f\left(x\right)\right|\le 2\mathrm{for}\mathrm{all}x\in (\mathrm{0,}2)$

If , i = 1, 2, 3 are polynomials in x such that  and

F(x) =  
then (x) at x = a is equal to

a) – 1

b) 0

c) 1

d) 2

If  then f (x) increases in

a) (−2, 2)

b) No value of x

c) (0, ∞)

d) (−∞, 0)

A curve passes through the point $\left(\mathrm{1,}\frac{\pi }{6}\right)$

. Let the slope of the curve at each point (x, y) is $\frac{y}{x}+\sec \left(\frac{y}{x}\right)$ , x > 0. Then the equation of the curve is

a) $\sin \left(\frac{y}{x}\right)=\mathrm{lnx}+\frac{1}{2}$

b) $\mathrm{cosec}\left(\frac{y}{x}\right)=lnx+2$

c) $\sec \left(\frac{2y}{x}\right)=\mathrm{tanx}+2$

d) $\cos \frac{2y}{x}=lnx+\frac{1}{2}$

The points  in the complex plane are the vertices of a parallelogram if and only if

a)

b)

c)

d) None of these