Question(s) from Search: IIT

The angle between the pair of tangents from a point P to the parabola y² = 4ax is 45°. Show that the locus of the point P is a hyperbola.

The integral $\underset{0}{\overset{\pi }{\int }}\sqrt{1+4{\sin }^2\frac{x}{2}-4\sin \frac{x}{2}}\mathrm{dx}$

is equal to

a) $\pi -4$

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

c) $4\sqrt{3}-4$

d) $4\sqrt{3}-4-\frac{\pi }{3}$

A box contains 24 identical balls of which 12 are white and 12 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the fourth time on the seventh draw is

a)

b)

c)

d)

Let F : ℝ → ℝ be a thrice differentiable function. Suppose that F(1) = 0, F(3) = −4 and F′(x) < 0 for all x ε (1, 3). Let f(x) = x F(x) for all x ε ℝ.The correct statement(s) is/are

a) f′(1) < 0

b) f(2) < 0

c) f′(x) ≠ 0 for every x ε (1, 3)

d) f′(x) = 0 for some x ε (1, 3)

Let A, B , C be three mutually independent events. Consider the two statements S₁ and S₂

S₁ : A and B ∪ Care independent

S₂  : A and B ∩ C are independent. Then

a) Both S₁ and S₂ are true

b) Only S₁ is true

c) Only S₂ is true

d) Neither S₁ nor S₂ is true

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. Equations of lines QR and RP are

a)

b)

c)

d)

Let f(x) = 7tan⁸x + 7tan⁶x – 3tan⁴x – 3tan²x for all $x\in \left(\frac{-\pi }{2},\frac{\pi }{2}\right)$

Then the correct expression(s) is (are)

a) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\mathrm{xf}\left(x\right)\mathrm{dx}=\frac{1}{12}$

b) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}f\left(x\right)\mathrm{dx}=0$

c) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\mathrm{xf}\left(x\right)\mathrm{dx}=\frac{1}{8}$

d) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}f\left(x\right)\mathrm{dx}=1$

Consider the lines
L₁: x + 3y – 5 = 0, L₂: 3x – ky – 1 = 0, L₃: 5x + 2y – 12 = 0.
Match the statement/expressions in column 1 with the statement/expression in column 2.

The number of quadratic polynomials f(x) with non-negative integer coefficients ≤ 3 satisfying f(0) = 0 and $\underset{0}{\overset{1}{\int }}f\left(x\right)\mathrm{dx}=1$

is

a) 8

b) 2

c) 4

d) 0

A function f : ℝ → ℝ, where ℝ is the set of real numbers, is defined by . Find the interval of values of α for which f is onto. Is the function one to one for α= 3? Justify your answer.

Let f : ℝ → ℝ be a function defined by $f\left(x\right)=\{\begin{array}{cc}[x]& x\le 2\\ 0& x>2\end{array}$

where [x] denotes the greatest integer less than or equal to x. If $I=\underset{-1}{\overset{2}{\int }}\frac{\mathrm{xf}\left(x^2\right)}{2+f(x+1)}\mathrm{dx}$ then the value of (4I – 1) is

a) 1

b) 3

c) 2

d) 0

Let f: [0, 2] → ℝ be a function which is continuous on [0, 2] and differentiable on (0, 2) with f(0) = 1. Let $F\left(x\right)=\underset{0}{\overset{x^2}{\int }}f\left(\sqrt{t}\right)\mathrm{dt}\mathrm{for}x\in [\mathrm{0,}2]$

. If F′(x) = f′(x) Ɐ x ∈ [0, 2] then F(2) equals

a) e² – 1

b) e⁴ – 1

c) e – 1

d) e²

(Multiple correct answers)

Let M and N are two events, the probability that exactly one of them occurs is

a) P (M) + P (N) − 2P (M ∩ N)

b) P (M) + P (N) − P ()

c)

d)

The area (in square units) of the region y² > 2x and x² + y² ≤ 4x, x ≥ 0, y > 0 is

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

b) $\pi -\frac{8}{3}$

c) $\pi -\frac{4\sqrt{2}}{3}$

d) $\frac{\pi }{2}-\frac{2\sqrt{2}}{3}$

Let f and g be real valued functions on (−1, 1) such that g’(x) is continuous, g(0) ≠ 0, g’(0) = 0, g’’(0) ≠ 0 and f(x) = g(x)sinx
Statement 1 -
Statement 2 – f’(0) = g(0)

a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1

b) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of statement 1

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true.

The area of the region $\left\{\left(x,y\right)\in R^2\mathrm{:}y>\sqrt{\left|x+3\right|},5y\le x+9\le 15\right\}$

is equal to

a) $\frac{1}{6}$

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

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

d) $\frac{5}{3}$

The area (in square units) bounded by the curves $y=\sqrt{x},2y-x+3=0$

, X – axis and lying in the first quadrant is

a) 9

b) 6

c) 18

d) $\frac{27}{4}$

One or more than one correct option

Let S be the area of the region enclosed by $y=e^{{-x}^2}$

, y = 0, x = 0 and x = 1, then

a) $S\ge \frac{1}{e}$

b) $S\ge 1-\frac{1}{e}$

c) $S\le \frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right)$

d) $S\le \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right)$

Show that the sum of the first n terms of the series
1² + 2.2² + 3² + 2.4² + 5² + 2.6² + .  .  .
is  when n is even, and  when n is odd.

Differentiate from first principles (or ab initio)

a) 2xcos(x² + 1)

b) xcos(x² + 1)

c) 2cosx(x² + 1)

d) 2xcosx(x² + 1) + sin(x² + 1)

One or more than one correct option

Let y(x) be a solution of the differential equation $\left(1+e^x\right)y^{\prime }+y{e}^x=1$

. If y(0) = 2, then which of the following statements is/are true?

a) y(−4) = 0

b) y(−2) = 0

c) y(x) has a critical point in the interval (−1, 0)

d) y(x) has no critical point in the interval

An urn contains two white and two black balls. A ball is drawn at random. If it is white it is not replaced in the urn. Otherwise it is placed along with the other balls of the same colour. The process is repeated. Find the probability that the third ball drawn is black?

Find the derivative with respect to x of the function

 at x =

a)

b)

c)

d)

The function y = f(x) is the solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{xy}}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}}$

in (−1, 1) satisfying f(0) = 0, then $\underset{\frac{-\sqrt{3}}{2}}{\overset{\frac{\sqrt{3}}{2}}{\int }}f\left(x\right)\mathrm{dx}$ is

a) $\frac{\pi }{3}-\frac{\sqrt{3}}{2}$

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

c) $\frac{\pi }{6}-\frac{\sqrt{3}}{4}$

d) $\frac{\pi }{6}-\frac{\sqrt{3}}{2}$

Solve