Question(s) from Search: IIT

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  

Let y′(x) + y(x) g′(x) = g(x) g′(x), y(0) = 0, x ∈ ℝ where f′(x) denotes $\frac{d}{\mathrm{dx}}f(x)$

and g(x) is a given non constant differentiable function on ℝ with g(0) = g(2) = 0. Then the value of y(2) is

a) 1

b) 0

c) 2

d) 4

For a positive integer n, define
 then

a) a(100) ≤ 100

b) a(100) > 100

c) a(200) ≤ 100

d) a(200) > 100

Let f:[0, 1] → ℝ (the set all real numbers)be a function. Suppose the function is twice differentiable, f(0) = f(1) = 0 and satisfiesf′′(x) – 2f′(x) + f(x) ≥ e^x, x ∈ [0, 1]If the function e^−x f(x) assumes its minimum in the interval [0, 1] at $x=\frac{1}{4}$

then which of the following is true?

a) $f^{\prime }\left(x\right)<f\left(x\right),\frac{1}{4}<x<\frac{3}{4}$

b) $f^{\prime }\left(x\right)>f\left(x\right),0<x<\frac{1}{4}$

c) $f^{\prime }\left(x\right)<f\left(x\right),0<x<\frac{1}{4}$

d) $f^{\prime }\left(x\right)<f\left(x\right),\frac{3}{4}<x<1$

There exists a function f(x) satisfying f (0) = 1,  and

f (x) > 0 for all x and

a)   for all x

b)  

c)   for all x

d)   for all x

Let k be an integer such that the triangle with vertices (k, −3k), (5, k) and (−k, 2) has area 28 square units. Then the orthocentre of the triangle is at the point

a) $\left(2,-\frac{1}{2}\right)$

b) $\left(\mathrm{1,}\frac{3}{4}\right)$

c) $\left(1,-\frac{3}{4}\right)$

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

If p is a natural number then prove that p^{n + 1} + (p + 1)^{2n – 1} is divisible by p² + p + 1 for every positive integer n.

A straight line L through the point (3, −2) is inclined at an angle of 60° to the line $\sqrt{3}x+y=1$

. If the line L also intersects the X- axis then the equation of L is

a) $y+\sqrt{3}x+2-3\sqrt{3}=0$

b) $y-\sqrt{3}x+2+3\sqrt{3}=0$

c) $\sqrt{3}y-x+3+2\sqrt{3}=0$

d) $\sqrt{3}y+x-3+2\sqrt{3}=0$

The orthocenter of the triangle formed by the lines
  lies in the quadrant number . . . . .

Prove by mathematical induction that
 for every positive integer n.

The sides of a rhombus are along the lines x – y + 1 = 0 and 7x – y – 5 = 0. If its diagonals intersect at (−1, −2) then which one of the following is a vertex of the rhombus?

a) $(-\mathrm{3,}-9)$

b) $(-\mathrm{3,}-8)$

c) $\left(\frac{1}{3},-\frac{8}{3}\right)$

d) $\left(\frac{-10}{3},\frac{-7}{3}\right)$