Question(s) from Search: IIT

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  

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

One or more than one correct option

A solution curve of the differential equation $\left(x^2+\mathrm{xy}+4x+2y+4\right)\frac{\mathrm{dy}}{\mathrm{dx}}-y^2=\mathrm{0,}x>0$

passes through the point (1, 3), then the solution curve

a) Intersects y = x + 2 exactly at one point

b) Intersects y = x + 2 exactly at two points

c) Intersects y = (x + 2)²

d) Does not intersect y = (x + 3)²

The value of

a) –1

b) 0

c) 1

d) i

e) None of these

Let U₁ = 1, U₂ = 1, U_{n + 2} = U_{n + 1} + U_n, n > 1. Use mathematical induction to show that
 
for all integers n > 1

Let f(x) = (1 – x)² sin²x + x²Consider the statementsStatement 1: There exists some x ∈ ℝ such that f(x) + 2x = 2(1 + x²)Statement 2: There exists some x ∈ ℝ such that 2f(x) + 1 = 2x(x + 1)

a) Both 1 and 2 are true

b) 1 is true and 2 is false

c) 1 is false and 2 is true

d) Both 1 and 2 are false

Let z and ω be two complex numbers such that |z| ≤ 1, |ω| ≤ 1 and   then z equals

a) 1 or i

b) i or –i

c) 1 or –1

d) i or –1

Given
 
 
Prove that
 

The coordinates of the in centre of the triangle that has the co ordinates of the mid points of its sides as (0, 1), (1, 1) and (1, 0) is

a) $2+\sqrt{2}$

b) $2-\sqrt{2}$

c) $1+\sqrt{2}$

d) $1-\sqrt{2}$

Using mathematical induction, prove that

 for n > 1