Question(s) from Search: IIT

Fill in the blank

The system of equations
 
 
 
will have a non-zero solution if real value of λ is given by …………

The function  is not one to one

a) True

b) False

For any real number x, let [x] denote the greater integer less than or equal to x. Let f be a real valued function defined on the interval [−10, 10] by $f\left(x\right)=\{\begin{array}{cc}x-[x]& \mathrm{if}\left[x\right]\mathrm{is}\mathrm{odd}\\ 1+\left[x\right]-x& \mathrm{if}\left[x\right]\mathrm{is}\mathrm{even}\end{array}$

then the value of $\frac{{\pi }^2}{10}\underset{-10}{\overset{10}{\int }}f\left(x\right)\mathrm{cosx\pi dx},$ is

a) 2

b) 0

c) 6

d) 4

Let  denotes the complement of an event E. Let E, F, G are pair wise independent events with P (G) > 0 and P (E ∩ F ∩ G) = 0 then  equals

a)

b)

c)

d)

Let A be a set of n distinct elements. Then find the total number of distinct functions from A to A is and out of these onto functions are .  .  .

$\underset{n\to \infty }{\lim }{\left(\frac{\left(n+1\right)\left(n+2\right)...3n}{n^{2n}}\right)}^{1/n}$

is equal to

a) $\frac{18}{e^4}$

b) $\frac{27}{e^2}$

c) $\frac{9}{e^2}$

d) $3\log 3-2$

(One or more correct answers)
For any two events in the sample space

a)  is always true

b)  does not hold

c) if A and B are independent

d)  if A and B are disjoint

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