Question(s) from Search: IIT

The area of the region $\left\{\left(x,y\right)\mathrm{:}x\ge \mathrm{0,}x+y\le \mathrm{3,}{x}^2<4y\wedge y\le 1+\sqrt{x}\right\}$

is

a) $\frac{59}{12}$

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

c) $\frac{78}{3}$

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

The total number of local maximum and minimum of the function
is

a) 0

b) 1

c) 2

d) 3

The area enclosed by the curve y = sinx + cosx and y = |cosx – sinx| over the interval $\left[\mathrm{0,}\frac{\pi }{2}\right]$

is

a) $4\left(\sqrt{2}-1\right)$

b) $2\sqrt{2}\left(\sqrt{2}-1\right)$

c) $2\left(\sqrt{2}-1\right)$

d) $2\sqrt{2}\left(\sqrt{2}+1\right)$

If  and b_n = 1 – a_n then find the least natural number n₀ such that b_n > a_n for all n ≥ n₀

If  are unit coplanar vectors then the scalar triple product  

a) 0

b) 1

c)

d)

One or more than one correct option

If the line x = α divides the area of the region R = {(x, y) ∈ ℝ² : x³ ≤ y ≤ x, 0 ≤ x ≤ 1 into two equal parts then

a) ${2\alpha }^4-{4\alpha }^2+1=0$

b) ${\alpha }^4+{4\alpha }^2-1=0$

c) $\frac{1}{2}<\alpha <1$

d) $0<\alpha <\frac{1}{2}$

The sides of a triangle inscribed in a given circle subtend angles α, β and γ at the centre. The minimum value of the Arithmetic mean of
 
 

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

If f(x) =  then on the interval [0, π]

a) tan  and  are both continuous

b) tan  and  are both discontinuous

c) tan  and  are both continuous

d) tan  is continuous but  is not

One or more than one correct option

A ray of light along $x+\sqrt{3}y=\sqrt{3}$

gets reflected upon reaching X- axis, the equation of the reflected ray is

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

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

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

d) $\sqrt{3}y=x-1$