Question(s) from Search: IIT

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

If  and  where 0 < x ≤1, then in this interval

a) Both f (x) and g (x) are increasing functions

b) Both f (x) and g (x) are decreasing functions

c) f (x) is an increasing function

d) g (x) is an increasing function

The number of common tangents to the circles x² + y² – 4x − 6y – 12 = 0 and x² + y² + 6x + 18y + 26 = 0 is

a) 1

b) 2

c) 3

d) 4

Let p ≥ 3 be an integer and α, β be the roots of x² – (p + 1) x + 1 = 0. Using mathematical induction show that αⁿ + βⁿ
i) is an integer
ii) and is not divisible by p.

The function  is not differentiable at

a) – 1

b) 0

c) 1

d) 2

One or more than one correct option

Let RS be a diameter of the circle x² + y² = 1 where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and the tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersect a line drawn through Q parallel to RS at a point E. Then the locus of E passes through the point(s)

a) $\left(\frac{1}{3},\frac{1}{\sqrt{3}}\right)$

b) $\left({\frac{1}{4},}^{}\frac{1}{2}\right)$

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

d) $\left(\frac{1}{4},-\frac{1}{2}\right)$

If x is not an integral multiple of 2π use mathematical induction to prove that
 

A circle passing through (1, −2) and touching the axis of X at (3, 0) also passes through the point

a) (−5, 2)

b) (2, −5)

c) (5, −2)

d) (−2, 5)

The circles  and  intersect each other in distinct points if

a) r < 2

b) r > 8

c) 2 < r < 8

d) 2 ≤ r ≤ 8

Prove by induction that
P_n = Aαⁿ + Bβⁿ for all n ≥ 1
Where α and β are roots of the quadratic equation
x² – (1 – P) x – P (1 – P) = 0,
P₁ = 1, P₂ = 1 – P², .  .  .,
P_n = (1 – P) P_{n – 1} + P (1 – P) P_{n – 2}  n ≥ 3,
and ,