Question(s) from Search: IIT

One or more than one correct option

Consider the family of circles whose centre lies on the straight line y = x. If the family of circles is represented by the differential equation Py′′ + Qy′ + 1 = 0 where P, Q are functions of x, y and y′ $\left(\mathrm{where}{y}^{\prime }=\frac{\mathrm{dy}}{\mathrm{dx}},y^{\prime \prime }=\frac{d^{2}y}{d{x}^2}\right)$

, then which of the following statements is/are true?

a) P = y + x

b) P = y – x

c) P + Q = 1 – x + y + y′ + (y′)²

d) P − Q = x + y − y′ − (y′)²

Find  at x = , when

a) 0

b) 1

c) – 1

d) 2

One or more than one correct options

Let f : (0, ∞) → ℝ be a differentiable function such that $f^{\prime }\left(x\right)=2-\frac{f(x)}{x}$

for all x ∈ (0, ∞) and f(1) ≠ 1. Then

a) $\underset{x\to 0^{+}{f}^{\prime }\left(\frac{1}{x}\right)=1}{\lim }$

b) $\underset{x\to 0^{+}x{f}^{\prime }\left(\frac{1}{x}\right)=2}{\lim }$

c) $\underset{x\to 0^{+}{x^{2}f}^{\prime }x=0}{\lim }$

d) $\left|f\left(x\right)\right|\le 2\mathrm{for}\mathrm{all}x\in (\mathrm{0,}2)$

If , i = 1, 2, 3 are polynomials in x such that  and

F(x) =  
then (x) at x = a is equal to

a) – 1

b) 0

c) 1

d) 2

If  then f (x) increases in

a) (−2, 2)

b) No value of x

c) (0, ∞)

d) (−∞, 0)

A curve passes through the point $\left(\mathrm{1,}\frac{\pi }{6}\right)$

. Let the slope of the curve at each point (x, y) is $\frac{y}{x}+\sec \left(\frac{y}{x}\right)$ , x > 0. Then the equation of the curve is

a) $\sin \left(\frac{y}{x}\right)=\mathrm{lnx}+\frac{1}{2}$

b) $\mathrm{cosec}\left(\frac{y}{x}\right)=lnx+2$

c) $\sec \left(\frac{2y}{x}\right)=\mathrm{tanx}+2$

d) $\cos \frac{2y}{x}=lnx+\frac{1}{2}$

The points  in the complex plane are the vertices of a parallelogram if and only if

a)

b)

c)

d) None of these

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]Which of the following is true?

a) $f\left(x\right)<\infty$

b) $\frac{-1}{2}<f\left(x\right)<\frac{1}{2}$

c) $\frac{-1}{4}<f\left(x\right)<1$

d) $-\infty <f\left(x\right)<0$

If ω(≠1) is a cube root of unity and  then A and B are respectively

a) 0, 1

b) 1, 1

c) 1, 0

d) – 1, 1

If (1 + x)ⁿ = C₀ + C₁x + C₂x² + .  .  . + C_nxⁿ, then show that the sum of the products of the C_j’s is taken two at a time represented by
 is equal to

Let a, b, c and d be non-zero real numbers. If the point of intersection of lines 4ax + 2ay + c = 0 and 5bx + 2by + d = 0 lie in the fourth quadrants and is equidistant from the two axes, then

a) 2bc – 3ad = 0

b) 2bc + 3ad = 0

c) 2ad – 3bc = 0

d) 3bc + 2ad = 0

One or more than one correct option

Let α, λ, μ ∈ ℝ. Consider the system of linear equations αx + 2y = λ 3x – 2y = μWhich of the following statements is/are correct?

a) If α = −3, then the system has infinitely many solutions for all values of λ and μ

b) If α ≠ −3, then the system of equations has a unique solution for all values of λ and μ

c) If λ + μ = 0, then the system has infinitely many solutions for α = −3

d) If λ + μ ≠ 0, then the system has no solution for α = −3

Let  and f = R – [R] where [ ] denotes the greatest integer function. Prove that Rf = 4^{2n + 4}

One or more than one correct option

Circle(s) touching X – axis at a distance 3 from the origin and having an intercept of length $2\sqrt{7}$

on Y – axis is/are

a) x² + y² – 6x + 8y + 9 = 0

b) x² + y² – 6x + 7y + 9 = 0

c) x² + y² – 6x − 8y + 9 = 0

d) x² + y² – 6x − 7y + 9 = 0

Using induction or otherwise, prove that for any non-negative integers m, n, r and k
 

Let V be the volume of the parallelepiped formed by the vectors  and . If a_r, b_r, c_r where r = 1, 2, 3 are non-negative real numbers and , show that V ≤ L³

One or more than one correct option

A circle S passes through the point (0, 1) and is orthogonal to the circles (x – 1)² + y² = 16 and x² + y² = 1, then

a) Radius of S is 8

b) Radius of S is 7

c) Centre of S is (−7, 1)

d) Centre of S is (−8, 1)

The locus of the midpoint of a chord of the circle  which subtend a right angle at the origin is

a)

b)

c)

d)

If n is a positive integer and 0 ≤ v < π then show that

A tangent PT is drawn to the circle x² + y² = 4 at the point $P\left(\sqrt{3},1\right)$

. A straight line L, perpendicular to PT is tangent to the circle (x – 3)² + y² = 1A possible equation of L is

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

b) $x+\sqrt{3}y=1$

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

d) $x+\sqrt{3}y=5$

Let 0 < A_i < π for i = 1, 2, .  .  . n. Use mathematical induction to prove that
 
where n ≥ 1 is a natural number.

The centre of those circles which touch the circle x² + y² – 8x – 8y = 0, externally and also touch the X- axis, lie on

a) A circle

b) An ellipse which is not a circle

c) A hyperbola

d) A parabola

Solve

 for every 0 < α, β < 2.