Question(s) from Search: IIT

Let f be a non-negative function defined on the interval [0, 1]. If $\underset{0}{\overset{x}{\int }}\sqrt{1-{\left(f^{\prime }(t)\right)}^2}\mathrm{dt}=\underset{0}{\overset{x}{\int }}f\left(t\right)\mathrm{dt},0\le x\le 1$

and f(0) = 0, then

a) $f\left(\frac{1}{2}\right)<\frac{1}{2}\wedge f\left(\frac{1}{3}\right)>\frac{1}{3}$

b) $f\left(\frac{1}{2}\right)>\frac{1}{2}\wedge f\left(\frac{1}{3}\right)>\frac{1}{3}$

c) $f\left(\frac{1}{2}\right)<\frac{1}{2}\wedge f\left(\frac{1}{3}\right)<\frac{1}{3}$

d) $f\left(\frac{1}{2}\right)>\frac{1}{2}\wedge f\left(\frac{1}{3}\right)<\frac{1}{3}$

(One or more correct answers)
If E and F are independent events such that 0 < P (E) < 1 and 0 < P (F) < 1 then

a) E and F are mutually exclusive

b) E and  are independent

c)  are independent

d)

Match the following
Let  

Let p be the first of the n Arithmetic Means between two numbers and q be the first of n Harmonic Means between the same numbers. Then show that q does not lie between p and

a) – 1

b) 2

c) 1 + e⁻¹

d) None of these

One or more than one correct answer

Let P and Q be distinct points on the parabola y² = 2x such that the circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of triangle OPQ is $3\sqrt{2}$

then which of the following is/are the coordinates of P?

a) $\left(\mathrm{4,}2\sqrt{2}\right)$

b) $\left(\mathrm{9,}3\sqrt{2}\right)$

c) $\left(\frac{1}{4},\frac{1}{\sqrt{2}}\right)$

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

The area (in square units) of the region described by A = {(x, y) : x² + y² ≤ 1 and y² ≤ 1 – x} is

a) $\frac{\pi }{2}+\frac{4}{3}$

b) $\frac{\pi }{2}-\frac{4}{3}$

c) $\frac{\pi }{2}-\frac{2}{3}$

d) $\frac{\pi }{2}+\frac{2}{3}$

If the straight line x = b divides the area enclosed by y = (1 – x)² , y = 0 and x = 0 into two parts R₁ (0 ≤ x ≤ b) and R₂ (b ≤x ≤ 1) such that $R_1-R_2=\frac{1}{4}$

then b equals

a) $\frac{3}{4}$

b) $\frac{1}{2}$

c) $\frac{1}{3}$

d) $\frac{1}{4}$

Let f(x) be differentiable on the interval (0, ∞) such that f (1) = 1 and  for each x > 0. Then f(x) is

a)

b)

c)

d)

If y = y(x) satisfies the differential equation $8\sqrt{x}\sqrt{9+\sqrt{x}}\mathrm{dy}={\left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)}^{-1}\mathrm{dx},x>0$

and $y\left(0\right)=\sqrt{7}$ Then y(256) =

a) 16

b) 3

c) 9

d) 80

A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4 and the probability that the lot contains exactly three defective articles is 0.6. Articles are drawn from the lot at random one by one without replacement and tested till defective articles are found. What is the probability that the testing will end at the 12^th testing?

If the curve y = f(x) passes through the point (1, −1) and satisfies the differential equation y(1 + xy) dx = xdy then $f\left(\frac{-1}{2}\right)$

is equal to

a) $\frac{-2}{5}$

b) $\frac{-4}{5}$

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

d) $\frac{4}{5}$

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