Question(s) from Search: IIT

Cosine of angle of intersection of curve y = 3^{x – 1}lnx and y = x^x – 1 is

Let A =

 
AU₁ =  , AU₂ =  and AU₃ =

a) −1

b) 0

c) 1

d) 3

If f : [1, ∞) → [2, ∞) is given by  then  equals

a)

b)

c)

d)

For the primitive differential equation
 

then  is

a) 3

b) 5

c) 1

d) 2

Consider the system of linear equations
 
 
 
Find the value of θ for which the systems of equations have non-trivial solutions.

The set of all solutions of the equation

Multiple choices with one or more than one correct answers
  then

a) x = f(y)

b) f(1) = 3

c) y increases with x for x < 1

d) f is a rational function of x

Given  and f(x) = cosx – x(x + 1). Find the range of f (A).

Multiple choices

If the first and  term of an Arithmetic Progression, a Geometric Progression and a Harmonic Progression are equal and their n^th term are a, b, c respectively then

a)

b)

c)

d)

Show that the value of  wherever defined, never lies between  and 3.

Let  where A, B, C are real numbers. Prove that if f(n) is an integer whenever n is an integer, then the numbers 2A, A + B and C are all integers. Conversely prove that if the numbers 2A, A + B and C all integers then f(n) is an integer whenever n is an integer.

Let  and  be three non-zero vectors such that c is a unit vector perpendicular to both the vectors a and b and the angle between the vectors a and b is  then
 is equal to

a) 1

b)

c)

d) None of these

Does there exist a Geometric Progression containing 27, 8 and 12 as three of its terms? If it exists, how many such progressions are possible?

The values of  lies in the interval .  .  .

If  and  then (gof)(x) is equal to

If 0 < x < 1, then  is equal to

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)