Question(s) from Search: IIT

One or more than one correct options

Let F : ℝ → (0, 1) be a continuous function. Then which of the following function(s) has (have) the value zero at some point in the interval (0, 1)?

a) $e^x-\underset{0}{\overset{x}{\int }}f\left(t\right)\mathrm{sint}\mathrm{dt}$

b) $f\left(x\right)+\underset{0}{\overset{\frac{\pi }{2}}{\int }}f\left(t\right)\mathrm{sint}\mathrm{dt}$

c) $x-\underset{0}{\overset{\frac{\pi }{2}-x}{\int }}f\left(t\right)\mathrm{cost}\mathrm{dt}$

d) $x^9-f(x)$

Consider a branch of the hyperbola
 
with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of triangle ABC is

a)

b)

c)

d)

One or more than one correct options

The value(s) of $\underset{0}{\overset{1}{\int }}\frac{x^4{(1-x)}^4}{{1+x}^2}\mathrm{dx}$

is (are)

a) $\frac{22}{7}-\pi$

b) $\frac{2}{105}$

c) $0$

d) $\frac{71}{15}-\frac{3\pi }{2}$

 =  

where t² = cot²x – 1

a) True

b) False

$\frac{\left(\underset{-1/2}{\overset{1/2}{\int }}\cos 2x\log \frac{1+x}{1-x}\mathrm{dx}\right)}{\left(\underset{0}{\overset{1/2}{\int }}\cos 2x\log \frac{1+x}{1-x}\right)}$

equals

a) 8

b) 2

c) 4

d) 0

Fill in the blank

The system of equations
 
 
 
will have a non-zero solution if real value of λ is given by …………

The function  is not one to one

a) True

b) False

For any real number x, let [x] denote the greater integer less than or equal to x. Let f be a real valued function defined on the interval [−10, 10] by $f\left(x\right)=\{\begin{array}{cc}x-[x]& \mathrm{if}\left[x\right]\mathrm{is}\mathrm{odd}\\ 1+\left[x\right]-x& \mathrm{if}\left[x\right]\mathrm{is}\mathrm{even}\end{array}$

then the value of $\frac{{\pi }^2}{10}\underset{-10}{\overset{10}{\int }}f\left(x\right)\mathrm{cosx\pi dx},$ is

a) 2

b) 0

c) 6

d) 4

Let  denotes the complement of an event E. Let E, F, G are pair wise independent events with P (G) > 0 and P (E ∩ F ∩ G) = 0 then  equals

a)

b)

c)

d)

Let A be a set of n distinct elements. Then find the total number of distinct functions from A to A is and out of these onto functions are .  .  .

$\underset{n\to \infty }{\lim }{\left(\frac{\left(n+1\right)\left(n+2\right)...3n}{n^{2n}}\right)}^{1/n}$

is equal to

a) $\frac{18}{e^4}$

b) $\frac{27}{e^2}$

c) $\frac{9}{e^2}$

d) $3\log 3-2$

(One or more correct answers)
For any two events in the sample space

a)  is always true

b)  does not hold

c) if A and B are independent

d)  if A and B are disjoint

Match the following
Let the function defined in column 1 have domain  and range (−∞ ∞)

Let a, b, c be real numbers such that
 

Then ax² + bx + c = 0 has

a) No root in (0, 2)

b) At least one root in (0, 2)

c) A double root in (0, 2)

d) Two imaginary roots

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
 
 

The value of $\sum _{k=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{(k-1)\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{\mathrm{k\pi }}{6}\right)}$

a) $3-\sqrt{3}$

b) $2\left(3-\sqrt{3}\right)$

c) $2\left(\sqrt{3}-1\right)$

d) $2\left(2+\sqrt{3}\right)$

Let y(x) be the solution of the differential equation $\left(\mathrm{xlnx}\right)\frac{\mathrm{dy}}{\mathrm{dx}}+y=2\mathrm{xlnx},\left(x\ge 1\right)$

. Given that y = 1 when x = 1, then y(e) is equal to

a) e

b) 0

c) 2

d) 2e

If C_r stands for  then the sum of the series
 
where n is a positive integer, is equal to

a) 0

b) (−)^n/2(n + 1)

c) (−)^n/2 (n + 2)

d) None of these

Let T > 0 be a fixed real number. Suppose f is a continuous function such that for all x  ℝ, f(x + T) = f(x). If  then the value of  is

a)

b)

c) 3I

d) 6I