Question(s) from Search: IIT

Multiple choices
If f(x) =  where [x] stands for the greatest integer function then

a)

b)

c)

d)

A circle C of radius 1 is inscribed in an equilateral triangle PQR. The point of contacts of C with its sides PQ, QR and RP are D, E, F respectively. The line PQ is given by  and the point D is . Further, it is given that the origin and the centre of C are on the same side of the line PQ. Points E and F are given by

a)

b)

c)

d)

One or more than one correct options

If $I=\sum _{k=1}^{98}\underset{k}{\overset{k+1}{\int }}\frac{\left(k+1\right)}{x(x+1)}\mathrm{dx}$

then

a) $I>{\log }_{e}99$

b) $I<{\log }_{e}99$

c) $I<\frac{49}{50}$

d) $I>\frac{49}{50}$

ConsiderL₁: 2x + 3y + p – 3 = 0; L₂: 2x + 3y + p + 3 = 0 where p is a real number and C : x² + y² + 6x – 10y + 30 = 0

Statement 1 – If the line L₁ is a chord of the circle C then L₂ is not always a diameter of C.

Statement 2 - If the line L₁ is a diameter of the circle C then L₂ is not a chord of the circle.
Which of the following four statements is true?

a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation for statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true

One or more than one correct options

If $I_n=\underset{-\pi }{\overset{\pi }{\int }}\frac{\sin \mathrm{nx}}{\left(1+n^x\right)\mathrm{sinx}}\mathrm{dx},n=\mathrm{0,}\mathrm{1,}\mathrm{2,}...$

then

a) $I_n=I_{n+2}$

b) $\sum _{n=1}^{10}{I}_{2n+1}=10\pi$

c) $\sum _{n=1}^{10}{I}_{2n}=0$

d) $I_n=I_{n+1}$

If E and F are events with P (E) ≤ P (F) and P (E ∩ F) > 0 then

a) occurrence of E ⇒ occurrence of F

b) occurrence of F ⇒ occurrence of E

c) non-occurrence of E ⇒ non-occurrence of F

d) none of the above occurrences hold

 =  

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 (in square units) bounded by the curves $y=\sqrt{x},2y-x+3=0$

, X – axis and lying in the first quadrant is

a) 9

b) 6

c) 18

d) $\frac{27}{4}$

One or more than one correct option

Let S be the area of the region enclosed by $y=e^{{-x}^2}$

, y = 0, x = 0 and x = 1, then

a) $S\ge \frac{1}{e}$

b) $S\ge 1-\frac{1}{e}$

c) $S\le \frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right)$

d) $S\le \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right)$

Show that the sum of the first n terms of the series
1² + 2.2² + 3² + 2.4² + 5² + 2.6² + .  .  .
is  when n is even, and  when n is odd.

Differentiate from first principles (or ab initio)

a) 2xcos(x² + 1)

b) xcos(x² + 1)

c) 2cosx(x² + 1)

d) 2xcosx(x² + 1) + sin(x² + 1)

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?