Question(s) from Search: IIT

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 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