Question(s) from Search: IIT

Let P be a point on the ellipse . Let the line parallel to Y–axis passing through P meets the circle  at the point Q such that P and Q are on the same side of the X–axis. For two positive real numbers r and s find the locus of the point R on PQ such that PˆR : RˆQ = r : s and P varies over the ellipse.

Find the area bounded by the curves
x² = y, x² = − y and y² = 4x – 3

a) 1

b) 3

c) 1/3

d) 1/9

Let E = {1, 2, 3, 4} and F = {1, 2}, then the number of onto functions from E to F is

a) 14

b) 16

c) 12

d) 8

For a twice differentiable function f(x), g(x) is defined as  If for a < b < c < d < e, f(a) = 0, f(b) = 2, f(c) = − 1, f(d) = 2, f(e) = 0 then find the maximum number of zeros of g(x).

a) 1

b) 2

c) 3

d) 6

Find the equation of the normal to the curve

The larger of cos (lnθ) and ln (cosθ) if  is

a) cos(lnθ)

b) ln(cosθ)

For any real t, ,  is a point on the hyperbola x² – y² = 1. Find the area bounded by the hyperbola and the line joining the centre to the points corresponding to t₁ and –t₁.

The integral $\underset{\pi /4}{\overset{\pi /2}{\int }}{\left(2\mathrm{cosecx}\right)}^{17}\mathrm{dx}$

is equal to

a) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}2{\left(e^u+e^{-u}\right)}^{16}\mathrm{du}$

b) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}{\left(e^u+e^{-u}\right)}^{17}\mathrm{du}$

c) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}{\left(e^u-e^{-u}\right)}^{17}\mathrm{du}$

d) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}2{\left(e^u+e^{-u}\right)}^{}\mathrm{du}$

X and Y are two sets and f : X → Y. If  then the true statement is

a)

b)

c) ,

d)

Let a and b are non-zero real numbers. Then the equation
(ax² + by² + c) (x² – 5xy + 6y²) = 0 represents

a) Four straight lines when c = 0 and a, b are of the same sign

b) Two straight lines and a circle when a = b and c is of sign opposite to that of a.

c) Two straight lines and a hyperbola when a and b are of the same sign

d) A circle and an ellipse when a and b are of the same sign and c is of sign opposite to that of a.

Statement 1: The value of the integral $\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}\frac{\mathrm{dx}}{1+\sqrt{\tan x}}$

is equal toStatement 2: $\underset{a}{\overset{b}{\int }}f\left(x\right)\mathrm{dx}=\underset{a}{\overset{b}{\int }}f\left(a+b-x\right)\mathrm{dx}$

a) Statement 1 is correct, statement 2 is correct. Statement 2 is correct explanation of statement 1

b) Statement 1 is correct, statement 2 is correct. Statement 2 is not a correct explanation of statement 1

c) Statement 1 is correct, statement 2 is false

d) Statement 1 is incorrect, statement 2 is correct

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) $3\log 3-2$