Question(s) from Search: IIT

Let A_n be the area bounded by the curve y = (tanx)ⁿ and the line
x = 0, y = 0 and . Prove that for  . Hence deduce that
 

Consider the circle x² + y² = 9 and the parabola y² = 8x. They intersect P and Q in the first and fourth quadrants respectively. Tangents to the circle at P and Q intersect the X–axis at R and tangents to the parabola at P and Q intersect the X- axis at S. The radius of the incircle of △PQR is

a) 4

b) 3

c)

d) 2

ABCD is a rhombus. The diagonals AC and BD intersect at the point M and satisfy BD = 2AC. If the points D and M represent the complex numbers 1 + i and (2 – i) respectively then find the complex number x + iy represented by A.

a)  

b)  

c)  

d)  

Find all possible values of b > 0, so that the area of the bounded region enclosed between the parabolas  and  is maximum.

a) b = 1

b) b ≥ 1

c) b ≤ 1

d) 0 < b < 1

Let f(x) = sinx and g(x) = ln|x|. If the ranges of the composition function fog and gof are R₁ and R₂ respectively then

a)

b) ,

c)

d)

Let C₁ and C₂ be the graph of the function y = x² and y = 2x respectively. Let C₃ be the graph of the function
y = f (x), 0 ≤ x ≤ 1, f (0) = 0. Consider a point P on C₁. Let the lines through P, parallel to the axes meet C₂ and C₃ at Q and R respectively (see figure). If for every position of P (on C₁) the area of the shaded regions OPQ and OPR are equal, determine the function f(x).

a) x² – 1

b) x³ – 1

c) x³ – x²

d) 1 + x² + x³

A hemispherical tank of radius 2 meters is initially full of water and has an outlet of 12cm² cross section area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law  where g(t) and h(t) are respectively the velocity of the flow through the outlet and the height of the water level above the outlet at the time t, and g is the acceleration due to gravity. Find the time it takes to empty the tank. (Hint: Form a differential equation by relating the decrease of water level to the outflow).

a)

b)

c)

d)

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