Question(s) from Search: IIT

For any integer n, the integral
 has the value

a) π

b) 1

c) 0

d) None of these

The area (in square units) of the region described by (x, y) : y² < 2x and y ≥ 4x – 1 is

a) $\frac{7}{32}$

b) $\frac{9}{32}$

c) $\frac{3}{2}$

d) $\frac{5}{3}$

Let f: [−1, 2] → [0, ∞) be a continuous function such that f(x) = f(1 –x), Ɐ x ∈ [−1, 2]. If $R_1=\underset{-1}{\overset{2}{\int }}xf\left(x\right)\mathrm{dx}$

and $R_2$ are the area of the region bounded by y = f(x), x = −1, x = 2 and the X- axis. Then

a) R₁ = 2R₂

b) R₁ = 3R₂

c) 2R₁ = R₂

d) 3R₁ = R₂

If $\left(2+\mathrm{sinx}\right)\frac{\mathrm{dy}}{\mathrm{dx}}+\left(y+1\right)\mathrm{cosx}=0\wedge y\left(0\right)=1$

, then $y\left(\frac{\pi }{2}\right)$ is equal to

a) $\frac{1}{3}$

b) $\frac{-2}{3}$

c) $\frac{-1}{3}$

d) $\frac{4}{3}$

One or more than one correct option

Consider the family of circles whose centre lies on the straight line y = x. If the family of circles is represented by the differential equation Py′′ + Qy′ + 1 = 0 where P, Q are functions of x, y and y′ $\left(\mathrm{where}{y}^{\prime }=\frac{\mathrm{dy}}{\mathrm{dx}},y^{\prime \prime }=\frac{d^{2}y}{d{x}^2}\right)$

, then which of the following statements is/are true?

a) P = y + x

b) P = y – x

c) P + Q = 1 – x + y + y′ + (y′)²

d) P − Q = x + y − y′ − (y′)²

Find  at x = , when

a) 0

b) 1

c) – 1

d) 2

Let f : (0, ∞) → ℝ and  If  then f(4) equals

a)

b) 7

c) 4

d) 2

Let $f\mathrm{:}\left[\frac{1}{2},1\right]\to R$

(the set of all real numbers) be a positive, non-constant and differentiable function such that $f^{\prime }\left(x\right)<2f\left(x\right)$ and $f\left(\frac{1}{2}\right)=1$ . Then the value of $\underset{1/2}{\overset{1}{\int }}f\left(x\right)\mathrm{dx}$ lies in the interval

a) $\left(\mathrm{2e-1,}2e\right)$

b) $\left(e-\mathrm{1,}2e-1\right)$

c) $\left(\frac{e-1}{2},e-1\right)$

d) $\left(\mathrm{0,}\frac{e-1}{2}\right)$

The smallest positive integer n for which  is

a) 8

b) 12

c) 12

d) None of these

Let the population of rabbits arriving at time t be governed by the differential equation $\frac{dp(t)}{\mathrm{dt}}=\frac{1}{2}p\left(t\right)-200$

. If p(0) = 100, then p(t) is equal to

a) 400 – 300e^t/2

b) 300 – 200e^−t/2

c) 600 – 500e^t/2

d) 400 – 300e^−t/2

If z = x + iy and ω =  then |ω| =1 implies that in the complex plane

a) z lies on the imaginary axis

b) z lies on the real axis

c) z lies on unit circle

d) none of these

For a positive integer n, define
 then

a) a(100) ≤ 100

b) a(100) > 100

c) a(200) ≤ 100

d) a(200) > 100

Let f:[0, 1] → ℝ (the set all real numbers)be a function. Suppose the function is twice differentiable, f(0) = f(1) = 0 and satisfiesf′′(x) – 2f′(x) + f(x) ≥ e^x, x ∈ [0, 1]If the function e^−x f(x) assumes its minimum in the interval [0, 1] at $x=\frac{1}{4}$

then which of the following is true?

a) $f^{\prime }\left(x\right)<f\left(x\right),\frac{1}{4}<x<\frac{3}{4}$

b) $f^{\prime }\left(x\right)>f\left(x\right),0<x<\frac{1}{4}$

c) $f^{\prime }\left(x\right)<f\left(x\right),0<x<\frac{1}{4}$

d) $f^{\prime }\left(x\right)<f\left(x\right),\frac{3}{4}<x<1$

There exists a function f(x) satisfying f (0) = 1,  and

f (x) > 0 for all x and

a)   for all x

b)  

c)   for all x

d)   for all x

Let k be an integer such that the triangle with vertices (k, −3k), (5, k) and (−k, 2) has area 28 square units. Then the orthocentre of the triangle is at the point

a) $\left(2,-\frac{1}{2}\right)$

b) $\left(\mathrm{1,}\frac{3}{4}\right)$

c) $\left(1,-\frac{3}{4}\right)$

d) $\left(\mathrm{2,}\frac{1}{2}\right)$

If p is a natural number then prove that p^{n + 1} + (p + 1)^{2n – 1} is divisible by p² + p + 1 for every positive integer n.

A straight line L through the point (3, −2) is inclined at an angle of 60° to the line $\sqrt{3}x+y=1$

. If the line L also intersects the X- axis then the equation of L is

a) $y+\sqrt{3}x+2-3\sqrt{3}=0$

b) $y-\sqrt{3}x+2+3\sqrt{3}=0$

c) $\sqrt{3}y-x+3+2\sqrt{3}=0$

d) $\sqrt{3}y+x-3+2\sqrt{3}=0$

The orthocenter of the triangle formed by the lines
  lies in the quadrant number . . . . .

Prove by mathematical induction that
 for every positive integer n.

The sides of a rhombus are along the lines x – y + 1 = 0 and 7x – y – 5 = 0. If its diagonals intersect at (−1, −2) then which one of the following is a vertex of the rhombus?

a) $(-\mathrm{3,}-9)$

b) $(-\mathrm{3,}-8)$

c) $\left(\frac{1}{3},-\frac{8}{3}\right)$

d) $\left(\frac{-10}{3},\frac{-7}{3}\right)$

Let  and  intersect the line
 at P and Q respectively. Bisector of the acute angle between L₁ and L₂ intersects L₃ in R.
Statement 1 – The ratio PR : RQ equals  because
Statement 2 – In any triangle, bisector of an angle divides the triangle into two similar triangles.
The question contains Statement 1(assertion) and Statement 2(reason). Of these statements, mark correct choice if

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.

Prove that  is an integer for every positive integer.

If f is a continuous function with  as |x| → ∞ then show that every line y = mx intersects the curve .

Show, by vector method, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of point of concurrency in terms of position vectors of the vertices.

The C be a circle with the centre at (1, 1) and radius 1. If T is the circle centred at (0, k) passing through origin and touches the circle C externally, then the radius of T is equal to

a) $\frac{\sqrt{3}}{\sqrt{2}}$

b) $\frac{\sqrt{3}}{2}$

c) $\frac{1}{2}$

d) $\frac{1}{4}$