Question(s) from Search: IIT

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) $\frac{1}{4}$

If AB is a diameter of a circle and C is any point on the circumference of the circle then

a) The area of ΔABC is maximum when it is isosceles

b) The area of ΔABC is minimum when it is isosceles

c) The perimeter of ΔABC is minimum when it is isosceles

d) None of these

Let f : ℝ → ℝ be any function. Define g : ℝ → ℝ by g (x) = |f (x)| for all x. Then g is

a) Onto if f is onto

b) One to one if f is one to one

c) Continuous if f is continuous

d) Differentiable if f is differentiable