Question(s) from Search: IIT

Consider the points
P: (−sin (β – α), cosβ)
Q: (cos (β – α), sinβ)
R: (−cos{(β – α) + θ}, sin (β – θ))
where 0 < α, β, θ <  then

a) P lies on the line segment RQ

b) Q lies on the line segment PR

c) R lies on the line segment QP

d) P, Q, R are non–collinear

One or more than one correct options

The options with the values of α and L that satisfy the equation $\frac{\underset{0}{\overset{4\pi }{\int }}{e}^t\left[{\sin }^6\mathrm{\alpha t}+{\cos }^4\mathrm{\alpha t}\right]\mathrm{dt}}{\underset{0}{\overset{\pi }{\int }}{e}^t\left[{\sin }^6\mathrm{\alpha t}+{\cos }^4\mathrm{\alpha t}\right]\mathrm{dt}}=L$

is/are

a) $\alpha =\mathrm{2,}L=\frac{e^{4\pi }-1}{e^{\pi }-1}$

b) $\alpha =\mathrm{2,}L=\frac{e^{4\pi }+1}{e^{\pi }+1}$

c) $\alpha =\mathrm{4,}L=\frac{e^{4\pi }-1}{e^{\pi }-1}$

d) $\alpha =\mathrm{4,}L=\frac{e^{4\pi }+1}{e^{\pi }+1}$

The number of points in the interval $\left[-\sqrt{13},\sqrt{13}\right]$

in which $f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)$ attains its maximum value is

a) 8

b) 2

c) 4

d) 0

If the integers m and n are chosen at random between 1 and 100 then the probability that a number of form  is divisible by 5, equals

a)

b)

c)

d)

Show that the integral
 =

where y = x^1/6

If $\alpha =\underset{0}{\overset{1}{\int }}{e}^{\left(9x+{3\tan }^{-1}x\right)}\left(\frac{12+{9x}^2}{1+x^2}\right)\mathrm{dx}$

Where ${\tan }^{-1}x$ takes only principal values then the value of $\left({\log }_e\left|1+\alpha \right|-\frac{3\pi }{4}\right)$ is

a) 6

b) 9

c) 8

d) 11

The intercept on X axis made by the tangent to the curve $y=\underset{0}{\overset{x}{\int }}\left|t\right|\mathrm{dt},t\in R$

which is parallel to the line y = 2x are equal to

a) ±1

b) ±2

c) ±3

d) ±4

The common tangent to the curve x² + y² = 2 and the parabola y² = 8x touch the circle at the points P, Q and the parabola at the points R, S. Then the area (in square units) of the quadrilateral PQRS is

a) 3

b) 6

c) 9

d) 15

(One or more correct answers)
Let 0 < P (A) < 1, 0 < P (B) < 1 and P (A ∪ B) = P (A) + P (B) – P (A ∩ B) then

a) P (B/A) = P (B) – P (A)

b) P (Aʹ – Bʹ) = P (Aʹ) – P (Bʹ)

c) P (A U B)ʹ = P (Aʹ) P (Bʹ)

d) P (A/B) = P (A)

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.