Question(s) from Search: IIT

A tangent to the ellipse x² + 4y² = 4 meets the ellipse x² + 2y² = 6 at P and Q. Prove that tangents at P and Q of the ellipse x² + 2y² = 6 are at right angles.

Let f(θ) = sinθ (sinθ + sin3θ) then f(θ)

a) ≥ 0 only when θ ≥ 0

b)  ≤ 0 for all real θ

c)  ≥ 0 for all real θ

d) ≤ θ only when θ ≤ 0

Let y = f(x) is a cubic polynomial having maximum at x = − 1 and  has a minimum at x = 1 and f(−1) = 10, f(1) = − 6. Find the cubic polynomial and also find the distance between the points which are maxima or minima.

a)

b)

c)

d)

Each of the following four inequalities given below define a region in the XY–plane. One of these four regions does not have the following property: For any two points (x₁, y₁) and (x₂, y₂) in the region, point  is also in the region. The inequality defining the region that does not have this property is

a) x² + 2y² ≤ 1

b) max (|x|, |y|) ≤ 1

c) x² – y² ≥ 1

d) y² – x ≤ 0

The domain of definition of the function           is

a)  

b)  

c)  

d)  

The set of values of x which ln(1 + x) ≤ x is equal to .  .  .  .

a) (−∞, −1)

b) (−1, 0)

c) (0, 1)

d) (1, ∞)

For any positive integers m, n (with n ≥ m), we are given that
  
Deduce that
  

If A and B are two independent events such that P (A) > 0 and P (B) ≠ 1 then  is equal to

a)

b)

c)

d)

If,  then g(f(x)) is invertible in the domain

a)

b)

c)

d)

Evaluate

a)

b)

c)

d)

Tangents are drawn to the circle  from a point on the hyperbola . Find the locus of the midpoint of the chord of contact.

The value of the integral $\underset{-\pi /2}{\overset{\pi /2}{\int }}\left(x^2+\log \frac{\pi -x}{\pi +x}\right)\mathrm{cosx}\mathrm{dx}$

is equal to

a) $0$

b) $\frac{{\pi }^2}{2}-4$

c) $\frac{{\pi }^2}{2}+4$

d) $\frac{{\pi }^2}{2}$

Show that the integral of sinxsin2xsin3x + sec²xcos²2x + sin⁴xcos⁴x is

Let P (x₁, y₁) and Q (x₂, y₂), y₁ < 0, y₂ < 0 be the end points of the latus rectum of the ellipse x² + 4y² = 4. The equations of the parabolas with latus rectum PQ are

a)

b)

c)

d)

Let F : ℝ → ℝ be a thrice differentiable function. Suppose that F(1) = 0, F(3) = −4 and F′(x) < 0 for all x ε (1, 3). Let f(x) = x F(x) for all x ε ℝ.If $\underset{1}{\overset{3}{\int }}{x}^2{F}^{\prime }\left(x\right)\mathrm{dx}=-12$

and $\underset{1}{\overset{3}{\int }}{x}^3{F^{\prime \prime }\left(x\right)\mathrm{dx}=40}^{}$ , then the correct expression is/are

a) $9f^{\prime }\left(3\right)+f^{\prime }\left(1\right)-32=0$

b) $\underset{1}{\overset{3}{\int }}f\left(x\right)\mathrm{dx}=12$

c) $9f^{\prime }\left(3\right)-f^{\prime }\left(1\right)+32=0$

d) $\underset{1}{\overset{3}{\int }}f\left(x\right)\mathrm{dx}=-12$

 =

a) +c

b) +c

c) +c

d)

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)