Question(s) from Search: IIT

A curve y = f(x) passes through the point P:(1, 1). The equation to the normal at (1, 1) to the curve y = f(x) is (x – 1) + a(y – 1)  = 0 and the slope of the tangent at any point on the curve is proportional to the ordinate of the point. Determine the equation of the curve. Also obtain the area bounded by the Y–axis, the curve and the normal at P.

a)

b) y = ;

c)  ;

d)

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 ratio of areas of the triangle PQS and PQR is

a)

b) 1:2

c)

d) 1:8

Let a + b = 4 where a < 2 and let g(x) be a differentiable function. If  for all x, prove that  increases as (b – a) increases.

A and B are two separate reservoirs of water. Capacity of reservoir A is double the capacity of reservoir B. Both the reservoirs are filled completely with water, their inlets are closed and then water is released simultaneously from both the reservoirs. The rate of flow of water out of each reservoir at any instant of time is proportionate to the quantity of water in the reservoir at the time. One hour after the water is released the quantity of water in reservoir A is   times the quantity of water in reservoir B. After how many hours do both the reservoirs have the same quantity of water?

a)

b)

c) ln2

d)  

The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse  is

a)  square units

b)

c)  square units

d) 27 square units

The function f(x) = |px – q|+ r|x|, x  when p > 0, q > 0, r > 0 assumes minimum value only on one point if

a)  p ≠ q

b)  r ≠ q

c)  r ≠ p

d)  p = q = r

Let b ≠ 0 and j = 0, 1, 2, .  .  . , n. Let S_j be the area of the region bounded by Y–axis and the curve
.

Show that S₀, S₁, S₂, .  .  .  , S_n are in geometric progression. Also find the sum for a = − 1 and b = π.

a)

b)

c)

d)

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