Question(s) from Search: IIT

Sketch the curves and identify the region bounded by
 

A window of perimeter (including the base of the arch) is in the form of a rectangle surmounted by a semicircle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass. The clear glass transmits three times as much light per square meter as the coloured glass. What is the ratio for the sides of the rectangle so that the window transmits the maximum light?

a)

b)

c)

d)

Let be a line in the complex plane where  is the complex conjugate of b. If a point  is the deflection of a point  through the line, show that .

Let

Find all possible values of b such that f(x) has the smallest value at x = 1.

a) (−2, ∞)

b) (−2, −1)

c) (1, ∞)

d) (−2, −1) ∪ (1, ∞)

Use mathematical induction for
 
to prove that
I_m = mπ, m = 0, 1, 2 .  .  .  .

Determine the points of maxima and minima of the function
  where b ≥ 0 is a constant.

a) Minima at x = x₁, maxima at x = x₂

b) Minima at x = x₂, maxima at x = x₁

c) Minima at x = x₁, x₂, no maxima

d) Maxima at x =x₁, x₂, no minima

where x_{1 =}   and x₂ =   

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 radius of the circum circle of △PRS is

a) 5

b)

c) 3

d)

Let  where 0 ≤ x ≤ 1. Determine the area bounded by y = f (x), X–axis, x = 0 and x = 1.

a)

b)

c)

d)

Which of the following function is periodic?

a) f(x) = x – [x] where [x] denotes the greatest integer less than equal to the real number x

b)

c) f(x) = x cos(x)

d) None of these

A curve C has the property that the tangent drawn at any point P on C meets the co-ordinate axes at A and B, and P is the mid-point of AB. The curve passes through the point (1, 1). Determine the equation of the curve.

a) x²y = 1

b) x = y

c) xy = 1

d) x² = y

Let –1 ≤ p ≤ 1. Show that the equation 4x³ – 3x – p = 0 has a unique root in the interval  and identify it.

a) p

b) p/3

c)

d)

Find the coordinates of all points P on the ellipse , for which the area of △PON is maximum where O denotes the origin and N the feet of perpendicular from O to the tangent at P.

Determine the equation of the curve passing through origin in the form  which satisfies the differential equation

If α, β are roots of  and γ, δ are roots of  then evaluate  in terms of p, q, r, s.

If p(x) = 51x¹⁰¹ – 2323x¹⁰⁰ – 45x + 1035, using Rolle’s theorem prove that at least one root lies between .

For what values of m does the system of equations 3x + my = m, 2x – 5y = 20 have solutions satisfying x > 0, y > 0?

a) m ε (

b) m ε (

c) m ε ( ∪ (

d) m ε (

Given

 
and f(x) is a quadratic polynomial. V is a point of maximum of f(x) and ‘A’ is the point where f(x) cuts the X–axis. ‘B’ is a point such that AB subtends a right angle at V. Find the area between chord AB and f(x).

a) 125

b) 125/2

c) 125/3

d) 125/6

The area enclosed within the curve |x| + |y| = 1 is .  .  .

a) 1

b)

c)

d) 2

Let a hyperbola pass through the foci of the ellipse  . The transverse and conjugate axes of the hyperbola coincide with the major and minor axes of the given ellipse. Also the product of the eccentricity of the given ellipse and hyperbola is 1 then

a) Equation of the hyperbola is

b) Equation of the hyperbola is

c) Focus of the hyperbola is (5, 0)

d) Vertex of the hyperbola is

The integral $\underset{2}{\overset{4}{\int }}\frac{{\mathrm{logx}}^2}{{\mathrm{logx}}^2+\log \left(x^2-12x+36\right)}\mathrm{dx}$

is equal to

a) 2

b) 4

c) 1

d) 6

Fifteen coupons are numbered 1, 2, 3, .  .  .   ., 15 respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9 is

a)

b)

c)

d) None of these

Match the statement of column 1 and the properties of column 2

The value of the integral $\underset{\sqrt{\log 2}}{\overset{\sqrt{\log 3}}{\int }}\frac{x\sin x^2}{\sin x^2+\sin \left(\log 6-x^2\right)}\mathrm{dx}$

is equal to

a) $\frac{1}{4}\log \frac{3}{2}$

b) $\frac{1}{2}\log \frac{3}{2}$

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

d) $\frac{1}{6}\log \frac{3}{2}$

Let g(x) be a function of x defined on (−1, 1). If the area of the equilateral triangle with two of its vertices as (0, 0) and [x, g(x)] is , then the function g(x) is

a)

b)

c)

d) None of the above

Show that the integral of   is