Question(s) from Search: IIT

Let the complex numbers  are vertices of an equilateral triangle. If  be the circumcentre of the triangle, then prove that

A two metre long object is fired vertically upwards from the mid-point of two locations A and B, 8 metres apart. The speed of the object after t seconds is given by  metres per second. Let α and β be the angles subtended by the objects A and B respectively after one and two seconds. Find the value of cos(α − β).

a)

b)

c)

d)

The point (α, β, γ) lies on the plane .
Let a =  . . . . .

Investigate for maxima and minima the function
 

a) Local maximum at x = 1, 7/5, 2

b) Local minimum at x = 1, 7/5, 2

c) Local maximum at x = 1, 2. Local minimum at x =  7/5

d) Local maximum at x = 1. Local minimum at x =  7/5

Sides a, b, c of a triangle ABC are  in arithmetic progression and  then
 

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)

ABCD is a rhombus. The diagonals AC and BD intersect at the point M and satisfy BD = 2AC. If the points D and M represent the complex numbers 1 + i and (2 – i) respectively then find the complex number x + iy represented by A.

a)  

b)  

c)  

d)  

Find all possible values of b > 0, so that the area of the bounded region enclosed between the parabolas  and  is maximum.

a) b = 1

b) b ≥ 1

c) b ≤ 1

d) 0 < b < 1

Let f(x) = sinx and g(x) = ln|x|. If the ranges of the composition function fog and gof are R₁ and R₂ respectively then

a)

b) ,

c)

d)

Let C₁ and C₂ be the graph of the function y = x² and y = 2x respectively. Let C₃ be the graph of the function
y = f (x), 0 ≤ x ≤ 1, f (0) = 0. Consider a point P on C₁. Let the lines through P, parallel to the axes meet C₂ and C₃ at Q and R respectively (see figure). If for every position of P (on C₁) the area of the shaded regions OPQ and OPR are equal, determine the function f(x).

a) x² – 1

b) x³ – 1

c) x³ – x²

d) 1 + x² + x³

A hemispherical tank of radius 2 meters is initially full of water and has an outlet of 12cm² cross section area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law  where g(t) and h(t) are respectively the velocity of the flow through the outlet and the height of the water level above the outlet at the time t, and g is the acceleration due to gravity. Find the time it takes to empty the tank. (Hint: Form a differential equation by relating the decrease of water level to the outflow).

a)

b)

c)

d)

Let P be a point on the ellipse . Let the line parallel to Y–axis passing through P meets the circle  at the point Q such that P and Q are on the same side of the X–axis. For two positive real numbers r and s find the locus of the point R on PQ such that PˆR : RˆQ = r : s and P varies over the ellipse.

Find the area bounded by the curves
x² = y, x² = − y and y² = 4x – 3

a) 1

b) 3

c) 1/3

d) 1/9

Let E = {1, 2, 3, 4} and F = {1, 2}, then the number of onto functions from E to F is

a) 14

b) 16

c) 12

d) 8

For a twice differentiable function f(x), g(x) is defined as  If for a < b < c < d < e, f(a) = 0, f(b) = 2, f(c) = − 1, f(d) = 2, f(e) = 0 then find the maximum number of zeros of g(x).

a) 1

b) 2

c) 3

d) 6

Find the equation of the normal to the curve

The larger of cos (lnθ) and ln (cosθ) if  is

a) cos(lnθ)

b) ln(cosθ)

For any real t, ,  is a point on the hyperbola x² – y² = 1. Find the area bounded by the hyperbola and the line joining the centre to the points corresponding to t₁ and –t₁.

The integral $\underset{\pi /4}{\overset{\pi /2}{\int }}{\left(2\mathrm{cosecx}\right)}^{17}\mathrm{dx}$

is equal to

a) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}2{\left(e^u+e^{-u}\right)}^{16}\mathrm{du}$

b) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}{\left(e^u+e^{-u}\right)}^{17}\mathrm{du}$

c) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}{\left(e^u-e^{-u}\right)}^{17}\mathrm{du}$

d) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}2{\left(e^u+e^{-u}\right)}^{}\mathrm{du}$

X and Y are two sets and f : X → Y. If  then the true statement is

a)

b)

c) ,

d)