Question(s) from Search: IIT

If  for all positive x where a > 0 and b > 0 then

a) 9ab² ≥ 4c³

b) 27ab² ≥ 4c³

c) 9ab² ≤ 4c³

d) 27ab² ≤ 4c³

Let ABCD be a square with side of length 2 units. C₂ is the circle through the vertices A, B, C, D and C₁ is the circle touching all the sides of the square ABCD. L is a line through A.

If P is a point on C₁ and Q is another point on C₂, then  is equal to

a) 0.75

b) 1.25

c) 1

d) 0.5

If a continuous function f defined on the real line ℝ, assumes positive and negative values in ℝ then the equation f(x) = 0 has a root in ℝ. For example, it is known that if a continuous function f on ℝ is positive at some points and its minimum value is negative then the equation f(x) = 0 has a root in ℝ. Consider the function f(x) =  for all real x where k is a real constant.

The positive value of k for which  has only one root is

a)

b) 1

c) e

d) ln2

Let . Find the intervals in which λ should lie in order that f(x) has exactly one minimum and exactly one maximum.

a)

b)

c)

d)

Consider a circle with centre lying on the focus of the parabola  such that it touches the directrix of the parabola. Then a point of intersection of the circle and parabola is

a) or

b)

c)

d)

Find the equation of the plane at a distance  from the point  and containing the line
 .

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)

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 .