Question(s) from Search: IIT

If a and b are real numbers between 0 and 1 such that the points  form an equilateral triangle then a is equal to .  .  .  .

a)  

b)  

c)  

d)  

Let E be the ellipse  and C be the circle . Let P and Q be the points (1, 2) and (2, 1) respectively. Then

a) Q lies inside C but outside E

b) Q lies outside both C and E

c) P lies inside both C and E

d) P lies inside C but outside E

Let a, b, c be the sides of a triangle where a ≠ c and λ ε R. If roots of the equation  are real then

a)

b)

c)

d)

Find the value of the determinant

where a, b, c are respectively p^th, q^th and r^th term of a harmonic

progression.

a) 0

b) 1

c) ½

d) None of the above

If tangents are drawn to the ellipse  then the locus of the mid-points of the intercepts made by the tangents between the coordinate axes is

a)

b)

c)

d)

Let S is the set of all real x, such that  is positive, then S contains

a)

b)

c)

d)

Let pλ⁴ + qλ³ + rλ² + sλ + t =  be an identity in λ where p, q, r, s, t are constants. Find the value of t.

a) 0

b) +1

c) –1

d) ±1

Let P be a variable point on the ellipse  with foci F₁ and F_2. . If A is the area of  then the maximum value of A is  . . . . .

A spherical rain drop evaporates at a rate proportional to its surface area at any instant. The differential equation giving the rate of change of the radius vector of the rain drop is . . . . .

The value of the determinant  is …………

a) 0

b) 1

c) a² + b² + c² – abc

d) a² + b² + c² – 3abc

The equation  represents

a) An ellipse

b) A hyperbola

c) A circle

d) None of these

Find all the real values of x which satisfy  and  .

For the hyperbola  which of the following remains constant with change in α

a) Abscissae of vertices

b) Abscissae of focii

c) Eccentricity

d) Directrix

The value of  is

a) 0

b) 1

c) 2

d) 4

For a ≤ 0, determine all real roots of the equation
 

If a, b, c, d and p are distinct real numbers such that
 
then a, b, c, d

a) Are in Arithmetic Progression

b) Are in Geometric Progression

c) Are in Harmonic Progression

d) Satisfy ab = cd

e) Satisfy none of these

Suppose f(x) is a function satisfying the following conditions

i) f(0) = 2, f(1) = 1

ii) f has a minimum value at x = 5/2 and

iii) for all x

 
where a, b are constants. Determine the constants a and b, and the function f(x).

a)

b)

c)

d)

Solve  

If  are in Geometric Progression then
 are in

a) Arithmetic Progression

b) Geometric Progression

c) Harmonic Progression

d) None of these

Let  for n ≥ 2 and
 

Then equals

a)

b)

c)

d)

India played two matches each with Australia and West indies. In any match the probability of India getting the points 0, 1, and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least seven points is

a) 0.8730

b) 0.0875

c) 0.0625

d) 0.0250

Let T_n denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If  then n equals

a) 5

b) 7

c) 6

d) 4

Three of the vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral equals

a)

b)

c)

d)

Let a, b, c be real numbers with a ≠ 0 and let α, β be roots of the equation . Express the roots of  in terms of α, β.

Suppose a, b, c are in Arithmetic Progression and  are in Geometric Progression. If  then the value of a is

a)

b)

c)

d)