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)

Let λ and α be real. Find the set of all values of λ for which the system of linear equations
 
 
 
has a non-trivial solution. For λ = 1 find the value of α.

Let f be a one–one function with domain {x, y, z} and range {1, 2, 3}. It is given that exactly one of the following statements is true and remaining statements are false f (1) = 1, f (y) ≠ 1, f (z) ≠ 2. Determine  

The value of . Given that a, x, y, z, b are in Arithmetic Progression while the value of . If a, x, y, z, b are in Harmonic Progression then find a and b.

Let {x} and [x] denote the fractional and integral part of a real number x respectively. Solve 4{x} = x + [x]

If S₁, S₂, .  .  .  .,S_n are the sums of infinite geometric series whose first terms are 1, 2, 3,   .  .  ., n and whose common ratios are  respectively, then find the value of

If  are three non–coplanar vectors, then

  equals

a) 0

b)

c)

d)

Let a, b are real positive numbers. If a, A₁, A₂, b are in Arithmetic Progression, a, G₁, G₂, b are in Geometric Progression and a, H₁, H₂, b are in Harmonic Progression show that
 

a) True

b) False