Question(s) from Search: IIT

Prove that there exists no complex number z such that  and .

Three normals with slopes  are drawn from a point P not on the axis of the parabola . If  results in the locus of P being a part of the parabola, find the value of α.

Find the value of the expression

1.(2−ω)(2−+ 2.(3−ω)(3−+ … (n−1).(n−ω)(n−

where ω is an imaginary cube root of unity.

a) n(n−1)(+3n+4)

b) n(n+1)(+3n+4)

c) n(n−1)(+n+1)

d) n(n+1)(+n+1)

If  are positive real numbers whose product is a fixed number c then the minimum value of  is

a)

b)

c)

d)

If three complex numbers are in arithmetic progression then they lie on a circle in the complex plane.

a) True

b) False

A solution of the differential equation  is

a)  y = 2

b)  y = 2x

c)  

d)  2

For all x,  then the interval in which a lies is

a) a <

b)

c)

d)

Let the three digit numbers A28, 3B9 and 62C where A, B, C are integers between 0 and 9, be divisible by a fixed number k. Show that the determinant
 

is divisible by k.

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  .

If the line  touches the hyperbola  then the point of contact is

a)

b)

c)

d)

Let  then one of the possible value of k is

a) 1

b) 2

c) 4

d) 16

Two events A and B have probabilities 0.25 and 0.50 respectively. The possibility of both A and B occur simultaneously is 0.14 then the probability that neither A nor B occur is

a) 0.39

b) 0.25

c) 0.11

d) None of these

Find the set of all x for which

Sum of the first n terms of the series  is

a) 2ⁿ – n – 1

b) 1 – 2^{− n}

c) n + 2^{− n} – 1

d) 2ⁿ + 1