Question(s) from Search: IIT

Find the shortest distance of the point (0, c) from the parabola
y = x², where 0 ≤ c ≤ 5.

a)

b)

c)

d)

Both roots of the equation

( x – b) ( x – c) + (x – c) ( x – a) + (x – a) (x – b) = 0 are always

a) positive

b) negative

c) real

d) none of these

a) – 1

b) 0

c) 1

d) 2

If  is purely real where ω = α + iβ, β ≠ 0 and z ≠ 1 then the set of real values of z is

a)  

b)  

c)  

d)  

Two vertices of an equilateral triangle are (- 1, 0) and (1, 0) and its third vertex lies above the X–axis, the equation of circumcircle is . . .

Two towns A and B are 60 meters apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all the 200 students is to be as small as possible then the school should be built at

a) Town B

b) 45 km from town A

c) Town A

d) 45 km from town B

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 line y = x meets y =  for k ≤ 0 at

a) No point

b) One point

c) Two points

d) More than two points

(One or more than one correct answer)
Let  and  be complex numbers such that  and . If has positive real part and  has negative imaginary part, then  may be

a) Zero

b) Real and positive

c) Real and negative

d) None of these

If then ab + bc + ca lies in the interval

a)  

b)  

c)  

d)  

Find the values of x and y for which the following equation is satisfied

a) x = y = −1

b) x = y = 3

c) x = 1, y = 3

d) x = 3, y = −1

The equation of the directrix of the parabola y² + 4y + 4x +2 = 0 is

a) x = − 1

b) x = 1

c)

d)

Let α, β be roots of the equation (x – a) (x – b) = c, c ≠ 0. Then the roots of the equation (x – α) (x – β) + c = 0 are

a) a, c

b) b, c

c) a, b

d) a + c, b + c

If = x + iy then

a) x = 3, y = 1

b) x = 1, y = 3

c) x = 0, y = 3

d) x = 0, y = 0

It is given that n is an odd integer greater than 3 and not a multiple of 3. Prove that  is a factor of  

The focal chord of  is tangent to  then the possible value of the slope of this chord are

a)   

b)  

c)   

d)   

If p, q ε {1, 2, 3, 4}. The number of equations of the form  having real roots is

a) 15

b) 9

c) 7

d) 8

If A =  and B = then the value of α for which A² = B is

a) 1

b) −1

c) 4

d) No real values

If   then show that |z| = 1.

Suppose that the normals drawn at three different points on the

parabola  pass through the point (h, 0). Show that h > 2.

Show that the locus of a point that divides a chord of slope 2 of the parabola  internally in the ratio 1:2 is a parabola. Find its vertex.

Let α, β be the roots of  and γ, δ roots of . If α, β, γ, δ are in geometric progression then the integral values of p and q respectively are

a) −2, −32

b) −2, 3

c) −6, 3

d) −6, −32

For what values of k does the following system of equations possess a non-trivial solution over the set of rationals? Find all the solutions.

x + y – 2z = 0

2x – 3y + z = 0

x – 5y + 4z = k

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)