Question(s) from Search: IIT

Let f(x) =

If f is continuous for all x, then k is equal to

a) 3

b) 5

c) 7

d) 9

Fill in the blank

The domain of the function f (x) =  is

a) [− 2, − 1]

b) [1, 2]

c) [− 2, − 1] ⋃ [1, 2]

d) None of the above

 

 

Then

a) 0

b) 1

c) 2

d) 4

The complex numbers  satisfying  are the vertices of the triangle which is

a) of zero area

b) right angle isosceles

c) equilateral

d) obtuse angled isosceles

Let x and y be two real variables such that x > 0 and xy = 1. Find the minimum value of x + y.

a) 1

b) 2

c) 3

d) 4

ABC is an isosceles triangle in a circle of radius r. If AB = AC and h is the altitude from A to BC then the triangle ABC has perimeter , area A = . . . . .

Also  . . . . .

Let f(x) = x|x|. The set of points where f(x) is twice differentiable is .  .  .  .

a) ℝ

b) 0

c) ℝ − {0, 1}

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

The expression  is a polynomial of degree

a) 5

b) 6

c) 7

d) 8

If f(x) =
then f(100) equals

a) 0

b) 1

c) 100

d) −100

Show that the area of the triangle on the argand diagram formed by the complex numbers z, iz, z + iz is   .

The angle between the tangents drawn from the point (1, 4) to the parabola  is

a)

b)

c)

d)

The equation  has

a) No solution

b) One solution

c) Two solutions

d) More than two solutions

If the system of equations

x + ay = 0

az + y = 0

ax + z = 0

has infinite solutions then the value of a is

a) −1

b) 1

c) 0

d) No real values