Question(s) from Search: IIT

Multiple choice

Let h(x) = f(x) – (f(x))² + (f(x))³ for every real number x, then

a) h increases whenever f is increasing

b) h is increasing whenever f is decreasing

c) h is decreasing whenever f is decreasing

d) nothing can be said in general

From the origin chords are drawn to the circle . The equation of the locus of the mid points of these chords is . . . . .

If then equals

a)

b)

c)

d)

The area of the triangle formed by the tangents from the point (4, 3) to the circle  and the line joining their point of contact is .

L =  = .  .  .  .

a) – 1

b) 0

c) 1

d) 2

Let  then the value of  is

a) 3ω

b) 3ω(ω – 1)

c) 3ω²

d) 3ω(1 – ω)

The area of triangle formed by the positive X–axis and the normal and tangent to the circle  at  is . . . . . .

Intercepts on the line y = x by the circle  is AB. Equation of the circle with AB as diameter is  . . . . .

The number of real solutions of the equation | x |² – 3 | x | + 2 = 0 is

a) 4

b) 1

c) 3

d) 2

Match the following

Let the function defined in column 1 has domain

a) i) → A, ii) → B

b) i) → A, ii) → C

c) i) → C, ii) → A

d) i) → B, ii) → C

A man walks a distance of three units from the origin towards north-east (N direction. From there he walks a distance of 4 units towards north–west (N direction to reach a point P. Then the position of P in the argand plane is

a)

b)

c)

d)

Find the coordinates of the point on the curve  where the tangent to the curve has the greatest slope.

a) (0, 0)

b)

c)

d)

If p, q, r are any real numbers, then

a) Max ( p, q ) < max ( p, q, r )

b) Min ( p, q ) =  

c) Max ( p, q ) < min ( p, q, r )

d) none of these

Show that, if
a, b, c, d ε ℝ

The equation of the common tangent touching the circle
 and the parabola , above X–axis is

a)

b)

c)

d)

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

Let z and ω be two complex numbers such that |z| ≤ 1 and |w| ≤ 1 then show that .

Sketch the region bounded by the curves y = x² and  . Find the area.

a)

b)

c)

d)

Find the equation of the normal to the curve  which passes through the point (1, 2).

(Multiple choices)
The determinant
  is equal to zero if

a) a, b, c are in arithmetic progression

b) a, b, c are in geometric progression

c) a, b, c are in harmonic progression

d) α is a root of the equation ax² + bx + c = 0

e) x – α is a factor of ax² + 2bx + c