Question(s) from Search: IIT

If  where
. Given F(5) = 5, then f(10) is equal to

a) 5

b) 10

c) 0

d) 15

Subjective problems
Let .  Find all real values of x for which y takes real values.

a) [− 1, 2)

b)  [3, ∞)

c) [− 1, 2) ∪ [3, ∞)

d) None of the above

Let R be the set of real numbers and f : R → R be such that for all x and y in R, . Prove that f(x) is constant.

If f₁(x) and f₂(x) are defined by domains D₁ and D₂ respectively then f₁(x) + f₂(x) is defined as on D₁ ⋂ D₂

a) True

b) False

If  then the domain of f(x) is

The real numbers x₁, x₂, x₃ satisfying the equation x³ – x² + βx + γ = 0 are in Arithmetic Progression. Find the interval in which β and γ lie.

Let p, q, r be three mutually perpendicular vectors of the same magnitude. If x satisfies the equation p  ((x – q)  p) + q ((x – r)  q) + r  ((x – p)  r) = 0 then x is given by

a)

b)

c)

d)

Let f(x) be a non constant differentiable function defined on (−∞, ∞) such that f(x) = f(1 – x) and  then

a)  vanishes at twice an (0, 1)

b)

c)

d)

Let and a unit vector c be coplanar. If c is perpendicular to a then c is equal to

a)

b)

c)

d)

Number of solutions of  lying in the interval  is

a) 0

b) 1

c) 2

d) 3

If three complex numbers are in Arithmetic Progression, then they lie on a circle in a complex plane.

a) True

b) False

Multiple choice

The vector  is

a) A unit vector

b) Makes an angle  with the vector

c) Parallel to vector

d) Perpendicular to the vector

A₁, A₂, …… , A_n are the vertices of  a regular polygon with n sides and O is the centre. Show that
 

If A, B, C are such that |B| = |C|. Prove that

Let u and v be unit vectors. If w is a vector such that , then prove that  and that equality holds if and only if  is perpendicular to

Let n be an odd integer. If sin nθ =  for every value of θ, then

a) = 1, = 3

b) = 0, = n

c) = −1, = n

d) = 1, =

Evaluate

a)

b)

c)

d)

One or more than one correct option

The circle C₁ : x² + y² = 3 with centre at O intersect the parabola x² = 2y at the point P in the first quadrant. Let the tangent to the circle C₁ at P touches other two circles C₂ and C₃ at R₂ and R₃ respectively. Suppose C₂ and C₃ have equal radii $2\sqrt{3}$

and centres Q₂ and Q₃ respectively. If Q₂ and Q₃ lie on the Y- axis, then

a) $Q_2{Q}_3=12$

b) $R_2{R}_3=4\sqrt{6}$

c) $\mathrm{area}\mathrm{of}△{}_2{R}_3\mathrm{is}R\sqrt{2}$

d) $\mathrm{area}\mathrm{of}△{\mathrm{PQ}}_2{Q}_3\mathrm{is}4\sqrt{2}$

Let f : ℝ → ℝ be a function defined by f (x) =  . The set of points where f (x) is not differentiable is

a) }

b)

c) {0, 1}

d)

The circle passing through the point (−1, 0) and touching the Y – axis at (0, 2) also passes through the point

a) $\left(\frac{-3}{2},0\right)$

b) $\left(\frac{-5}{2},0\right)$

c) $\left(\frac{-3}{2},\frac{5}{2}\right)$

d) $(-\mathrm{4,}0)$

Let a, b, c be positive real numbers such that b² – 4ac > 0 and let α₁ = c. Prove by induction that
 

Is well defined and  for n=1, 2, …

Here well defined means that the denominator in the expression of  is not zero.

Let O be the vertex and Q be any point on the parabola x² = 8y. If the point P divides the line segment $\widehat{\mathrm{OQ}}$

internally in the ratio 1 : 3 then the locus of P is

a) x² = y

b) y² = x

c) y² = 2x

d) x² = 2y

Solve the following equation for x
 

a) −1

b)

c) 0

d) −1 and

If f is a differentiable function satisfying  for all n ≥ 1,

n  I then

a)

b)

c)

d)  is not necessarily zero

Evaluate