Question(s) from Search: IIT

If α is a repeated root of a quadratic equation f(x) = 0 and A(x), B(x), C(x) be polynomials of degree 3, 4, 5 respectively, Then show that
 

is divisible by f(x) where prime denotes the derivatives.

The differential equation  determines a family of circles with

a) Variable radii and a fixed centre ( 0, 1)

b) Variable radii and a fixed centre ( 0, -1)

c) Fixed radius and a variable centre along the X-axis

d) Fixed radius and a variable centre along the Y-axis

Prove that for all values of θ
 = 0

If   and  , then show that
 

A = , B = , U = , V =

If AX = U has infinitely many solutions, prove that BX = V has no unique solution. Also prove that if afd ≠ 0 then BX = V has no solution. X is a vector.

If , for every real number x, then the minimum value of f

a) does not exist because f is unbounded

b) is not attained even though f is bounded

c) is equal to 1

d) is equal to –1

Let u (x) and v (x) satisfy the differential equations and  where p (x), f (x) and g (x) are continuous functions. If u (x₁) > v (x₁) for some x₁ and f (x) > g (x) for all x > x₁, prove that at any point (x, y) where x > x₁ does not satisfy the equations y = u (x) and y = v (x)

The function  is defined by then  is

a)

b)

c)

d) None of these

  is

Suppose  for x ≥ . If g(x) is the function whose graph is the reflection of f(x) with respect to the line y = x then g(x) equals

a)

b)

c)

d)

Domain of definition of the function   for real values of x is

a)

b)

c)

d)

Let λ and α be real. Find the set of all values of λ for which the system of linear equations
 
 
 
has a non-trivial solution. For λ = 1 find the value of α.

Let f be a one–one function with domain {x, y, z} and range {1, 2, 3}. It is given that exactly one of the following statements is true and remaining statements are false f (1) = 1, f (y) ≠ 1, f (z) ≠ 2. Determine  

The value of . Given that a, x, y, z, b are in Arithmetic Progression while the value of . If a, x, y, z, b are in Harmonic Progression then find a and b.

Let {x} and [x] denote the fractional and integral part of a real number x respectively. Solve 4{x} = x + [x]

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