Question(s) from Search: IIT

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 S₁, S₂, .  .  .  .,S_n are the sums of infinite geometric series whose first terms are 1, 2, 3,   .  .  ., n and whose common ratios are  respectively, then find the value of

If  are three non–coplanar vectors, then

  equals

a) 0

b)

c)

d)

Let a, b are real positive numbers. If a, A₁, A₂, b are in Arithmetic Progression, a, G₁, G₂, b are in Geometric Progression and a, H₁, H₂, b are in Harmonic Progression show that
 

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, =

The points with position vectors  and  are collinear for all real values of k.

a) True

b) False

Multiple choices
Let and  (x is measured in radians) then x lies in the interval

a)

b)

c)

d)

If  

and the vectors (1, a, a²), (1, b, b²), (1, c, c²) are non-coplanar then the product abc is

Let  and c be two vectors perpendicular to each other in the XY–plane. All vectors in the same plane having projections 1 and 2 along b and c respectively, are given by

 lies between –4 and 10.

a) True

b) False