Question(s) from Search: IIT

For all x ε ( 0, 1 )

a)

b) ln (1 + x) < x

c) sinx > x

d) lnx > x

Given x = cy + bz, y = az + cx, z = bx + ay where x, y, z are not all zero, prove that  a² + b² + c² + 2abc = 1

Let and  are two complex numbers such that  then prove that .

The number of values of k for which the system of equations

(k + 1) x + 8y = 4k

kx + ( k + 3 ) y = 3k – 1

has infinitely many solutions is

a) 0

b) 1

c) 2

d) Infinity

Without expanding a determinant at any stage show that
 = Ax + B

where A, B are non-zero constants

True/False
If the complex numbers  represent the vertices of an equilateral triangle with  then .

a) True

b) False

The order of the differential equation whose general solution is given by  is

a) 5

b) 4

c) 3

d) 2

If f (x) =

a) f (x) is a strictly increasing function

b) f (x) has a local maxima

c) f (x) is a strictly decreasing function

d) f (x) is bounded

Let Δa =
Then show that  = c, a constant.

For any two complex numbers  and any real numbers  is equal to .  .  .  .

a)  

b)  

c)  

d)  

The locus of a variable point whose distance from  is  times its distance from the line  is

a) Ellipse

b) Parabola

c) Hyperbola

d) None of these

If  and  then  equals

a)

b)

c)

d) 1

The second degree polynomial satisfying f (0) = 0, f (1) = 1,  for all x ε [0, 1] is

a)

b) No such polynomial

c)

d)

For a > 0, d > 0, find the value of the determinant
 

a) 0

b) 1

c)

d)

Multiple choices

For real x, the function  will assume all real values provided

a)

b)

c)

d)

If the matrix A is equal to where a, b, c are real positive numbers, abc = 1 and A^TA = I then find the value of a³ + b³ + c³.

a) 1

b) 2

c) 3

d) 4

Prove if α, β are roots of the equation  and γ, δ are roots of  then show that
 

The number of values of x where the function  attains its maximum is

a) 0

b) 1

c) 2

d) infinite

The domain of the definition of the function y given by the equation  is

a) 0 < x < 1

b) 0 ≤ x ≤ 1

c) ∞ < x ≤ 0

d) ∞ < x ≤ 1

Solution of the differential equation is

Let A =

If U₁, U_2, U₃ are column matrices satisfying
AU₁ =  , AU₂ =  and AU₃ =

and U is a 3 x 3 matrix whose columns are U₁, U_2, U₃  then the value of [ 3  2  0 ] U  is

a)

b)

c)

d)

Let f(x) =   , x ≠  then for what value of α, f(f(x)) = x

a)

b)

c)

d)

If  and  then f is

a) One-one and onto

b) One-one but not onto

c) Onto but not one-one

d) Neither one-one nor onto

If
and
Then f – g is

a) Neither one to one nor onto

b) One to one and onto

c) One to one and into

d) Many one and onto

Let a, b, c, d be real numbers in geometric progression. If u, v, w satisfy the system of equations

 
 
 
Then show that the roots of the equation
 
 
and  are reciprocal of each other.