Question(s) from Search: IIT

For all x ε ( 0, 1 )

a)

b) ln (1 + x) < x

c) sinx > x

d) lnx > x

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

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.

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
 

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the value of the determinant chosen is positive is

a)

b)

c)

d)

If one root of  is equal to the power of the other then show that
 

If a, b, c are in Geometric Progression then the equations  
have a common root if  are in

a) Arithmetic Progression

b) Geometric Progression

c) Harmonic Progression

d) None of these

Solve for x in the following equation

Let T_r be the r^th term of an Arithmetic Progression for  If for some positive integers m, n we have  and  then T_mn equals

a)

b)

c)

d)

Consider an infinite geometric series with first term a and common ratio r. If its sum is four and the second term is  then

a)

b)

c)

d)

For every positive integer n, prove that
 
Hence or otherwise prove that
 
Where [ ] denotes greatest integer not exceeding x.

If  are positive real numbers whose product is a fixed number c then the minimum value of
  is

a)

b)

c)

d)

Let a and b the roots of the equation  and those of  are c and d, then find the value of a + b + c + d when a ≠ b ≠ c ≠ d.

Fill in the blanks

If  is a root of the equation  where p and q are real then (p, q)  …………

If the m^th, n^th and p^th term of an Arithmetic Progression and a Geometric Progression are equal and are x, y, z then prove that
 

Fill in the blank

If the quadratic equation
  and  have a common root then the numerical value of a + b is …………

Fill in the blank

The sum of the real roots of the equation
 is ………..

If  are in Arithmetic Progression, determine the value of x.

The number  is

a) an integer

b) a rational number

c) an irrational number

d) a prime number

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of four consecutive terms of it, prove that the resulting sum is square of an integer.