Let a, b, c be real numbers, a ≠ 0. If α is a root of β is a root of  and 0 < α < β then the equation  has a root γ that always satisfies

a) γ =

b) γ =

c) γ = α

d) α < γ < β

Let α, β be roots of the equation (x – a) (x – b) = c, c ≠ 0. Then the roots of the equation (x – α) (x – β) + c = 0 are

a) a, c

b) b, c

c) a, b

d) a + c, b + c

The expression  is a polynomial of degree

a) 5

b) 6

c) 7

d) 8

If p, q, r are positive and are in arithmetic progression the roots of the quadratic  are all real for

a)

b)

c)

d)

The equation  has

a) No solution

b) One solution

c) Two solutions

d) More than two solutions

The roots of the equation  are real and less than 3, then

a) a < 2

b) 2 < a < 3

c) 3 ≤ a ≤ 4

d) a > 4

If α and β (α < β) are roots of the equation  where c < 0 < b then

a) 0 < α < β

b) α < 0 < β < | α |

c) α < β < 0

d) α < 0 < | α | < β

If a, b, c, d are positive real numbers such that a + b + c + d = 2 then M = ( a + b ) ( c + d ) satisfies

a) 0 ≤ M ≤ 1

b) 1 ≤ M ≤ 2

c) 2 ≤ M ≤ 3

d) 3 ≤ M ≤ 4

For the equation  if one of the roots is square of the other then p is equal to

a)

b)

c) 3

d)

For all x ε ( 0, 1 )

a)

b) ln (1 + x) < x

c) sinx > x

d) lnx > x

Let α, β be the roots of  and γ, δ roots of . If α, β, γ, δ are in geometric progression then the integral values of p and q respectively are

a) −2, −32

b) −2, 3

c) −6, 3

d) −6, −32

Let f(x) =  and m(b) be the minimum value of f(x). As b varies, range of m(b) is

a)

b) [ 0,

c) [

d)

The number of solutions of  is

a) 3

b) 1

c) 2

d) 0

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

a)

b)

c)

d)

The set of all real numbers x for which  is

a)

b)

c)

d)

If ε then  is always greater than or equal to

a) 2 tan

b) 1

c) 2

d)

For all x,  then the interval in which a lies is

a) a <

b)

c)

d)

If one root is square of the other root of the equation  then the relation between p and q is

a)

b)

c)

d)

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)

Let a, b, c be the sides of a triangle where a ≠ c and λ ε R. If roots of the equation  are real then

a)

b)

c)

d)

Let α, β be roots of the equation are the roots of the equation  then the value of r is equal to

a)

b)

c)

d)

Multiple choices

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

a)

b)

c)

d)