Let S is the set of all real x, such that  is positive, then S contains

a)

b)

c)

d)

The equation  has

a) At least one real solution

b) Exactly three real solutions

c) Has exactly one irrational solution

d) Complex roots

Show that square of  is a rational number.

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

Show that for for any triangle with sides a, b, c
3 (ab + bc + ac) ≤ (a + b + c)² < 4 (ab + bc + ca)

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

Find all the real values of x which satisfy  and  .

If a > 0, b > 0, c > 0, prove that  

Solve for x

For a ≤ 0, determine all real roots of the equation
 

Find the set of all x for which

Let  be roots of the equations and  respectively. If the system of equations  and  have non-trivial solutions then prove that

Solve for x in the following equation

Solve  

If α, β are roots of  and  are roots of  for some constant δ, then prove that
 

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

Let a, b, c be real numbers with a ≠ 0 and let α, β be roots of the equation . Express the roots of  in terms of α, β.

If  is the area of a triangle with sides a, b, c then show that
 .
Also show that equality occurs if a = b = c

 where a, b ε R then find the value of a for which equation has unequal roots for all values of b.

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.

If a < b < c < d then the roots of the equation
  
are real and distinct.

a) True

b) False

If x and y are positive real numbers and m and n are any positive integers then

a) True

b) False

Fill in the blanks

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