I) Form the equations whose roots are:

a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

II) Prove that the roots of the following equations are real:

i)

ii)

III) If the equation has equal roots, find the values of m.

IV) For what values of m will the equation have equal roots?

V) For what value of m will the equation have roots equal in magnitude but opposite in sign?

VI) Prove that the roots of the following equations are rational:

(i)

(ii)

VII) If are the roots of the equation , find the values of

(i)

(ii)

(iii)

VIII) Find the value of:

(a) when

(b) when

(c) when

IX) If and are the roots of form the equation whose roots are and /

X) Prove that the roots of are always real.

XI) If are the roots of , find the value of (i) (ii)

XII) Find the condition that one root of shall be n times the other.

XIII) If are the roots of form the equation whose roots are and .

XIV) Form the equation whose roots are the squares of the sum and of the differences of the roots of .

XV) Discuss the signs of the roots of the equation

XVI) If a, b and c are odd integers, prove that the roots of the equation cannot be rational numbers.

XVII) Given that the equation has four real positive roots, prove that (a) (b) , where equality holds, in each case, if and only if the roots are equal.

XVIII) Let be a quadratic polynomial in which a and b are integers. Given any integer n, show that there is an integer M such that .

Cheers,

Nalin Pithwa.

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