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.