Q1 of 168 Page 7

Divide 12 into two parts such that their product is 32.

Let the first number be ‘X’, so the other number will be ’(12–X)’.

∵ X (12 – X) = 32

12X – X² = 32

X² – 12X + 32 = 0

On factorising further,

X² – 4X – 8X + 32 = 0

X(X – 4) – 8(X – 4) = 0

(X – 4)(X – 8) = 0

So, X = 4 or 8

∴ The numbers are 4 and 8.

Find the values of k, for which the given equation has real roots:

x² + k(4x + k — 1) + 2 = 0

Prove that the equation x²(a² + b²) + 2x(ac + bd) + (c² + d²)= 0 has no real root, if ad ≠ bc.

Two numbers differ by 3 and their product is 504. Find the numbers.

Find two consecutive positive integers, the sum of whose squares is 365.