iGCSE Mathematics (0580) :E1.3 Calculate with squares, square roots, cubes and cube roots and other powers and roots of numbers..iGCSE Style Questions Paper 4

(a) Ans: \(\frac{10x}{(x-3)(x+2)}\) or \(\frac{10x}{x^2 – x – 6}\)

Combine the fractions using a common denominator \((x-3)(x+2)\):

\(\frac{(x+3)(x+2) – (x-2)(x-3)}{(x-3)(x+2)} = \frac{x^2 + 5x + 6 – (x^2 – 5x + 6)}{(x-3)(x+2)} = \frac{10x}{x^2 – x – 6}\).

(b) Ans: \(k = 14\)

Simplify the equation using exponent rules:

\(2^{12 – \frac{k}{2}} = 2^5 \implies 12 – \frac{k}{2} = 5 \implies \frac{k}{2} = 7 \implies k = 14\).

(c) Ans: \(2y^3 – 3y^2 – 23y + 12\)

First expand \((y+3)(y-4) = y^2 – y – 12\), then multiply by \((2y-1)\):

\((y^2 – y – 12)(2y – 1) = 2y^3 – y^2 – 2y^2 + y – 24y + 12 = 2y^3 – 3y^2 – 23y + 12\).

(d) Ans: \(x = \frac{3}{y – 1}\)

Rearrange the equation to solve for \(x\):

\(x = \frac{3 + x}{y} \implies xy = 3 + x \implies xy – x = 3 \implies x(y – 1) = 3 \implies x = \frac{3}{y – 1}\).