(a) Ans: $125x^9$
First, simplify $25^{\frac{3}{2}} = 5^3 = 125$ and $x^{6 \times \frac{3}{2}} = x^9$.
Multiply them to get $125x^9$.
(b) Ans: $6^{n-2}$
The sequence is geometric with common ratio $6$.
The first term is $\frac{1}{6} = 6^{-1}$, so the $n$th term is $6^{n-2}$.
(c) Ans: $3x^3 + 2x^2 – 37x + 12$
First expand $(x+4)(x-3) = x^2 + x – 12$.
Multiply by $(3x-1)$ and simplify to get $3x^3 + 2x^2 – 37x + 12$.
(d)(i)
Multiply through by $(x-2)$ to eliminate the fraction.
Expand and simplify to get $2x^2 + x – 3 = 0$.
(d)(ii) Ans: $x = 1$ or $x = -\frac{3}{2}$
Factorise $2x^2 + x – 3 = (2x+3)(x-1)$.
Solve for $x$ to get the roots.
(e)
Surface area of cylinder: $8\pi x^2$.
Surface area of hemisphere: $75\pi y^2$.
Equate and solve for $x^2$ to get $x^2 = \frac{75y^2}{8}$.
