SPM Additional Mathematics 2018, Paper 1 (Question 17 – 19)

Question 17 (4 marks):
$\begin{array}{l}\left(\text{a}\right)\text{ Given }P={\log }_{a}Q\text{, state the }\\ \text{conditions of }a\text{.}\\ \left(\text{b}\right)\text{ Given }{\log }_{3}y=\frac{2}{{\log }_{xy}3}\text{, express }\\ y\text{ in terms of }x\text{.}\end{array}$

Solution:
(a)
a > 0, a ≠ 1

(b)
$\begin{array}{l}{\log }_{3}y=\frac{2}{{\log }_{xy}3}\\ \frac{{\log }_{xy}y}{{\log }_{xy}3}=\frac{2}{{\log }_{xy}3}\\ {\log }_{xy}y=2\\ y={\left(xy\right)}^2\\ y=x^2{y}^2\\ \frac{1}{x^2}=\frac{y^2}{y}\\ y=\frac{1}{x^2}\end{array}$

Question 18 (3 marks):
Diagram 8 shows the graph y = a (x – p)² + q, where a, p and q are constants. The straight line y = –8 is the tangent to the curve at point H.

Diagram 8

(a) State the coordinates of H.
(b) Find the value of a.


Solution:
(a)
$\begin{array}{l}\text{Coordinate }x\text{ of }H\\ =\frac{-1+7}{2}\\ =\frac{6}{2}\\ =3\\ \\ \text{Thus, coordinates of }H=\left(3,-8\right).\end{array}$

(b)
$\begin{array}{l}y=a{\left(x-p\right)}^2+q\\ y=a{\left(x-3\right)}^2+\left(-8\right)\\ y=a{\left(x-3\right)}^2-8\mathrm{\dots \dots \dots }\left(1\right)\\ \\ \text{Substitute }\left(7,0\right)\text{ into }\left(1\right):\\ 0=a{\left(7-3\right)}^2-8\\ 0=16a-8\\ 16a=8\\ a=\frac{1}{2}\end{array}$

Question 19 (3 marks):
Faizal has a rectangular plywood with a dimension 3x metre in length and 2x metre in width. He cuts part of the plywood into a square shape with sides of x metre to make a table surface.
Find the range of values of x if the remaining area of the plywood is at least (x² + 4) metre².

Solution:



Area of plywood – area of square ≥ (x² + 4)
3x(2x) – x² ≥ x² + 4
6x² – x² – x² ≥ 4
4x² ≥ 4
x² – 1 ≥ 0
(x + 1)(x – 1) ≥ 0
x ≤ –1 or x ≥ 1
Thus, x ≥ 1 (length is > 0)