2.12.6 Quadratic Functions, SPM Practice (Paper 1 Question 26 – 30)

Question 26 (SPM 2018 – 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)

Question 27 (SPM 2018 – 3 marks):
It is given that the curve y = (p – 2)x² – x + 7, where p is a constant, intersects with the straight line y = 3x + 5 at two points.
Find the range of values of p.

Solution:
$\begin{array}{l}y=\left(p-2\right)x^2-x+7\mathrm{\dots \dots \dots }\left(1\right)\\ y=3x+5\mathrm{\dots \dots \dots \dots \dots \dots \dots \dots \dots }\left(2\right)\\ \\ \text{Substitute }\left(1\right)\text{ into }\left(2\right):\\ \left(p-2\right)x^2-x+7=3x+5\\ \left(p-2\right)x^2-4x+2=0\\ a=\left(p-2\right),b=-4,c=2\\ b^2-4ac>0\\ {\left(-4\right)}^2-4\left(p-2\right)\left(2\right)>0\\ 16-8p+16>0\\ -8p>-32\\ 8p<32\\ p<4\end{array}$

Question 28 (SPM 2018 – 3 marks):
It is given that the quadratic equation hx² – 3x + k = 0, where h and k are constants has roots β and 2β.
Express h in terms of k.

Solution:
$\begin{array}{l}h{x}^2-3x+k=0\\ a=h,b=-3,c=k\\ \\ \text{SOR}=-\frac{b}{a}=-\frac{\left(-3\right)}{h}=\frac{3}{h}\\ \\ \text{POR}=\frac{c}{a}=\frac{k}{h}\\ \\ \text{Given roots}=\beta \text{ and 2}\beta .\\ \text{SOR}=\beta +\text{2}\beta =3\beta ;\\ \text{POR}=\beta \left(\text{2}\beta \right)=2{\beta }^2\\ \frac{3}{h}=\text{3}\beta \\ \beta =\frac{1}{h}\mathrm{\dots \dots \dots \dots ..}\left(1\right)\\ \frac{k}{h}=2{\beta }^2\mathrm{\dots \dots \dots ..}\left(2\right)\\ \\ \text{Substitute }\left(1\right)\text{ into }\left(2\right):\\ \frac{k}{h}=2{\left(\frac{1}{h}\right)}^2\\ \frac{k}{h}=\frac{2}{h^2}\\ \frac{h^2}{h}=\frac{2}{k}\\ h=\frac{2}{k}\end{array}$

Question 29  (SPM 2015 – 3 marks):
The graph of a quadratic function f(x) =px² – 8x + q, where p and q are constants, has a maximum point.
(a) Given p is an integer such that -2 < p < 2, state the value of p.
(b) Using the answer from 3(a), find the value of q when the graph touches the x-axis at one point.
[3 marks]


Answer:
(a) p = -1

(b)
$$ \begin{aligned} & f(x)=-x^2-8 x+q \\ & a=-1, b=-8, c=q \\ & b^2-4 a c=0 \\ & (-8)^2-4(-1) q=0 \\ & 64+4 q=0 \\ & 4 q=-64 \\ & q=-16 \end{aligned} $$

Question 30 (SPM 2015 – 2 marks):
It is given -7 is one of the roots of the quadratic equation (x + k)² = 16, where k is a constant.
Find the values of k.
[2 marks]

Answer:
$$ \begin{aligned} (x+k)^2= & 16 \\ \sqrt{(-7+k)^2} & = \pm \sqrt{16} \\ -7+k & = \pm 4 \\ -7+k & =4 \\ k & =4+7=11 \\ -7+k & =-4 \\ k & =-4+7=3 \end{aligned} $$