CAIE A-Level Mathematics AS 1.1.5 Quadratic Form Equations Questions

Practise interpreting quadratic graphs and transformations within the AS mathematics syllabus.

Exam points

interpret key features of quadratic graphs
apply graph transformations to quadratic functions
use graphs to solve quadratic equations

[Maximum number: 3]

Hence find the exact solutions of the equation $3 x^{4}-12 x^{2}-15=0$ .

$\left(x^{2}-2\right)^{2}=9$ leading to $x^{2}-2= \pm 3$
Must be $x^{2}$ unless substitution is clear.

$x^{2}=-1$ or $x^{2}=5$
Marking guidance:

Allow omission of -1 if $\pm 3$ seen.

$x= \pm \sqrt{5}$
B1 SC if M1M1 not awarded. Ignore $\pm \mathrm{i}, \mathrm{i},-\mathrm{i}, \sqrt{ }-1$ .
Use of calculator with no working scores 0/3.

Alternative method for Question 1(b)

$3 x^{4}-12 x^{2}-15=0$ leading to $3\left(x^{2}-5\right)\left(x^{2}+1\right)[=0]$ $x^{2}=-1$ or $x^{2}=5$
Allow omission of -1 if factors seen. Factorising or other
valid method.

$x= \pm \sqrt{5}$
B1 SC if M1M1 not scored. Ignore $\pm \mathrm{i}, \mathrm{i},-\mathrm{i}, \sqrt{ }-1$ .
Use of calculator with no working scores 0/3.

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CAIE A-Level Mathematics AS 1.1.5 Quadratic Form Equations Questions question 1