CAIE A-Level Mathematics AS 1.1 Quadratics Questions

Practise completing the square, solving quadratic equations and inequalities, using the discriminant, eliminating simultaneous variables and recognising quadratic form.

Exam points

complete the square to expose the vertex, range or exact roots of a quadratic
use b² − 4ac with strict or non-strict inequalities to classify real roots
substitute or eliminate variables and solve the resulting quadratic for every valid solution

Covered syllabus points

1.1.1—Completing the square 1.1.2—Quadratic discriminant 1.1.3—Quadratic equations 1.1.4—Simultaneous equations 1.1.5—Quadratic-form equations

[Maximum number: 3]

The function f is defined by $\mathrm{f}(x)=10+6 x-x^{2}$ for $x \in \mathbb{R}$ .

By completing the square, find the range of f .

Express f(x) as: $a-(x-3)^{2}$ or $a-(3-x)^{2}$ where $a= \pm 19$ or $\pm 1$
OE

If the form $-f(x)=\left(x^{2}-6 x-10\right)$ is used the form must be returned to $f(x)=\ldots$

Completed square form must give $-x^{2}$ . Answers must come from completion of the square (not calculus or graphs).

$19-(3-x)^{2}$ or $19-(x-3)^{2}$
OE

$f(x) \leqslant 19$ or $y \leqslant 19$ with $\leqslant$ , not <
or $-\infty<\mathrm{f}(x) \leqslant 19$ or $-\infty \leqslant \mathrm{f}(x) \leqslant 19$
or $(-\infty, 19]$ or $[-\infty, 19]$

A1 FT

Using their constant following the award of M1.
SC B1 answer only or answer from a method not
involving completion of the square.

[Maximum number: 3]

(a)

Express $4 x^{2}-24 x+p$ in the form $a(x+b)^{2}+c$ , where a and b are integers and c is to be given in terms of the constant p.

[ 2 ]

4(x-3)^2

a=4,\quad b=-3

c=p-36

(b)

Hence or otherwise find the set of values of p for which the equation $4 x^{2}-24 x+p=0$ has no real roots.

[ 1 ]

p-36>0 leading to p>36 or $24^{2}-4 \times 4 p\langle 0 \Rightarrow p\rangle 36$ or 36<p
Marking guidance:

Allow $(36, \infty)$ or $36<p<\infty$ . Consider final answer only.

[Maximum number: 1]

Figure for Question 3 — CAIE A-Level Mathematics AS

The function f is defined by $\mathrm{f}(x)=\frac{2}{x^{2}}+4$ for x<0. The diagram shows the graph of y=f(x).

Solve the equation f(x)=4.5.

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question 3