Edexcel A-Level Mathematics AS P1.1 Algebra and Functions Questions

Practise P1 algebra and functions through indices, surds, quadratics, inequalities, intersections and graph-based reasoning.

Exam points

simplify powers or surds before solving equations in exact form
use quadratics, discriminants and completed square forms to find ranges or constants
read or define graph regions using inequalities and intersection coordinates

Covered syllabus points

P1.1.1 - Laws of indices P1.1.2 - Use and manipulate surds P1.1.3 - Quadratic functions and their graphs P1.1.4 - Discriminant of a quadratic P1.1.5 - Completing the square P1.1.6 - Solve simultaneous equations P1.1.7 - Interpreting linear and quadratic inequalities P1.1.8 - Graphing linear and quadratic inequalities P1.1.9 - Solving linear and quadratic inequalities P1.1.10 - Algebraic manipulation P1.1.11 - Graphs of functions P1.1.12 - Effect of simple

[Maximum number: 3]

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

By substituting $p=2^{x}$ , show that the equation

2 \times 4^{x}-2^{x+3}=17 \times 2^{x-1}-4

can be written in the form

4 p^{2}-33 p+8=0

With $p=2^x$ : $4^x=p^2,\ 2^{x+3}=8p,\ 2^{x-1}=\frac{p}{2}$ .

$2p^2-8p=\frac{17p}{2}-4$ $\Rightarrow4p^2-33p+8=0$

B1 M1 A1*

[Maximum number: 5]

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

(a)

Show that $\frac{\sqrt{180}-\sqrt{80}}{\sqrt{5}}$ is an integer and find its value.

[ 2 ]

Writes $\sqrt{180}$ as $6 \sqrt{5}$ or $\sqrt{80}$ as $4 \sqrt{5}$
Concludes working $\frac{6 \sqrt{5}-4 \sqrt{5}}{\sqrt{5}}=\frac{2 \sqrt{5}}{\sqrt{5}}=2$
(2)

(b)

Simplify

\frac{4 \sqrt{5}-5}{7-3 \sqrt{5}}

giving your answer in the form $a+b \sqrt{5}$ where a and b are rational numbers.

[ 3 ]

$\frac{4\sqrt5-5}{7-3\sqrt5} =\frac{4\sqrt5-5}{7-3\sqrt5}\times\frac{7+3\sqrt5}{7+3\sqrt5}$ $=\frac{28\sqrt5-35+12(\sqrt5)^2-15\sqrt5}{49-9\times5}$ $=\frac{25}{4}+\frac{13}{4}\sqrt5$
Marking guidance for this part:
M1 for a correct attempt to rationalise by multiplying by $7+3\sqrt5$ .
dM1 for a surd-free denominator and an attempt to multiply out the numerator with four terms.
A1 for $\frac{25}{4}+\frac{13}{4}\sqrt5$ .

[Maximum number: 6]

Figure 2

Figure 2 shows a sketch of the straight line l and the curve C.
Given that l cuts the y-axis at -12 and cuts the x-axis at 4, as shown in Figure 2,

Given that C
- has equation y=f(x) where f(x) is a quadratic expression
- has a minimum point at (7,-18)
- cuts the x-axis at 4 and at k, where k is a constant

(a)

deduce the value of k,

[ 1 ]

(b)

find f(x).

[ 3 ]

y=A(x-4)(x-10) or $y=C(x-7)^2-18$
Using (7,-18): $-18=A(7-4)(7-10)\Rightarrow A=2$

OR using (4,0): $0=C(4-7)^2-18\Rightarrow C=2$ y=2(x-4)(x-10), or $y=2(x-7)^2-18$

(c)

The region R is shown shaded in Figure 2.
Use inequalities to define R.

[ 2 ]

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)

Question (c)