A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 7: Algebraic methods · Lesson 3

Mathematical proof

75 marks

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Quick questions to start.

A1
Last lesson
Use the factor theorem to show that $(x - 2)$ is a factor of $x^{3} - 3 x^{2} - 4 x + 12$ .
[2 marks]
A2
Needed today
Write $x^{2} - 18 x + 86$ in the form $(x - p)^{2} + q$ .
[2 marks]
Answer
A3
From chapter 2
Find the discriminant of $3 x^{2} - 5 x + 4$ . Hence state the number of real roots of $3 x^{2} - 5 x + 4 = 0$ .
[2 marks]
Answer
A4
From chapter 5
A line passes through $(- 1, 4)$ and $(3, - 4)$ . Find the gradient of a line perpendicular to it.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Prove that $(2 x + 3) (x - 4) (x + 1)$ $\equiv 2 x^{3} - 3 x^{2} - 17 x - 12$ .

Hint Start from the left-hand side. Multiply two of the brackets first, then the third. Do not work on both sides at once.

B1
Prove that $(2 x + \frac{1}{x})^{3}$ $\equiv 8 x^{3} + 12 x + \frac{6}{x} + \frac{1}{x ^{3}}$ .
[3 marks]

Example 2

Prove that the sum of the squares of any three consecutive integers is always $2$ more than a multiple of $3$ .

Hint Call the integers

n - 1

n

and

n + 1

, where

n

is an integer. Expand and collect.

B2
Prove that $1$ more than the product of two consecutive even numbers is always the square of an odd number.
[3 marks]

Example 3

The equation $k x^{2} + 4 k x + 3 = 0$ , where $k$ is a constant, has no real roots. Prove that $0 \leq k < \frac{3}{4}$ .

Hint No real roots means

b^{2} - 4 a c < 0

. That rule needs a quadratic, so look at

k = 0

on its own.

B3
The equation $p x^{2} + 6 x - 2 = 0$ , where $p$ is a non-zero constant, has two distinct real roots. Prove that $p > - \frac{9}{2}$ .
[4 marks]

Example 4

Prove that every square number is either a multiple of $3$ or $1$ more than a multiple of $3$ .

Hint Every integer

n

3 k

3 k + 1

3 k + 2

for some integer

k

. Square each case.

B4
Prove that $n^{2} + n + 1$ is not divisible by $5$ for any integer $n$ .
[4 marks]

Practice

A-Level

C1
Prove that $(2 x - 1)^{2} > 8 x - 12$ for all real values of $x$ .
[3 marks]

C2
Prove that the points $A (- 2, 1)$ , $B (4, 3)$ , $C (5, 0)$ and $D (- 1, - 2)$ are the vertices of a rectangle, and that $A B C D$ is not a square.
[4 marks]
Not drawn accurately
C3
Prove that, when $n$ is an odd integer, $n^{2} - 1$ is a multiple of $8$ .
[4 marks]

Exam-style questions

Edexcel exam style

D1
Mia wants to prove that $x + \frac{9}{x} \geq 6$ for all positive real $x$ . She writes: $x + \frac{9}{x} \geq 6 \Rightarrow x^{2} + 9 \geq 6 x \Rightarrow x^{2} - 6 x + 9 \geq 0 \Rightarrow (x - 3)^{2} \geq 0$ , which is true. So $x + \frac{9}{x} \geq 6$ .
The statement is true but Mia's proof is not valid.
(a)
Explain why Mia's argument is not a valid proof.
[1 mark]
(b)
Write out a correct proof that $x + \frac{9}{x} \geq 6$ for all $x > 0$ . State clearly where you use the fact that $x > 0$ .
[4 marks]
(c)
Use a counter-example to show that the inequality is not true for all real $x \neq = 0$ .
[2 marks]
Total for D1: 7 marks

D2
The circle $C$ has equation $x^{2} + y^{2} = 125$ . The line $l$ has equation $y = 2 x + c$ , where $c$ is a constant.
Not drawn accurately
(a)
Show that the $x$ -coordinates of any points where $l$ meets $C$ satisfy $5 x^{2} + 4 c x + c^{2} - 125 = 0$ .
[2 marks]
(b)
Hence prove that $l$ is a tangent to $C$ when $c = 25$ , and find the coordinates of the point where it touches $C$ .
[5 marks]
Answer $(\dots\dots, \dots\dots)$
(c)
Find the set of values of $c$ for which $l$ meets $C$ at two distinct points.
[3 marks]
Answer $\dots\dots$
Total for D2: 10 marks
D3
In this question $n$ is an integer.
(a)
Prove that $n^{3} + 3 n^{2} + 2 n$ is a multiple of $6$ .
[4 marks]
(b)
Hence prove that $n^{3} + 3 n^{2} + 8 n$ is a multiple of $6$ .
[2 marks]
(c)
Sam says: " $n^{3} + 3 n^{2} + 2 n$ is always a multiple of $12$ ." Show that Sam is wrong.
[1 mark]
(d)
Prove that, when $n$ is even, $n^{3} + 3 n^{2} + 2 n$ is a multiple of $24$ .
[3 marks]
Total for D3: 10 marks
D4
$f (x) = x^{2} + (k - 3) x + 3 - 2 k$ , where $k$ is a constant.
(a)
Show that the discriminant of $f (x)$ is $k^{2} + 2 k - 3$ .
[2 marks]
(b)
Hence find the set of values of $k$ for which $f (x) = 0$ has no real roots.
[3 marks]
Answer $\dots\dots$
(c)
$g (x) = x^{2} + (k - 3) x - 2 k$ . Prove that $g (x) = 0$ has two distinct real roots for every value of $k$ .
[4 marks]
Total for D4: 9 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Prove that $n^{5} - n$ is divisible by $30$ for every integer $n$ .
[6 marks]
Hint 1 · What to try
$30 = 2 \times 3 \times 5$ . Factorise $n^{5} - n$ as far as you can and look for each of the three primes.
Hint 2 · The first line
$n^{5} - n = n (n^{4} - 1) = (n - 1) n (n + 1) (n^{2} + 1)$ . Three consecutive integers give the $2$ and the $3$ .
Hint 3 · The full method
For $5$ , try the five cases $n = 5 k, 5 k \pm 1, 5 k \pm 2$ . The first three make $n$ , $n - 1$ or $n + 1$ a multiple of $5$ . For $n = 5 k \pm 2$ , $n^{2} + 1 = 25 k^{2} \pm 20 k + 5$ , a multiple of $5$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can prove an identity by working from one side to the other.
I can prove a statement by deduction, by exhaustion or by completing the square.
I can use the discriminant to prove a result about a constant.
I can disprove a statement with one counter-example.

Answers and mark scheme

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