1.01a Proof: structure of mathematical proof and logical steps

6

1. Prove the identity \(\left( \frac { 1 } { \cos x } - \tan x \right) \left( \frac { 1 } { \sin x } + 1 \right) \equiv \frac { 1 } { \tan x }\).
2. Hence solve the equation \(\left( \frac { 1 } { \cos x } - \tan x \right) \left( \frac { 1 } { \sin x } + 1 \right) = 2 \tan ^ { 2 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

2 Prove the identity $$\frac { 1 + \sin x } { \cos x } + \frac { \cos x } { 1 + \sin x } \equiv \frac { 2 } { \cos x }$$

2 Prove the identity $$\tan ^ { 2 } x - \sin ^ { 2 } x \equiv \tan ^ { 2 } x \sin ^ { 2 } x$$

1 Prove the identity $$\cot \theta - \tan \theta \equiv 2 \cot 2 \theta$$

6 It is given that \(\tan 3 x = k \tan x\), where \(k\) is a constant and \(\tan x \neq 0\).

1. By first expanding \(\tan ( 2 x + x )\), show that $$( 3 k - 1 ) \tan ^ { 2 } x = k - 3$$
2. Hence solve the equation \(\tan 3 x = k \tan x\) when \(k = 4\), giving all solutions in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
3. Show that the equation \(\tan 3 x = k \tan x\) has no root in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\) when \(k = 2\).

9. An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays \(\pounds 149\) in the first month, \(\pounds 147\) in the second month, \(\pounds 145\) in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
2. Find the amount Sean repays in the 21st month. Over the \(n\) months, he repays a total of \(\pounds 5000\).
3. Form an equation in \(n\), and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0 .$$
4. Solve the equation in part (c).
5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.

5. (i) Use algebra to prove that for all \(x \geqslant 0\) $$3 x + 1 \geqslant 2 \sqrt { 3 x }$$ (ii) Show that the following statement is not true.
"The sum of three consecutive prime numbers is always a multiple of 5 "

10. (i) Prove by counter example that the statement
"if \(p\) is a prime number then \(2 p + 1\) is also a prime number" is not true.
(ii) Use proof by exhaustion to prove that if \(n\) is an integer then $$5 n ^ { 2 } + n + 12$$ is always even.

1. A student was asked to prove by exhaustion that
   if \(n\) is an integer then \(2 n ^ { 2 } + n + 1\) is not divisible by 3

The start of the student's proof is shown in the box below. Consider the case when \(n = 3 k\) $$2 n ^ { 2 } + n + 1 = 18 k ^ { 2 } + 3 k + 1 = 3 \left( 6 k ^ { 2 } + k \right) + 1$$ which is not divisible by 3 Complete this proof.

1. (i) Use a counter example to show that the following statement is false

$$\text { " } n ^ { 2 } + 3 n + 1 \text { is prime for all } n \in \mathbb { N } \text { " }$$ (ii) Use algebra to prove by exhaustion that for all \(n \in \mathbb { N }\) $$\text { " } n ^ { 2 } - 2 \text { is not a multiple of } 4 \text { " }$$

3. (i) Use algebra to prove that for all real values of \(x\) $$( x - 4 ) ^ { 2 } \geqslant 2 x - 9$$ (ii) Show that the following statement is untrue. $$2 ^ { n } + 1 \text { is a prime number for all values of } n , n \in \mathbb { N }$$

3. (i) Prove that for all single digit prime numbers, \(p\), $$p ^ { 3 } + p \text { is a multiple of } 10$$ (ii) Show, using algebra, that for \(n \in \mathbb { N }\) $$( n + 1 ) ^ { 3 } - n ^ { 3 } \text { is not a multiple of } 3$$

1. (i) A student writes the following statement:
   "When \(a\) and \(b\) are consecutive prime numbers, \(a ^ { 2 } + b ^ { 2 }\) is never a multiple of 10 "
   Prove by counter example that this statement is not true.
   (ii) Given that \(x\) and \(y\) are even integers greater than 0 and less than 6 , prove by exhaustion, that

$$1 < x ^ { 2 } - \frac { x y } { 4 } < 15$$

9. (a) Prove that for all positive values of \(x\) and \(y\), $$\frac { x + y } { 2 } \geqslant \sqrt { x y }$$ (b) Prove by counter-example that this inequality does not hold when \(x\) and \(y\) are both negative.
(1)
\includegraphics[max width=\textwidth, alt={}, center]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-29_61_54_2608_1852}

1. Given that \(a , b\) and \(c\) are integers greater than 0 such that

• \(c = b + 2\)
• \(a + b + c = 10\)

Prove, by exhaustion, that the product of \(a , b\) and \(c\) is always even.
You may use the table below to illustrate your answer. You may not need to use all rows of this table.

1. Given that \(a , b\) and \(c\) are integers greater than 0 such that

• \(c = 3 a + 1\)
• \(a + b + c = 15\) prove, by exhaustion, that the product \(a b c\) is always a multiple of 4
  You may use the table below to illustrate your answer.

You may not need to use all rows of this table.

Given \(n \in \mathbb { N }\), prove, by exhaustion, that \(n ^ { 2 } + 2\) is not divisible by 4 . \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-12_2658_1943_111_118}

1. (a) Show that

$$\cot x - \cot 2 x \equiv \operatorname { cosec } 2 x , \quad x \neq \frac { n \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve for \(0 \leqslant \theta \leqslant \pi\) $$\operatorname { cosec } \left( 3 \theta + \frac { \pi } { 3 } \right) + \cot \left( 3 \theta + \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 3 } }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7. (a) Show that

1. \(\frac { \cos 2 x } { \cos x + \sin x } \equiv \cos x - \sin x , \quad x \neq \left( n - \frac { 1 } { 4 } \right) \pi , n \in \mathbb { Z }\),
2. \(\frac { 1 } { 2 } ( \cos 2 x - \sin 2 x ) \equiv \cos ^ { 2 } x - \cos x \sin x - \frac { 1 } { 2 }\).
   (b) Hence, or otherwise, show that the equation $$\cos \theta \left( \frac { \cos 2 \theta } { \cos \theta + \sin \theta } \right) = \frac { 1 } { 2 }$$ can be written as $$\sin 2 \theta = \cos 2 \theta$$ (c) Solve, for \(0 \leqslant \theta < 2 \pi\), $$\sin 2 \theta = \cos 2 \theta$$ giving your answers in terms of \(\pi\).

1. (a) By writing \(\sin 3 \theta\) as \(\sin ( 2 \theta + \theta )\), show that

$$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$ (b) Given that \(\sin \theta = \frac { \sqrt { } 3 } { 4 }\), find the exact value of \(\sin 3 \theta\).

1. 1. Prove that

$$\sec ^ { 2 } x - \operatorname { cosec } ^ { 2 } x \equiv \tan ^ { 2 } x - \cot ^ { 2 } x$$ (ii) Given that $$y = \arccos x , \quad - 1 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant \pi ,$$

1. express arcsin \(x\) in terms of \(y\).
2. Hence evaluate \(\arccos x + \arcsin x\). Give your answer in terms of \(\pi\).

1. (a) Prove that

$$2 \cot 2 x + \tan x \equiv \cot x \quad x \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(- \pi \leqslant x < \pi\), $$6 \cot 2 x + 3 \tan x = \operatorname { cosec } ^ { 2 } x - 2$$ Give your answers to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-14_2258_47_315_37}