Proof

Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x^2 = 25$$ [2]

1 Celia states that \(n ^ { 2 } + 2 n + 10\) is always odd when \(n\) is a prime number. Prove that Celia's statement is false.

Prove or disprove each of the following statements:

1. If \(n\) is an integer, then \(3n^2 - 11n + 13\) is a prime number. [2]
2. If \(x\) is a real number, then \(x^2 - 8x + 17\) is positive. [2]
3. If \(p\) and \(q\) are irrational numbers, then \(pq\) is irrational. [2]

Prove that, when \(n\) is an integer, \(n^3 - n\) is always even. [3]

Prove that, when \(n\) is an integer, \(n^3 - n\) is always even. [3]

1. Show that, if \(n\) is a positive integer, then \((x^n - 1)\) is divisible by \((x - 1)\). [1]
2. Hence show that, if \(k\) is a positive integer, then \(2^{8k} - 1\) is divisible by 17. [4]

Prove that \(x - 10 < x^2 - 5x\) for all real values of \(x\). [4]

Use the triangle in Fig. 4 to prove that \(\sin^2 \theta + \cos^2 \theta = 1\). For what values of \(\theta\) is this proof valid? [3] \includegraphics{figure_4}

Given that \(y \in \mathbb{R}\), prove that $$(2 + 3y)^4 + (2 - 3y)^4 \geq 32$$ Fully justify your answer. [6 marks]

1 Is the following statement true or false? Justify your answer. $$x ^ { 2 } = 4 \text { if and only if } x = 2$$

Which of these statements is correct? Tick one box. [1 mark] \(x = 2 \Rightarrow x^2 = 4\) \(x^2 = 4 \Rightarrow x = 2\) \(x^2 = 4 \Leftrightarrow x = 2\) \(x^2 = 4 \Rightarrow x = -2\)

1 Explain why each of the following statements is false. State in each case which of the symbols ⟹, ⟸ or ⇔ would make the statement true.

1. ABCD is a square ⇔ the diagonals of quadrilateral ABCD intersect at \(90 ^ { \circ }\)
2. \(x ^ { 2 }\) is an integer \(\Rightarrow x\) is an integer

Prove by contradiction the following proposition: When \(x\) is real and positive, \(x + \frac{81}{x} \geq 18\). [4]

11 Line 8 states that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\) for \(a\), \(b \geqslant 0\). Explain why the result cannot be extended to apply in each of the following cases.

1. One of the numbers \(a\) and \(b\) is positive and the other is negative.
2. Both numbers \(a\) and \(b\) are negative.

10

1. 
   10
2. 
   10
3. Given that \(a\) and \(b\) are distinct positive numbers, use proof by contradiction to prove that $$\frac { a } { b } + \frac { b } { a } > 2$$ \section*{END OF SECTION A
   TURN OVER FOR SECTION B}

Prove by contradiction that \(\sqrt{7}\) is irrational. [5]

1. Use proof by contradiction to prove that \(\sqrt { 7 }\) is irrational.
   (You may assume that if \(k\) is an integer and \(k ^ { 2 }\) is a multiple of 7 then \(k\) is a multiple of 7 )

1. (i) Find, as natural logarithms, the solutions of the equation

$$\mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } + 15 = 0$$ (ii) Use proof by contradiction to prove that \(\log _ { 2 } 3\) is irrational.

12 Prove by contradiction that 3 is the only prime number which is 1 less than a square number.

1. (i) Show that \(k ^ { 2 } - 4 k + 5\) is positive for all real values of \(k\).
   (ii) A student was asked to prove by contradiction that "There are no positive integers \(x\) and \(y\) such that \(( 3 x + 2 y ) ( 2 x - 5 y ) = 28\) " The start of the student's proof is shown below.

Assume that positive integers \(x\) and \(y\) exist such that $$\left. \begin{array} { c } ( 3 x + 2 y ) ( 2 x - 5 y ) = 28 \\ \text { If } 3 x + 2 y = 14 \text { and } 2 x - 5 y = 2 \\ 3 x + 2 y = 14 \\ 2 x - 5 y = 2 \end{array} \right\} \Rightarrow x = \frac { 74 } { 19 } , y = \frac { 22 } { 19 } \text { Not integers }$$ Show the calculations and statements needed to complete the proof.

12 Prove by contradiction that 3 is the only prime number which is 1 less than a square number.

Prove that the sum of a rational number and an irrational number is always irrational. [5 marks]

4 Millie is attempting to use proof by contradiction to show that the result of multiplying an irrational number by a non-zero rational number is always an irrational number. Select the assumption she should make to start her proof.
Tick ( \(\checkmark\) ) one box. Every irrational multiplied by a non-zero rational is irrational. □ Every irrational multiplied by a non-zero rational is rational. □ There exists a non-zero rational and an irrational whose product is irrational. □ There exists a non-zero rational and an irrational whose product is rational. □

Prove that the sum of a rational number and an irrational number is always irrational. [5 marks]

5 Prove that, for integer values of \(n\) such that \(0 \leq n < 4\) $$2 ^ { n + 2 } > 3 ^ { n }$$

5 A student's attempt to prove by contradiction that there is no largest prime number is shown below.
If there is a largest prime, list all the primes.
Multiply all the primes and add 1.
The new number is not divisible by any of the primes in the list and so it must be a new prime. The proof is incorrect and incomplete.
Write a correct version of the proof.

1 Prove that the product of consecutive integers is always even.

Prove that \(n\) is a prime number greater than \(5 \Rightarrow n^4\) has final digit \(1\) [5 marks]

3. Prove by contradiction that there is no greatest odd integer.

Given that \(n\) is an integer such that \(1 \leq n \leq 4\), prove that \(2n^2 + 5\) is a prime number. [3]

16. Prove by contradiction that if \(n ^ { 2 }\) is a multiple of \(3 , n\) is a multiple of 3 .

Shona makes the following claim. "\(n\) is an even positive integer greater than \(2 \Rightarrow 2^n - 1\) is not prime" Prove that Shona's claim is true. [4]

5 Positive integers \(a , b\) and \(c\) are said to form a Pythagorean triple if \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).

1. Given that \(t\) is an integer greater than 1 , show that \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\) form a Pythagorean triple.
2. The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\).

2. $$f ( x ) = 1 - \frac { 3 } { x + 2 } + \frac { 3 } { ( x + 2 ) ^ { 2 } } , x \neq - 2$$

1. Show that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 1 } { ( x + 2 ) ^ { 2 } } , x \neq - 2\).
2. Show that \(x ^ { 2 } + x + 1 > 0\) for all values of \(x\).
3. Show that \(\mathrm { f } ( x ) > 0\) for all values of \(x , x \neq - 2\).

The diagram below shows a circle and four triangles. \(AB\) is a diameter of the circle. \(C\) is a point on the circumference of the circle. Triangles \(ABK\), \(BCL\) and \(CAM\) are equilateral. Prove that the area of triangle \(ABK\) is equal to the sum of the areas of triangle \(BCL\) and triangle \(CAM\). [5 marks]