Algebraic inequality proof

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 "

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)
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7

1. Show that
   (A) \(( x - y ) \left( x ^ { 2 } + x y + y ^ { 2 } \right) = x ^ { 3 } - y ^ { 3 }\),
   (B) \(\left( x + \frac { 1 } { 2 } y \right) ^ { 2 } + \frac { 3 } { 4 } y ^ { 2 } = x ^ { 2 } + x y + y ^ { 2 }\).
2. Hence prove that, for all real numbers \(x\) and \(y\), if \(x > y\) then \(x ^ { 3 } > y ^ { 3 }\). Section B (36 marks)

1. (i) Given that \(p\) and \(q\) are integers such that

use algebra to prove by contradiction that at least one of \(p\) or \(q\) is even.
(ii) Given that \(x\) and \(y\) are integers such that

• \(x < 0\)
• \(( x + y ) ^ { 2 } < 9 x ^ { 2 } + y ^ { 2 }\) show that \(y > 4 x\)

12 Lines 5 and 6 outline the stages in a proof that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\). Starting from \(( a - b ) ^ { 2 } \geqslant 0\), give a detailed proof of the inequality of arithmetic and geometric means.

16 Show that the expression \(a \left( \frac { x _ { P } + x _ { Q } } { 2 } \right) ^ { 2 } + b \left( \frac { x _ { P } + x _ { Q } } { 2 } \right) + c - a \left( \frac { x _ { P } - x _ { Q } } { 2 } \right) ^ { 2 }\) is equivalent to \(a x _ { P } x _ { Q } + b \left( \frac { x _ { P } + x _ { Q } } { 2 } \right) + c\), as given in lines 15 and 16 .

4 In this question you must show detailed reasoning.
Show that \(\frac { 1 } { \sqrt { 10 } + \sqrt { 11 } } + \frac { 1 } { \sqrt { 11 } + \sqrt { 12 } } + \frac { 1 } { \sqrt { 12 } + \sqrt { 13 } } = \frac { 3 } { \sqrt { 10 } + \sqrt { 13 } }\).

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 \(p \geq -1\), prove by induction that, for all integers \(n \geq 1\), $$(1 + p)^n \geq 1 + np$$ [6 marks]

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

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