OCR MEI Paper 3 2018 June — Question 12 3 marks

Exam BoardOCR MEI
ModulePaper 3 (Paper 3)
Year2018
SessionJune
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof
TypeAlgebraic inequality proof
DifficultyStandard +0.3 This is a guided algebraic proof starting from a given inequality (a-b)²≥0, requiring students to expand, rearrange, and manipulate to reach the AM-GM inequality. While it involves multiple algebraic steps and division by 2, the proof is well-scaffolded with a clear starting point and standard algebraic manipulation techniques. It's slightly easier than average as it's a classic textbook proof with explicit guidance on where to begin.
Spec1.01a Proof: structure of mathematical proof and logical steps
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.
Question 12:
AnswerMarks Guidance
AnswerMark Guidance
\((a-b)^2 \geq 0 \Rightarrow a^2 - 2ab + b^2 \geq 0\)B1 Squaring bracket
\(a^2+b^2 \geq 2ab \Rightarrow a^2+2ab+b^2 \geq 4ab\)B1 Adding \(2ab\) to each side
\((a+b)^2 \geq 4ab\)
\(a+b \geq \sqrt{4ab} \Rightarrow a+b \geq 2\sqrt{ab} \Rightarrow \frac{a+b}{2} \geq \sqrt{ab}\)B1 Square root and correct completion 
## Question 12:

| Answer | Mark | Guidance |
|--------|------|----------|
| $(a-b)^2 \geq 0 \Rightarrow a^2 - 2ab + b^2 \geq 0$ | B1 | Squaring bracket |
| $a^2+b^2 \geq 2ab \Rightarrow a^2+2ab+b^2 \geq 4ab$ | B1 | Adding $2ab$ to each side |
| $(a+b)^2 \geq 4ab$ | | |
| $a+b \geq \sqrt{4ab} \Rightarrow a+b \geq 2\sqrt{ab} \Rightarrow \frac{a+b}{2} \geq \sqrt{ab}$ | B1 | Square root and correct completion |

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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.

\hfill \mbox{\textit{OCR MEI Paper 3 2018 Q12 [3]}}
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