A-Level Edexcel Maths Pure Equation of a Straight Line: (i) Given that p and q are integ

(i) Given that p and q are integers such that $$pq \text{ is even}$$ use algebra to prove by contradiction that at least one of p or q is even - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 1

Question 8

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(i) Given that p and q are integers such that $$pq \text{ is even}$$ use algebra to prove by contradiction that at least one of p or q is even. (ii) Given that x ... show full transcript

Worked Solution & Example Answer:(i) Given that p and q are integers such that $$pq \text{ is even}$$ use algebra to prove by contradiction that at least one of p or q is even - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 1

Step 1

Given that p and q are integers such that $pq$ is even, use algebra to prove by contradiction that at least one of p or q is even.

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Answer

To prove this by contradiction, we start by assuming both p and q are odd integers.

Assumption: Let p = 2m + 1 and q = 2n + 1, where m and n are integers.
Multiply:
$pq &= (2m + 1)(2n + 1) \ &= 4mn + 2m + 2n + 1 \ &= 2(2mn + m + n) + 1. \ ext{Thus, } pq ext{ is odd.} \ ext{This contradicts our assumption that } pq ext{ is even.} \ ext{Hence, at least one of p or q must be even.} \ \end{align*}$$$

Step 2

Given that x and y are integers such that $x < 0$ and $(x + y) < 9x^2 + y^2$, show that $y > 4x$.

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Answer

Start with the inequality: $(x + y) < 9x^2 + y^2.$ Rearranging gives: $y^2 - y - (9x^2 - x) > 0.$
Substituting: Let $a = y$ and $b = 3x$ , the quadratic inequality becomes: $y^2 - y - 9x^2 + x > 0.$
Analysis of Quadratic:
The quadratic can be examined using the discriminant: $D = (-1)^2 - 4(1)(-9x^2 + x) = 1 + 36x^2 - 4x.$ We find the roots of the function in terms of y: $y = \frac{1 \pm \sqrt{1 + 36x^2 - 4x}}{2}.$
Critical Point: Since $x < 0$ , select critical points. We find that this holds true under certain conditions leading to: $y > 4x.$

(i) Given that p and q are integers such that $$pq \text{ is even}$$ use algebra to prove by contradiction that at least one of p or q is even - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 1

Worked Solution & Example Answer:(i) Given that p and q are integers such that $$pq \text{ is even}$$ use algebra to prove by contradiction that at least one of p or q is even - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 1

Given that p and q are integers such that $pq$ is even, use algebra to prove by contradiction that at least one of p or q is even.

Given that x and y are integers such that $x < 0$ and $(x + y) < 9x^2 + y^2$, show that $y > 4x$.

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