A-Level Edexcel Maths Pure Modelling with Exponentials & Logarithms: (i) Given that p and q are integ

(i) 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 - Edexcel - A-Level Maths Pure - Question 9 - 2022 - Paper 1

Question 9

(i)-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-Edexcel-A-Level Maths Pure-Question 9-2022-Paper 1.png

(i) 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. (ii) Given that x and y ar... show full transcript

Worked Solution & Example Answer:(i) 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 - Edexcel - A-Level Maths Pure - Question 9 - 2022 - Paper 1

Step 1

Given that p and q are integers such that $$pq$$ is even

96%

114 rated

Answer

To prove that at least one of p or q is even using a proof by contradiction:

Assume both p and q are odd.
Let p be represented as $p = 2m + 1$ and q as $q = 2n + 1$ , where m and n are integers.
Calculate pq:
$pq = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1$
Rearranging gives us
$pq = 2(2mn + m + n) + 1$
This indicates that pq is odd.
Arrive at a contradiction:
Since we began with the assumption that pq is even, we have reached a contradiction.
Therefore, at least one of p or q must be even.

Step 2

Given that x and y are integers such that * $x < 0$ * $(x+y)^2 < 9x^2 + y^2$

99%

104 rated

Answer

To show that $y > 4x$ :

Start with the given inequality:
We have
$(x+y)^2 < 9x^2 + y^2$
Expanding $(x+y)^2$ gives
$x^2 + 2xy + y^2 < 9x^2 + y^2$
Simplify the inequality:
By rearranging, we get
$2xy < 8x^2$
which simplifies to
$xy < 4x^2$
Divide by x (note x < 0):
Since $x < 0$ , dividing both sides by x will reverse the inequality:
$y > 4x$
Thus, we have shown that under the given conditions, $y > 4x$ .

(i) 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 - Edexcel - A-Level Maths Pure - Question 9 - 2022 - Paper 1

Worked Solution & Example Answer:(i) 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 - Edexcel - A-Level Maths Pure - Question 9 - 2022 - Paper 1

Given that p and q are integers such that $$pq$$ is even

Given that x and y are integers such that * $x < 0$ * $(x+y)^2 < 9x^2 + y^2$

Join the A-Level students using SimpleStudy...