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10. (i) Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4 - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 1
Question 12
10. (i) Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4. (ii) "Given x ∈ ℝ, the value of |3x − 28| is greater than or equal to the value of (x − 9)." State, ... show full transcript
Worked Solution & Example Answer:10. (i) Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4 - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 1
Step 1
Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4.
Answer
To prove that for all natural numbers n, the expression n² + 2 is not divisible by 4, we can analyze the form that n can take.
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Consider Residues Modulo 4:
Natural numbers can be congruent to 0, 1, 2, or 3 modulo 4. -
Evaluate Each Case:
- If n ≡ 0 (mod 4):
Then n² ≡ 0 (mod 4) and n² + 2 ≡ 2 (mod 4) which is not divisible by 4. - If n ≡ 1 (mod 4):
Then n² ≡ 1 (mod 4) and n² + 2 ≡ 3 (mod 4) which is not divisible by 4. - If n ≡ 2 (mod 4):
Then n² ≡ 0 (mod 4) (as 2² = 4 is divisible by 4) and n² + 2 ≡ 2 (mod 4) which is not divisible by 4. - If n ≡ 3 (mod 4):
Then n² ≡ 1 (mod 4) and n² + 2 ≡ 3 (mod 4) which is not divisible by 4.
- If n ≡ 0 (mod 4):
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Conclusion:
In all cases, n² + 2 is not divisible by 4. Therefore, the statement is proven for all natural numbers n.
Step 2
State, giving a reason, if the above statement is always true, sometimes true or never true.
Answer
The statement "|3x − 28| is greater than or equal to (x − 9)" is sometimes true.
Reasoning:
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Breaking Down the Absolute Value: The expression |3x − 28| can have two cases:
- Case 1: 3x − 28 ≥ 0, then |3x − 28| = 3x − 28.
- Case 2: 3x − 28 < 0, then |3x − 28| = -(3x − 28) = 28 − 3x.
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Numerical Example:
If we substitute x = 10:- |3(10) − 28| = |30 − 28| = 2.
- (10 − 9) = 1.
Here, 2 ≥ 1 is true.
If we substitute x = 8:
- |3(8) − 28| = |24 − 28| = 4.
- (8 − 9) = -1.
Here, 4 ≥ -1 is true.
However, for certain values of x, such as x = 9.5:
- |3(9.5) − 28| = |28.5 − 28| = 0.5.
- (9.5 − 9) = 0.5
Here, 0.5 = 0.5, which fulfills 'greater than or equal to'.
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Conclusion:
The statement is sometimes true as it depends on the value of x.