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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 2
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 2
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 it based on the parity of n.
Case 1: n is even
If n is even, we can express n as n = 2k for some integer k. Then:
This shows that n² + 2 is always even but not divisible by 4 since 2(2k² + 1) has exactly one factor of 2, making it not divisible by 4.
Case 2: n is odd
If n is odd, we can express n as n = 2k + 1 for some integer k. Then:
This can be expressed as 4(k² + k) + 3, which indicates that n² + 2 is congruent to 3 mod 4. Hence, it is not divisible by 4 either.
Conclusion
Since in both cases, whether n is even or odd, n² + 2 is not divisible by 4, we have proven that for all n ∈ ℕ, n² + 2 is indeed not divisible by 4.
Step 2
State, giving a reason, if the above statement is always true, sometimes true or never true.
Answer
To analyze the statement given, "Given x ∈ ℝ, the value of |3x - 28| is greater than or equal to the value of (x - 9)," we can evaluate specific numerical values.
Testing the Statement
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Example where the statement is true: Let x = 10.
- Calculate |3(10) - 28| = |30 - 28| = 2
- And (10 - 9) = 1 In this case, 2 is indeed greater than 1.
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Example where the statement is false: Let x = 9.5.
- Calculate |3(9.5) - 28| = |28.5 - 28| = 0.5
- And (9.5 - 9) = 0.5 Here, 0.5 is equal to 0.5, hence not greater.
Conclusion
This shows the statement is sometimes true because there exist values of x for which the condition holds and others where it does not, making it conditionally valid.