[Maximum mark: 6]
Question:
Consider any three consecutive integers, n – 1, n and n + 1.
(a) Prove that the sum of these three integers is always divisible by 3.
▶️Answer/Explanation
Ans: (n – 1) + n + (n + 1)
= 3n
which is always divisible by 3
(b) Prove that the sum of the squares of these three integers is never divisible by 3.
▶️Answer/Explanation
Ans: (n -1)2 +n2 + (n+1)2 (=n2 – 2n + 1 + n2 + n2 + 2n + 1)
attempts to expand either (n -1)2 or (n+1)2 ( do not accept n2 – 1 or n2 + 1)
= 3n2 + 2
demonstrating recognition that 2 is not divisible by 3 or \(\frac{2}{3}\) seen after correct expression divided by 3
3n2 is divisible by 3 and so 3n2 + 2 is never divisible by 3 OR the first term is divisible by 3, the second is not
OR \(3\left ( n^{2}+\frac{2}{3} \right ) OR \frac{3n^{2} + 2}{3} = n^{2} + \frac{2}{3}\) hence the sum of the squares is never divisible by 3
Question:
[Maximum mark: 8] [without GDC]
(a) Prove the identity \(2x^{3}+7x^{2}-14x+5\equiv (x+5)(2x^{2}-3x+1)\) [1]
▶️Answer/Explanation
Ans: RHS to LHS (easy)
(b) Given that \(3x^{3}+13x^{2}-3x+35\equiv (x+5)(ax^{2}+bx+c)\) find the values of \(a,b, c\) . [2]
▶️Answer/Explanation
Ans: RHS = \(ax^{3}+bx^{2}+cx+5ax^{2}+5bx+5c=ax^{3}+(b+5a)x^{2}+(c+5b)x+5c\)
Compare with LHS: a = 3, b + 5a =13⇔b = −2 , and 5c = 35⇔c = 7
(c) Given that \(ax^{3}+bx^{2}-23x+c\equiv (x+5)(2x^{2}+dx+2)\) find the values of \(a,b, c, d\) . [2]
▶️Answer/Explanation
Ans: RHS = \(2x^{3}+dx^{2}+2x+10x^{2}+5dx+10=2x^{3}+(d+10)x^{2}+(2+5d)x+10\)
Compare with LHS: a = 2 , b = d +10 , 2 + 5d = −23 and c =10
Thus a = 2 , b = 5 , c =10 and d = −5
Question
[Maximum mark: 4] [without GDC]
Prove the following statement:
For all integers \(n\) , if n2+ 2n+5 is odd then \(n\) is even.
▶️Answer/Explanation
Ans.
Suppose that the result is not true. That is \(n = 2k + 1\) is odd. Then
\(n^{2}+2n+5=(2k+1)^{2}+2(2k+1)+5=4k^{2}+4k+1+4k+2+7=4k^{2}+8k+10\)
which is even. Contradiction.
Question:
Consider two consecutive positive integers, n and n + 1 .
Show that the difference of their squares is equal to the sum of the two integers.
▶️Answer/Explanation
Ans:
diff. of squares =\(\left(n+1\right)^2-n^2\)
=\(\left(n+1+n\right)\left(n+1-n\right)\)
= \(\left(n+1+n\right)\left(n+1-n\right)\)
=2n+1
=n+1+n
=sum of 2 integers