SSCE HSC Mathematics Extension 2 Applications of mathematical induction: Prove that if a is any odd integ

Question

Prove that if a is any odd integer, then a^2 - 1 is divisible by 8.

Use mathematical induction to prove that \binom{2n}{n} < 2^{n-2}, for all integers n \geq 5.

For the complex numbers z and w, it is known that arg(\frac{z}{w}) = -\frac{\pi}{2}.
Find \left| \frac{z}{z+w} \right|.

The following argument attempts to prove that 0 = 1.

We evaluate \int \frac{1}{x} dx using the method of integration by parts.

\int \frac{1}{x} dx = \int 1 \cdot \frac{1}{x} dx 
= \frac{1}{x} \cdot x - \int \frac{1}{x^2} dx 
= 1 + \int \frac{1}{x} dx

So we have \int \frac{1}{x} dx = 1 + \int \frac{1}{x} dx.

We may now subtract \int \frac{1}{x} dx from both sides to show that 0 = 1.

Explain what is wrong with this argument.

The diagram shows triangle OQA.

The point P lies on OA such that OP:OA = 3:5.

The point B lies on OQ such that OB:OQ = 1:3.

The point R is the intersection of AB and PQ.

The point T is chosen on AQ so that O, R and T are collinear.

Let a = \overline{OA}, b = \overline{OB} and \overline{PR} = k\overline{PQ} where k is a real number.
(i) Show that \overline{OR} = \frac{3}{5}(1 - k)a + 3kb.

Writing \overline{AR} = h\overline{AB}, where h is a real number, it can be shown that \overline{OR} = (1 - h)a + hb. (Do NOT prove this.)

(ii) Show that k = \frac{1}{6}.

(iii) Find \overline{OT} in terms of a and b.

Prove that if a is any odd integer, then a^2 - 1 is divisible by 8 - HSC - SSCE Mathematics Extension 2 - Question 14 - 2024 - Paper 1

Worked Solution & Example Answer:Prove that if a is any odd integer, then a^2 - 1 is divisible by 8 - HSC - SSCE Mathematics Extension 2 - Question 14 - 2024 - Paper 1

Find \overline{OT} in terms of a and b.

Join the SSCE students using SimpleStudy...