Use the Question 11 Writing Booklet. (a) Find $(i + 6j) + (2i - 7j)$. (b) Expand and simplify $(2a - b)^4$. (c) Use the substitution $u = x + 1$ to find $\int \sqrt{x+1} \, dx$. (d) A committee containing 5 men and 3 women is to be formed from a group of 10 men and 8 women. In how many different ways can the committee be formed? (e) A spherical bubble is moving up through a liquid. As it rises, the bubble gets bigger and its radius increases at the rate of 0.2 mm/s. At what rate is the volume of the bubble increasing when its radius reaches 0.6 mm? Express your answer in mm³/s rounded to one decimal place. (f) Evaluate $\int_0^{\sqrt{3}} \frac{1}{\sqrt{4 - x^2}} \, dx$. (g) By factorising, or otherwise, solve $2\sin^3x + 2\sin^2x - \sin x - 1 = 0$ for $0 \leq x \leq 2\pi$. (h) The roots of $x^4 - 3x^3 + 6x^2 + \alpha x + \beta = 0$ are $\alpha, \beta, \gamma$ and $\delta$. What is the value of $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}$?

SSCE HSC Mathematics Extension 1 Fundamental counting principle: Use the Question 11 Writing Book

Use the Question 11 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 11 - 2021 - Paper 1

Question 11

Use the Question 11 Writing Booklet. (a) Find $(i + 6j) + (2i - 7j)$. (b) Expand and simplify $(2a - b)^4$. (c) Use the substitution $u = x + 1$ to find $\int \sq... show full transcript

Worked Solution & Example Answer:Use the Question 11 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 11 - 2021 - Paper 1

Step 1

Find $(i + 6j) + (2i - 7j)$

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Answer

To find the sum of the vectors, combine the respective components:

(i + 6j) + (2i - 7j) = (1 + 2)i + (6 - 7)j = 3i - j.

Thus, the answer is $3i - j$ .

Step 2

Expand and simplify $(2a - b)^4$

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Answer

Using the binomial theorem, we expand:

(2a - b)^4 = \sum_{k=0}^4 \binom{4}{k} (2a)^{4-k} (-b)^k.

Calculating each term gives:

For $k=0$ : $16a^4$
For $k=1$ : $-32a^3b$
For $k=2$ : $24a^2b^2$
For $k=3$ : $-8ab^3$
For $k=4$ : $b^4$

Thus, combining all terms:

(2a - b)^4 = 16a^4 - 32a^3b + 24a^2b^2 - 8ab^3 + b^4.

Step 3

Use the substitution $u = x + 1$ to find $\int \sqrt{x+1} \, dx$

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Answer

Making the substitution $u = x + 1$ means $x = u - 1$ which gives:

\int \sqrt{u} \, du = \frac{2}{3}u^{3/2} + C = \frac{2}{3}(x+1)^{3/2} + C.

Step 4

A committee containing 5 men and 3 women is to be formed from a group of 10 men and 8 women. In how many different ways can the committee be formed?

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Answer

To form the committee:

Choose 5 men from 10:

\binom{10}{5}.

Choose 3 women from 8:

\binom{8}{3}.

Total ways to form the committee:

\binom{10}{5} \times \binom{8}{3} = 252 \times 56 = 14112.

Step 5

At what rate is the volume of the bubble increasing when its radius reaches 0.6 mm?

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Answer

The volume $V$ of a sphere is given by:

V = \frac{4}{3}\pi r^3.

Using the chain rule:

\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}.

Calculating:

\frac{dV}{dr} = 4\pi r^2 = 4\pi (0.6)^2 = 4\pi (0.36).

Thus:

\frac{dV}{dt} = 4\pi (0.36) \cdot 0.2 = 0.144\pi \text{ mm}^3/s \approx 0.4524 \text{ mm}^3/s ext{ (to one decimal place)}.

Step 6

Evaluate $\int_0^{\sqrt{3}} \frac{1}{\sqrt{4 - x^2}} \, dx$

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Answer

This integral can be solved using a trigonometric substitution: $\sin^{-1}(\frac{x}{2})$ gives:

\int_0^{\sqrt{3}} \frac{1}{\sqrt{4 - x^2}} \, dx = \sin^{-1}(\frac{\sqrt{3}}{2}) - \sin^{-1}(0) = \frac{\pi}{3} - 0 = \frac{\pi}{3}.

Step 7

By factorising, or otherwise, solve $2\sin^3x + 2\sin^2x - \sin x - 1 = 0$ for $0 \leq x \leq 2\pi$

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Answer

Rearranging gives:

\sin x(2\sin^2 x + 2\sin x - 1) = 0.

Thus:

One solution is $\sin x = 0$ : $x = 0, \pi, 2\pi.$
For the quadratic, use the quadratic formula:

2\sin^2 x + 2\sin x - 1 = 0 \implies \sin x = \frac{-2 \pm \sqrt{4 + 8}}{4} = \frac{-2 \pm 2\sqrt{3}}{4} = \frac{-1 \pm \sqrt{3}}{2}.

Giving solutions:

$\sin x = \frac{-1 + \sqrt{3}}{2}$ and $\sin x = \frac{-1 - \sqrt{3}}{2}.

Solutions exist for: - $\frac{\pi}{3}, \frac{2\pi}{3}$.

Step 8

What is the value of $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}$?

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Answer

Using Viète's formulas, for the polynomial $x^4 - 3x^3 + 6x^2 + \alpha x + \beta = 0$ : $\alpha + \beta + \gamma + \delta = 3,$ where \alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta = 6. Formally: $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta} = \frac{\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta}{\alpha\beta\gamma\delta}.$

Use the Question 11 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 11 - 2021 - Paper 1

Worked Solution & Example Answer:Use the Question 11 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 11 - 2021 - Paper 1

Find $(i + 6j) + (2i - 7j)$

Expand and simplify $(2a - b)^4$

Use the substitution $u = x + 1$ to find $\int \sqrt{x+1} \, dx$

A committee containing 5 men and 3 women is to be formed from a group of 10 men and 8 women. In how many different ways can the committee be formed?

At what rate is the volume of the bubble increasing when its radius reaches 0.6 mm?

Evaluate $\int_0^{\sqrt{3}} \frac{1}{\sqrt{4 - x^2}} \, dx$

By factorising, or otherwise, solve $2\sin^3x + 2\sin^2x - \sin x - 1 = 0$ for $0 \leq x \leq 2\pi$

What is the value of $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}$?

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