Q: Solve the equation: $$(\log_{10} x)^2$ - \log_{10} x - 12 = 0.$$

$$x = 10$$^{4}$$ \text{ or } x = 10$$^{-3}$

Question 1

Sally rode $$2.1 km on $$her bike, then walked $$3.0 km $$with her sister, and finally drove $$20.4 km in a $$car. What is the total distance Sally travelled, expressed in standard scientific notation?

Accepted Answer

$$2.45 	imes 10^1$ km

Question 2

On an island, the ratio of adult men ($$M) to $$adult women ($$W) is$$ $$2:3$$, and the ratio of adults ($$M+W) to $$children under $$18$$ ($$C) is$$ $$5:4. $$What is the ratio of adult women to children under $$18$$?

Accepted Answer

$$3:8$$

Question 3

Rewrite $$\sqrt{6x^4$y^3$$} \over $$2x^2y$$^{1/2}$$ in the simplest index form.

Accepted Answer

$$\dfrac{\sqrt{6}x^2$y^{3/2}$$}{$$2x^2y$$^{1/2}$$} = \dfrac{\sqrt{6}y}{2}$$

Question 4

Solve the equation: $$(\log_{10} x)^2$ - \log_{10} x - 12 = 0.$$

Accepted Answer

$$x = 10$$^{4}$$ 	ext{ or } x = 10$$^{-3}$

Question 5

Solve the inequality $$-2x^2 + 3$$ \u003e -1$$ and express your answer in interval notation.

Accepted Answer

$$x \in (-1, 2)$$

Question 6

Factorise and simplify the expression $$\dfrac{x^2$ - 3x - 18}{9 - x^2$}.$$

Accepted Answer

$$-\dfrac{x-6}{x+3}$$

Question 7

A swimming pool has a cross-section that is a trapezium with parallel sides $$4 m $$and $$2 m$$, height $$3 m$$, and length $$8 m. $$What is the capacity of the pool in litres?

Accepted Answer

$$72,000 L$$

Question 8

Find the $$15th $$term of the arithmetic sequence $$\dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \ldots$$ and the sum of its first $$15$$ terms.

Accepted Answer

$$\dfrac{1}{16}$$, $$\approx 2.71$$

Question 9

A number $$a is $$reduced to $$\dfrac{1}{5} of $$its value every second. If $$a = 189.23 	imes 10^6$, after how many minutes will the number be less than 1$?

Accepted Answer

$$\approx 6$$ 	ext{ minutes}

Question 10

A population of kangaroos is modelled by $$K_t = 32e$$^{0.06t}$$, where t$ is in months. How many years did it take for the population to reach 500$?

Accepted Answer

$$\approx 4.5$$ 	ext{ years}

Question 11

Given $$P_n = P_0$ \left(1 + \dfrac{r}{100}\right)^n$$, a person owes $$25,000$$ after $$18$$ months, starting with $$3,250$$ and making no payments. What was the monthly rate $$r$$ charged?

Accepted Answer

$$r \approx 13.2\%$$

Question 12

Find the area of the circle that remains after the sector $$OAB ($$with radius $$5 cm $$and angle $$1.6$$ radians) is removed. Give your answer to $$4$$ decimal places.

Accepted Answer

$$54.6344 cm$^2$$

Question 13

Find the value in degrees of the angle $$	heta in $$the diagram below, where $$	an 	heta = \dfrac{3}{4}. $$Use $$2$$ decimal places.

Accepted Answer

$$36.87^\circ$$

Question 14

Find $$\vec{c} = \vec{b} - 3\vec{a}$$, where $$\vec{a} = 2\mathbf{i} + 3\mathbf{j}$$ and $$\vec{b} = 5\mathbf{i} - \mathbf{j}.$$

Accepted Answer

$$-1\mathbf{i} - 10\mathbf{j}$$

Question 15

An ellipse is represented by $$\dfrac{(x+2)^2$}{9} + \dfrac{(y-4)^2$}{4} = 1. $$What are the coordinates of the centre?

Accepted Answer

$$(-2, 4)$$

Question 16

Complete the square for $$y^2 - 6y$.

Accepted Answer

$$(y-3)^2 - 9$

Question 17

Calculate $$\int_{-3}^{3} (x^2$ + 1) dx.$$

Accepted Answer

$$24 ($$since $$\int_{-3}^{3} x^2$ dx = 18$$ and $$\int_{-3}^{3} 1 dx = 6)$$

Question 18

The position $$x(t) of a $$spring is given by $$x(t) = 2e$$^{-t}$$ \cos(3t). $$What is the velocity of the spring at $$t = 2$$? (Use $$4$$ decimal places.)

Accepted Answer

$$-1.3152$$

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