Question 1

An adult blue whale has a mass of $$1.5 	imes 10^5$ kg. The mass of a sardine is 150g, and there are on average 4$$ 	imes $$10^3$$ sardines in a school. How many schools of sardines would it take to equal the mass of an adult blue whale?

Accepted Answer

$$2.5 	imes 10^3$

Question 2

A cereal mixture contains rice, wheat, and corn in the ratio $$2:3:5. $$How many grams of rice are there in a $$1.5 kg $$bag?

Accepted Answer

$$300 g$$

Question 3

Rewrite $$\dfrac{\sqrt[3]{27} a^2$ b$$^{-3}$$ c$$^{2/3}$$}{6 \sqrt{a^3$} \sqrt[3]{b^2$}} in $$the simplest index form.

Accepted Answer

$$\dfrac{3 a$$^{1/2}$$ c$$^{2/3}$$}{6 b$$^{7/3}$$}$$

Question 4

Solve the equation $$2^{2x} - 10(2^x) + 16 = 0$.

Accepted Answer

$$x = 2 or$$ $$x = 4$$

Question 5

Solve the inequality $$\dfrac{1}{x-9} \leq 1. $$Express your answer in interval notation.

Accepted Answer

$$(-\infty, 4] \cup (9, \infty)$$

Question 6

Factorise and simplify $$\dfrac{x^2$ - 16}{x^2$ + 2x + 8}.$$

Accepted Answer

$$\dfrac{(x-4)(x+4)}{x^2$ + 2x + 8}$$

Question 7

Find the volume of fully compacted concrete that would fill a cone of base diameter $$0.4 m $$and height $$75 cm. $$Give your answer to the nearest cubic centimetre.

Accepted Answer

$$3142 cm$^3$$

Question 8

In a zombie apocalypse, the number of people who newly become zombies doubles each week. If $$5$$ people become zombies in the first week, how many zombies will there be after $$12$$ weeks?

Accepted Answer

$$20480$$

Question 9

A gossip spreads according to the model $$P_t = 4e$$^{0.75t}$$, where P$ is the number of people who have heard it and t$ is the time in hours. How many people would have heard the gossip after 5$ hours?

Accepted Answer

$$4e^{0.75 	imes 5}$$

Question 10

Find the area of the circle below that remains after the sector $$OAB$$ has been removed. The circle has radius $$3.5 cm $$and the sector has angle $$1.2$$ radians. Use $$2 d.p. in $$your answer.

Accepted Answer

$$30.55 cm$^2$$

Question 11

Mike is flying a kite. The string from Mike to the kite is $$12 m $$and the distance from Mike to a rock that is directly beneath the kite is $$5 m. $$Find the height at which the kite is flying.

Accepted Answer

$$10.9 m$$

Question 12

A conic section is represented by the equation $$\dfrac{(x-4)^2$}{4} + \dfrac{(y+2)^2$}{2} = 1. $$What type of conic section does it represent?

Accepted Answer

Ellipse

Question 13

Complete the square for $$2x^2 - 4x$.

Accepted Answer

$$2(x-1)^2 - 2$

Question 14

Calculate $$\int x^4$ - \dfrac{3}{4x^2$} \, dx.$$

Accepted Answer

$$\dfrac{x^5$}{5} - \dfrac{3}{4x}$$

Question 15

Find the value of $$\int_{-1}^{2} 4 - x^2$ \, dx.$$

Accepted Answer

$$7$$

Question 16

The height of a ball thrown vertically in the air is given by $$h(t) = h_0$ + v_0$ t - \dfrac{gt^2$}{2}$$, where $$g = 9.8 m/s$^2. By $$using the first derivative, determine the time when the ball reaches its highest position.

Accepted Answer

$$t = \dfrac{v_0$}{g}$$

Question 17

Given $$f(x) = (1 - x^3$) \sin(3x)$$, find $$f'(x).$$

Accepted Answer

$$-3x^2$$ \sin(3x) + (1 - $$x^3) 3$$ \cos(3x)$$

Question 18

Find a unit vector in the direction of $$\vec{b} = 3\vec{i} + 4\vec{j}$$ and write it in terms of $$\vec{i}$$ and $$\vec{j}.$$

Accepted Answer

$$0.6\vec{i} + 0.8\vec{j}$$

Question 19

If $$\cos(	heta) = \dfrac{\sqrt{2}}{4}$$, what are the two exact values of $$\sin(	heta)$$?

Accepted Answer

$$\pm \dfrac{\sqrt{14}}{4}$$

Precalculus and Introductory Calculus: Comprehensive Study Guide

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