Q: Solve the inequality $$3 - x^2$ \leq -1.$$

$$x \geq 2$$ \text{ or } x \leq -2$

Q: By finding the second derivative of $$f(x) = x^3$ + 3x$$, determine the nature of the stationary points at $$x = 1$$ and $$x = -1.$$

$$x = 1$$ \text{ is a maximum, } x = -1$ \text{ is a minimum}

Question 1

The distance from the Earth to the Sun is approximately 149,600,000 km. Express this distance in meters using scientific notation.

Accepted Answer

$$1.496 	imes 10$$^{11}$$ m$

Question 2

Make $$x$$ the subject of the equation $$r = \frac{x}{x+2y} + 4p.$$

Accepted Answer

$$x = \frac{8py - 2ry}{r - 4p - 1}$$

Question 3

Suppose you bought an article for $$24.50$$ and the total bill after tax was $$27.32. $$What was the rate of tax you paid, expressed as a percentage to 1 decimal place?

Accepted Answer

$$11.5\%$$

Question 4

Rewrite $$\left(\frac{x^3$ u^3$ y$$^{-3}$$}{y^3$ u^3$ x^3$}\$$right)^{\frac{1}$${3}} in $$its simplest index form.

Accepted Answer

$$\sqrt[3]{\frac{x}{y}}$$

Question 5

Solve the equation $$5^{1-x} = 25$ for x$.

Accepted Answer

$$x = -1$$

Question 6

Solve the inequality $$3 - x^2$ \leq -1.$$

Accepted Answer

$$x \geq 2$$ 	ext{ or } x \leq -2$

Question 7

A cone has a base radius of $$2 cm $$and height $$3 cm. $$How many litres of water can it contain? Give your answer in standard scientific notation to 4 decimal places.

Accepted Answer

$$1.2566 L$$

Question 8

Using Heron's formula, find the area of the base of a pyramid with sides $$1.5 cm$$, $$2 cm$$, and $$3 cm. $$Give your answer to 4 decimal places.

Accepted Answer

$$1.332 cm$^2$$

Question 9

Find the $$12th $$term of the arithmetic sequence $$3, 5.5, 8, \ldots.$$

Accepted Answer

$$30.5$$

Question 10

The value of a product over time is modeled by $$P_t = 21000e$$^{-0.15t}$$, where t$ is in years. What is the value after 5$ years?

Accepted Answer

$$9919.70$$

Question 11

The amount of radioactive material left in a substance at time $$t (in $$days) is given by $$N_t = N_0$ e$$^{-kt}$$. If N_0$ = 500$$, $$k = 0.13863$$, and $$N_t = 16$$, how many days does it take for the substance to reduce to $$16 g$$?

Accepted Answer

$$25$$ days

Question 12

Find the area of the sector OAB in a circle of radius $$6 cm $$and angle $$	heta = 2$$ radians.

Accepted Answer

$$6 cm$^2$$

Question 13

Given a right triangle with sides $$2.5 cm $$and $$4 cm$$, find the angle $$\alpha in $$radians to 2 decimal places.

Accepted Answer

$$0.90$$ 	ext{ rad}$

Question 14

Find the value in degrees of angle $$B in a $$triangle with sides $$8 cm$$, $$10 cm$$, and $$5 cm $$using the cosine rule.

Accepted Answer

$$29.69^{\circ}$$

Question 15

Given the vector $$\vec{a} = -2\hat{i} - \sqrt{4}\hat{j}$$, find a unit vector in the direction of $$\vec{a} in $$surd form.

Accepted Answer

$$\frac{-2}{\sqrt{5}}\hat{i} - \frac{2}{\sqrt{5}}\hat{j}$$

Question 16

A circle is represented by $$(x+2)^2$$ + $$(y-4)^2 = 50$. What are the coordinates of the centre and the radius?

Accepted Answer

Centre $$(-2, 4)$$, radius $$\sqrt{50}$$

Question 17

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

Accepted Answer

$$-\frac{2}{x} - \frac{3}{4}x^4$ + C$$

Question 18

Find the coordinates $$(x, y) of $$the stationary points of $$f(x) = x^3$ + 3x.$$

Accepted Answer

$$(1, 4)$$ and $$(-1, 0)$$

Question 19

By finding the second derivative of $$f(x) = x^3$ + 3x$$, determine the nature of the stationary points at $$x = 1$$ and $$x = -1.$$

Accepted Answer

$$x = 1$$ 	ext{ is a maximum, } x = -1$ 	ext{ is a minimum}

Precalculus Comprehensive Study Notes: Algebra, Functions, Trigonometry, Sequences, Geometry, and Calculus

Study Guide - Practice Questions

Study Guide - Flashcards