Math – Tuteee

1

One-step equations: add and subtract

4 questions

Solve:
$x + 1 = 2$
Solve:
$x - 2 = 4$
Solve:
$x + 5 = 2$
Solve:
$x - 7 = -2$

2

One-step equations: multiply and divide

4 questions

Solve:
$3x = 12$
Solve:
$\frac{x}{5} = 2$
Solve:
$-2x = 8$
Solve:
$4x = -20$

3

Two-step equations

5 questions

Solve:
$2x + 1 = 7$
Solve:
$5x - 3 = 12$
Solve:
$3x + 4 = 1$
Solve:
$4 - 2x = 10$
Solve:
$\frac{x}{2} + 1 = 4$

4

Variables on both sides

4 questions

Solve:
$3x + 2 = x + 8$
Solve:
$5x - 3 = 2x + 9$
Solve:
$2x + 7 = 5x - 2$
Solve:
$4 - x = 2x + 1$

5

Expanding brackets

5 questions

Expand:
$3(x + 2)$
Expand:
$4(x - 3)$
Expand:
$x(x + 4)$
Expand:
$2x(3x - 1)$
Expand and simplify:
$2(x + 3) + 3(x - 1)$

6

Equations with brackets

5 questions

Solve:
$2(x + 3) = 14$
Solve:
$3(x - 2) = 9$
Solve:
$4(x - 1) = 2x$
Solve:
$2(x + 3) + x = 15$
Solve:
$2(x + 3) = 3(x - 1)$

7

Brackets with negatives

4 questions

Solve:
$-2(x - 4) = 6$
Solve:
$10 - 3(x + 1) = 1$
Solve:
$3(x - 2) - 2(x - 1) = 5$
Solve:
$2(3x - 1) - 3(x + 2) = 1$

8

Equations with fractions

5 questions

Solve:
$\frac{x}{2} + 3 = 5$
Solve:
$\frac{6}{x} = 3$
Solve:
$\frac{x}{2} + x = 6$
Solve:
$\frac{x}{2} + \frac{x}{3} = 5$
Solve:
$\frac{x}{3} + \frac{x}{4} = 7$

9

Fraction equations: expression numerators

5 questions

Solve:
$\frac{x + 1}{2} = 3$
Solve:
$\frac{2x - 7}{3} = 5$
Solve:
$\frac{x + 1}{2} + \frac{x - 1}{3} = 6$
Solve:
$\frac{x + 4}{2} = \frac{2x - 1}{3}$
Solve:
$\frac{2x + 1}{3} - \frac{x}{2} = 1$

10

Expanding binomial products

5 questions

Expand:
$(x + 2)(x + 3)$
Expand:
$(x - 1)(x + 4)$
Expand:
$(x + 3)^{2}$
Expand:
$(x - 2)^{2}$
Expand:
$(x + 5)(x - 5)$

11

Factorising: common factors

4 questions

Factorise:
$6x + 9$
Factorise:
$x^{2} + 4x$
Factorise:
$6x^{2} - 9x$
Factorise:
$4x^{2} + 6x$

12

Factorising quadratics

5 questions

Factorise:
$x^{2} + 7x + 12$
Factorise:
$x^{2} - 8x + 15$
Factorise:
$x^{2} + 2x - 15$
Factorise:
$x^{2} - x - 12$
Factorise:
$x^{2} - 25$

13

Factorising: harder quadratics

4 questions

Factorise:
$4x^{2} - 9$
Factorise:
$2x^{2} + 5x - 3$
Factorise:
$3x^{2} - 10x + 8$
Factorise fully:
$2x^{2} + 8x + 6$

14

Quadratic equations by factorising

5 questions

Solve:
$(x - 2)(x + 6) = 0$
Solve:
$x^{2} - 6x = 0$
Solve:
$x^{2} - 25 = 0$
Solve:
$x^{2} + 7x + 12 = 0$
Solve:
$2x^{2} + 5x - 3 = 0$

15

Quadratic equations: rearrange first

4 questions

Solve:
$x^{2} = 5x + 14$
Solve:
$(x + 1)(x - 2) = 4$
Solve:
$x + \frac{6}{x} = 5$
Solve:
$x(x + 2) = 2x + 9$

16

The square-root method

4 questions

Solve:
$x^{2} = 49$
Solve:
$2x^{2} = 50$
Solve:
$(x - 1)^{2} = 9$
Solve in exact form:
$(x + 2)^{2} = 5$

17

Completing the square

6 questions

What number completes the square?
$x^{2} + 6x + \Box$
Write in the form $(x + a)^{2} + b$ :
$x^{2} + 6x + 1$
Solve in exact form by completing the square:
$x^{2} - 6x + 4 = 0$
Solve in exact form by completing the square:
$x^{2} + 3x + 1 = 0$
Solve in exact form by completing the square:
$2x^{2} - 8x + 3 = 0$
Complete the square to solve for $x$ :
$ax^{2} + bx + c = 0$

18

The quadratic formula

5 questions

Use the quadratic formula to solve in exact form:
$x^{2} + 3x + 1 = 0$
Use the quadratic formula to solve in exact form:
$x^{2} - 5x + 2 = 0$
Use the quadratic formula to solve:
$2x^{2} + 7x + 3 = 0$
Use the quadratic formula to solve in exact form:
$3x^{2} + 5x + 1 = 0$
Solve in exact form:
$x^{2} - x - 1 = 0$

19

The discriminant

5 questions

Find the discriminant:
$x^{2} + 3x + 5 = 0$
How many real roots does this equation have?
$x^{2} + 3x + 5 = 0$
Use the discriminant to describe the roots of:
$x^{2} - 8x + 16 = 0$
Find $k$ so that this equation has equal roots:
$x^{2} - 4x + k = 0$
Find the values of $k$ for which this equation has equal roots:
$x^{2} + kx + 9 = 0$

20

Index laws

5 questions

Simplify:
$x^{3} \times x^{4}$
Simplify:
$\frac{x^{7}}{x^{3}}$
Simplify:
$(x^{2})^{5}$
Simplify:
$(2x^{3})^{2}$
Simplify:
$\frac{x^{5} \times x^{2}}{x^{4}}$

21

Negative and fractional indices

5 questions

Evaluate:
$5^{0}$
Evaluate:
$2^{-3}$
Evaluate:
$9^{\frac{1}{2}}$
Evaluate:
$8^{\frac{2}{3}}$
Write with a negative index:
$\frac{1}{x^{3}}$

22

Index equations

6 questions

Solve:
$3^{x} = 9$
Solve:
$2^{x} = \frac{1}{8}$
Solve:
$4^{x} = 2$
Solve:
$9^{x} = 27$
Solve:
$2^{x+1} = 16$
Solve:
$25^{x} = \frac{1}{5}$

23

Simplifying surds

4 questions

Simplify:
$\sqrt{12}$
Simplify:
$\sqrt{50}$
Simplify:
$\sqrt{72}$
Simplify:
$2\sqrt{27}$

24

Operations with surds

4 questions

Simplify:
$\sqrt{2} \times \sqrt{8}$
Simplify:
$\sqrt{3} + 2\sqrt{3}$
Simplify:
$\sqrt{50} - \sqrt{8}$
Expand and simplify:
$(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})$

25

Rationalising and surd equations

5 questions

Rationalise the denominator:
$\frac{1}{\sqrt{2}}$
Rationalise the denominator and simplify:
$\frac{6}{\sqrt{3}}$
Solve:
$x\sqrt{2} = \sqrt{18}$
Solve:
$\sqrt{x - 1} = 3$
Solve in exact form:
$x + \sqrt{8} = \sqrt{50}$

26

Simultaneous equations

5 questions

Solve simultaneously: $x + y = 10$ and $x - y = 4$ .
Solve simultaneously: $y = 2x - 1$ and $3x + y = 9$ .
Solve simultaneously: $2x + 3y = 7$ and $x + y = 3$ .
Solve simultaneously: $3x + 2y = 16$ and $2x - y = 6$ .
Find the x-values where $y = x^{2}$ meets $y = 2x + 3$ .

27

Function notation and evaluation

6 questions

If $f(x) = 2x + 1$ , find $f(3)$ .
If $f(x) = x^{2} - 4$ , find $f(-2)$ .
Find the zero of $f(x) = 2x - 6$ .
$f(x) = x + 1$ for $x < 2$ and $f(x) = 3x$ for $x \ge 2$ . Find $f(0)$ .
Using the same $f$ , with $f(x) = x + 1$ for $x < 2$ and $f(x) = 3x$ for $x \ge 2$ , find $f(2)$ .
If $f(x) = 2x + 1$ and $g(x) = x^{2}$ , find $f(g(3))$ .

28

Domain and range

5 questions

Is the relation $\{(1, 2), (1, 3), (2, 4)\}$ a function?
State the domain:
$f(x) = \sqrt{x}$
State the domain:
$f(x) = \frac{1}{x - 3}$
State the range:
$y = x^{2} + 2$
State the domain:
$f(x) = \sqrt{x - 4}$

29

Gradient and intercepts

5 questions

State the gradient:
$y = 3x - 2$
Find the gradient of the line through $(1, 2)$ and $(3, 8)$ .
Find the y-intercept of the line $y = 2x - 8$ .
Find the x-intercept of the line $y = 2x - 8$ .
Find the gradient of a line perpendicular to $y = 2x + 1$ .

30

Finding line equations

5 questions

Find the equation of the line with gradient 2 and y-intercept 3.
Find the equation of the line with gradient 4 through $(1, 5)$ .
Find the equation of the line through $(2, 7)$ and $(4, 13)$ .
Find the equation of the line through $(0, -1)$ parallel to $y = 2x + 5$ .
Find the equation of the line through $(2, -3)$ perpendicular to $y = \frac{x}{2} + 1$ .

31

Linear inequalities

5 questions

Solve:
$2x + 1 > 7$
Solve:
$3x - 4 \le 11$
Solve:
$-2x > 6$
Solve:
$5 - x < 2$
Solve:
$1 < 2x - 3 \le 5$

32

Parabola features

6 questions

Does $y = -2x^{2} + 3$ open up or down?
Find the y-intercept:
$y = x^{2} - 4x + 1$
Find the axis of symmetry:
$y = x^{2} - 4x + 1$
Find the vertex:
$y = x^{2} - 4x + 1$
Find the minimum value:
$y = x^{2} - 2x + 5$
Find the x-intercepts:
$y = x^{2} - 2x - 8$

33

Quadratic inequalities

5 questions

Solve:
$x^{2} - 9 < 0$
Solve:
$x^{2} > 16$
Solve:
$(x - 1)(x - 4) > 0$
Solve:
$x^{2} - 5x + 6 \le 0$
Solve:
$x^{2} \le 4x$

34

Cubics and intersections

5 questions

State the x-intercepts:
$y = (x - 1)(x + 2)(x - 3)$
Find the y-intercept:
$y = (x - 1)(x + 2)(x - 3)$
Does $y = -x^{3}$ rise or fall from left to right?
The curve $y = kx^{3}$ passes through $(2, 16)$ . Find $k$ .
Solve:
$x^{3} = 27$

35

Reciprocal functions and asymptotes

4 questions

For $f(x) = \frac{1}{x}$ , find $f(4)$ .
What value does $y = \frac{1}{x}$ approach as $x \to \infty$ ?
Write the equation of the vertical asymptote:
$y = \frac{1}{x + 3}$
State the equations of both asymptotes:
$y = \frac{1}{x - 2} + 3$

36

Circles and semicircles

5 questions

Write the equation of the circle with centre $(0, 0)$ and radius 6.
State the centre and radius:
$(x - 2)^{2} + (y + 5)^{2} = 49$
Write the equation of the circle with centre $(1, -2)$ and radius 3.
Find the centre and radius:
$x^{2} + y^{2} - 6x + 4y - 12 = 0$
State the domain of the semicircle:
$y = \sqrt{9 - x^{2}}$

37

Absolute value functions

5 questions

Evaluate:
$|-7| + |3|$
Solve:
$|x| = 5$
Solve:
$|2x - 3| = 7$
Solve:
$|2x - 1| = -3$
Solve:
$|x - 4| < 3$

38

Transformations, even and odd functions

6 questions

Write the equation of $y = \sqrt{x}$ after translating it up 2 units.
Write the equation of $y = x^{2}$ after translating it 3 units to the right.
Write the equation of $y = x^{2}$ after reflection in the x-axis.
State the vertex:
$y = (x + 2)^{2} - 3$
Is $f(x) = x^{4} - 3x^{2}$ even, odd or neither?
Is $f(x) = x^{3} - x$ even, odd or neither?

39

Right-triangle trigonometry: sides

5 questions

A right-angled triangle has legs 6 and 8. Find the length of the hypotenuse.
In a right-angled triangle the hypotenuse is 10 and one angle is $30^{\circ}$ . Find the length of the side opposite that angle.
In a right-angled triangle the hypotenuse is 8 and one angle is $60^{\circ}$ . Find the length of the side adjacent to that angle.
In a right-angled triangle one angle is $45^{\circ}$ and the side adjacent to it is 7. Find the length of the side opposite it.
From a point 50 m from the base of a tower, the angle of elevation to the top is $35^{\circ}$ . Find the height of the tower, to the nearest metre.

40

Right-triangle trigonometry: angles and bearings

4 questions

A right-angled triangle has the side opposite angle $\theta$ equal to 7 and hypotenuse 14. Find $\theta$ .
In a right-angled triangle the side opposite angle $\theta$ is 5 and the side adjacent is 5. Find $\theta$ .
In a right-angled triangle the side adjacent to angle $\theta$ is 4 and the hypotenuse is 8. Find $\theta$ .
A hiker walks 5 km east, then 5 km north. Find the three-figure bearing of the hiker from the starting point.

41

Exact values

5 questions

Find the exact value:
$\sin 30^{\circ}$
Find the exact value:
$\cos 45^{\circ}$
Find the exact value:
$\tan 60^{\circ}$
Find the exact value:
$\sin 60^{\circ}$
Find the exact value:
$\tan 30^{\circ}$

42

Angles of any magnitude

6 questions

Find the exact value:
$\sin 150^{\circ}$
Find the exact value:
$\cos 120^{\circ}$
Find the exact value:
$\tan 225^{\circ}$
Find the exact value:
$\sin 300^{\circ}$
Find the exact value:
$\sin 180^{\circ}$
Find the exact value:
$\cos(-60^{\circ})$

43

Radians

5 questions

Convert to radians, in exact form:
$60^{\circ}$
Convert $\frac{3\pi}{4}$ radians to degrees.
Find the exact value:
$\sin\frac{\pi}{6}$
Find the exact value:
$\cos \pi$
Find the exact value:
$\tan\frac{\pi}{4}$

44

Arc length and sectors

5 questions

Find the exact arc length of a sector with radius 6 and angle $\frac{\pi}{3}$ .
Find the exact area of a sector with radius 6 and angle $\frac{\pi}{3}$ .
Find the exact perimeter of a sector with radius 6 and angle $\frac{\pi}{3}$ .
Find the arc length of a sector with radius 10 and angle 0.8 radians.
An arc of length 6 subtends an angle $\theta$ at the centre of a circle of radius 4. Find $\theta$ in radians.

45

The sine rule

4 questions

In triangle $ABC$ , $a = 10$ , angle $A = 40^{\circ}$ and angle $B = 70^{\circ}$ . Find side $b$ , to 1 decimal place.
In triangle $ABC$ , $a = 8$ , angle $A = 60^{\circ}$ and angle $B = 45^{\circ}$ . Find side $b$ , to 1 decimal place.
In triangle $ABC$ , $a = 12$ , angle $A = 45^{\circ}$ and angle $B = 30^{\circ}$ . Find side $b$ in exact form.
In triangle $ABC$ , $a = 5$ , $b = 10$ and angle $A = 30^{\circ}$ . Find angle $B$ .

46

The cosine rule and triangle area

5 questions

A triangle has sides 8 and 6 with an included angle of $30^{\circ}$ . Find its area.
A triangle has sides 4 and 6 with an included angle of $45^{\circ}$ . Find its exact area.
A triangle has sides $b = 7$ and $c = 5$ with included angle $A = 60^{\circ}$ . Find side $a$ , in exact form.
A triangle has sides 3, 5 and 7. Find the size of the largest angle.
A triangle has sides 5, 6 and 7. Find the size of the angle opposite the side of length 7, to the nearest degree.

47

Trigonometric identities

5 questions

Find the exact value of cosec $30^{\circ}$ .
Find the exact value of cot $45^{\circ}$ .
Simplify:
$\sin(90^{\circ} - \theta)$
If $\sin\theta = \frac{3}{5}$ and $\theta$ is acute, find the exact value of $\cos\theta$ .
Simplify $\sin\theta$ × cosec $\theta$ .

48

Trigonometric equations

5 questions

Solve for $0^{\circ} \le x < 360^{\circ}$ :
$\sin x = \frac{1}{2}$
Solve for $0^{\circ} \le x < 360^{\circ}$ :
$\cos x = -\frac{1}{2}$
Solve for $0^{\circ} \le x < 360^{\circ}$ :
$\tan x = 1$
Solve for $0 \le x \le 2\pi$ :
$\sin x = \frac{1}{2}$
Solve for $0 \le x \le 2\pi$ :
$2\cos x = \sqrt{3}$

49

Trig graphs: amplitude and period

5 questions

State the amplitude:
$y = 3\sin x$
State the period in radians:
$y = \sin(2x)$
State the period in radians:
$y = 2\cos(\frac{x}{2})$
State the range:
$y = 2\sin x$
State the maximum value:
$y = 4\sin x + 1$

50

Trig graphs: transformations

5 questions

State the centre line:
$y = \sin x + 3$
The graph of $y = \cos x$ is translated up 2 units. Write the equation of the new graph.
Find the maximum and minimum values:
$y = 2\sin x + 1$
State the amplitude and period:
$y = 5\sin(3x)$
The graph of $y = \sin x$ is translated $\frac{\pi}{2}$ to the right. Write the equation of the new graph.

51

Exponential functions

5 questions

If $f(x) = 2^{x}$ , find $f(3)$ .
If $f(x) = 5e^{x}$ , find $f(0)$ .
Write the equation of the horizontal asymptote:
$y = 2^{x} + 3$
Solve:
$3^{x} = 81$
State the gradient of the tangent to $y = e^{x}$ at $x = 0$ .

52

Introduction to logarithms

5 questions

Evaluate:
$\log_{2} 8$
Evaluate:
$\log_{10} 1000$
Evaluate:
$\log_{5} 1$
Evaluate:
$\log_{2} \frac{1}{8}$
Evaluate:
$\ln e$

53

Logarithm laws

5 questions

Evaluate:
$\log_{2} 40 - \log_{2} 5$
Write as a single logarithm:
$\log_{2} 5 + \log_{2} 3$
Express as a single logarithm:
$2\log_{3} 5 - \log_{3} 5$
Evaluate (base 10):
$\log 5 + \log 2$
Evaluate:
$\ln(e^{3})$

54

Exponential equations

5 questions

Solve:
$2^{x} = 32$
Solve:
$2^{x-1} = 16$
Solve:
$3^{2x} = 81$
Solve giving the exact answer:
$e^{x} = 5$
Solve giving the exact answer:
$3^{x} = 20$

55

Logarithmic equations

4 questions

Solve:
$\log_{2} x = 5$
Solve:
$\log_{3} x = -2$
Solve:
$\ln x = 3$
Solve:
$\log_{3} x + \log_{3}(x - 2) = 1$

56

Graphs of exponentials and logarithms

4 questions

State the domain:
$y = \log_{10} x$
Find the x-intercept of $y = \log_{2} x$ .
Write the equation of the horizontal asymptote:
$y = e^{x} - 2$
State the domain:
$y = \ln(x - 3)$

57

Exponential growth and decay

5 questions

A quantity satisfies $\frac{dQ}{dt} = -0.04Q$ . Is this exponential growth or decay?
A population is modelled by $Q = 500e^{0.03t}$ . State the initial population.
A sample decays according to $Q = Ae^{kt}$ . Write down the differential equation it satisfies.
A quantity satisfies $\frac{dQ}{dt} = 0.2Q$ . Find the rate of change $\frac{dQ}{dt}$ at the instant when $Q = 50$ .
A population is modelled by $Q = 100e^{0.05t}$ . Find the exact time $t$ at which $Q = 200$ .

58

Exponential and logarithmic models

4 questions

On the Richter scale, how many times greater is the ground-shaking amplitude of a magnitude 7 earthquake than a magnitude 5 one?
A sound has intensity $10^{4}$ times the reference intensity $I_{0}$ . Find its level in decibels, using $dB = 10\log_{10}(I/I_{0})$ .
A sound becomes 100 times more intense. By how many decibels does its level increase?
A population grows so that $P = P_{0}e^{kt}$ . It increases from 200 to 500 over 5 years. Find the exact value of $k$ .

59

Sequences and the general term

5 questions

A sequence has $a_{n} = 2n + 1$ . Find $a_{5}$ .
List the first four terms of the sequence:
$a_{n} = n^{2}$
A sequence has $a_{1} = 3$ and $a_{n+1} = a_{n} + 4$ . Find $a_{4}$ .
For $a_{n} = 3n$ , find the partial sum $S_{3}$ .
Evaluate:
$\sum_{k=1}^{4} (2k - 1)$

60

Arithmetic sequences

5 questions

State the common difference of the arithmetic sequence $3, 7, 11, \ldots$
Find the 10th term of the arithmetic sequence $3, 7, 11, \ldots$
An arithmetic sequence has first term 100 and common difference $-7$ . Find the 15th term.
Which term of the arithmetic sequence $5, 8, 11, \ldots$ is equal to 50?
An arithmetic sequence has 3rd term 12 and 8th term 27. Find the first term $a$ and common difference $d$ .

61

Arithmetic series

4 questions

Find the sum of the first 20 terms of the arithmetic series $5 + 8 + 11 + \ldots$
Find the sum of the first 20 terms of the arithmetic series $2 + 4 + 6 + \ldots$
Find the sum:
$2 + 5 + 8 + \ldots + 50$
An arithmetic sequence has 3rd term 12 and 8th term 27. Find the sum of the first 20 terms.

62

Geometric sequences

5 questions

State the common ratio of the geometric sequence $3, 6, 12, \ldots$
Find the 6th term of the geometric sequence $3, 6, 12, 24, \ldots$
Find the 6th term of the geometric sequence $2, -6, 18, \ldots$
A geometric sequence has common ratio 3 and third term 18. Find the first term.
For the geometric sequence with $a = 2$ and $r = 3$ , find $n$ such that $a_{n} = 162$ .

63

Geometric series

4 questions

Find the sum of the first 5 terms of the geometric series $2 + 6 + 18 + \ldots$
Find the sum of the first 6 terms of the geometric series $1 + 2 + 4 + \ldots$
Find the sum of the first 8 terms of the geometric series $3 + 6 + 12 + \ldots$
A geometric sequence has first term 5 and common ratio 2. Find the first term that exceeds 1000.

64

The limiting sum

5 questions

For what values of the common ratio $r$ does a geometric series have a limiting sum?
Find the limiting sum of $8 + 4 + 2 + 1 + \ldots$
Find the limiting sum of $9 + 6 + 4 + \ldots$
Express the recurring decimal as a fraction in simplest form:
$0.7777\ldots$
Find the limiting sum of the geometric series with first term 24 and common ratio $\frac{3}{4}$ .

65

Loans and repayments

5 questions

A $5000 loan charges 1% interest per month. After one month's interest, a $300 repayment is made. Find the amount still owing.
A $1000 loan is charged 10% interest per year. A $400 repayment is made at the end of the year. Find the balance owing after that repayment.
Continuing from a $700 balance owing on a loan at 10% p.a., another $400 is repaid at the end of the next year. Find the new balance owing.
A loan is repaid in 60 equal monthly repayments of $500. Find the total amount paid.
A $1000 loan is charged 10% per annum and repaid in 2 equal instalments of $M$ , one at the end of each year, reducing the balance to $0. Find $M$ , to the nearest cent.

66

Annuities

4 questions

$1000 is paid into an account at the end of each year, earning 5% p.a. compound interest. How much interest has the first $1000 (deposited at the end of year 1) earned by the end of year 3?
$1000 is invested at the end of each year for 3 years at 5% p.a. Find the future value just after the third payment.
$2000 is invested at the end of each year for 3 years at 10% p.a. Find the future value just after the third payment.
$500 is invested at the end of each year for 4 years at 6% p.a. Using the GP sum formula, find the future value just after the fourth payment, to the nearest cent.

67

Rates of change and gradients

5 questions

An object travels 150 metres in 20 seconds. Find its average speed in m/s.
For $y = x^{2}$ , find the average rate of change from $x = 1$ to $x = 3$ .
For $y = x^{2} + 1$ , find the gradient of the chord from $x = 2$ to $x = 5$ .
For $y = x^{2}$ , the chord from $x = 2$ to $x = 2 + h$ has gradient $4 + h$ . What does this gradient approach as $h \to 0$ ?
On $y = x^{2}$ , the gradient of the chord from $x = 3$ to $x = 3 + h$ simplifies to $6 + h$ . State the gradient of the tangent at $x = 3$ .

68

The derivative from first principles

4 questions

Use first principles to differentiate:
$f(x) = 3x + 2$
Use first principles to differentiate:
$f(x) = x^{2}$
Use first principles to differentiate:
$f(x) = x^{2} + x$
Given $f(x) = x^{2}$ , use $f'(x) = 2x$ to find the gradient of the tangent at $x = 3$ .

69

The power rule

5 questions

Differentiate:
$y = 7$
Differentiate:
$y = x^{5}$
Differentiate:
$y = 3x^{4}$
Differentiate:
$y = x^{3} - 2x^{2} + 5x - 7$
If $f(x) = 2x^{3}$ , find $f'(2)$ .

70

Negative and fractional powers

4 questions

Differentiate:
$y = \frac{1}{x}$
Differentiate:
$y = \frac{1}{x^{2}}$
Differentiate:
$y = \sqrt{x}$
Differentiate:
$y = 4\sqrt{x}$

71

Tangents and normals

5 questions

For $y = x^{2}$ , find the gradient of the tangent at the point $(2, 4)$ .
Find the equation of the tangent to $y = x^{2}$ at $(2, 4)$ .
Find the point on $y = x^{2}$ where the tangent has gradient 6.
For $y = x^{2}$ , find the gradient of the normal at $(2, 4)$ .
Find the equation of the normal to $y = x^{2}$ at $(2, 4)$ .

72

The chain rule

5 questions

Differentiate:
$y = (2x + 1)^{5}$
Differentiate:
$y = (1 - 2x)^{4}$
Differentiate:
$y = (x^{2} + 1)^{3}$
Differentiate:
$y = \sqrt{3x + 1}$
Differentiate:
$y = \frac{1}{x^{2} + 3}$

73

The product rule

5 questions

Differentiate $y = (x + 1)(x^{2} + 2)$ using the product rule.
Differentiate $y = (x^{2} + 1)(x - 3)$ using the product rule.
Differentiate:
$y = x(2x + 1)^{4}$
Differentiate, giving your answer in factored form:
$y = x^{2}(x - 3)^{2}$
If $y = x(x - 2)^{3}$ , find $\frac{dy}{dx}$ at $x = 1$ .

74

The quotient rule

5 questions

Differentiate:
$y = \frac{x}{x + 1}$
Differentiate:
$y = \frac{2x - 1}{x + 3}$
Differentiate:
$y = \frac{x - 4}{x + 2}$
Differentiate, giving your answer in factored form:
$y = \frac{x^{2}}{x + 1}$
If $y = \frac{x + 1}{x - 1}$ , find $\frac{dy}{dx}$ at $x = 2$ .

75

Differentiating exponentials and logarithms

5 questions

Differentiate:
$y = e^{x}$
Find $\frac{dy}{dx}$ for $y = e^{3x}$ .
Differentiate:
$y = \ln x$
Differentiate:
$y = \ln(2x + 1)$
Differentiate:
$y = x^{2}\ln x$

76

Differentiating trigonometric functions

5 questions

Differentiate:
$y = \sin x$
Find $\frac{dy}{dx}$ for $y = \cos x$ .
Differentiate:
$y = \sin(3x)$
Differentiate:
$y = x + \sin x$
Differentiate:
$y = \tan(2x)$

77

Stationary points and concavity

6 questions

Find the x-coordinate of the stationary point:
$y = x^{2} - 4x + 1$
For $y = x^{2} - 4x + 1$ , find the values of $x$ for which the function is increasing.
Find the x-coordinates of the stationary points:
$y = x^{3} - 3x$
For $y = x^{3} - 3x$ , classify the stationary point at $x = 1$ as a maximum or minimum.
Find the second derivative:
$y = x^{4}$
Find the point of inflection:
$y = x^{3} - 6x^{2} + 5$

78

Optimisation and motion

6 questions

A quantity is modelled by $P = 16x - x^{2}$ . Find the value of $x$ that maximises $P$ .
Two positive numbers add to 20. Find their maximum possible product.
A rectangle has perimeter 40 cm. Find its maximum possible area.
A particle has displacement $x = t^{2} - 4t$ metres at time $t$ seconds. Find its velocity $v$ .
A particle has velocity $v = 2t - 4$ . Find when it is at rest.
A particle has displacement $x = t^{3} - 6t^{2} + 9t$ metres. Find all times $t \ge 0$ when it is at rest.

79

Primitives

5 questions

Find the primitive:
$5$
Find the primitive:
$x^{3}$
Find the primitive:
$6x^{2}$
Find the primitive:
$6x^{2} - 4x + 1$
Given $f'(x) = 6x$ and $f(1) = 5$ , find $f(x)$ .

80

Primitives: harder powers

4 questions

Find the primitive:
$\frac{1}{x^{2}}$
Find the primitive:
$\sqrt{x}$
Find the primitive:
$(3x + 1)^{5}$
Find the primitive:
$(2x - 5)^{3}$

81

The definite integral

5 questions

Evaluate:
$\int_{2}^{5} 4 \,dx$
Evaluate:
$\int_{1}^{2} 3x^{2} \,dx$
Evaluate:
$\int_{1}^{3} (3x^{2} - 2x) \,dx$
Evaluate:
$\int_{1}^{2} \frac{1}{x^{2}} \,dx$
Given $\int_{0}^{3} f(x) \,dx = 7$ and $\int_{3}^{5} f(x) \,dx = 4$ , find $\int_{0}^{5} f(x) \,dx$ .

82

The trapezoidal rule

4 questions

Use one application of the trapezoidal rule to estimate $\int_{0}^{2} f(x) \,dx$ , given $f(0) = 3$ and $f(2) = 7$ .
Use the trapezoidal rule with two strips ( $h = 1$ ) to estimate $\int_{0}^{2} f(x) \,dx$ , given $f(0) = 1$ , $f(1) = 4$ and $f(2) = 9$ .
Use one application of the trapezoidal rule to estimate in exact form:
$\int_{1}^{3} \frac{1}{x} \,dx$
A function has $f(0) = 5$ , $f(2) = 8$ and $f(4) = 11$ . Use the trapezoidal rule with two strips ( $h = 2$ ) to estimate $\int_{0}^{4} f(x) \,dx$ .

83

Integrating exponentials

4 questions

Find:
$\int e^{x} \,dx$
Find:
$\int e^{2x} \,dx$
Find:
$\int 2e^{4x} \,dx$
Evaluate:
$\int_{0}^{1} e^{x} \,dx$

84

Integrating reciprocals

4 questions

Find:
$\int \frac{6}{x} \,dx$
Evaluate:
$\int_{1}^{3} \frac{1}{x} \,dx$
Find:
$\int \frac{4}{2x - 1} \,dx$
Find:
$\int \frac{2x}{x^{2} + 1} \,dx$

85

Integrating trigonometric functions

5 questions

Find:
$\int \sin x \,dx$
Find:
$\int (\sin x + \cos x) \,dx$
Find:
$\int \cos 2x \,dx$
Find:
$\int \sin 3x \,dx$
Evaluate:
$\int_{0}^{\pi/4} \sec^{2} x \,dx$

86

Areas under curves

4 questions

Find the area under $y = x^{2}$ from $x = 0$ to $x = 3$ .
Find the area between $y = x^{2} + 1$ , the x-axis, and the lines $x = 0$ and $x = 2$ .
Find the area enclosed between $y = x^{2} - 4$ , the x-axis, and the lines $x = 0$ and $x = 2$ .
Find the area enclosed by $y = 4 - x^{2}$ and the x-axis.

87

Areas between curves

4 questions

Find the area enclosed between $y = x$ and $y = x^{2}$ .
Find the area enclosed between $y = 2x$ and $y = x^{2}$ .
Find the total area between $y = x^{3}$ , the x-axis, and the lines $x = -1$ and $x = 1$ .
Find the area enclosed between $y = x + 2$ and $y = x^{2}$ .

88

Applications of integration

4 questions

A particle has velocity $v = 3t^{2}$ and initial displacement $x(0) = 1$ . Find $x$ as a function of $t$ .
Given $f'(x) = 2x + 1$ and $f(2) = 7$ , find $f(x)$ .
Water flows into a tank at a rate of $\frac{dV}{dt} = 10 - 2t$ litres per minute. Find the change in volume from $t = 0$ to $t = 3$ .
A particle has velocity $v = 6t^{2} + 4$ m/s. Find the distance it travels from $t = 0$ to $t = 2$ .

89

Mean, median and mode

5 questions

Find the mean of the data set:
$4, 7, 9, 10, 10$
Find the median of the data set:
$3, 8, 5, 12, 9$
Find the median of the data set:
$6, 2, 9, 4$
Find the mode of the data set:
$2, 5, 5, 7, 5, 8$
The mean of $5, 8, x, 11$ is 9. Find $x$ .

90

Range, quartiles and standard deviation

6 questions

Find the range of the data set:
$4, 9, 2, 15, 7$
Find the lower quartile ( $Q_1$ ) of the data set:
$3, 5, 7, 8, 10, 12, 15, 18$
A data set has five-number summary: minimum 4, $Q_1 = 10$ , median 14, $Q_3 = 20$ , maximum 30. Find the interquartile range.
Find the interquartile range of the data set:
$3, 5, 6, 8, 11, 12, 15$
A data set has $Q_1 = 20$ and $Q_3 = 32$ . Using the $1.5 \times \text{IQR}$ rule, find the value above which a data point is an outlier.
Find the standard deviation of the data set, correct to two decimal places:
$2, 4, 6, 8, 10$

91

Bivariate data and correlation

5 questions

On a scatterplot, the points rise from the lower-left to the upper-right. Describe the correlation between the two variables.
The correlation coefficient between two variables is $r = -0.85$ . Describe the strength and direction of the correlation.
State the largest possible value of Pearson's correlation coefficient $r$ .
The least-squares regression line for a data set is $y = 2x + 5$ . Use it to predict $y$ when $x = 10$ .
A regression line found from data with $x$ between 1 and 8 is used to predict $y$ at $x = 20$ . Is this interpolation or extrapolation?

92

Probability basics

5 questions

A fair die is rolled. Find the probability of rolling an even number.
The probability it rains tomorrow is 0.3. Find the probability it does not rain.
A bag contains 3 red and 5 blue marbles. One marble is drawn at random. Find the probability that it is red.
Two fair coins are tossed. Find the probability of getting two heads.
Two fair dice are rolled. Find the probability that the sum is 7.

93

Venn diagrams and the addition rule

5 questions

$A$ and $B$ are mutually exclusive with $P(A) = 0.25$ and $P(B) = 0.35$ . Find $P(A \cup B)$ .
$A$ and $B$ are mutually exclusive with $P(A) = 0.3$ and $P(A \cup B) = 0.9$ . Find $P(B)$ .
$P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \cap B) = 0.2$ . Find $P(A \cup B)$ .
$P(A) = 0.5$ , $P(B) = 0.6$ and $P(A \cup B) = 0.8$ . Find $P(A \cap B)$ .
In a class of 30 students, 18 play soccer, 12 play tennis and 5 play both. How many play neither?

94

Conditional probability

5 questions

$P(A \cap B) = 0.2$ and $P(B) = 0.5$ . Find $P(A|B)$ .
$P(A \cap B) = 0.18$ and $P(A) = 0.3$ . Find $P(B|A)$ .
Of 40 people, 24 own a phone, and 18 of those also own a laptop. Find the probability a phone owner owns a laptop.
A card is drawn from a standard deck. Given it is a face card (J, Q, K), find the probability it is a king.
$P(A|B) = 0.6$ and $P(A \cap B) = 0.3$ . Find $P(B)$ .

95

Independent events

4 questions

A fair die is rolled twice. Find the probability of getting a six both times.
$P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \cap B) = 0.2$ . Are $A$ and $B$ independent?
$A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.4$ . Find $P(A \cap B')$ .
$A$ and $B$ are independent with $P(A) = 0.6$ and $P(B) = 0.3$ . Find $P(A \cup B)$ .

96

Discrete random variables

5 questions

$X$ has $P(1) = 0.3$ , $P(2) = 0.4$ and $P(3) = k$ . Find $k$ .
A fair die is rolled and $X$ is the number shown. Find $P(X \ge 5)$ .
A fair die is rolled and $X$ is the number shown. Find E(X).
$X$ takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Find E(X).
$X$ takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Find Var(X).

97

Continuous random variables

4 questions

For a continuous random variable $X$ , state the value of $P(X = 3)$ .
$X$ is uniformly distributed on $[2, 10]$ . Write its probability density function $f(x)$ .
$X$ has $f(x) = \frac{1}{8}$ for $2 \le x \le 10$ . Find $P(4 \le X \le 7)$ .
$f(x) = kx$ is a probability density function on $[0, 4]$ . Find $k$ .

98

The normal distribution

5 questions

A normal distribution has mean 70 and standard deviation 5. Find the z-score of $x = 80$ .
A normal distribution has mean 70 and standard deviation 5. Find the z-score of $x = 60$ .
For a normal distribution, what percentage of data lies within two standard deviations of the mean?
$X \sim N(\mu, \sigma^{2})$ . Using the empirical rule, find $P(\mu \le X \le \mu + \sigma)$ .
$X \sim N(\mu, \sigma^{2})$ . Using the empirical rule, find the percentage of data more than two standard deviations above the mean.

99

Mixed paper I

6 questions

Solve:
$3(x - 4) = 2x + 1$
Simplify:
$\sqrt{75} - \sqrt{12}$
Find the equation of the line through $(-1, 2)$ and $(2, 11)$ .
Find the exact value:
$\cos 150^{\circ}$
Find the sum of the first 15 terms of the arithmetic series $4 + 7 + 10 + \ldots$
Differentiate:
$y = x^{3} - 4x^{2} + 2x$

100

Mixed paper II

6 questions

Use the quadratic formula to solve in exact form:
$2x^{2} - 4x - 1 = 0$
Differentiate:
$y = \frac{3x + 2}{x - 1}$
A population grows so that $P = P_{0}e^{kt}$ . It increases from 300 to 900 over 4 years. Find the exact value of $k$ .
A test score is normally distributed with mean 60 and standard deviation 10. Using the empirical (68-95-99.7) rule, find $P(X < 40)$ .
Find the area enclosed between $y = x^{2}$ and $y = x + 6$ .
Find and classify the stationary points:
$y = x^{3} - 3x^{2} - 9x + 5$