1. Expand: 3(x + 4)
3x + 12
Multiply everything inside the brackets by 3: 3 times x is 3x, and 3 times 4 is 12, so the answer is 3x + 12.
2. Expand: 2(x - 5)
2x - 10
Multiply both terms inside the brackets by 2: 2 times x is 2x, and 2 times negative 5 is negative 10, giving 2x - 10. A common slip is multiplying only the x and leaving the 5 alone.
3. Expand: 4(2x + 3)
8x + 12
Multiply both terms inside the brackets by 4: 4 times 2x is 8x, and 4 times 3 is 12, giving 8x + 12.
4. Expand: -2(x - 4)
-2x + 8
Negative 2 times x is negative 2x. Negative 2 times negative 4 is positive 8, because a negative times a negative is positive. So the answer is -2x + 8, not -2x - 8.
5. Expand: (x + 2)(x + 3)
x^2 + 5x + 6
Multiply every term in the first bracket by every term in the second: x times x is x^2, x times 3 is 3x, 2 times x is 2x, and 2 times 3 is 6. Add the like terms 3x and 2x to get 5x, so the answer is x^2 + 5x + 6.
6. Expand: (x - 3)(x + 5)
x^2 + 2x - 15
Multiply out: x times x is x^2, x times 5 is 5x, negative 3 times x is negative 3x, and negative 3 times 5 is negative 15. Combine 5x and negative 3x to get positive 2x, so the answer is x^2 + 2x - 15.
7. Factorise: 4x + 8
4(x + 2)
4 is the highest common factor of 4x and 8. Dividing each term by 4 gives x and 2, so 4x + 8 factorises to 4(x + 2). Check it by expanding back: 4 times x is 4x, 4 times 2 is 8.
8. Factorise: x^2 + 7x + 12
(x + 3)(x + 4)
Find two numbers that multiply to give 12 and add to give 7: those numbers are 3 and 4 (3 times 4 is 12, 3 plus 4 is 7). So it factorises to (x + 3)(x + 4). The pair (2 and 6) multiplies to 12 but adds to 8, not 7, so it does not fit.
9. Factorise: x^2 - 2x - 15
(x - 5)(x + 3)
Find two numbers that multiply to negative 15 and add to negative 2: negative 5 and positive 3 work, since negative 5 times 3 is negative 15 and negative 5 plus 3 is negative 2. So the factors are (x - 5)(x + 3).
10. Factorise: x^2 - 9
(x - 3)(x + 3)
This is a difference of two squares: x^2 - 9 is x^2 - 3^2, which always factorises to (x - 3)(x + 3). Check by expanding: x times x is x^2, x times 3 is 3x, negative 3 times x is negative 3x (they cancel), and negative 3 times 3 is negative 9.
11. Solve: x + 7 = 12
x = 5
Subtract 7 from both sides: x + 7 - 7 = 12 - 7, so x = 5.
12. Solve: 3x = 18
x = 6
Divide both sides by 3: 3x divided by 3 is x, and 18 divided by 3 is 6, so x = 6.
13. Solve: x/4 = 3
x = 12
Multiply both sides by 4 to undo the division: x = 3 times 4, which is 12.
14. Solve: 2x + 5 = 17
x = 6
First subtract 5 from both sides: 2x = 12. Then divide both sides by 2: x = 6. A common mistake is dividing before subtracting, which gives the wrong answer.
15. Solve: 3x - 4 = 2x + 9
x = 13
Get the x terms on one side by subtracting 2x from both sides: x - 4 = 9. Then add 4 to both sides: x = 13.
16. Solve: 5(x - 2) = 20
x = 6
Divide both sides by 5 first: x - 2 = 4. Then add 2 to both sides: x = 6. If you expand the brackets first you get 5x - 10 = 20, which leads to the same answer, x = 6.
17. What does the gradient of a straight-line graph measure?
How steep the line is
The gradient tells you how steep a line is: how much y changes for every one unit that x changes. It says nothing about length or area.
18. Find the gradient of the line joining (1, 2) and (3, 8).
3
Gradient is the change in y divided by the change in x. The change in y is 8 - 2 = 6, and the change in x is 3 - 1 = 2, so the gradient is 6 divided by 2, which is 3.
19. In the equation y = 2x + 5, what is the gradient?
2
A straight line written as y = mx + c has gradient m and y-intercept c. Here m = 2, so the gradient is 2.
20. In the equation y = 3x - 4, where does the line cross the y-axis?
(0, -4)
In y = mx + c, the number c is the y-intercept, the value of y when x = 0. Here c = -4, so the line crosses the y-axis at (0, -4).
21. Which gradient describes a line that slopes downhill from left to right?
A negative gradient
As x increases, y decreases on a downhill line, so the change in y divided by the change in x is negative. A gradient of zero is a flat, horizontal line instead.
22. Find the gradient of the line joining (2, 5) and (4, 1).
-2
Change in y is 1 - 5 = -4. Change in x is 4 - 2 = 2. Gradient is -4 divided by 2, which is -2. Because y went down as x went up, the gradient must come out negative.
23. Which pair of values satisfies both x + y = 7 and x - y = 3?
x = 5, y = 2
Adding the two equations eliminates y: (x + y) + (x - y) = 7 + 3, so 2x = 10, giving x = 5. Substituting back into x + y = 7 gives y = 2. Check the second equation too: 5 - 2 = 3, which matches.
24. Solve simultaneously: x + y = 10 and x - y = 2.
x = 6, y = 4
Add the equations to eliminate y: 2x = 12, so x = 6. Then substitute into x + y = 10: 6 + y = 10, so y = 4. Check the second equation: 6 - 4 = 2, which matches.
25. Solve simultaneously: y = x + 1 and 2x + y = 10.
x = 3, y = 4
Substitute y = x + 1 into the second equation: 2x + (x + 1) = 10, which simplifies to 3x + 1 = 10, so 3x = 9 and x = 3. Then y = x + 1 = 4.
26. Solve simultaneously: 3x + y = 11 and x + y = 5.
x = 3, y = 2
Subtracting the second equation from the first eliminates y: (3x + y) - (x + y) = 11 - 5, giving 2x = 6, so x = 3. Substitute into x + y = 5: 3 + y = 5, so y = 2.
27. Simplify: x^3 x x^2
x^5
When you multiply powers of the same base, you add the indices: x^3 times x^2 is x to the power of (3 + 2), which is x^5.
28. Simplify: (x^2)^3
x^6
When you raise a power to another power, you multiply the indices: (x^2)^3 is x to the power of (2 times 3), which is x^6. It is not 2 + 3, which is a common mix-up with the multiplying rule.
29. What is 2^-3 equal to?
1/8
A negative index means take the reciprocal of the positive power: 2^-3 = 1 / 2^3 = 1/8. It does not make the answer negative.
30. Simplify: (2x^3)^2
4x^6
Raise both the number and the power to the outer index: 2 squared is 4, and (x^3)^2 is x^(3 times 2) = x^6. So the answer is 4x^6, not 2x^6, because the 2 must be squared as well.
31. Write 0.006 in standard form.
6 x 10^-3
To write 0.006 as a number between 1 and 10 times a power of 10, use 6, then work out how many places the decimal point moved: from 0.006 to 6 it moved three places to the right, so the power is negative 3: 6 x 10^-3.
32. Which of these is 3.2 x 10^4 written as an ordinary number?
32,000
10^4 means move the decimal point 4 places to the right. Starting from 3.2, moving the decimal point 4 places gives 32,000.
33. A number in standard form is 5 x 10^-2. What is it as an ordinary number?
0.05
A negative power of 10 means the number is small: 10^-2 means move the decimal point 2 places to the left. Starting from 5, that gives 0.05.
34. Which of these is written correctly in standard form?
7.1 x 10^5
Standard form needs the first number to be between 1 and 10 (not 0.71 and not 71), and the base of the power must be 10, not 5. Only 7.1 x 10^5 satisfies both rules.
35. A right-angled triangle has shorter sides 6 cm and 8 cm. What is the hypotenuse?
10 cm
6^2 + 8^2 = 36 + 64 = 100. The square root of 100 is 10, so the hypotenuse is 10 cm. Adding the two sides directly (6 + 8 = 14) is a common mistake: Pythagoras uses squares, not the sides themselves.
36. A ladder (the hypotenuse) is 13 m long and reaches 12 m up a wall. How far is the base of the ladder from the wall?
5 m
Here the hypotenuse is 13 m and one shorter side is 12 m, so the missing side squared is 13^2 - 12^2 = 169 - 144 = 25. The square root of 25 is 5, so the base is 5 m from the wall.
37. Which equation correctly states Pythagoras' rule for a right-angled triangle with hypotenuse c and other sides a and b?
a^2 + b^2 = c^2
Pythagoras' rule states that the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides.
38. A rectangle is 12 cm long and 5 cm wide. What is the length of its diagonal?
13 cm
The diagonal of a rectangle is the hypotenuse of a right-angled triangle formed by the length and width: 12^2 + 5^2 = 144 + 25 = 169. The square root of 169 is 13, so the diagonal is 13 cm.
39. Does a triangle with sides 5, 12 and 13 have a right angle?
Yes, because 5^2 + 12^2 = 13^2
Check Pythagoras' rule with the longest side as the hypotenuse: 5^2 + 12^2 = 25 + 144 = 169, and 13^2 = 169. Since both sides match, the triangle does have a right angle.
40. In a right-angled triangle, which ratio is sin(angle)?
Opposite over hypotenuse
The three trig ratios in a right-angled triangle follow SOH CAH TOA. Sin is Opposite over Hypotenuse (SOH), so sin(angle) = opposite divided by hypotenuse.
41. In a right-angled triangle, which ratio is tan(angle)?
Opposite over adjacent
Using SOH CAH TOA, tan is Opposite over Adjacent (TOA), so tan(angle) = opposite divided by adjacent.
42. A right-angled triangle has an adjacent side of 8 cm and a hypotenuse of 10 cm. Which ratio should you use to find the angle?
cos(angle) = 8/10
Cos is Adjacent over Hypotenuse (CAH). Since we are given the adjacent side (8) and the hypotenuse (10), the correct ratio is cos(angle) = 8/10, not 10/8, which would be upside down.
43. A right-angled triangle has an opposite side of 6 cm and an adjacent side of 8 cm. Which calculation gives the angle?
tan^-1(6/8)
Since we know the opposite (6) and adjacent (8) but not the hypotenuse, tan is the right ratio: tan(angle) = opposite/adjacent = 6/8. To find the angle itself, use the inverse: angle = tan^-1(6/8).
44. A tree casts a shadow. The angle from the ground to the top of the tree is 40 degrees, and the shadow is 15 m long. Which calculation finds the tree's height?
15 x tan(40)
The shadow (15 m) is adjacent to the angle, and the height is opposite the angle. Tan(angle) = opposite/adjacent, so opposite = adjacent x tan(angle), giving height = 15 x tan(40).
45. What is the volume of a rectangular box 4 cm long, 3 cm wide and 2 cm high?
24 cm^3
Volume of a rectangular box is length times width times height: 4 x 3 x 2 = 24 cm^3.
46. What is the volume of a cube with side length 5 cm?
125 cm^3
A cube's volume is side length cubed: 5 x 5 x 5 = 125 cm^3.
47. What is the total surface area of a cube with side length 3 cm?
54 cm^2
A cube has 6 identical square faces. Each face has area 3 x 3 = 9 cm^2, and there are 6 faces, so the total surface area is 6 x 9 = 54 cm^2.
48. A cylinder has radius 3 cm and height 10 cm. Which expression gives its volume? (Use pi.)
pi x 3^2 x 10
The volume of a cylinder is the area of the circular base (pi times radius squared) multiplied by the height: pi x 3^2 x 10. A common mistake is using the radius instead of the radius squared, or using diameter instead of radius.
49. On a box plot, what does the whole length of the box and whiskers, from one end to the other, show?
The range of the data
The whiskers stretch from the minimum value to the maximum value, so the full length from end to end shows the range: the spread from smallest to largest result.
50. A scatter graph shows points that rise from bottom left to top right. What kind of correlation is this?
Positive correlation
When one variable increases as the other increases, the points trend upward from left to right, which is called positive correlation.
51. A scatter graph shows points that fall from top left to bottom right. What kind of correlation is this?
Negative correlation
When one variable decreases as the other increases, the points trend downward from left to right. This is negative correlation: for example, more time spent studying might correlate with fewer mistakes.
52. Two box plots compare test scores for class A and class B. Class A's box is entirely to the right of class B's box. What does this tell you?
Class A generally scored higher than class B
A box plot further to the right sits on higher values along the number line. Since the whole of class A's box sits to the right of class B's, class A's typical scores (including the median) were higher.
53. A bag has 3 red balls and 2 blue balls. A ball is picked, replaced, then a second ball is picked. What is the probability of red then red?
9/25
Because the ball is replaced, the two picks are independent and each has probability 3/5 of being red. Multiply along the branches of the tree: 3/5 x 3/5 = 9/25.
54. A bag has 3 red balls and 2 blue balls, with no replacement. What is the probability of picking red then blue?
3/10
The first pick is red with probability 3/5. Since the ball is not replaced, only 4 balls remain (2 red, 2 blue) for the second pick, so picking blue second has probability 2/4. Multiply along the tree: 3/5 x 2/4 = 6/20, which simplifies to 3/10.
55. A coin is flipped twice. Using a probability tree, what is the probability of getting exactly one head?
1/2
The two ways to get exactly one head are head-then-tail or tail-then-head. Each single outcome has probability 1/2 x 1/2 = 1/4 on the tree, and since there are two such branches, add them: 1/4 + 1/4 = 1/2.