Question1
- C1.15
Work out the number of months in 5 years.
▶️Answer/Explanation
60
Since there are 12 months in a year, we multiply
$
5 \times 12 = 60
$
60 months
Question2
2. C1.10
Write 3752 correct to the
(a) nearest 10
(b) nearest 100.
▶️Answer/Explanation
(a) 3750 cao
(b) 3800 cao
(a)
Look at the last digit (2).
Since 2 < 5, round down.
$
3752 \approx 3750
$
(b)
Look at the tens digit (5).
Since 5 or more, round up.
$
3752 \approx 3800
$
Question3
3: C1.16
Magazines cost $3.40 each.
Rosina has $15 to buy as many magazines as possible.
Complete the statement.
Rosina can buy …………….. magazines and will have $ …………….. left.
▶️Answer/Explanation
4 1.4[0]
Cost of one magazine: $\$3.40$
Rosina’s total money: $\$15$
$
\frac{15}{3.40} \approx 4.41
$
Since she can only buy whole magazines, she can afford 4 magazines.
$
4 \times 3.40 = 13.60
$
subtract from the total amount:
$
15 – 13.60 = 1.40
$
Rosina can buy 4 magazines and will have $1.40 left.
Question4
4: C4.5
Write down the mathematical name of a 4-sided shape that has rotational symmetry of order 2 and no lines of symmetry.
▶️Answer/Explanation
Parallelogram
It is a parallelogram (that is not a rectangle or rhombus).
- Rotational symmetry of order 2: It looks the same when rotated 180°.
- No lines of symmetry: Unlike a rectangle or rhombus, a general parallelogram doesn’t have any lines of symmetry.
Question5
5: C9.3
21 8 15 32 3 29 19 45 8
Calculate the mean of these numbers.
▶️Answer/Explanation
20
$
21 + 8 + 15 + 32 + 3 + 29 + 19 + 45 + 8 = 180
$
9 numbers in total.
$
\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}}
$
$
\text{Mean} = \frac{180}{9} = 20
$
Question6
6: C1.15
A train journey starts at 21 : 43.
It takes 8 hours and 32 minutes.
Find the time the journey finishes.
▶️Answer/Explanation
06 15 or 6:15am
Adding 8 hours to the start time
$
21:43 + 8\text{ hours} = 05:43 \text{(next day)}
$
adding 32 minutes to 05:43:
$
05:43 + 32 \text{ minutes} = 06:15
$
The journey finishes at 06:15 (6:15 AM).
Question7
7: C1.5
Write these numbers in order, starting with the smallest.
$\frac{15}{213}$ 0.071 0.7 7%
……………………. < ……………………. < ………………….. < …………………….
smallest
▶️Answer/Explanation
$7 \% \quad \frac{15}{213} \quad 0.071 \quad 0.7$
Convert to decimals
\( \frac{15}{213} \approx 0.0704 \)
\( 0.071 \) (already a decimal)
\( 0.7 \) (already a decimal)
\( 7\% = 0.07 \)
$
0.07 < 0.0704 < 0.071 < 0.7
$
Ordered List (Smallest to Largest)
$
7\% < \frac{15}{213} < 0.071 < 0.7
$
Question8
8: C1.4
Write the fraction $\frac{24}{84}$ in its simplest form.
▶️Answer/Explanation
$\frac{2}{7}$
Greatest Common Divisor of 24 and 84, which is 12.
divide both the numerator and denominator by 12:
$
\frac{24 \div 12}{84 \div 12} = \frac{2}{7}
$
Question9
9: C2.2
Simplify.
$3a-5b-a-6b$
▶️Answer/Explanation
$2a – 11b$
Combine terms with \(a\) and \(b\) separately
$
(3a – a) + (-5b – 6b)
$
$
= 2a – 11b
$
Question10
10: C2.5
The cost of hiring a bicycle, $\$C$, for y hours is given by the formula $C=12+3.5y.$
Maria pays $\$36.50$ to hire this bicycle. Work out the number of hours she hires the bicycle for.
▶️Answer/Explanation
7
$
C = 12 + 3.5y
$
Maria pays $\$36.50$, so substitute this into the equation:
$
36.50 = 12 + 3.5y
$
$
36.50 – 12 = 3.5y
$
$
24.50 = 3.5y
$
$
y = \frac{24.50}{3.5}
$
$
y = 7
$
Question11
11: C4.6
$a=\binom{3}{7}$ $b=\binom{-2}{5}$
Work out a-2b
▶️Answer/Explanation
$\binom{7}{-3}$
$
a = \binom{3}{7} = (3, 7)
$
$
b = \binom{-2}{5} = (-2, 5)
$
Multiply the vector \( b \) by 2
$
2b = 2(-2, 5) = (-4, 10)
$
$
a – 2b = (3, 7) – (-4, 10)
$
\( 3 – (-4) = 3 + 4 = 7 \)
\( 7 – 10 = -3 \)
$
a – 2b =\binom{7}{-3}
$
Question12
12(a): C4.6
12(b): C4.6
(a) The diagram shows a pair of parallel lines and a straight line.
Write down the geometrical reason why the value of x is 52.
(b) Find the value of y and write down the geometrical reason for your answer.
▶️Answer/Explanation
(a) Alternate angles
(b) 196
Angles at a point sum to 360
(a) It is because of the Corresponding Angles rule.
Corresponding angles are equal when a straight line crosses two parallel line
(b)
The angles around a point sum to 360°.
$
38^\circ + 126^\circ + y = 360^\circ
$
$
164^\circ + y = 360^\circ
$
$
y = 360^\circ – 164^\circ
$
$
y = 196^\circ
$
Question13
13: C5.4
Calculate the volume of a sphere with diameter 4.8cm.
[The volume, V, of a sphere with radius r is $V=\frac{4}{3}\pi r^3$]
▶️Answer/Explanation
57.9 or 57.90 to 57.91….
volume of a sphere is
$
V = \frac{4}{3}\pi r^3
$
The diameter is 4.8 cm, so the radius is 1/2.
$
r = \frac{4.8}{2} = 2.4 \, \text{cm}
$
$
V = \frac{4}{3}\pi (2.4)^3
$
$
V = \frac{4}{3} \times \pi \times 13.824
$
$
V = \frac{4 \times 13.824}{3} \times \pi
$
$
V = \frac{55.296}{3} \times \pi
$
$
V \approx 18.43\pi
$
$
V \approx 18.43 \times 3.14 \approx 57.9 \, \text{cm}^3
$
Question14
14: C1.9
By writing each number in the calculation correct to 1 significant figure, work out an estimate for the
value of
$\frac{6.7 \times 2.1}{18-5.9}$
▶️Answer/Explanation
$\frac{7 \times 2}{20-6}$
Round each number to 1 significant figure
6.7 → 7
2.1 → 2
18 → 20
5.9 → 6
$
= \frac{7 \times 2}{20 – 6}
$
$
= \frac{14}{14}
$
$
= 1
$
Question15
15: C8.1
Eric has four colours of paint.
The table shows the probability that he uses each colour.
Find the value of x.
▶️Answer/Explanation
0.22 oe
In probability, the sum of all probabilities must equal $1.$
$
0.3 + 0.35 + 0.13 + x = 1
$
$
0.3 + 0.35 + 0.13 = 0.78
$
$
0.78 + x = 1
$
$
x = 0.22
$
Question16
16: C2.4
Factorise completely.
$8x^2-20x$
▶️Answer/Explanation
4x(2x – 5)
$
8x^2 – 20x
$
the common factor is $4x$
$
= 4x(2x – 5)
$
Question17
17(a): C2.7
17(b): C2.7
(a) The nth term of a sequence is $10-n^2$
Write down the first three terms of this sequence.
(b) These are the first four terms of another sequence.
7 10 13 16
Find an expression for the nth term of this sequence.
▶️Answer/Explanation
(a) 9 6 1
(b) 3n + 4
(a)
The nth term formula is:
$
10 – n^2
$
For \( n = 1 \): \( 10 – 1^2 = 10 – 1 = 9 \)
For \( n = 2 \): \( 10 – 2^2 = 10 – 4 = 6 \)
For \( n = 3 \): \( 10 – 3^2 = 10 – 9 = 1 \)
First 3 terms
$
9, 6, 1
$
(b)
The sequence is
$
7, 10, 13, 16
$
An arithmetic sequence, where the first term is \( 7 \) and the common difference is \( 3 \).
Formula for the nth term of an arithmetic sequence is
$
a_n = a_1 + (n-1)d
$
\( a_1 = 7 \) (first term)
\( d = 3 \) (common difference)
$
a_n = 7 + (n-1)(3)
$
$
a_n = 7 + 3n – 3
$
$
a_n = 3n + 4
$
Question18
18: C1.10
The length, l metres, of a piece of wood is 3.6 metres, correct to the nearest 10 centimetres.
Complete this statement about the value of l.
▶️Answer/Explanation
3.55 3.65
The length of the wood is given as 3.6 meters, correct to the nearest 10 centimeters.
Since 10 centimeters = 0.1 meters, the length could be 0.05 meters above or below 3.6 meters.
$
\text{Half of 0.1 meters} = 0.05 \text{ meters}.
$
$
3.6 – 0.05 \leq l < 3.6 + 0.05
$
$
3.55 \leq l < 3.65
$
Question19
19: C1.8
Calculate $1\div (6.4 \times 10^{-5})$
Give your answer in standard form.
▶️Answer/Explanation
$1.5625 × 10^4$
$
1 \div (6.4 \times 10^{-5}) = \frac{1}{6.4 \times 10^{-5}}
$
$
= 1 \times \frac{1}{6.4} \times 10^5
$
$
= (0.15625) \times 10^5
$
Convert to standard form
$
= 1.5625 \times 10^4
$
Question20
20: C1.4
Without using a calculator, work out
$2\frac{1}{7}\div \frac{5}{9}$
You must show all your working and give your answer as a mixed number in its simplest form.
▶️Answer/Explanation
$\frac{15}{7} \times \frac{9}{5}$
$
2 \frac{1}{7} = \frac{15}{7}
$
Division is the same as multiplying by the reciprocal
$
\frac{15}{7} \div \frac{5}{9} = \frac{15}{7} \times \frac{9}{5}
$
$
= \frac{15 \times 9}{7 \times 5} = \frac{135}{35}
$
$
= \frac{27\times 5}{7\times 5}
$
$
= \frac{27}{7}
$
$
= 3 \frac{6}{7}
$
Question21
21: C6.2
The diagram shows a right-angled triangle.
Use the information in the diagram to write down and solve an equation to find the value of x.
▶️Answer/Explanation
$4x + 3x – 1 + 90 = 180$
Sum of the angles in a triangle is always 180°.
$
(4x)^{\circ} + (3x – 1 )^{\circ}+(90)^{\circ} = 180
$
$
4x + 3x – 1 + 90 = 180
$
$
7x + 89 = 180
$
$
7x = 180 – 89
$
$
7x = 91
$
$
x = \frac{91}{7}
$
$
x = 13
$
Question22
22: C6.2
The diagram shows a right-angled triangle.
Calculate the value of y.
▶️Answer/Explanation
30.4 or 30.36 to 30.37
Tangent of an angle in a right-angled triangle is
$
\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}
$
Opposite side (perpendicular) = 7.5 cm
Adjacent side (base) = 12.8 cm
$
\tan(y) = \frac{7.5}{12.8}
$
$
\tan(y) = 0.5859
$
$
y = \tan^{-1}(0.5859)
$
$
y \approx 30.3^\circ
$
Question23
23: C4.4
Triangle ABC is similar to triangle DEF.
Calculate the value of h.
▶️Answer/Explanation
$6.3$
Since the triangles are similar, the ratios of corresponding sides are equal
$
\frac{AB}{BC} = \frac{DE}{EF} = \frac{AC}{DF}
$
\( AB = 5.6 \, \text{cm} \)
\( BC = 7.2 \, \text{cm} \)
\( DE = h \, \text{cm} \)
\( EF = 8.1 \, \text{cm} \)
$
\frac{5.6}{7.2} = \frac{h}{8.1}
$
$
\frac{5.6}{7.2} = 0.7778
$
$
0.7778 = \frac{h}{8.1}
$
$
h = 0.7778 \times 8.1
$
$
h \approx 6.3 \, \text{cm}
$
Question24
24(a): C1.2
24(b): C1.2
$$
\begin{aligned}
& \mathscr{E}=\{\mathrm{a}, \mathrm{~b}, \mathrm{c}, \mathrm{~d}, \mathrm{e}, \mathrm{f}, \mathrm{~g}, \mathrm{~h}, \mathrm{i}, \mathrm{j}, \mathrm{k}\} \\
& F=\{\mathrm{f}, \mathrm{a}, \mathrm{c}, \mathrm{e}\} \\
& B=\{\mathrm{b}, \mathrm{a}, \mathrm{c}, \mathrm{k}\}
\end{aligned}
$$
(a) Complete the Venn diagram.
(b) Find $\mathrm{n}(F \cup B)$.
▶️Answer/Explanation
(b) 6
(a)
Universal set: \( \mathscr{E} = \{a, b, c, d, e, f, g, h, i, j, k\} \)
Set \( F \): \( \{f, a, c, e\} \)
Set \( B \): \( \{b, a, c, k\} \)
\( F \cap B = \{a, c\} \)
Elements only in \( F \): \( \{f, e\} \)
Elements only in \( B \): \( \{b, k\} \)
Elements not in \( F \cup B \): \( \{d, g, h, i, j\} \)
(b)
The union of two sets includes all unique elements from both sets
$F \cup B = \{f, a, c, e, b, k\}$
$n(F \cup B) = 6$
Question25
25(a): C2.5
25(b): C2.5
25(c): C2.5
At a cinema, an adult ticket costs $\$a$ and a child ticket costs $\$c$.
(a) Farah buys 3 adult tickets and 4 child tickets for $\$38.50$ .
Complete the equation.
$3a + 4c =$ ?
(b) Hana buys $6$ adult tickets and $5$ child tickets for $\$65.00$ .
Write down another equation in terms of a and c.
(c) Solve the two simultaneous equations to find the value of a and the value of c.
You must show all your working.
▶️Answer/Explanation
(a) 38.5
(b) 6a + 5c = 65
(c) Correctly eliminating one variable
[a =] 7.5
[c =] 4
(a)
Farah buys 3 adult tickets and 4 child tickets for $\$38.50.$
So the equation
$
3a + 4c = 38.50
$
(b)
Hana buys 6 adult tickets and 5 child tickets for $\$65.00.$
So the equation
$
6a + 5c = 65.00
$
(c)
1. \( 3a + 4c = 38.50 \)
2. \( 6a + 5c = 65.00 \)
Multiply the first equation by 2 to equate the coefficients of \( a \)
$
2(3a + 4c) = 2(38.50)
$
$
6a + 8c = 77.00
$
$
(6a + 8c) – (6a + 5c) = 77.00 – 65.00
$
$
3c = 12.00
$
$
c = 4.00
$
Substitute \( c = 4.00 \) into equation (1)
$
3a + 4(4) = 38.50
$
$
3a + 16 = 38.50
$
$
3a = 38.50 – 16
$
$
3a = 22.50
$
$
a = 7.50
$
The adult ticket price (\( a \)) is $\$7.50 $
The child ticket price (\( c \)) is $\$4.00$