Pack B — Fluency

Mathematics

Year 9 · Algebra, Equations and Lines

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–40

1.
State the gradient $m$ and the $y$ -intercept $c$ of the line $y = -2x + 5$ .
2.
Find the gradient of the line through $(0, 3)$ and $(5, 13)$ .
3.
Find the midpoint of the segment from $(-1, 3)$ to $(5, 7)$ .
4.
Find the distance between $(1, 2)$ and $(7, 10)$ .
5.
A line parallel to $y = -3x + 1$ . Find its gradient.
6.
A line perpendicular to $y = 3x + -1$ . Find its gradient.
7.
Complete the table of values for $y = -1x + 4$ at $x = -1, 0, 1, 2$ .
8.
Write the equation of the line with gradient -2 and $y$ -intercept 5.
9.
State the equation of the horizontal line through $(2, -2)$ and the vertical line through $(4, 5)$ .
10.
Find the $x$ - and $y$ -intercepts of $y = 3x + 6$ .
11.
Solve $y = 3x + -2$ and $y = 10$ for $x$ and $y$ .
12.
Solve $x + y = 15$ and $x - y = 3$ .
13.
Solve $y = 2x$ and $y = 3x - 4$ .
14.
Solve $y = 3x + -2$ and $y = -1x + 6$ .
15.
Use substitution: $y = 3x$ and $x + y = 12$ .
16.
Solve $2x + y = 10$ and $y = x + 1$ .
17.
Graphically: which point is on both lines $y = x + 2$ and $y = -x + 4$ ?
18.
Solve $x = 5$ and $y = 2x - 3$ .
19.
Solve $x + 2y = 8$ and $x - 2y = 0$ .
20.
Two equations are $y = x + 3$ and $y = 2x + 1$ . Find the intersection.
21.
Evaluate $3^4$ .
22.
Write $5^4 \times 5^3$ as a single power of $a$ .
23.
Write $\dfrac{a^p}{a^q}$ as a single power.
24.
Write 7500000 in standard form.
25.
Write the decimal 0.000 25 in standard form.
26.
Evaluate $(5)^0$ and $(5)^{-2}$ .
27.
Write $(a^p)^q$ as a single power.
28.
Simplify $x^p \times x^q$ .
29.
Write 0.5 × 10⁵ in standard form.
30.
Evaluate $5^0$ and $5^1$ .
31.
Reflect the point $(-2, 5)$ in the $x$ -axis.
32.
Reflect the point $(-2, 5)$ in the $y$ -axis.
33.
Translate $(2, -1)$ by the vector $(4, 3)$ . State the image.
34.
Rotate $(3, 0)$ by 90° anticlockwise about the origin. State the image.
35.
Triangle ABC has $A(1,1), B(2,1), C(1,3)$ . Enlarge by sf 2 from the origin. State $A', B', C'$ .
36.
Two figures are similar with scale factor 3. The smaller has length 5 cm. Find the corresponding length on the larger.
37.
Two triangles A and B are similar. A: sides 3, 4, 5. B: shortest side 6. Find B's other sides.
38.
Two triangles are congruent. State the four uniqueness rules.
39.
Reflect the point $(3, 4)$ in the line $y = x$ . State the image.
40.
Two similar shapes have linear scale factor 2. The smaller has area 16 cm². Find the larger area.

SilverQuestions 41–80

41.
Find the gradient through $(-2, 5)$ and $(4, -7)$ .
42.
Find the distance between $(1, 1)$ and $(4, 5)$ as a simplified surd.
43.
Find the equation of the line through $(0, -1)$ and $(4, 11)$ .
44.
Find the equation of the line parallel to $y = 3x + -1$ passing through $(2, 5)$ .
45.
Find the equation of the line perpendicular to $y = 3x + -2$ through $(6, 1)$ .
46.
Find the midpoint of $A(-5, -2)$ and $B(9, 8)$ with mixed signs.
47.
Find the equation of the line with gradient -2 through $(-1, 4)$ .
48.
Decide parallel / perpendicular / neither: (a) $y = 3x + 2$ and $y = 3x - 5$ . (b) $y = 4x + 1$ and $y = -x/4 + 5$ .
49.
Find the equation of the line perpendicular to $y = (1/3)x + 2$ passing through $(3, 1)$ .
50.
Find the equation of the line through (2, 4) and (5, 4).
51.
Solve $3x + 2y = 12$ and $x - y = 1$ by elimination.
52.
Solve $y = 4x - 7$ and $2x + y = 5$ .
53.
Solve $2x + y = 7$ and $3x - 2y = 21$ by elimination.
54.
Decide whether the system has unique, no, or infinite solutions: $x + y = 3$ and $2x + 2y = 6$ .
55.
Solve $y = -x + 6$ and $y = 2x - 3$ .
56.
Solve $2(x - 1) + y = 5$ and $3x + 2y = 14$ .
57.
Solve $\dfrac{x}{2} + y = 7$ and $x + 2y = 14$ .
58.
Find the intersection of $3x + 4y = 12$ and $x = 0$ .
59.
Solve $y = 2x + 3$ and $y = -x + 6$ graphically (i.e. find intersection).
60.
Two pens and three pencils cost £2.20. Five pens and one pencil cost £3.10. Find each price.
61.
Simplify $(a \times 10^p) \times (b \times 10^q)$ .
62.
Simplify $\dfrac{a \times 10^p}{b \times 10^q}$ .
63.
Simplify $\dfrac{x^p}{x^q}$ .
64.
Evaluate $a^{1/2}$ .
65.
Write $35 \times 10^5$ in standard form.
66.
Compute $5 \times 10^4 + 3 \times 10^3$ in standard form.
67.
Evaluate $a^{-1}$ .
68.
Simplify $(2x)^3$ .
69.
Convert: a light-year ≈ $9.5 \times 10^{12}$ km. Express in metres.
70.
Compute $(2 \times 10^3)^2$ .
71.
Two similar rectangles have areas 18 cm² and 162 cm². The smaller's shorter side is 3 cm. Find the larger's shorter side.
72.
A triangle has vertices $A(0,0), B(2,0), C(0,2)$ . Reflect in the $y$ -axis to find $A', B', C'$ .
73.
Translate $\triangle ABC$ with $A(1, 2), B(3, 2), C(2, 4)$ by vector $(2, -1)$ . State the image vertices.
74.
A square with vertices $A(1,1), B(3,1), C(3,3), D(1,3)$ is enlarged by factor 2 about the origin. State the image vertices.
75.
Rotate $\triangle ABC$ with $A(1,1), B(2,1), C(1,3)$ by 90° anticlockwise about the origin.
76.
Two similar triangles have a side ratio of $\dfrac{x+2}{15} = \dfrac{x}{12}$ . Find $x$ .
77.
Two similar prisms have linear scale factor 3. The smaller has volume 27 cm³. Find the larger volume.
78.
A triangle with vertices $(0,0), (4,0), (4,3)$ is reflected in the $x$ -axis. State the image area and verify it equals the original.
79.
Two triangles are congruent. State which rule(s) apply when given (a) 3 sides match, (b) 2 sides and the included angle match.
80.
A square with area 25 cm² is enlarged by scale factor 2. Find the new area and verify the rule.

GoldQuestions 81–120

81.
A line passes through $A(-2, 1)$ and $B(7, 4)$ . Find its equation.
82.
The point C divides AB in ratio 2:1, $A(-2, 1)$ , $B(7, 4)$ . Find C.
83.
Find the equation of the perpendicular bisector of A(1, 2) and B(5, 6).
84.
Find the $x$ - and $y$ -intercepts of $3x + 4y = 12$ .
85.
Does the point (3, 7) lie on the line $y = 2x + 1$ ? Justify.
86.
Find the equation of the line parallel to $2x + 3y = 6$ passing through $(0, 0)$ .
87.
Find the perimeter of triangle $A(0, 0)$ , $B(4, 3)$ , $C(8, 0)$ .
88.
A line $\ell_1$ has equation $y = mx + 2$ . It is perpendicular to $\ell_2: y = (1/4)x + 5$ . Find $m$ .
89.
Find the foot of the perpendicular from $P(5, 1)$ to the line $y = 2x - 1$ .
90.
A line through $(1, 5)$ is parallel to a line through $(0, 1)$ and $(4, 9)$ . Find its equation.
91.
Use equating $y = y$ method to solve $y = 2x + 1$ and $y = -x + 7$ .
92.
A coach hires a 32-seater bus and a 16-seater bus to take 224 students. Total cost £1900 (32-seater £250, 16-seater £150). Find how many of each.
93.
Concert ticket pricing: Friday 50 adult + 20 child = £790; Saturday 80 adult + 40 child = £1320. Find adult and child prices.
94.
A father is 4× his son's age. In 4 years, he will be 3× his son's age. Find their ages.
95.
Solve $\dfrac{x + y}{3} = 5$ and $\dfrac{x - y}{2} = 1$ .
96.
Solve $2x + y = 10$ and $x^2 + y = 14$ .
97.
A 2-burger meal costs £8.50; a 1-burger + 1-fries meal costs £6.20. Find the prices of a burger and fries.
98.
Three friends share an amount. Alice has £ $x$ , Bob has £ $y$ , Charles £ $z$ . $x + y = 50$ , $y + z = 60$ , $x + z = 70$ . Find each share.
99.
Find the value of $k$ for which $y = 3x + k$ passes through the intersection of $y = 2x + 1$ and $y = x + 4$ .
100.
Find the line through the intersection of $y = 2x + 1$ and $y = -x + 7$ that is parallel to $y = 4x - 5$ .
101.
Simplify $\dfrac{(x^p)^q}{x^r}$ .
102.
Write $35 \times 10^5$ in standard form.
103.
Evaluate $a^{1/2}$ where a = 25.
104.
UK population was $6.50 \times 10^7$ in 2016, growing 0.6% per year. Estimate the 2020 population (3 s.f.).
105.
Compute $(3 \times 10^4) \times (4 \times 10^{-2})$ in standard form.
106.
Convert $a \times 10^p$ to ordinary: $3.6 \times 10^{-3}$ .
107.
Evaluate $a^{2/3}$ where a = 8.
108.
Compute $\dfrac{6 \times 10^7}{2 \times 10^4}$ in standard form.
109.
Simplify $(2x^3)^2 \times (3x)^2$ .
110.
Order from smallest to largest: $3 \times 10^{-3}, 2 \times 10^{-4}, 4 \times 10^{-3}$ .
111.
A triangle ABC: $A(1,1), B(3,1), C(2,3)$ . Enlarge by sf 2 from origin. State image vertices.
112.
Two triangles share sides $a_1, b_1, c_1$ and $a_2, b_2, c_2$ with corresponding sides equal: $a_1 = a_2, b_1 = b_2$ , included angle equal. State the congruence rule.
113.
Triangle ABC is similar to DEF with sf $k = 5/3$ from ABC to DEF. AB = 6, AC = 9. Find DE and DF.
114.
Two similar shapes have areas 18 cm² and 50 cm². Find the linear scale factor.
115.
A triangle has sides 6 cm, 8 cm, 10 cm. A similar triangle has perimeter 36 cm. Find its sides.
116.
Triangle ABC has $A(0, 0), B(4, 0), C(0, 3)$ . Rotate 90° clockwise about origin. State the image vertices.
117.
Two similar prisms have surface areas $24$ cm² and $96$ cm². Find the volume ratio.
118.
A rectangle 4 × 6 is enlarged by sf 1.5. Find (a) new dimensions, (b) new area, (c) new perimeter.
119.
A triangle has vertices $A(1, 2), B(4, 2), C(4, 6)$ . Reflect in $y = x$ to get image $A', B', C'$ .
120.
A triangle PQR has $PQ = 6, QR = 8, RP = 10$ . Show ΔPQR is right-angled. Identify the right angle.

PlatinumQuestions 121–160

121.
A triangle has vertices A(1, 2), B(7, 4), C(4, 10). (a) Find the midpoint M of BC. (b) Find the equation of the median from A.
122.
Show that points P(-1, -3), Q(2, 3), R(5, 9) are collinear.
123.
Find the perpendicular distance from $P(5, 1)$ to the line $y = 2x - 1$ .
124.
A line passes through $(a, 0)$ and $(0, b)$ . Find its equation.
125.
Find the area of the triangle bounded by $y = x + 1$ , $y = -x + 5$ , $y = 0$ .
126.
A triangle has vertices $A(0, 0), B(4, 2), C(1, 5)$ . Verify whether it is right-angled.
127.
The point M(3, 2) is the midpoint of AB, where A = (1, 5). Find B.
128.
A quadrilateral has vertices A(0, 0), B(4, 0), C(5, 3), D(1, 3). Find the gradients of all 4 sides and identify the shape.
129.
Find the value of $k$ for which the lines $y = 2x + k$ and $y = 3x + 1$ intersect at a point on the $x$ -axis.
130.
The line $\ell$ has equation $3x - 4y = 12$ . Find (a) the slope, (b) the $x$ - and $y$ -intercepts, (c) the perpendicular distance from origin to $\ell$ .
131.
Solve $\dfrac{2x + 3y}{5} = 4$ and $3x - y = 7$ .
132.
Solve $2x + 3y = 19$ and $4x - y = 17$ using (a) substitution, (b) elimination.
133.
Solve $\dfrac{x + 1}{2} - \dfrac{y - 2}{3} = 1$ and $x + y = 5$ .
134.
Two mobile phone tariffs: A: £15 + 10p/min; B: £25 + 5p/min. Find the minutes for equal cost, then which is cheaper for 300 min.
135.
Find the values of $a$ and $b$ such that the system $ax + 2y = 5, 3x + by = 7$ has the solution $(1, 2)$ .
136.
A two-digit number has the digit sum 11 and is 27 greater than the number with reversed digits. Find the original number.
137.
Solve $3x - y = 7$ and $x^2 + y^2 = 25$ .
138.
A 2-variable system has solution $(2, 3)$ . Find the system if both lines pass through this point and one has slope 2 and the other slope $-1$ .
139.
A boat travels 36 km downstream in 2 hours and back upstream in 3 hours. Find the speed of the boat in still water and the speed of the current.
140.
Two mixtures: 60% milk and 40% water in mixture A; 20% milk and 80% water in B. How many litres of each to make 10 L of 50% milk-50% water?
141.
Solve $x^p = {rhs}$ .
142.
Evaluate $a^{-1/2}$ .
143.
A nearby star is $4.2$ light-years away. 1 light-year ≈ $9.46 \times 10^{12}$ km. Find distance in km (3 s.f.).
144.
Solve $x^{1/3} = n$ .
145.
A bacteria culture starts at $2.5 \times 10^4$ cells, doubling every hour. (a) After 1 hour. (b) After 5 hours. (c) When does it exceed $10^8$ ?
146.
Evaluate $27^{-2/3}$ .
147.
Solve $2^x = 32$ .
148.
Solve $x^4 = 81$ for real $x$ .
149.
Express $\sqrt[3]{27 a^9}$ in simplified index form.
150.
Compute $(10^3)^{1/3}$ and $(10^6)^{1/3}$ .
151.
A cylindrical glass of radius 6 cm and height 12 cm is full of water. Water is poured into a similar cylindrical glass (radius half of the first). To what height does the water rise?
152.
Triangle ABC is similar to triangle PQR with sf 2 from ABC to PQR. Triangle ABC has area 18 cm². Find PQR's area and perimeter ratio.
153.
A photo is enlarged so its area is doubled. Find the exact linear scale factor.
154.
A triangle is rotated 90° about the origin, then reflected in the $x$ -axis. The original is $A(1, 2)$ . Find the final image.
155.
Two similar cones have heights 6 and 9. The smaller has slant 8. Find the larger's slant.
156.
Triangle ABC has angles 60°, 80°, 40°. Triangle DEF has angles 60°, 80°, 40°. Are they similar? Are they congruent?
157.
A triangle has area 24 cm². It is enlarged by sf 2.5 from a point. Find the area of the image.
158.
Triangle ABC has $A(0,0), B(3,0), C(0,4)$ . It is enlarged by sf 2 from the centre $(1, 1)$ . Find the image vertices.
159.
Two similar triangles have linear scale factor 4:7. Their corresponding areas are in what ratio?
160.
A figure of area $A$ is transformed by rotation. Show that the area is preserved.