14 marks
Solve the simultaneous equations
x2 + y2 = 20
y = 2x
Show answer
x = 2, y = 4 and x = -2, y = -4
Weekly check: Grade 8
20 mixed questions from 12 grade 8 topics, 50 marks.
Type your answers below for an instant mark and a score, or print the sheet, mark it with the answers sheet, and use the solutions to see every step written out by hand. Set one a week and watch the score climb.
The questions
23 marks
OAB is a triangle. OA = a and OB = b.
M is the midpoint of OA and N is the midpoint of OB.
Show that MN is parallel to AB.
Show answer
MN = (12)(b - a) = (12)AB, so MN is parallel to AB
32 marks
The curve C has equation y = cos x° for 0 ≤ x ≤ 360.
Write down the coordinates of the minimum point of C.
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(180, -1)
44 marks
OAB is a triangle.
Vector OA = a and vector OB = b.
M is the midpoint of OA and N is the midpoint of OB.
Prove that MN is parallel to AB.
Show answer
MN = 12(b - a) = 12AB, so MN is a scalar multiple of AB and the lines are parallel
52 marks
Expand and simplify (5 + 2√3)(5 − 2√3)
Show answer
13
62 marks
The graph of y = sin x for 0° ≤ x ≤ 360° has a maximum point at (90, 1).
- Write down the coordinates of the maximum point of the graph of y = sin x − 3
- Write down the coordinates of the maximum point of the graph of y = sin (x − 30°)
Show answer
a) (90, -2) b) (120, 1)
72 marks
Solve (x − 3)(x + 7) ≥ 0
Show answer
x = 3
82 marks
The number of members of a gym is modelled by the rule
un+1 = 1.2un − 50
where un is the number of members after n years.
There are 500 members now, so u0 = 500.
Work out the number of members the model predicts after 2 years.
Show answer
610
93 marks
A runner's velocity-time graph is a smooth curve. The table gives her velocity v m/s at time t seconds.
t: 0, 10, 20, 30 v: 0, 4.2, 6.6, 7.8
Use the trapezium rule with 3 strips of equal width to estimate the distance the runner travels in the first 30 seconds.
Show answer
147 m
102 marks
ABCDEFGH is a cuboid with base ABCD and top face EFGH, where E is directly above A, F is directly above B, G is directly above C and H is directly above D.
AB = 3 cm, BC = 4 cm and CG = 12 cm.
Work out the length of AG.
Show answer
13 cm
112 marks
A circle has its centre at the origin O. The diameter of the circle is 16.
Find the equation of the circle.
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x2 + y2 = 64
122 marks
f(x) = x3 − 4x − 3
Show that the equation f(x) = 0 has a root between x = 2 and x = 3.
Show answer
f(2) = -3 and f(3) = 12, a change of sign
132 marks
Write x + 12 + 2x − 15 as a single fraction in its simplest form.
Show answer
(9x + 3)/10
143 marks
The temperature, T °C, of a cup of tea is plotted against time, t minutes, and the points form a smooth curve going down.
At t = 10 a tangent is drawn to the curve. The tangent passes through the points (2, 80) and (14, 56).
- Work out the gradient of the tangent.
- What does this gradient tell you about the tea at t = 10?
Show answer
a) -2 b) the temperature is falling at 2 °C per minute at t = 10
154 marks
The curve C has equation y = x2 − 4x + 7
The line L has equation y = 2x − 2
Show that L touches C at exactly one point, and find the coordinates of this point.
Show answer
x2 - 6x + 9 = (x - 3)2 = 0 has the repeated root x = 3, so exactly one point of intersection, (3, 4)
162 marks
Sketch the graph of y = tan x° for 0 ≤ x ≤ 360.
Show clearly where the graph crosses the x-axis and mark the values of x where the graph has asymptotes.
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Correct tan shape: branches through (0,0), (180,0), (360,0) with vertical asymptotes at x = 90 and x = 270
172 marks
The graph of y = f(x) has a turning point at (3, −5).
- Write down the coordinates of the turning point of the graph of y = f(x) + 4
- Write down the coordinates of the turning point of the graph of y = f(x − 2)
Show answer
a) (3, -1) b) (5, -5)
183 marks
The circle C has equation x2 + y2 = 61
- Show that the point A (5, −6) lies on C.
- Does the point B (4, 7) lie on C? You must justify your answer.
Show answer
a) 25 + 36 = 61, so A lies on the circle b) No: 16 + 49 = 65, which is not 61, so B does not lie on the circle
192 marks
Erin is asked to rationalise the denominator of 8√6
Here is her working.
8√6 = 8 × √6√6 × √6 = 8√636
Erin has made one mistake.
Explain the mistake Erin has made.
Show answer
√6 x √6 = 6, not 36. The denominator should be 6, giving 8√66 = 4√63.
202 marks
Rationalise the denominator of 12√3
Give your answer in its simplest form.
Show answer
4√3
Your score
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Every question here is original and written by Robinson Tuition, drawn from the grade 8 topic worksheets. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.