13 marks
Prove that x2 − 6x + 11 is greater than 0 for all values of x.
Show answer
(x - 3)2 + 2 >= 2 > 0
Weekly check: Grade 9
20 mixed questions from 8 grade 9 topics, 54 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
22 marks
A bag contains n counters. 3 of the counters are red.
A counter is taken at random, replaced, and then a second counter is taken at random.
Write down an expression, in terms of n, for the probability that both counters are red.
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9n2
33 marks
A regular hexagon is drawn inside a circle of radius r so that all six vertices lie on the circle. Each side of the hexagon is equal to r.
The perimeter of the hexagon is less than the circumference of the circle.
Use this to show that π > 3.
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6r 3
43 marks
y = −x2 + 6x − 4
By completing the square, find the coordinates of the turning point. State whether it is a maximum or a minimum.
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(3, 5) maximum
53 marks
By completing the square, find the coordinates of the minimum point of the curve
y = x2 + 10x − 2
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(-5, -27)
63 marks
Write x2 − 6x + 11 in the form (x − a)2 + b, where a and b are integers.
Hence find the coordinates of the turning point of the curve y = x2 − 6x + 11
Show answer
a) (x - 3)2 + 2 b) (3, 2)
73 marks
f(x) = 2x + 1 and g(x) = x − 4
Find (fg)−1(x).
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(x + 7)/2
82 marks
Work out the exact value of sin 60° × cos 30°.
Give your answer as a fraction.
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34
93 marks
A bag contains 2x green counters and (x + 4) yellow counters. A counter is taken at random.
The probability that it is green is 25.
Work out the value of x.
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x = 2
102 marks
A biased coin is flipped twice. The flips are independent.
The probability of getting heads on both flips is 0.64.
Work out the probability of getting heads on one flip of the coin.
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0.8
112 marks
A box contains n counters. 5 of the counters are green.
A counter is taken at random and not replaced. A second counter is then taken at random.
Write down an expression, in terms of n, for the probability that both counters are green.
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20n(n-1)
122 marks
Work out the exact value of (sin 30° + cos 60°) × tan 60°.
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√3
132 marks
Two shapes are mathematically similar. The smaller has area 12 cm2 and the larger has area 27 cm2. A side of the smaller shape is 4 cm long.
Work out the length of the matching side of the larger shape.
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6 cm
143 marks
Prove algebraically that the difference between the squares of two consecutive integers is equal to the sum of the two integers.
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(n + 1)2 - n2 = 2n + 1 = n + (n + 1)
153 marks
Prove algebraically that the sum of the squares of any three consecutive integers is always 2 more than a multiple of 3.
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(n - 1)2 + n2 + (n + 1)2 = 3n2 + 2
163 marks
y = 3x2 − 12x + 5
Write y in the form a(x + b)2 + c and write down the coordinates of the turning point.
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3(x - 2)2 - 7; (2, -7)
173 marks
AB is a diameter of a circle with centre O.
C is a point on the circle, not at A or B.
Prove that angle ACB = 90°.
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Using the isosceles triangles OAC and OBC, angle ACB = x + y where 2x + 2y = 180, so angle ACB = 90 degrees
183 marks
y = x2 + 6x + 4
By completing the square, find the coordinates of the turning point and say whether it is a maximum or a minimum.
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(-3, -5) minimum
193 marks
Prove algebraically that (3n + 1)2 − (3n − 1)2 is a multiple of 12 for every integer n.
Show answer
(3n + 1)2 - (3n - 1)2 = 12n, a multiple of 12
203 marks
A car starts from rest and accelerates uniformly to a velocity of 24 m/s in 8 seconds.
It then travels at a constant velocity of 24 m/s for a further 20 seconds.
- Work out the acceleration of the car during the first 8 seconds.
- Work out the total distance travelled by the car in the 28 seconds.
Show answer
a) 3 m/s2 b) 576 m
Your score
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Every question here is original and written by Robinson Tuition, drawn from the grade 9 topic worksheets. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.