Printable · GCSE Higher · ages 14-16

Algebra worksheet — GCSE Higher

Fifteen questions across the algebra statements at Higher tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.

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Algebra worksheet — GCSE Higher

Maths · GCSE Higher · Algebra · 15 questions

MathsUKwww.geekhero.co.uk

1.A ball is dropped and bounces. The height of each bounce after the first is 8 cm less than the bounce before it. The first bounce reaches 60 cm. Work out the height of the 5th bounce.
(a)40
(b)92
(c)20
(d)28
2.The equation x² − 7 = 0 has a positive root. Let f(x) = x² − 7. Given that x₁ = 2.6 and x₂ = 2.65, work out which of these is correct.
y = x² − 7
(a)x₁ closer: f(x₁) negative ⇒ nearer root
(b)x₁ closer: −0.24 is the smaller value
(c)x₂ is the exact root, since f(x₂) ≈ 0
(d)x₂ closer: f(x₂) = 0.0225, nearer to 0
3.A circle has centre O(0, 0) and equation x² + y² = 169. The point Q has coordinates (10, 11). Work out which of these gives the correct position of Q together with correct working.
(a)Outside: OQ² = 221 > r² = 169
(b)Outside: OQ = 21 (10 + 11) > r = 13
(c)Inside: OQ ≈ 14.87 < r² = 169
(d)Inside: OQ² = 221 < (2r)² = 676
4.Make x the subject of the formula y = 3(x + 2).
(a)x = (y − 2)/3
(b)x = y/3 − 2
(c)x = 3y − 2
(d)x = y/3 + 2
5.The iterative formula xₙ₊₁ = xₙ³ − 2 is used repeatedly, starting from x₀ = 2. Which of these describes what happens to the sequence of values as n increases?
(a)It stays constant, since x₀ = 2 is a fixed point.
(b)It oscillates between two fixed values.
(c)It converges to a fixed value close to 2.
(d)It diverges, growing rapidly without limit.
6.A straight line passes through the points (2, 7) and (5, 16). Work out the value of y when x = 0.
(a)1
(b)−1
(c)3
(d)4
7.Work out the maximum value of y = sin x, and the smallest positive value of x, in degrees, at which it occurs.
y = sin(x)
(a)1 at x = 180°
(b)1 at x = 90°
(c)90 at x = 1
(d)1 at x = 0°
8.The iterative formula xₙ₊₁ = √(2xₙ + 15) is used repeatedly, starting from x₀ = 1. Work out the value that xₙ approaches, correct to 2 decimal places.
(a)1.00
(b)5.00
(c)17.00
(d)4.99
9.A student attempts to prove that the product of two consecutive integers is always even: (i) Let the two consecutive integers be n and n + 1. (ii) Since n(n + 1) is even, one of n and n + 1 must be an even number. (iii) Therefore, n(n + 1) is even. At which statement does the proof first assume the very fact it is trying to prove?
(a)Statement (iii)
(b)Statement (i)
(c)Statement (ii)
(d)The proof does not assume anything it is trying to prove
10.A graph has equation y = −2x² + 5. Which statement about its shape is correct?
y = -2x² + 5
(a)It is U-shaped, since x² is always positive.
(b)It is a straight line, because of the constant term +5.
(c)It repeatedly rises and falls, like a wave shape.
(d)It is n-shaped, since the x² coefficient is negative.
11.f(x) = x³ − 5x − 6. Given that f(2.6) = −1.424 and f(2.7) = 0.183, work out what this shows about the equation x³ − 5x − 6 = 0.
y = x
(a)It has a solution between x = 2.6 and x = 2.7
(b)There is no root in this interval
(c)It has a solution between x = −2.6 and x = −2.7
(d)It shows that x = 2.6 is a solution of the equation
12.Solve the inequality 5x − 3 > 2x + 9.
(a)x > 4
(b)x > 2
(c)x < 4
(d)x > 36
13.The nth term of a sequence is n² + 2n − 4. Work out the 7th term.
(a)63
(b)31
(c)59
(d)44
14.A plumber charges a £40 call-out fee plus £25 for each hour worked. The total charge, £C, for a job lasting h hours is given by C = 40 + 25h. Work out the charge for a job lasting 3 hours, and name what kind of statement C = 40 + 25h is.
(a)£65, and C = 40 + 25h is an expression
(b)£115, and C = 40 + 25h is a formula
(c)£195, and C = 40 + 25h is a formula
(d)£115, and C = 40 + 25h is an expression
15.A circle has centre (0, 0) and passes through the point (12, 35). Work out the equation of the circle.
(a)x² + y² = 1225
(b)x² + y² = 1369
(c)x² + y² = 144
(d)x² + y² = 2209

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