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 · Calculator · 15 questions

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1.A rectangular photo has length (x + 3) cm and width x cm. Its area is 40 cm². Work out the value of x.
(a)x = −8
(b)x = 5
(c)x = 5 or x = −8
(d)x = 4
2.The graph of P = 480 ÷ V shows how the pressure, P pascals, of a gas depends on its volume, V cm³, at constant temperature. Use the relationship to work out the volume when the pressure is 32 pascals, and then work out by how much the volume increases when the pressure drops to 24 pascals.
(a)8 cm³
(b)5 cm³
(c)15 cm³
(d)20 cm³
3.The point (20, 21) lies on the circle x² + y² = 841, which has centre O(0, 0). The tangent to the circle at (20, 21) crosses the x-axis at P and the y-axis at Q. Work out the area of triangle OPQ, correct to 1 decimal place.
(a)842.0
(b)2.0
(c)1684.0
(d)580.7
4.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.
5.Two brothers have ages that add up to 30. The elder brother is 6 years older than the younger brother. Work out the age of the younger brother.
(a)24
(b)18
(c)15
(d)12
6.Solve 5x² − 15x = 0.
(a)x = 0 or x = 15
(b)x = 3
(c)x = 0 or x = 3
(d)x = 5 or x = 3
7.By completing the square, find the turning point of the curve y = x² + 8x − 3.
y = x² + 8x − 3
(a)x = −8, y = −67
(b)x = −4, y = 19
(c)x = 4, y = −19
(d)x = −4, y = −19
8.The equation x² − 3x − 7 = 0 can be solved using the iterative formula xₙ₊₁ = √(3xₙ + 7). The starting value is x₀ = 4, so x₁ is the value after the formula has been used once. Work out x₃ correct to 3 decimal places.
(a)4.535
(b)4.481
(c)13.149
(d)4.521
9.A recipe gives the cooking time, in minutes, for a chicken as T = 40m + 20, where m is the mass in kg. A chicken has a mass of 1.8 kg. Work out the cooking time in hours and minutes.
(a)1 hour 12 minutes
(b)1 hour 48 minutes
(c)1 hour 40 minutes
(d)1 hour 32 minutes
10.A car is decelerating. The tangent to its velocity-time graph at t = 12 seconds passes through the points (8, 22) and (16, 6), where velocity is in m/s and time is in seconds. Work out the gradient of this tangent, in m/s².
(a)−2 m/s²
(b)−16 m/s²
(c)2 m/s²
(d)−0.5 m/s²
11.A plumber charges a call-out fee of £30 plus £25 per hour worked. Work out the total charge for a job that takes 3 hours.
(a)£165
(b)£105
(c)£75
(d)£90
12.A circle has centre (0, 0) and equation x² + y² = 50. Work out the length of the diameter of the circle, correct to 1 decimal place.
(a)100.0
(b)25.0
(c)14.1
(d)7.1
13.A cyclist's speed rises from rest to a maximum — quickly at first, then more and more slowly — so her speed-time graph is a curve that is concave down (its gradient decreases as time goes on). A trapezium estimate is made for the distance travelled during this phase, using two points on the curve. Is the trapezium estimate an overestimate or an underestimate of the true distance, and why?
(a)Underestimate: the chord lies below the curve
(b)Overestimate: the trapezium rule always overestimates
(c)Exact: a trapezium and a curve enclose equal areas
(d)Overestimate: the chord lies above the curve
14.A quadratic graph y = ax² + bx + c has its turning point on the y-axis. Which statement about its roots must be true?
(a)The two roots must both be positive.
(b)The graph has no roots at all.
(c)The graph must touch the x-axis at exactly one point.
(d)If the graph has two real roots, they are equal and opposite in value, so they sum to zero.
15.Points A(−1, 2) and B(5, 8) are the endpoints of a line segment. Work out the equation of the perpendicular bisector of AB.
(a)x + y = 4
(b)x + y = 12
(c)x − y = −3
(d)x + y = 7

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