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.
1.Explain why the equation 4x + 3 = 4x + 10 has no solution for x.
(a)Subtracting 4x gives 3 = 10, which is never true.
(b)x would have to be negative, but x must be positive.
(c)It's true for every x, so there are infinite solutions.
(d)x = 7, since 10 − 3 = 7.
2.Which expression means 'y squared, multiplied by 3'?
(a)3y²
(b)y³
(c)(3y)²
(d)3 + y²
3.Write y = x² + 12x + 40 in the form (x + a)² + b, and hence write down the minimum value of y.
y = x² + 12x + 40
(a)−6
(b)36
(c)4
(d)40
4.A company's cost, in £, for producing x items is shown on a graph. The tangent to the curve at x = 50 passes through (30, 400) and (70, 800). Interpret the gradient of this tangent in the context of the company's costs.
(a)The cost decreases by about £10 per extra item
(b)The cost increases by about £400 per extra item
(c)The cost increases by about £0.10 per extra item
(d)The cost increases by about £10 per extra item
5.f(x) = x + 2 and g(x) = x². Work out the value of x for which fg(x) = gf(x).
y = x + 2
(a)1.5
(b)0
(c)−0.5
(d)no solution
6.A line segment has one endpoint at (−6, 2) and its midpoint at (−1, 5). Work out the coordinates of the other endpoint.
(a)(−3.5, 3.5)
(b)(5, 3)
(c)(4, 5)
(d)(4, 8)
7.A car's journey is represented by a velocity-time graph. The total distance travelled, found from the area under the graph, is 350 metres, and the total time for the journey is 25 seconds. Work out the car's average speed for the whole journey, in m/s.
(a)375 m/s
(b)14 m/s
(c)8750 m/s
(d)0.07 m/s
8.Solve 4x² − 9 = 0.
(a)x = 3/2 or x = −3/2
(b)x = 3/2 only
(c)x = 3 or x = −3
(d)x = 9/4 or x = −9/4
9.The function f(x) = x² for all real values of x has no inverse function, but g(x) = x² for x ≥ 0 does have one. Which statement correctly explains this?
y = x²
(a)Restricting any domain always creates an inverse
(b)Squares can never be reversed, under any conditions
(c)g's outputs are positive; f's could be negative
(d)g is one-to-one: f(3) = f(−3), removed by x ≥ 0
10.A line has equation y = −4x + 1. Work out the equation of the line perpendicular to it that passes through the point (0, 3).
y = -4x + 1
(a)y = −4x + 3
(b)y = (−1/4)x + 3
(c)y = 4x + 3
(d)y = (1/4)x + 3
11.A geometric sequence begins 80, 40, 20, 10, ... Work out the next term in the sequence.
(a)2.5
(b)0
(c)20
(d)5
12.Meera writes the statement 3(x + 4) = 3x + 12. Which of these correctly describes what she has written, with a reason?
(a)An identity, true for every value of x
(b)A formula, relating two expressions
(c)An equation, because it has an equals sign
(d)An equation, solved by one value of x
13.The graph of y = f(x) has a minimum point at (4, −1). Write down the coordinates of the corresponding turning point on the graph of y = −f(x).
(a)(4, 1)
(b)(4, −1)
(c)(−4, 1)
(d)(−4, −1)
14.The nth term of a sequence is 3n² + 2n − 1. Work out the 10th term.
(a)260
(b)319
(c)320
(d)279
15.Three vertices of a rectangle are (−4, −1), (2, −1) and (2, 3). The sides of the rectangle are parallel to the axes. Write down the coordinates of the fourth vertex.