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A Level H1 Mathematics Algebra Functions Quiz

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Questions

A-Level Maths H1 Quiz - Algebra Functions

Name: ________________________
Class: ________________________
Date: ________________________
Score: ______ / 50

Duration: 1 hour 15 minutes
Total Marks: 50

Instructions:

Answer ALL questions in the spaces provided.
Show all working clearly. Marks are awarded for method, not just final answers.
Unless otherwise stated, give non-exact answers to 3 significant figures.
You may use an approved graphing calculator (without CAS).
The number of marks is given in brackets [ ] at the end of each question or part question.

Section A: Functions and Their Properties (Questions 1–5)

[10 marks]

1. The function f is defined by f(x) = e^(2x) − 3 for x ∈ ℝ.

(a) Find the value of f(0). [1]

(b) Find the value of x for which f(x) = 5, giving your answer in exact form. [2]

2. A function g is defined by g(x) = ln(2x + 1) for x > −½.

(a) Evaluate g(2), giving your answer correct to 3 significant figures. [1]

(b) Find g⁻¹(x), the inverse function of g, and state its domain. [3]

3. The graph of y = f(x) is shown below. The curve has a maximum point at (1, 4) and passes through the origin.

[Assume a sketch is provided showing a curve with maximum at (1, 4), passing through (0, 0) and (2, 0)]

(a) Write down the value of f(1). [1]

(b) State the values of x for which f(x) = 0. [1]

(c) Explain why f is not a one-one function for x ∈ ℝ. [1]

4. The functions h and k are defined by: h(x) = 3x − 2, for x ∈ ℝ k(x) = x², for x ∈ ℝ

Find, in its simplest form: (a) hk(x) [1] (b) kh(x) [1]

5. A function p is defined by p(x) = 2e^(−x) + 1 for x ∈ ℝ.

(a) Write down the equation of the horizontal asymptote of the graph of y = p(x). [1]

(b) State the range of p. [1]

Section B: Equations and Inequalities (Questions 6–10)

[12 marks]

6. Solve the equation e^(2x) − 4e^x + 3 = 0, giving your answers in exact form. [3]

7. Solve the inequality 2x² − 5x − 3 > 0. [3]

8. Find the values of k for which the equation x² + kx + 9 = 0 has two distinct real roots. [2]

9. Solve the simultaneous equations: y = 2x + 1 y = x² − x + 3 [3]

10. Find the range of values of x for which the curve y = x² + 4x + 1 lies above the line y = 2x − 3. [3]

Section C: Applications of Functions (Questions 11–15)

[14 marks]

11. The population of a bacterial culture, P thousand, t hours after the start of an experiment, is modelled by the equation P = 5e^(0.2t).

(a) Find the initial population. [1]

(b) Find the time taken for the population to reach 20 thousand, giving your answer correct to the nearest minute. [3]

(c) Find the rate of change of the population when t = 5. [2]

12. A company's profit, $P thousand, is related to the number of units produced, x hundred, by the equation P = 50 ln(x + 1) − 2x, for x ≥ 0.

(a) Find the profit when 400 units are produced. [1]

(b) Find the value of x that maximises the profit. [3]

(c) Hence find the maximum profit, correct to the nearest thousand dollars. [1]

13. The value of an investment, $V, after t years is given by V = 10000e^(0.05t).

(a) Find the value of the investment after 10 years. [1]

(b) Find the number of years for the investment to double in value. [2]

14. The temperature T °C of a cooling liquid t minutes after being removed from a heat source is modelled by T = 25 + 75e^(−0.1t).

(a) State the room temperature according to this model. [1]

(b) Find the temperature of the liquid after 5 minutes. [1]

(c) Find the time taken for the temperature to fall to 40 °C. [2]

15. A curve has equation y = e^x − 2x.

(a) Find the coordinates of the stationary point on the curve, giving your answers in exact form. [3]

(b) Determine the nature of this stationary point. [2]

Section D: Graphs and Transformations (Questions 16–20)

[14 marks]

16. The diagram shows the graph of y = f(x). The curve has a minimum point at (−1, −2) and passes through the points (0, 0) and (2, 0).

[Assume a sketch is provided]

On separate diagrams, sketch the graphs of: (a) y = f(x) + 3 [2] (b) y = f(x − 1) [2]

In each case, indicate clearly the coordinates of the minimum point and the points where the curve meets the axes.

17. The function f is defined by f(x) = e^x − 1 for x ∈ ℝ.

(a) Sketch the graph of y = f(x), indicating clearly any asymptotes and the coordinates of any points where the curve meets the axes. [3]

(b) On the same diagram, sketch the graph of y = f⁻¹(x). [2]

18. The graph of y = ln x is transformed by a stretch parallel to the y-axis with scale factor 2, followed by a translation of 3 units in the positive x-direction.

Find the equation of the resulting graph. [2]

19. A function g is defined by g(x) = 2 − e^(−x) for x ∈ ℝ.

(a) Find g(0) and state the equation of the asymptote of the graph of y = g(x). [2]

(b) Sketch the graph of y = g(x), indicating clearly the asymptote and the coordinates of any points where the curve meets the axes. [3]

20. The curve C has equation y = 3 − 2e^(−x).

(a) State the equation of the asymptote of C. [1]

(b) Find the exact coordinates of the point where C crosses the x-axis. [2]

(c) Sketch the graph of C, indicating clearly the asymptote and the coordinates of any points where the curve meets the axes. [2]

END OF QUIZ

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Answers

A-Level Maths H1 Quiz - Algebra Functions — ANSWER KEY

Total Marks: 50

Section A: Functions and Their Properties (Questions 1–5)

1. f(x) = e^(2x) − 3

(a) f(0) = e^(2×0) − 3 = e⁰ − 3 = 1 − 3 = −2 [M1, A1 — 1 mark]

(b) f(x) = 5 ⇒ e^(2x) − 3 = 5 ⇒ e^(2x) = 8 ⇒ 2x = ln 8 ⇒ x = ½ ln 8 = ln(8^(½)) = ln(√8) = ln(2√2) [M1 for setting up equation, A1 for exact form — 2 marks]

Accept x = ½ ln 8, x = ln √8, or equivalent exact forms.

2. g(x) = ln(2x + 1), x > −½

(a) g(2) = ln(2×2 + 1) = ln 5 ≈ 1.609 ≈ 1.61 (3 s.f.) [B1 — 1 mark]

(b) Let y = ln(2x + 1) ⇒ e^y = 2x + 1 ⇒ 2x = e^y − 1 ⇒ x = ½(e^y − 1) ∴ g⁻¹(x) = ½(e^x − 1) [M1 for swapping and rearranging, A1 for correct expression]

Domain of g⁻¹: x ∈ ℝ (since range of g is ℝ) [B1 — 3 marks total]

3. Graph of y = f(x) with maximum at (1, 4), passing through (0, 0) and (2, 0).

(a) f(1) = 4 [B1 — 1 mark]

(b) f(x) = 0 when x = 0 and x = 2 [B1 — 1 mark]

(c) f is not one-one because there exist two different x-values (e.g., x = 0 and x = 2) that give the same y-value (y = 0). A horizontal line y = k for 0 < k < 4 would intersect the graph at two points. [B1 for valid reasoning — 1 mark]

4. h(x) = 3x − 2, k(x) = x²

(a) hk(x) = h(k(x)) = h(x²) = 3(x²) − 2 = 3x² − 2 [B1 — 1 mark]

(b) kh(x) = k(h(x)) = k(3x − 2) = (3x − 2)² = 9x² − 12x + 4 [B1 — 1 mark]

5. p(x) = 2e^(−x) + 1

(a) As x → ∞, e^(−x) → 0, so p(x) → 1. Horizontal asymptote: y = 1 [B1 — 1 mark]

(b) Since e^(−x) > 0 for all x, 2e^(−x) > 0, so p(x) > 1. Range: p(x) > 1 (or (1, ∞)) [B1 — 1 mark]

Section B: Equations and Inequalities (Questions 6–10)

6. e^(2x) − 4e^x + 3 = 0 Let u = e^x ⇒ u² − 4u + 3 = 0 ⇒ (u − 1)(u − 3) = 0 ⇒ u = 1 or u = 3 ⇒ e^x = 1 ⇒ x = ln 1 = 0 ⇒ e^x = 3 ⇒ x = ln 3 ∴ x = 0 or x = ln 3 [M1 for substitution, M1 for solving quadratic, A1 for both exact answers — 3 marks]

7. 2x² − 5x − 3 > 0 Factorise: (2x + 1)(x − 3) > 0 Critical values: x = −½, x = 3 Sign analysis: positive when x < −½ or x > 3 ∴ x < −½ or x > 3 [M1 for factorising, M1 for critical values and sign analysis, A1 — 3 marks]

8. x² + kx + 9 = 0 has two distinct real roots when discriminant > 0. Δ = k² − 4(1)(9) = k² − 36 > 0 ⇒ k² > 36 ⇒ k < −6 or k > 6 [M1 for discriminant condition, A1 — 2 marks]

9. y = 2x + 1 and y = x² − x + 3 Equate: 2x + 1 = x² − x + 3 ⇒ 0 = x² − 3x + 2 ⇒ 0 = (x − 1)(x − 2) ⇒ x = 1 or x = 2 When x = 1: y = 2(1) + 1 = 3 When x = 2: y = 2(2) + 1 = 5 ∴ Solutions: (1, 3) and (2, 5) [M1 for equating, M1 for solving quadratic, A1 for both coordinate pairs — 3 marks]

10. Curve y = x² + 4x + 1 lies above line y = 2x − 3 when: x² + 4x + 1 > 2x − 3 ⇒ x² + 2x + 4 > 0 ⇒ (x + 1)² + 3 > 0 Since (x + 1)² ≥ 0 for all real x, (x + 1)² + 3 ≥ 3 > 0 for all real x. ∴ The curve lies above the line for all real values of x. [M1 for setting up inequality, M1 for completing square/analysing, A1 — 3 marks]

Section C: Applications of Functions (Questions 11–15)

11. P = 5e^(0.2t)

(a) Initial population (t = 0): P = 5e⁰ = 5 thousand [B1 — 1 mark]

(b) P = 20 ⇒ 5e^(0.2t) = 20 ⇒ e^(0.2t) = 4 ⇒ 0.2t = ln 4 ⇒ t = 5 ln 4 ≈ 6.93147 hours = 6 hours + 0.93147 × 60 minutes ≈ 6 hours 56 minutes [M1 for setting up, M1 for solving, A1 for correct time — 3 marks]

(c) dP/dt = 5 × 0.2 × e^(0.2t) = e^(0.2t) When t = 5: dP/dt = e^(0.2×5) = e¹ = e ≈ 2.72 thousand per hour [M1 for differentiation, A1 — 2 marks]

12. P = 50 ln(x + 1) − 2x, x ≥ 0

(a) 400 units ⇒ x = 4 (since x is in hundreds) P = 50 ln(5) − 2(4) = 50 ln 5 − 8 ≈ 50(1.60944) − 8 = 80.472 − 8 = 72.472 ≈ 72.5 thousand dollars (3 s.f.) [B1 — 1 mark]

(b) dP/dx = 50/(x + 1) − 2 Set dP/dx = 0: 50/(x + 1) − 2 = 0 ⇒ 50/(x + 1) = 2 ⇒ x + 1 = 25 ⇒ x = 24 Second derivative: d²P/dx² = −50/(x + 1)² < 0 for all x ≥ 0, so maximum. ∴ Profit is maximised when x = 24 (2400 units) [M1 for differentiation, M1 for setting to zero and solving, A1 — 3 marks]

(c) Maximum profit: P = 50 ln(25) − 2(24) = 50 ln 25 − 48 ≈ 50(3.21888) − 48 = 160.944 − 48 = 112.944 ≈ 113 thousand dollars (nearest thousand) [B1 — 1 mark]

13. V = 10000e^(0.05t)

(a) t = 10: V = 10000e^(0.5) ≈ 10000 × 1.64872 = 16487.2 ≈ $16,500 (3 s.f.) [B1 — 1 mark]

(b) Double value: 20000 = 10000e^(0.05t) ⇒ 2 = e^(0.05t) ⇒ ln 2 = 0.05t ⇒ t = (ln 2)/0.05 ≈ 0.693147/0.05 = 13.8629 ≈ 13.9 years (3 s.f.) [M1 for setting up, A1 — 2 marks]

14. T = 25 + 75e^(−0.1t)

(a) As t → ∞, e^(−0.1t) → 0, so T → 25. Room temperature = 25 °C [B1 — 1 mark]

(b) t = 5: T = 25 + 75e^(−0.5) ≈ 25 + 75(0.60653) = 25 + 45.4898 = 70.4898 ≈ 70.5 °C [B1 — 1 mark]

(c) T = 40: 40 = 25 + 75e^(−0.1t) ⇒ 15 = 75e^(−0.1t) ⇒ e^(−0.1t) = 0.2 ⇒ −0.1t = ln 0.2 ⇒ t = −10 ln 0.2 ≈ −10(−1.60944) = 16.0944 ≈ 16.1 minutes [M1 for setting up, A1 — 2 marks]

15. y = e^x − 2x

(a) dy/dx = e^x − 2 Stationary point when dy/dx = 0: e^x − 2 = 0 ⇒ e^x = 2 ⇒ x = ln 2 y-coordinate: y = e^(ln 2) − 2 ln 2 = 2 − 2 ln 2 Coordinates: (ln 2, 2 − 2 ln 2) [M1 for differentiation, M1 for solving, A1 for exact coordinates — 3 marks]

(b) d²y/dx² = e^x At x = ln 2: d²y/dx² = e^(ln 2) = 2 > 0 ∴ The stationary point is a minimum. [M1 for second derivative, A1 for conclusion — 2 marks]

Section D: Graphs and Transformations (Questions 16–20)

16. Original: minimum at (−1, −2), passes through (0, 0) and (2, 0).

(a) y = f(x) + 3: Translation 3 units up. - Minimum point: (−1, 1) - x-intercepts: f(x) + 3 = 0 ⇒ f(x) = −3. From graph, this occurs at two points (by symmetry, approximately x ≈ −2 and x ≈ 3, but exact values depend on the specific f(x); accept reasonable estimates based on the given sketch). - y-intercept: (0, 3) [B1 for correct minimum, B1 for correct intercepts — 2 marks]

(b) y = f(x − 1): Translation 1 unit right. - Minimum point: (0, −2) - x-intercepts: (1, 0) and (3, 0) - y-intercept: f(−1) = −2 ⇒ (0, −2) [B1 for correct minimum, B1 for correct intercepts — 2 marks]

Mark according to sketch accuracy and correct labelling of key points.

17. f(x) = e^x − 1

(a) Graph of y = e^x − 1: - Horizontal asymptote: y = −1 (as x → −∞) - y-intercept: (0, e⁰ − 1) = (0, 0) - x-intercept: e^x − 1 = 0 ⇒ e^x = 1 ⇒ x = 0, so (0, 0) - Shape: exponential curve, increasing, passing through origin, approaching y = −1 as x → −∞ [B1 for asymptote, B1 for intercept, B1 for correct shape — 3 marks]

(b) f⁻¹(x) = ln(x + 1), domain x > −1. Graph is reflection of y = f(x) in the line y = x. - Vertical asymptote: x = −1 - Passes through (0, 0) - Shape: logarithmic curve, increasing, defined for x > −1 [B1 for correct reflection, B1 for asymptote and intercept — 2 marks]

18. y = ln x Stretch parallel to y-axis, scale factor 2: y = 2 ln x Translation 3 units in positive x-direction: y = 2 ln(x − 3) Equation: y = 2 ln(x − 3) [M1 for stretch, A1 for final equation — 2 marks]

19. g(x) = 2 − e^(−x)

(a) g(0) = 2 − e⁰ = 2 − 1 = 1 As x → ∞, e^(−x) → 0, so g(x) → 2. Asymptote: y = 2 [B1 for g(0), B1 for asymptote — 2 marks]

(b) Sketch: - Horizontal asymptote: y = 2 (as x → ∞) - As x → −∞, e^(−x) → ∞, so g(x) → −∞ - y-intercept: (0, 1) - x-intercept: 2 − e^(−x) = 0 ⇒ e^(−x) = 2 ⇒ −x = ln 2 ⇒ x = −ln 2 ≈ −0.693 - Shape: increasing curve, crossing x-axis at (−ln 2, 0), y-axis at (0, 1), approaching y = 2 from below as x → ∞ [B1 for asymptote, B1 for intercepts, B1 for correct shape — 3 marks]

20. C: y = 3 − 2e^(−x)

(a) As x → ∞, e^(−x) → 0, so y → 3. Asymptote: y = 3 [B1 — 1 mark]

(b) Crosses x-axis when y = 0: 3 − 2e^(−x) = 0 ⇒ 2e^(−x) = 3 ⇒ e^(−x) = 1.5 ⇒ −x = ln 1.5 ⇒ x = −ln 1.5 Coordinates: (−ln 1.5, 0) [M1 for setting up, A1 for exact form — 2 marks]

(c) Sketch: - Horizontal asymptote: y = 3 - y-intercept: (0, 3 − 2) = (0, 1) - x-intercept: (−ln 1.5, 0) ≈ (−0.405, 0) - Shape: increasing curve, crossing axes as above, approaching y = 3 from below as x → ∞, and y → −∞ as x → −∞ [B1 for asymptote and intercepts, B1 for correct shape — 2 marks]

END OF ANSWER KEY

Marking notes: Award method marks (M1) for correct approach even if final answer contains arithmetic errors. Award accuracy marks (A1) only for fully correct answers. Where exact answers are required, decimal approximations should not be accepted unless the question explicitly permits them. For graph sketches, look for correct shape, labelled asymptotes, and correctly plotted key points.