Examination-Type Questions

📋 A Note on Preparation and Past Papers

For the final stage of your revision, practicing with past examination papers is one of the most effective ways to prepare.

• You can legally access official Cambridge O Level Additional Mathematics (4037) past papers and specimen papers from reputable educational websites. Use these under fair usage guidelines for educational purposes.
• The questions in this section are original and designed to mirror the style, difficulty, and format of the Cambridge examination. They are NOT taken from official Cambridge papers.

✏️ Section A: Short Questions

1. Functions

(a) The function \(f\) is defined by \(f(x) = 3x - 2\) for \(x \in \mathbb{R}\). Find \(f^{-1}(x)\). [2]

(b) The function \(g\) is defined by \(g(x) = x^2 + 4\) for \(x \in \mathbb{R}, x \ge 0\). Explain why \(g^{-1}\) exists. [1]

(c) Find and simplify an expression for \(fg(x)\) where \(f(x) = 2x + 1\) and \(g(x) = \frac{x-3}{2}\). [2]

2. Quadratic Functions

The quadratic function is \(f(x) = 2x^2 - 8x + 11\).

(a) Express \(f(x)\) in the form \(a(x-h)^2 + k\). [3]

(b) Write down the minimum value of \(f(x)\). [1]

(c) The equation \(f(x) = k\) has no real roots. Find the range of values of \(k\). [2]

3. Simultaneous Equations

Solve the simultaneous equations:

Show all your working. [4]

4. Logarithmic and Exponential Functions

(a) Solve the equation \(\log_2 x + \log_2 (x-2) = 3\). [4]

(b) The population of a town is given by \(P(t) = 5000(1.02)^t\), where \(t\) is the number of years after 2020. Find the year when the population reaches 7000. [3]

5. Straight Line Graphs

The line \(L\) passes through the points \((2, 5)\) and \((8, 17)\).

(a) Find the gradient of \(L\). [1]

(b) Find the equation of the line \(M\) which is perpendicular to \(L\) and passes through the midpoint of the segment joining the two points. [4]

6. Trigonometry

(a) Given that \(\cos x = \frac{5}{13}\) and \(0^\circ < x < 90^\circ\), find the exact value of \(\tan x\). [2]

(b) Solve the equation \(3\tan x = 2\) for \(0^\circ \le x \le 360^\circ\). Give your answers correct to one decimal place. [3]

7. Permutations and Combinations

A committee of 4 people is to be chosen from 6 men and 5 women.

(a) How many different committees are possible? [2]

(b) How many committees contain exactly 2 women? [2]

(c) How many committees contain at least 2 men? [3]

8. Series

(a) The second term of a geometric progression is 12 and the fifth term is 96. Find the common ratio and first term. [3]

(b) Find the sum to infinity of the geometric progression \(20 + 5 + \frac{5}{4} + \cdots\). [2]

9. Differentiation and Integration

(a) Differentiate \(y = 2x^3 \ln x\) with respect to \(x\). [3]

(b) Evaluate \(\int_0^{\pi/2} \cos 2x \, dx\). [3]

📝 Section B: Longer, Exam-Style Questions

10. Equations, Inequalities and Graphs

(a) Solve the inequality \(|2x - 1| \le |x + 3|\). [4]

(b) The cubic polynomial \(p(x) = x^3 + ax^2 + bx - 20\) has a factor of \((x+1)\) and leaves a remainder of \(-16\) when divided by \((x-2)\).

(i) Find the value of \(a\) and of \(b\). [5]

(ii) Hence, factorise \(p(x)\) completely and solve the equation \(p(x) = 0\). [3]

11. Calculus (Max/Min Problem)

A rectangular storage box with a square base and an open top is to have a volume of \(32 \text{ cm}^3\).

(a) Show that the surface area, \(A\), of the box is given by \(A = x^2 + \frac{128}{x}\), where \(x\) is the side length of the square base. [3]

(b) Find, using calculus, the minimum surface area of the box. [4]

12. Coordinate Geometry of the Circle

The circle \(C\) has equation \(x^2 + y^2 - 6x + 4y - 12 = 0\).

(a) Find the centre and radius of \(C\). [3]

(b) Show that the line \(y = 2x - 1\) is a tangent to \(C\). [4]

(c) Find the point of contact of this tangent. [2]

13. Vectors

The position vectors of points A and B are \(\mathbf{a} = (2, 3)\) and \(\mathbf{b} = (8, 11)\).

(a) Find the vector \(\overrightarrow{AB}\). [1]

(b) Find the unit vector in the direction of \(\overrightarrow{AB}\). [2]

(c) Point C lies on AB such that AC:CB = 3:1. Find the position vector of C. [3]

14. Trigonometry (Identities)

Prove that:

(a) \(\sec^2 x - \tan^2 x = 1\) [2]

(b) \((\sin x + \cos x)^2 = 1 + \sin 2x\) [3]

15. Series (Binomial Theorem)

(a) Expand \((2x - 3)^4\) in ascending powers of \(x\). [4]

(b) Find the coefficient of \(x^3\) in the expansion of \((1 - 2x)^5\). [2]

📚 Past Paper Reference Guide

Use this table to find official past papers for targeted practice on specific topics.

Reference Key:

• Paper 1 (Non-Calculator): Questions from these papers test fundamental skills and algebraic manipulation.
• Paper 2 (Calculator): Questions from these papers often involve more complex problem-solving and require use of a calculator for numerical work.