Question
Bank

Unit 1 › Functions and Graphs › Polynomials

If f(x) = 3x² − 2x + 5, what is f′(x)?

Unit 1 › Trigonometric Functions › Exact values

The exact value of cos(60°) is:

Unit 2 › Sequences and Series › Geometric sequences

A geometric sequence has first term 3 and common ratio 2. What is the 6th term?

Unit 2 › Exponential Functions › Logarithms

The value of log₂(64) is:

Unit 1 › Trigonometric Functions › Period and amplitude

The period of the function y = 3sin(2x) is:

Unit 3 › Further Differentiation › Chain rule

The derivative of y = (3x + 1)⁵ is:

Unit 3 › Integration › Definite integrals

The value of ∫₁³ (2x − 1) dx is:

Unit 1 › Counting and Probability › Conditional probability

A bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?

Unit 2 › Introduction to Differential Calculus › Stationary points

Find the coordinates of the stationary point of f(x) = x² − 8x + 15. Determine whether it is a local maximum or minimum, justifying your answer.

Unit 2 › Exponential Functions › Exponential equations

Solve the equation 3^(2x − 1) = 27 for x.

Unit 3 › Integration › Area under a curve

Find the exact area enclosed between the curve y = x² − x and the x-axis.

Unit 2 › Sequences and Series › Arithmetic series

An arithmetic sequence has first term 7 and common difference 4. Find the sum of the first 20 terms.

Unit 3 › Further Differentiation › Product rule

Use the product rule to differentiate y = x²eˣ. Hence find the equation of the tangent to the curve at x = 1.

Unit 4 › Logarithmic Function › Differentiating logarithms

Differentiate y = ln(x² + 3x) and find the gradient of the curve at x = 1.

Unit 1 › Counting and Probability › Binomial probability

A biased coin has P(heads) = 0.6. The coin is tossed 8 times. Find the probability of obtaining exactly 5 heads, correct to 4 decimal places.

Unit 3 › Further Differentiation and Applications › Optimisation

A rectangular pen is built against an existing wall using only three sides of fencing. The total fencing available is 120 m. Let x be the length (in metres) of each side perpendicular to the wall. (a) Show that the enclosed area is A = x(120 − 2x). (b) Find the value of x that maximises the area. (c) Calculate the maximum area. (d) Explain why x must satisfy 0 < x < 60.

Unit 2 › Exponential Functions › Growth and decay

The number of bacteria in a culture after t hours is modelled by N = 200e^(0.3t). (a) State the initial number of bacteria. (b) Find the population after 6 hours, to the nearest bacterium. (c) Find the rate of growth at t = 6 hours. (d) Find the time for the population to reach 5000 bacteria, correct to 2 decimal places.

Unit 3 › Integration › Kinematics

A particle moves in a straight line with velocity v(t) = 3t² − 12t + 9 m/s for t ≥ 0. (a) Find the acceleration function a(t). (b) Find all times when the particle is momentarily at rest. (c) Find the displacement of the particle from t = 0 to t = 4. (d) Find the total distance travelled from t = 0 to t = 4.

Unit 4 › Continuous Random Variables › Normal distribution

The heights of adults at a gym are normally distributed with mean 168 cm and standard deviation 10 cm. (a) Find P(X > 180). (b) Find P(155 < X < 178). (c) The gym has 400 members. How many are expected to be shorter than 160 cm? (d) The tallest 5% qualify for a high-reach programme. Find the minimum qualifying height.

Unit 1 › Functions and Graphs › Composite functions

Let f(x) = x + 3 and g(x) = x² − 1. (a) Find f(g(x)) and g(f(x)). (b) Find the value(s) of x for which f(g(x)) = g(f(x)). (c) Solve f(g(x)) = 3. (d) State the domain and range of g(f(x)).