A Level Maths Paper 3

Question 1 [2 marks]

Sequences, Series and the Binomial Expansion

An arithmetic sequence has first term 5 and common difference 3.

Find the 20th term.

Question 2 [2 marks]

Coordinate Geometry

Find the equation of the straight line that passes through the points (1, 4) and (5, -8), giving your answer in the form y = mx + c.

Question 3 [3 marks]

Integration

Evaluate the definite integral of (3x^2 - 2x + 1) with respect to x between x = -1 and x = 2.

Question 4 [3 marks]

Vectors

Find the magnitude of the vector v = 5i - 12j, and find a unit vector in the same direction as v.

Question 5 [4 marks]

Differentiation

Using differentiation from first principles, show that the derivative of x^2 is 2x.

Question 6 [4 marks]

Exponentials and Logarithms

Solve the equation 5^(2x - 1) = 30, giving your answer to 3 significant figures.

Question 7 [4 marks]

Proof and Algebraic Methods

Without using a calculator, find the quotient and remainder when 2x^3 - 3x^2 + 4x - 1 is divided by (x - 1).

Question 8 [5 marks]

Forces and Newton's Laws

A block of mass 6 kg lies on a rough horizontal surface. The coefficient of friction between the block and the surface is 0.25. A horizontal force of 20 N is applied to the block.

Find the acceleration of the block. Use g = 9.8 m/s^2.

Question 9 [5 marks]

Moments

A non-uniform plank AB has length 5 m and weight 120 N, and rests horizontally on two supports at A and B. When a block of weight 60 N is placed at a point 2 m from A, the reaction at A is 100 N.

Find the reaction at B in this situation, and find the distance of the centre of mass of the plank from A.

Question 10 [5 marks]

Trigonometry

Express 3 sin(x) + 4 cos(x) in the form R sin(x + alpha), where R > 0 and 0 < alpha < 90 deg, giving the value of alpha to 1 decimal place.

Hence state the maximum value of 3 sin(x) + 4 cos(x), and the smallest positive value of x, in degrees, at which it occurs.

Question 11 [5 marks]

Probability and the Binomial Distribution

Two fair six-sided dice are thrown, and the sum of the two scores is recorded.

Find the probability that the sum is a prime number, given that at least one die shows a 5.

Question 12 [6 marks]

Sampling and Data Presentation

A college has 900 students: 340 study Sciences, 260 study Humanities and 300 study Arts. A stratified sample of 45 students is to be selected by subject, and within Sciences, systematic sampling is then used to choose the required number from a numbered list of all 340 Science students.

Find the number of Science students needed in the sample, and find the sampling interval that should be used for the systematic sampling.

Question 13 [6 marks]

Kinematics

A particle P moves in a straight line such that its displacement s metres from a fixed point O at time t seconds (t >= 0) is given by s = t^3 - 7t^2 + 10t.

Find the two values of t (other than t = 0) at which P is at O, and find the velocity of P at the larger of these two times.

Question 14 [6 marks]

The Normal Distribution and Hypothesis Testing

A seed supplier claims that 40% of a particular type of seed germinate within one week of planting. A gardener believes the true proportion is lower and plants a random sample of 25 of the seeds.

Using the binomial distribution X ~ B(25, 0.4), find the critical region for a test of H0: p = 0.4 against H1: p < 0.4 at the 5% significance level, stating the actual significance level. Given that 6 of the 25 seeds germinate within a week, state the conclusion of the test.