A Level Maths Paper 1

Question 1 [2 marks]

Proof and Algebraic Methods

Without using a calculator, show that sqrt(45) + sqrt(20) can be written as k*sqrt(5), stating the value of k.

Question 2 [3 marks]

Integration

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

Question 3 [3 marks]

Forces and Newton's Laws

A box of mass 4 kg rests on a horizontal surface.

Find the normal reaction force on the box. Use g = 9.8 m/s^2.

Question 4 [3 marks]

The Normal Distribution and Hypothesis Testing

The random variable X ~ N(75, 12^2).

Find P(65 < X < 90).

Question 5 [3 marks]

Vectors

Points A and B have position vectors a = 3i + 2j and b = 9i - 4j relative to a fixed origin O.

Find the position vector of the point P that divides AB such that AP:PB = 1:2.

Question 6 [4 marks]

Differentiation

Without using a calculator, find dy/dx for y = sin(3x) + cos(2x), and find the exact gradient of the curve at x = pi/6.

Question 7 [4 marks]

Moments

A light rod AB has length 6 m. Weights of 50 N and 70 N are attached at A and B respectively.

Find the distance from A at which the rod must be pivoted for it to balance horizontally.

Question 8 [4 marks]

Exponentials and Logarithms

Without using a calculator, solve the equation e^(2x) - 4e^x + 3 = 0, giving both values of x as exact values.

Question 9 [4 marks]

Trigonometry

Without using a calculator, prove the identity sec^2(x) - sin^2(x) sec^2(x) = 1.

Question 10 [5 marks]

Kinematics

A particle P moves along the x-axis such that its displacement from a fixed point O at time t seconds is given by x = 2t^3 - 15t^2 + 24t + 5 for t >= 0.

Find the velocity and the acceleration of P when t = 3, and determine whether P is accelerating or decelerating at this instant.

Question 11 [5 marks]

Sequences, Series and the Binomial Expansion

The second term of a geometric sequence is 6 and the fifth term is 162.

Find the first term and the common ratio, and hence find the sum of the first 6 terms.

Question 12 [4 marks]

Coordinate Geometry

Using coordinates, find the area of the triangle with vertices A(1, 2), B(7, 4) and C(3, 8).

Question 13 [4 marks]

Sampling and Data Presentation

A sample of 12 pairs of data gives a product moment correlation coefficient of r = 0.62. The critical value for a sample of size 12 at the 5% significance level (one-tail) is 0.497.

Test, at the 5% significance level, whether this provides evidence of positive correlation between the variables in the population.

Question 14 [6 marks]

Probability and the Binomial Distribution

A random variable X ~ B(n, 0.25).

Given that P(X = 2) = P(X = 3), find the value of n.

Question 15 [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.