A Level Maths Short Paper A

Question 1 [2 marks]

Differentiation

Without using a calculator, differentiate y = 4x^3 - 7x^2 + 2 with respect to x.

Question 2 [3 marks]

Trigonometry

Without using a calculator, given that sin(theta) = 3/5 where theta is acute, find the exact values of cos(theta) and tan(theta).

Question 3 [3 marks]

Integration

Evaluate the definite integral of (3x^2 + 2) with respect to x between x = 1 and x = 4.

Question 4 [3 marks]

Coordinate Geometry

A circle has equation x^2 + y^2 - 6x + 4y - 12 = 0.

Find the centre and radius of the circle.

Question 5 [4 marks]

Sequences, Series and the Binomial Expansion

Without using a calculator, find the first three terms, in ascending powers of x, of the binomial expansion of (1 + 2x)^6, simplifying each coefficient.

Question 6 [4 marks]

Differentiation

A curve has equation y = (4x - 1)/(x + 2).

Find dy/dx using the quotient rule, and find the gradient of the curve at the point where x = 1.

Question 7 [5 marks]

Proof and Algebraic Methods

Without using a calculator, expand and simplify (3 + sqrt(5))(4 - sqrt(5)), giving your answer in the form a + b*sqrt(5), stating the values of a and b.

Question 8 [5 marks]

Exponentials and Logarithms

The mass, in grams, of a radioactive substance is modelled by M = 80 e^(-0.05t), where t is the time in days after measurements began.

Find the mass after 20 days, and find the time taken for the mass to halve, giving your answer to 1 decimal place.

Question 9 [5 marks]

Sequences, Series and the Binomial Expansion

In the expansion of (1 + 3x)(2 - x)^5 in ascending powers of x, find the coefficient of x^2.

Question 10 [6 marks]

Exponentials and Logarithms

A quantity y is modelled by y = A x^n, where A and n are constants.

Given that y = 20 when x = 2, and y = 312.5 when x = 5, find the values of A and n.