OCR A Level Maths A Pure Mathematics June 2022 question paper H240/01

Give your answer correct to 3 significant figures.

State whether the value from part (a) is an under-estimate or an over-estimate, giving a

reason for your answer.

Explain how the trapezium rule could be used to obtain a more accurate estimate.

Given that a and b are real numbers, find a counterexample to disprove the statement that,

A student writes the statement that sin x ° = 0.5 ⟺ x ° = 30 °.

Explain why this statement is incorrect.

Write a corrected version of this statement.

Prove that the sum of four consecutive multiples of 4 is always a multiple of 8.

The diagram shows the curves y = x² - 2x + 1 and y =-x² + 6x - 5. This diagram is repeated in the Printed Answer Booklet.

On the diagram in the Printed Answer Booklet, draw the line y = 2 x - 2.

Write 2x² + 6x + 7 in the form p(x + q)² + r, where p, q and r are constants.

State the coordinates of the minimum point on the graph of y = 2x² + 6x + 7.

Hence deduce

• the minimum value of 2 tan²θ + 6 tan θ + 7,

• the smallest positive value of θ, in degrees, for which the minimum value occurs.

The graph of y = 2^x can be transformed to the graph of y = 2^x+4 either by a translation or by a stretch.

Give full details of the translation.

Give full details of the stretch.

In this question you must show detailed reasoning.

Solve the equation log₂(8x) = 1 - log₂(1 - x).

Find the first four terms in the expansion of (3 + 2x)⁵ in ascending powers of x.

Hence determine the coefficient of y³ in the expansion of (3 + 2y + 4y²)⁵.

A curve has equation 2x³ + 6xy - 3y² = 2.

Show that there are no points on this curve where the tangent is parallel to y = x.

Substance A is decaying exponentially such that its mass is m grams at time t minutes. Find the missing values of m and t in the following table.

Substance B is also decaying exponentially, according to the model m = 160e^-0.055t, where m grams is its mass after t minutes.

Determine the value of t for which the mass of substance B is half of its original mass.

Determine the rate of decay of substance B when t = 15.

State whether substance A or substance B is decaying at a faster rate, giving a reason for your

answer.

Show that θ = 1.25 sin θ.

The equation θ = 1.25 sin θ has only one root for θ > 0.

This root can be found by using the iterative formula θ_n+1 = 1.25 sin θ_n with a starting value of θ₁ = 0.5.

• Write down the values of θ₂, θ₃ and θ₄.

• Hence find the value of this root correct to 3 significant figures.

Draw a suitable diagram to show why using an iterative process with the formula

θ_n+1 = sin^-1(0.8θ_n) does not converge to the root found in (b).

Find the equation of this curve, giving your answer in the form e^3y = f(x).

Show that, when x = e², the y-coordinate of this curve can be written as

y = a + ln(be³ + c), where a, b and c are constants to be determined.

Show that the equation of the tangent at P can be written as y =-2p²x + 4p.

The tangent to this curve at P crosses the x-axis at the point A and the normal to this curve at P crosses the x-axis at the point B.

Show that the ratio PA: PB is 1:2p².