WMA12 Pure Mathematics 2 (P2): January 2022 Paper 01

The table below shows corresponding values of x and y for

The values of y are given to 3 decimal places.

Using the trapezium rule with all the values of y in the given table,

(a)obtain an estimate for

giving your answer to 2 decimal places.[3]

(b)Using your answer to part (a) and making your method clear, estimate

(i)^6−x 2 dx ∫₅

(ii)3 + 2^5−x dx ( ) ∫₅[4]

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve C has equation

(a)Find dydx , giving each term in simplest form.[2]

(b)Hence find the coordinates of the stationary point of C.[4]

² y

(c)Find ddx₂ and hence determine the nature of the stationary point of C.[2]

(a)Find the first 4 terms, in ascending powers of x, of the binomial expansion of

where k is a non‑zero constant. Give each term in simplest form.[4]

In the expansion of f(x), the constant term is 3 times the coefficient of x.

(b)Find the value of k.[3]

Using the laws of logarithms, solve

[5]

where A and B are integers.

Given that

• when f(x) is divided by (x – 1) the remainder is k

• when f(x) is divided by (x + 1) the remainder is –10k

• k is a constant

(a)show that

[3]

Given also that (3x – 2) is a factor of f(x),

(b)find the value of A and the value of B.[3]

(c)Hence find the quadratic expression g(x) such that

[2]

The points P(23, 14), Q(15, –30) and R(–7, –26) lie on the circle C, as shown in Figure 1.

(a)Show that angle PQR = 90°[2]

(b)Hence, or otherwise, find

(i)the centre of C,

(ii)the radius of C.[3]

Given that the point S lies on C such that the distance QS is greatest,

(c)find an equation of the tangent to C at S, giving your answer in the form ax + by + c = 0, where a, b and c are integers to be found.[3]

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

(i)Solve, for –90° < x < 90°, the equation

giving your answers to one decimal place.[4]

(ii)Solve, for 0 < θ < 2π, the equation

giving your answers to 3 significant figures.[4]

A metal post is repeatedly hit in order to drive it into the ground.

Given that

• on the 1st hit, the post is driven 100 mm into the ground

• on the 2nd hit, the post is driven an additional 98 mm into the ground

• on the 3rd hit, the post is driven an additional 96 mm into the ground

• the additional distances the post travels on each subsequent hit form an

(a)show that the post is driven an additional 62 mm into the ground with the 20th hit.[1]

(b)Find the total distance that the post has been driven into the ground after 20 hits.[2]

Given that for each subsequent hit after the 20th hit

• the additional distances the post travels form a geometric sequence with

• on the 22nd hit, the post is driven an additional 60 mm into the ground

(c)find the value of r, giving your answer to 3 decimal places.[2]

After a total of N hits, the post will have been driven more than 3 m into the ground.

(d)Find, showing all steps in your working, the smallest possible value of N.[4]

• the curve C with equation y = x – x²

• the line l with equation y = mx, where m is a constant and 0 < m < 1

The line and the curve intersect at the origin O and at the point P.

(a)Find, in terms of m, the coordinates of P.[2]

The region R₁, shown shaded in Figure 2, is bounded by C and l.

(b)Show that the area of R₁ is

[5]

The region R₂, also shown shaded in Figure 2, is bounded by C, the x‑axis and l.

Given that the area of R₁ is equal to the area of R₂

(c)find the exact value of m.[3]

(i)Prove by counter example that the statement

is not true.[1]

(ii)Use proof by exhaustion to prove that if n is an integer then

is always even.[4]