WMA12 Pure Mathematics 2 (P2): Oct/Nov 2022 Paper 01

Given that a, b and c are integers greater than 0 such that

• c = b + 2

• a + b + c = 10

Prove, by exhaustion, that the product of a, b and c is always even.

You may use the table below to illustrate your answer.[3]

A curve C has equation y = f(x) where

and k is a constant.

Given that when f(x) is divided by (4x – 5) the remainder is 24332

(a)show that k = 25[2]

(b)Find the first three terms, in ascending powers of x, of the binomial expansion of

giving each term in simplest form.[3]

Using the solution to part (b) and making your method clear,

(c)find the gradient of C at the point where x = 0[2]

A sequence a₁, a₂, a₃, … is defined by

Find the exact values of

(a)(i)a₁

(ii)a₂

(iii)a₃[3]

(b)Hence find the exact value of

You must make your method clear.[4]

The weight of a baby mammal is monitored over a 16‑month period.

The weight of the mammal, w kg, is given by

where t is the age of the mammal in months and a is a constant.

Given that the weight of the mammal was 10 kg when t = 3

(a)show that a = 1.072 correct to 3 decimal places.[3]

Using a = 1.072

(b)find an equation for t in terms of w[3]

(c)find the value of t when w = 15, giving your answer to 3 significant figures.[2]

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

(a)Show that the equation

can be written as

[3]

(b)Hence solve for – π2  x  π2

[5]

The curve C₁ has equation y = f(x).

A table of values of x and y for y = f(x) is shown below, with the y values rounded to 4 decimal places where appropriate.

(a)Use the trapezium rule with all the values of y in the table to find an approximation for

giving your answer to 3 decimal places.[3]

Figure 1 Figure 2

The region R, shown shaded in Figure 1, is bounded by

• the curve C₁

• the curve C₂ with equation y = 2 – 14 x²

• the line with equation x = 2

• the y‑axis

The region R forms part of the design for a logo shown in Figure 2.

The design consists of the shaded region R inside a rectangle of width 2 and height 3

Using calculus and the answer to part (a),

(b)calculate an estimate for the percentage of the logo which is shaded.[4]

The curve C has equation

(a)Write the equation of C in the form

where a, b, c and d are fully simplified constants.[3]

The curve C has three turning points.

Using calculus,

(b)show that the x coordinates of the three turning points satisfy the equation

[3]

Given that the x coordinate of one of the turning points is 1

(c)find, using algebra, the exact x coordinates of the other two turning points.

[3]

A geometric sequence has first term a and common ratio r

Given that S_∞ = 3a

(a)show that r = 23[2]

Given also that

where u_k is the k^th term of this sequence,

(b)find the value of S₁₀ giving your answer to one decimal place.[5]

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

• the curve C₁ with equation y = x³– 5x²+ 3x + 14

• the circle C₂ with centre T

The point T is the minimum turning point of C₁

Using Figure 3 and calculus,

(a)find the coordinates of T[3]

The curve C₁ intersects the circle C₂ at the point A with x coordinate 2

(b)Find an equation of the circle C₂[3]

The line l shown in Figure 3, is the tangent to circle C₂ at A

(c)Show that an equation of l is

[3]

The region R, shown shaded in Figure 3, is bounded by C₁, l and the y‑axis.

(d)Find the exact area of R.[3]

Given a = log2 3

(i)write, in simplest form, in terms of a,

(a)log2 9

(b)log2 3 16 [3]

(ii)Solve

giving your answer, in simplest form, in terms of a.[4]