WMA12 Pure Mathematics 2 (P2): January 2023 Paper 01

Figure 1 shows a sketch of part of the curve with equation y = f (x)

The table below shows some corresponding values of x and y for this curve.

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)Use your answer to part (a) to estimate

(i)f (x) − 2 dx ( ) ∫₋₁

(ii)f (x − 2) dx ∫₁[3]

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

Solutions based entirely on calculator technology are not acceptable.

A brick is in the shape of a cuboid with width x cm, length 3x cm and height h cm, as shown in Figure 2.

The volume of the brick is 972 cm³

(a)Show that the surface area of the brick, S cm², is given by

[3]

(b)Find dSdx[1]

(c)Hence find the value of x for which S is stationary.[2]

(d)Find d Sdx₂ and hence show that the value of x found in part (c) gives the minimum

value of S.[2]

(e)Hence find the minimum surface area of the brick.[1]

(a)Find the first 4 terms, in ascending powers of x, of the binomial expansion of f (x). Give each term in simplest form.[4]

Given that, in the binomial expansion of f (x), the coefficients of x, x² and x³ are the first 3 terms of an arithmetic progression,

(b)find, using algebra, the possible values of k.

(Solutions relying entirely on calculator technology are not acceptable.)[3]

(i)Using the laws of logarithms, solve

[3]

(ii)Given that

find y in terms of a, giving your answer in simplest form.[3]

where p and q are constants and p > 0

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

(a)show that

[2]

Given also that when f (x) is divided by (x + p) the remainder is 9

(b)show that

[2]

(c)Hence find the value of p and the value of q.[3]

(d)Hence find a quadratic expression g(x) such that

[2]

The circle C has equation

(a)Find

(i)the coordinates of the centre of C,

(ii)the exact radius of C.[3]

The point P lies on C.

Given that the tangent to C at P has equation x + 2y + 10 = 0

(b)find the coordinates of P[4]

(c)Find the equation of the normal to C at P, giving your answer in the form y = mx + c where m and c are integers to be found.[3]

A geometric sequence has first term a and common ratio r, where r > 0

Given that

• the 3rd term is 20

• the 5th term is 12.8

(a)show that r = 0.8[1]

(b)Hence find the value of a.[2]

Given that the sum of the first n terms of this sequence is greater than 156

(c)find the smallest possible value of n.

[4]

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

Solutions based entirely on calculator technology are not acceptable.

(i)Solve, for – π2 < x < π, the equation

giving your answers, in radians, to 2 decimal places.[4]

(ii)Solve, for 0 < θ < 360°, the equation

giving your answers, in degrees, to one decimal place.[5]

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

Solutions based entirely on calculator technology are not acceptable.

• the curve C with equation y = x²– 4x + 5

• the line l with equation y = 2

The curve C intersects the y‑axis at the point D.

(a)Write down the coordinates of D.[1]

The curve C intersects the line l at the points E and F, as shown in Figure 3.

(b)Find the x coordinate of E and the x coordinate of F.[2]

Shown shaded in Figure 3 is

• the region R₁ which is bounded by C, l and the y‑axis

• the region R₂ which is bounded by C and the line segments EF and DF

area of R₁ Given that = k, where k is a constant, area of R₂

(c)use algebraic integration to find the exact value of k, giving your answer as a simplified fraction.[5]

A student was asked to prove by exhaustion that

The start of the student’s proof is shown in the box below.

Complete this proof.[4]