WMA12 Pure Mathematics 2 (P2): May/June 2019 Paper 01

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

Find the value of

(a)(i)a₂

(ii)a₁₀₇[2]

(b)(2a – 1) ∑_n[2]

A circle C has equation

(a)(i)the coordinates of the centre of C,

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

The point P(5, 4) lies on C.

(b)Find the equation of the tangent to C at P, writing your answer in the form y = mx + c, where m and c are constants to be found.[4]

(i)Use algebra to prove that for all real values of x

[3]

(ii)Show that the following statement is untrue.

[1]

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

[4]

(b)Given that x is small, so terms in x⁴ and higher powers of x may be ignored, show

where a and b are constants to be found.[3]

A company makes a particular type of watch.

The annual profit made by the company from sales of these watches is modelled by the equation

where P is the annual profit measured in thousands of pounds and £x is the selling price of the watch.

According to this model,

(a)find, using calculus, the maximum possible annual profit.[6]

(b)Justify, also using calculus, that the profit you have found is a maximum.[2]

Given (x – 3) is a factor of f(x),

(a)show that k = 9[2]

(b)Using algebra and showing each step of your working, fully factorise f(x).[4]

(c)Solve, for 0 - ș  360q, the equation

giving your answers to one decimal place.[2]

Kim starts working for a company.

Ɣ In year 1 her annual salary will be £16 200

Ɣ In year 10 her annual salary is predicted to be £31 500

Model A assumes that her annual salary will increase by the same amount each year.

(a)According to model A, determine Kim’s annual salary in year 2.[3]

Model B assumes that her annual salary will increase by the same percentage each year.

(b)According to model B, determine Kim’s annual salary in year 2. Give your answer to the nearest £10[3]

(c)Calculate, according to the two models, the difference between the total amounts that Kim is predicted to earn from year 1 to year 10 inclusive. Give your answer to the nearest £10[3]

(i)Find the exact solution of the equation

giving your answer in the form a + b log₂3, where a and b are constants to be found.[4]

(ii)Using the laws of logarithms, solve

[5]

(a)Show that the equation

can be written in the form

[4]

(b)Hence solve, for 0 - x  π2

giving your answers, where appropriate, to 2 decimal places.[4]

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

(a)find the range of values of x for which f(x) is increasing,[4]

(b)show that ⎛ 36⎝ + 2x − 13⎞⎠ dx = 0

[4]

The point P(2, 0) and the point Q(6, 0) lie on C.

Given ⎛ 36⎝ + 2x − 13⎞⎠ dx = −8 ⎜ ⎟

(c)(i)state the value of ⎛ 36⎝ + 2x − 13⎞⎠ dx

(ii)find the value of the constant k such that ⎛ 36⎝ + 2x + k⎞⎠ dx = 0

[3]