WMA12 Pure Mathematics 2 (P2): Oct/Nov 2021 Paper 01

The first three terms, in ascending powers of x, of the binomial expansion of (1 + kx)¹⁶ are

where k and p are constants.

(a)Find, in simplest form,

(i)the value of k

(ii)the value of p[3]

Using the value of k found in part (a),

(b)find the term in x² in the expansion of g (x).[3]

A sequence is defined by

where k is a positive constant.

(a)Find, in terms of k, an expression for u₃[2]

Given that u = 117

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

Figure 1 shows a sketch of part of the curve with equation y = log₁₀ x

The region R, shown shaded in Figure 1, is bounded by the curve, the line with equation x = 2, the x-axis and the line with equation x = 14

Using the trapezium rule with four strips of equal width,

(a)show that the area of R is approximately 10.10[3]

(b)Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of R.[1]

(c)Using the answer to part (a) and making your method clear, estimate the value of

(i)log₁₀ x dx ∫₂

(ii)log₁₀ 100x³ dx ∫₂[4]

(a)State the value of the remainder when f (x) is divided by (2x − 3)[1]

(b)Use the factor theorem to show that (x − 3) is a factor of f (x)[2]

(c)Hence,

(i)factorise f (x)

(ii)show that the equation f (x) = 0 has only one real root.[5]

A company that owned a silver mine

• extracted 480 tonnes of silver from the mine in year 1 • extracted 465 tonnes of silver from the mine in year 2 • extracted 450 tonnes of silver from the mine in year 3

and so on, forming an arithmetic sequence.

(a)Find the mass of silver extracted in year 14[2]

After a total of 7770 tonnes of silver was extracted, the company stopped mining.

Given that this occurred at the end of year N,

(b)show that

[3]

(c)Hence, state the value of N.[1]

(i)The circle C₁ has equation

(a)Find the coordinates of the centre of C₁[2]

(b)State the possible range in values for k.[2]

(ii)The point P( p, 0) , the point Q(−2, 10) and the point R(8, −14) lie on a different circle, C₂

Given that

• p is a positive constant • QR is a diameter of C₂

find the exact value of p.[4]

(i)A geometric sequence has first term 4 and common ratio 6 Given that the n^th term is greater than 10¹⁰⁰, find the minimum possible value of n.[3]

(ii)A different geometric sequence has first term a and common ratio r.

Given that

• the second term of the sequence is −6 • the sum to infinity of the series is 25

(a)show that

[3]

(b)Write down the solutions of

[1]

Hence,

(c)state the value of r, giving a reason for your answer,[1]

(d)find the sum of the first 4 terms of the series.[2]

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

Solutions relying entirely on calculator technology are not acceptable.

Figure 2 shows a sketch of part of the curve C with equation

The point M is the maximum turning point of C and is shown in Figure 2.

Given that the x coordinate of M is 2

(a)show that k = 28[3]

(b)Determine the range of values of x for which y is increasing.[2]

The line l passes through M and is parallel to the x-axis.

The region R, shown shaded in Figure 2, is bounded by the curve C, the line l and the y-axis.

(c)Find, by algebraic integration, the exact area of R.[5]

(a)Prove that for all positive values of x and y,

[3]

(b)Prove by counter-example that this inequality does not hold when x and y are both negative.[1]

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

Solutions relying entirely on calculator technology are not acceptable.

(i)Solve, for − π2 < x < π2

[5]

(ii)Solve, for 0 < θ < 360°

giving your answers, as appropriate, to one decimal place.[5]