WMA12 Pure Mathematics 2 (P2): May/June 2022 Paper 01

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

Give each coefficient as an integer.[4]

Figure 1 shows the graph of

where x is in radians.

The table below shows some values of x and y for this graph, with values of y given to 3 decimal places.

(a)Complete the table above, giving values of y to 3 decimal places.[2]

(b)Use the trapezium rule with all the y values in the completed table to find, to 2 decimal places, an estimate for

[3]

(c)Use your answer to part (b) to find an estimate for

[3]

(i)Show that the following statement is false:

[2]

(ii)Given that the points A(1, 0) , B(3, −10) and C(7, −6) lie on a circle, prove that AB is a diameter of this circle.[5]

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

Given that a and b are positive constants, solve the simultaneous equations

[6]

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

Solutions relying entirely on calculator technology are not acceptable.

Solve, for −180° < θ  180° , the equation

[6]

In a geometric sequence u₁ , u₂ , u₃ , …

• the common ratio is r • u₂+ u₃ = 6 • u₄= 8

(a)Show that r satisfies

[3]

Given that the geometric sequence has a sum to infinity,

(b)find u₁[3]

(c)find S_∞[2]

where A and B are constants.

Given that

• (x + 2) is a factor of f(x)

• f (x) dx = 176 ∫₃

find the value of A and the value of B.[7]

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

Solutions relying entirely on calculator technology are not acceptable.

A curve has equation

(a)Find dydx[3]

(b)Hence find the coordinates of the stationary points of the curve.[5]

A scientist is using carbon‑14 dating to determine the age of some wooden items.

The equation for carbon‑14 dating an item is given by

• N grams is the amount of carbon‑14 currently present in the item • k grams was the initial amount of carbon‑14 present in the item • t is the number of years since the item was made • λ is a constant, with 0 < λ < 1

(a)Sketch the graph of N against t for k = 1[2]

Given that it takes 5700 years for the amount of carbon‑14 to reduce to half its initial value,

(b)show that the value of the constant λ is 0.999878 to 6 decimal places.[2]

Given that Item A

• is known to have had 15 grams of carbon‑14 present initially • is thought to be 3250 years old

(c)calculate, to 3 significant figures, how much carbon‑14 the equation predicts is currently in Item A.[2]

Item B is known to have initially had 25 grams of carbon‑14 present, but only 18 grams now remain.

(d)Use algebra to calculate the age of Item B to the nearest 100 years.[3]

The circle C has centre X (3, 5) and radius r

The line l has equation y = 2x + k , where k is a constant.

(a)Show that l and C intersect when

[3]

Given that l is a tangent to C,

(b)show that 5r²= (k + p)² , where p is a constant to be found.[3]

• cuts the y‑axis at the point A • touches the circle C at the point B

as shown in Figure 2.

Given that AB = 2r

(c)find the value of k[6]