WMA11 Pure Mathematics 1 (P1): May/June 2021 Paper 01

The curve C has equation

(a)Find dydx , giving your answer in simplest form.[4]

The point P (4, 3) lies on C.

(b)Find the equation of the normal to C at the point P. Write your answer in the form ax + by + c = 0, where a, b and c are integers to be found.[4]

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

where a is a positive constant.

Given f(–1) = 32

(a)(i)show that the only possible value for a is 3

(ii)Using a = 3 solve the equation

[5]

(b)Hence find all real solutions of

(i)3y + 26y^2/3 – 9y^1/3 = 0

(ii)3(9³ ) + 26(9² ) – 9(9 ) = 0[5]

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

Figure 1 shows the plan view of a flower bed. The flowerbed is in the shape of a triangle ABC with

● AB = p metres

● AC = q metres

● BC = 2 2 metres

● angle BAC = 60°

(a)Show that

[2]

Given that side AC is 2 metres longer than side AB, use algebra to find

(b)(i)the exact value of p,

(ii)the exact value of q.[5]

Using the answers to part (b),

(c)calculate the exact area of the flower bed.[2]

Find

writing each term in simplest form.[6]

The share value of two companies, company A and company B, has been monitored over a 15-year period.

The share value P_A of company A, in millions of pounds, is modelled by the equation

where t is the number of years after monitoring began.

The share value P_B of company B, in millions of pounds, is modelled by the equation

where t is the number of years after monitoring began.

Figure 2 shows a graph of both models.

Use the equations of one or both models to answer parts (a) to (d).

(a)Find the difference between the share value of company A and the share value of company B at the point monitoring began.[2]

(b)State the maximum share value of company A during the 15-year period.[1]

(c)Find, using algebra and showing your working, the times during this 15-year period when the share value of company A was greater than the share value of company B.[4]

(d)Explain why the model for company A should not be used to predict its share value when t = 20[1]

The curve C has equation y = f(x), x > 0

Given that

● C passes through the point P (8, 2)

● fʹ (x) = + 3 − 2 x

(a)find the equation of the tangent to C at P. Write your answer in the form y = mx + c, where m and c are constants to be found.[3]

(b)Find, in simplest form, f(x).[5]

The line l₁ has equation 4y + 3x = 48

The line l₁ cuts the y-axis at the point C, as shown in Figure 3.

(a)State the y coordinate of C.[1]

The point D (8, 6) lies on l₁

The line l₂ passes through D and is perpendicular to l₁

The line l₂ cuts the y-axis at the point E as shown in Figure 3.

(b)Show that the y coordinate of E is − 143[3]

A sector BCE of a circle with centre C is also shown in Figure 3.

Given that angle BCE is 1.8 radians,

(c)find the length of arc BE.[3]

The region CBED, shown shaded in Figure 3, consists of the sector BCE joined to the triangle CDE.

(d)Calculate the exact area of the region CBED.[3]

The curve C₁ has equation

(a)Write 3x²+ 6x + 9 in the form

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

The point P is the minimum point of C₁

(b)Deduce the coordinates of P.[1]

A different curve C₂ has equation

where A, B, C and D are constants.

Given that C₂

● passes through P

● intersects the x-axis at –4, –2 and 3

(c)find, making your method clear, the values of A, B, C and D.[5]

Figure 4 shows a sketch of the curve with equation

The line l, shown in Figure 4, is an asymptote to y = tan x

(a)State an equation for l.[1]

A copy of Figure 4, labelled Diagram 1, is shown on the next page.

(b)(i)On Diagram 1, sketch the curve with equation

stating the equation of the horizontal asymptote of this curve.

(ii)Hence, giving a reason, state the number of solutions of the equation

in the region –2π  x  2π[4]

(c)State the number of solutions of the equation tan x = 1x + 1 in the region

(i)0  x  40π

(ii)–10π  x  52π[2]