WMA11 Pure Mathematics 1 (P1): May/June 2025 Paper 01

Given that

express each of the following in the form kxⁿ where k and n are fully simplified constants.

(a)p^1/2[1]

(b)( pq)^–1[2]

(c)pq²[2]

The curve C has equation

(a)Find dydx writing your answer in simplest form.[2]

The point P lies on C.

Given that

• the x coordinate of P is 2^k where k is a constant • the gradient of C at the point P is 16

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

Find

writing your answer in simplest form.[5]

Figure 1 shows a sketch of part of the curve with equation

where x is measured in degrees.

The points P and Q lie on the curve and are shown in Figure 1.

(a)State the coordinates of P.[1]

(b)State the coordinates of Q.[2]

(c)State the number of solutions of the equation

(i)4 cos x = 3 in the interval 0 < x  18 000°

(ii)5 + 4 cos x = 1 in the interval –720° < x  720°

(iii)4 cos x – 3 = 1 in the interval –1080°  x  1080°[3]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

The line l₁ has equation

The line l₂ passes through the origin and is perpendicular to l₁

(a)Find an equation for l₂[2]

The lines l₁ and l₂ intersect at the point P.

(b)Use algebra to find the coordinates of P.[3]

(c)Hence find the shortest distance from l₁ to the origin.

Write your answer as a fully simplified surd.[2]

The shaded area in Figure 2 shows the plan view of a helicopter landing pad.

The area consists of the major sector AOB of a circle centre O joined to a triangle AOC.

Given that

• AO = OB = 15 m • BC = 2 m • CBO is a straight line • angle ACO = 0.6 radians

(a)show that angle COA is 1.847 radians to 3 decimal places.[3]

(b)Find the total area of the helicopter landing pad. Give your answer in m² to 3 significant figures.[3]

(c)Find the perimeter of the helicopter landing pad. Give your answer in metres to 3 significant figures.[3]

Figure 3 shows a sketch of part of the curve C₁

Given that C₁

• has equation y = f (x) where f (x) is a cubic function • touches the x-axis at the origin and cuts the x-axis at x = 4 • passes through the point (10, 120)

(a)find f (x)[3]

The curve C₂ has equation y = 1.2x (8 – x)

On the following page there is a copy of Figure 3 called Diagram 1.

(b)On Diagram 1 sketch a graph of the curve C₂[2]

(c)Use algebra to find the coordinates of the points where C₁ and C₂ intersect. Show each stage of your working.[5]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

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

The point P (4, 12) lies on the curve.

Given that

• f′ (x) = 3 x + kx² where k is a constant

• the equation of the tangent to the curve at P has equation y = 10x + c where c is a constant

(a)(i)show that k = 14

(ii)find the value of c[4]

(b)Hence find the value of f′′ (x) at P.[3]

(c)Find f (x).[4]

y

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Figure 4 shows a sketch of

• the graph C₁ with equation y = 2x

• the graph C₂ with equation y = 12 – x

(a)Describe fully the single transformation that would transform

(i)the graph with equation y = x onto C₁

(ii)the graph with equation y = – x onto C₂[4]

The graphs C₁ and C₂ meet at the point P, as shown in Figure 4.

(b)(i)Show that the x coordinate of P is a solution of

(ii)Hence find, in simplest form, the exact coordinates of P.[6]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

(a)Find the exact solutions to the given equation for k = 4.5[3]

(b)Find the set of possible values of k for which the given equation has no real roots.[4]