WMA11 Pure Mathematics 1 (P1): January 2026 Paper 01

Find, in simplest form,

dy

(a) dx[3]

(b)y dx ∫[3]

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

(a)Given

[2]

(b)Hence solve the simultaneous equations

writing the answers in the form p + q 3 , where p and q are integers to be found.[4]

(i)Find the range of values of z for which

giving the answer in simplest form.[2]

(ii)

Figure 1 shows a plot of

• the line with equation y = x + 5

• the curve with equation y = x (x – 3)

On the opposite page there is a copy of Figure 1 labelled Diagram 1. On Diagram 1 represent the following inequality graphically.

[3]

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

The point A (–1, 8) and the point B (7, 2) lie on the line L₁

(a)Find an equation for L₁ in the form ax + by + c = 0 , where a, b and c are integers.[3]

The line L₂

• passes through A and is perpendicular to L₁

• cuts the x-axis at the point C

(b)Find the coordinates of C.[3]

(c)Hence find the area of triangle ABC.[3]

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

Figure 2 shows the logo for the tourist board of an island.

The logo consists of a sector BODCB of a circle centre O joined to a triangle AOD.

Given that

• ABO is a straight line

• OD = 3 cm

• AD = 8 cm

• angle AOD = 2.5 radians

(a)show that angle ODA is 0.415 radians to 3 significant figures.[3]

(b)Find the total area of the logo, in cm², to one decimal place.[3]

(c)Find the perimeter of the logo, ABCDA, in cm, to one decimal place.[3]

Figure 3 shows a sketch of the curve C with equation y = f (x) where f (x) is a cubic function in x.

The curve C

• cuts the x-axis at (–2, 0) and cuts the y-axis at (0, 10)

• touches the x-axis at (5, 0)

as shown in Figure 3.

(a)Deduce the roots of the equation

(i)f x = 0

(ii)f (x – 3) = 0[2]

(b)Find an expression for f (x). You should leave your answer in factorised form.[3]

The curve C intersects the straight line y = 10 (x + 2) at exactly three points.

(c)Use algebra to find the exact x coordinates of the three points of intersection.

[4]

The curve C has equation

(a)Show that dydx can be written in the form

where p, q and r are constants to be found.[5]

(b)Hence state

(i)the minimum value of dydx

(ii)the value of x at which this minimum value occurs.[2]

Given that S is the point on C at which the gradient is a minimum,

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

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

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

A point P lies on C.

Given that

• f ′ (x) = 4x +₂ – 4 x

• the equation of the tangent to C at P is y = 10x – 6 3

(a)(i)verify that 3 is a possible x coordinate of P,

(ii)find, using algebra, the other possible x coordinate of P.[6]

Given that the x coordinate of P is 3

(b)find an equation of the normal to C at P.[2]

(c)Find f (x) , writing your answer in simplest form. You must show each stage of your working.[4]

Figure 4 shows a sketch of part of the graph of the trigonometric function with equation y = f (x).

(a)Write down an expression for f (x).[2]

Copies of Figure 4 (labelled Diagram 1 and Diagram 2) can be found on the following pages.

(b)(i)On Diagram 1 sketch a graph of the curve with equation

(ii)Hence find the number of solutions of the equation

[3]

(c)(i)On Diagram 2 sketch a graph of the curve with equation

(ii)Hence find the number of solutions of the equation

giving a reason for your answer.[4]

(a)

Only use this copy if you need to redraw your graph.

Only use this copy if you need to redraw your graph.