WMA11 Pure Mathematics 1 (P1): Oct/Nov 2025 Paper 01

The curve C has equation

(a)Find dydx writing the answer in simplest form.[3]

The point P  1 4 , 5 2  lies on C.  

(b)Find the equation of the tangent to C at P, writing your answer in the form y = mx + c where m and c are integers.[3]

Find

writing the answer in simplest form.[5]

The sides of a triangle are in the ratio 2 : 4 : 5

(a)Find the size of the largest angle of the triangle, in degrees, to one decimal place.[2]

Given that the area of the triangle is 140 cm²

(b)find the length of the shortest side, in cm, to one decimal place.[3]

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

(i)Using the laws of indices, solve

[3]

(ii)Find the real roots of the equation

[4]

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

(a)Express 2x²– 16x + 50 in the form

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

Figure 1 shows a sketch of the curve C with equation y = 2x²– 16x + 50

Given that in Figure 1

• M is the minimum point on C • line l passes through the origin and intersects C at the points M and P

(b)find, using algebra and showing your working, the x coordinate of P.[5]

The region R is shown shaded in Figure 1.

(c)Use inequalities to fully define R.[2]

(a)Sketch the graph of the curve C with equation

where k is a positive constant.

On your sketch show

• the coordinates of any points where C cuts the coordinate axes • the equation of the vertical asymptote to C[4]

The straight line l has equation

Given that there is at least one point of intersection between l and C,

(b)find the range of possible values of k.[5]

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

Figure 2 shows the plan view of a design for a swimming pool.

The design consists of a sector POS of a circle centre O joined to a major sector QORTQ of a different circle, also with centre O.

Given that

• angle POS is 1.65 radians • the area of sector POS is 30 m² • PQ = 2.8 m

(a)show that, to 3 significant figures, OQ = 8.83 m[3]

(b)Find the total surface area of the swimming pool in m² to the nearest integer.[3]

(c)Find the total perimeter of the swimming pool in metres to 2 significant figures.[3]

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

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

Given that

• f(x)  2x   k, where k is a constant

• the equation of the tangent to the curve at x = 2 is y  5x  3 2

(a)find the exact value of k.[2]

(b)Find an equation of the normal to the curve at x = 2[2]

(c)Find f(x), writing your answer in simplest form.[4]

Figure 3 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]

The point P lies on y = f(x) and is shown in Figure 3.

(b)State the coordinates of the point to which P is transformed when the graph of y = f(x) is transformed to the graph with equation

  

(i)y  f x     6 [2]

(ii)y   12 f (x)[2]

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

Figure 4 shows a sketch of the curve C with equation

The line l is the normal to C at the point P where x = 0

The line l also intersects C at points Q and R as shown in Figure 4.

(a)Find, using algebra, the x coordinate of point Q.[6]

The point T lies on C.

Given that

• the tangent to C at T is parallel to l • the x coordinate of T is positive

(b)find, using algebra, the exact x coordinate of T.[4]