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

The curve C, shown in Figure 1,

• has equation y = f (x) , x ∈ 

• cuts the x-axis at –1

• has a maximum turning point at P (2, 5)

• has a horizontal asymptote with equation y = 3

The curve C has no other turning points or asymptotes.

(a)Find the coordinates of the point to which P is transformed when the curve with equation y = f (x) is transformed to the curve with equation

(i)y = f (x) + 7

(ii)y = 3 f (x)[2]

Given that the line with equation y = k , where k is a constant, cuts or meets C exactly once,

(b)state the range of possible values of k.[2]

(c)Write down the solution of the equation

[1]

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

A vessel that contained some water started to leak from a hole in its base.

The volume of water in the vessel 25 minutes after the leak started was 6 m³

The volume of water in the vessel 64 minutes after the leak started was 3.3 m³

The volume of water, V m³ , in the vessel t minutes after the leak started, is modelled by the equation

where a and b are constants.

(a)Find the value of a and the value of b.[4]

(b)Using the equation of the model,

(i)find the initial volume of water in the vessel,

(ii)find the time taken, after the leak started, for the vessel to empty. Give your answer to the nearest minute.[3]

Figure 2 shows part of the graph of the trigonometric function with equation y = f (x) , where x is measured in degrees.

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

(b)State the number of solutions of the equation

(i)f (x) = 2 in the interval –720°  x  720°

(ii)f (x) = –3 in the interval –720°  x  720°[2]

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)Solve the equation

giving the answer in the form a + b 3 , where a and b are rational numbers.[4]

(a)Find y dx writing the answer in simplest form. ∫[3]

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

(ii)Hence find the exact solutions of the equation dydx = 3

[4]

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

The curve C₁ has equation y = (x + 5)(3x + 2)(2x – 5)

The curve C₂ has equation y = 3x²– 33x – 50

Use algebra to find the x coordinates of the points of intersection of C₁ and C₂[5]

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

(a)Sketch the curve C with equation

State on your sketch

• the equation of the vertical asymptote

• the coordinates of the point of intersection of C with the y-axis[3]

The straight line l has equation y = mx – 4 , where m is a constant.

Given that l cuts C at least once,

(b)(i)show that

(ii)find the range of values of m.[6]

Figure 3 shows a sketch of a line l and a quadratic curve C.

Given that l passes through (–3, 0) and (2, 18)

(a)find an equation for l in the form y = mx + c where m and c are constants.[3]

Given that

• C and l intersect at the points (–3, 0) and (2, 18)

• C crosses the y-axis at (0, –6)

(b)find an equation for C.[4]

The region R, shown shaded in Figure 3, is bounded by C, l and the y-axis.

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

Figure 4 shows the outline of a sign that is used to advertise a bird sanctuary.

The sign is composed of a triangle CPQ joined to a sector QCRTQ of a circle, centre C.

Given that

• angle QPR = 0.8 radians

• PQ = 0.5 m

• PC = 1.84 m

• PRC is a straight line

(a)find the radius, CQ, of the sector, in metres to 3 decimal places.[2]

(b)Hence show that angle PCQ is 0.236 radians to 3 decimal places.[2]

(c)Find the total area of the sign, giving your answer in m² to one decimal place.[3]

(d)Find the total perimeter of the sign, giving your answer in metres to one decimal place.[2]

The curve C has equation y = f (x) .

Given that

k • f ′ (x) = x(x − 3)5 where k is a constant

• the point P with x coordinate 4 lies on C

• the equation of the normal to C at P is y = – 54 x + 2

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

(b)find the value of k,[2]

(c)find f (x) , writing the answer in simplest form.[5]