WMA11 Pure Mathematics 1 (P1): May/June 2026 Paper 01

Find

giving the answer in simplest form.[5]

The line l₁ has equation y  10  3x

The line l₂

• is perpendicular to l₁

• intersects l₁ at the point P with x coordinate 4

(a)Find an equation for l₂ writing the answer in the form ax  by  c  0 , where a, b and c are integers.[3]

Given that

• line l₁ cuts the y-axis at the point M

• line l₂ cuts the y-axis at the point N

(b)find the area of triangle MPN, making your method clear.[3]

A triangular plot of land, ABC, has area 60 m²

Given

• AB = 15 m

• AC = 10 m

• angle BAC   

(a)show that sin   45[1]

Given also that   is the largest angle in the triangle,

(b)find the length of the side BC, in metres, to 3 significant figures,[3]

(c)find the size of the smallest angle in the triangle, in degrees, to one decimal place.[2]

Solutions relying on calculator technology are not acceptable.

(a)Solve the equation

[5]

(b)(i)Show that the equation

can be written as

(ii)Hence, using algebra, show that x = 16 is the only solution of the equation

[5]

The curve C has equation

(a)Write 2x²  12x  7 in the form a x  b  c , where a, b and c are constants.[3]

The point P is the maximum point on C.

(b)Find the coordinates of P.[2]

Figure 1 shows a sketch of the curve C and a line l.

The line l passes through the point P and the origin O.

The region R, shown shaded in Figure 1, is bounded by C, l and the y-axis.

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

Figure 2 shows a sketch of the curve C with equation y = f(x).

The curve C

• touches the x-axis at − 32

• cuts the x-axis at 4

• cuts the y-axis at −12

Given that f(x) is a cubic function,

(a)find f(x). You may leave the answer in a factorised form.[3]

The line with equation y  2x  12 cuts C at 0, 12 and at two further points, P and Q.

(b)Use algebra to find the exact x coordinates of P and Q. Give each answer in simplest form.

(Solutions relying entirely on calculator technology are not acceptable.)[4]

Figure 3 shows a sector AOB of a circle, centre O.

The radius OA of the sector is r cm and angle AOB is θ radians.

Given that

• the area of sector AOB is 1 cm²

• the perimeter of sector AOB is 8 cm

(a)write down two different equations involving r and θ .[2]

Given that r > 2

(b)solve the two simultaneous equations in part (a) to find the exact value of r and the exact value of θ . Write each answer in the form a + b c where a, b and c are integers.

[5]

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

The point P with x coordinate 14 lies on C.

Given that

• f (x)  10 ₂

• the tangent to C at P has equation y  kx  2 , where k is a constant

(a)(i)find the value of k

(ii)find the equation of the normal to C at P. Give the answer in the form y  mx  c , where m and c are constants.[6]

(b)Find f(x) .[5]

The curve C has equation y  sin  x   8  ¹  

Figure 4 shows a sketch of C₁ in the interval 0  x  2

The point P is a minimum turning point on C₁ and is shown in Figure 4.

(a)State the x coordinate of P.[1]

(b)State the number of roots of the equation

in the interval 0  x  101 , giving a reason for your answer.[2]

The curve C has equation y  5sin  x   8  ²  

The point Q is the minimum turning point on C₂ in the interval 0  x  2

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

The curve C has equation

where k is a positive constant.

(a)Sketch a graph of C. Show on your sketch the coordinates of the point where C crosses a coordinate axis.[3]

The line l has equation x  y  12

Given that l is a tangent to C,

(b)show that

[4]

The line l meets C at a point P.

Given that P lies below the x-axis,

(c)find the coordinates of P.

(Solutions relying entirely on calculator technology are not acceptable.)[4]