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

Find

giving the answer in simplest form.[5]

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

A tree was planted.

Exactly 2 years after it was planted, the height of the tree was 2. 70 m.

Exactly 7 years after it was planted, the height of the tree was 4. 50 m.

Given that t years after it was planted, the height of the tree, H metres, is modelled by the equation

where a and b are constants,

(a)find the complete equation of the model.[4]

(b)Hence find, according to the model, the height of the tree when it was planted.[2]

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

In triangle ABC,

• AC = 12 cm

• AB = 8 cm

• angle BCA  32

• angle CBA   

(a)Find both possible values for θ , giving each answer to 2 decimal places.[3]

Given that θ > 90

(b)find, to 3 significant figures, the length of BC,[2]

(c)find, to 3 significant figures, the area of the triangle.[2]

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

(i)Using the laws of indices, solve

[3]

(ii)Find the real roots of the equation

[4]

In this question you must show detailed reasoning. Solutions relying on calculator technology are not acceptable.

Figure 1 shows a sketch of part of the curve with equation y  x³  2x  3

The point P(2, 15) lies on the curve.

(a)Use calculus to find the gradient of the tangent to the curve at P.[2]

The point Q with x coordinate 2 + h where h > 0 also lies on the curve.

(b)Find, in terms of h, the gradient of the straight line PQ. Give the answer in simplest form.[4]

(c)Explain briefly the relationship between the answer to part (b) and the answer to part (a).[1]

Figure 2 Figure 3

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

Figure 2 shows

• a sector DOBCD of a circle with centre O

• a right-angled triangle OAB with the right angle at A

A company’s logo, shown shaded in Figure 3, is formed when the right-angled triangle is removed from the sector of the circle.

Given that

• OB = OD 9 cm=

• AB = 7.3 cm

(a)find the size of angle BOA, in radians, to 3 decimal places,[2]

(b)find the area of the logo, in cm ² , to 3 significant figures,[3]

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

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

The point P with x coordinate 4 lies on C.

Given that

k  x • f(x) ₂ where k is a constant

• the normal to C at P has equation y  2x  5  0

(a)(i)show that k = 18

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

(b)Find an expression for f (x) in terms of x, giving the answer in simplest form.[5]

In this question you must show detailed reasoning. Solutions relying on calculator technology are not acceptable.

Figure 4 shows the curve C₁ with equation y = f (x) where f (x) is a cubic function in x.

The curve C₁

• cuts the x-axis at 5 and the y-axis at 20

• touches the x-axis at −4

as shown in Figure 4.

(a)Deduce the range of values of x for which f (x)  0[2]

(b)Find an expression for f (x) in terms of x. Give the answer in a fully factorised form.[3]

The curve C₂ has equation y  3xx  5

(c)Find the exact x coordinates of all points of intersection of C₁ and C₂[4]

Figure 5 shows a sketch of the curve C with equation

where x is measured in degrees and k is a constant.

The curve C

• cuts the x-axis at 250°

• has a minimum point at P

as shown in Figure 5.

(a)State the value of k.[1]

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

(c)Solve the equation

for values of x in the range 3600  x  3960[2]

(d)Find the coordinates of the point to which P is mapped when C is transformed to the curve with equation

(i)y  4cos x  k

(ii)y  cos x  3k[2]

y

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

Figure 6 shows a sketch of

• the curve C with equation x²  2 y²  6x  y  1  0

• the line l with equation y  2x  k where k is a constant

Given that k can vary,

(a)show that when l touches C

[4]

Given further that

• k < 0

• l touches C at the point P

(b)find the exact coordinates of P[5]