WMA11 Pure Mathematics 1 (P1): May/June 2024 Paper 01

Find

giving each term in simplest form.[3]

(i)Given that m = 2ⁿ, express each of the following in simplest form in terms of m.

(a)2ⁿ⁺³[1]

(b)16³ⁿ[2]

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

Solutions relying on calculator technology are not acceptable.

Solve the equation

giving your answer in the form p + q 3 where p and q are integers.[3]

Figure 1 shows a sketch of the curve with equation y = f(x).

The curve passes through the points (–1, 0) and (0, 2) and touches the x‑axis at the point (3, 0).

On separate diagrams, sketch the curve with equation

(a)y = f(x + 3)[3]

(b)y = f(–3x)[3]

On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.

The curve C₁ has equation

and the curve C₂ has equation

where k is a constant.

Given that C₁ and C₂ meet at a single point P

(a)show that

[3]

(b)Hence find the coordinates of P[3]

Figure 2 shows the plan view of a garden.

The shape of the garden ABCDEA consists of a triangle ABE and a right‑angled triangle BCD joined to a sector BDE of a circle with radius 6 m and centre B.

The points A, B and C lie on a straight line with AB = 10.8 m

Angle BCD = π2 radians, angle EBD = 1.3 radians and AE = 12.2 m

(a)Find the area of the sector BDE, giving your answer in m²[2]

(b)Find the size of angle ABE, giving your answer in radians to 2 decimal places.[2]

(c)Find the area of the garden, giving your answer in m² to 3 significant figures.[3]

Solutions relying on calculator technology are not acceptable.

Figure 3 shows

• the line l with equation y – 5x = 75

• the curve C with equation y = 2x² + x – 21

The line l intersects the curve C at the points P and Q, as shown in Figure 3.

(a)Find, using algebra, the coordinates of P and the coordinates of Q.[4]

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

(b)Use inequalities to define the region R.[3]

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

and k is a constant.

(a)Find, in simplest form,

(i)f ′(x)

(ii)f ′′(x)[3]

The curve with equation y = f ′(x) intersects the curve with equation y = f ′′(x) at the points A and B.

Given that the x coordinate of A is 5

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

(c)Hence find the coordinates of B.[3]

The curve C₁ has equation

(a)Sketch the graph of C₁ showing the coordinates of any points of intersection with the coordinate axes.[3]

The curve C₂ has equation y = Ax where A is a constant.

(b)Show that the x coordinates of the points of intersection of C₁ and C₂ satisfy the equation

[1]

(c)Hence find the range of possible values of A for which C₁ meets C₂ at 4 distinct points.[3]

Given that

• the point A has coordinates (4, 2)

• the point B has coordinates (15, 7)

• the line l₁ passes through A and B

(a)find an equation for l₁, giving your answer in the form px + qy + r = 0 where p, q and r are integers to be found.[3]

The line l₂ passes through A and is parallel to the x‑axis.

The point C lies on l₂ so that the length of BC is 5 5

(b)Find both possible pairs of coordinates of the point C.[4]

(c)Hence find the minimum possible area of triangle ABC.[2]

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

Given that

• f ′(x) = 6x –

• the point P(4, 12) lies on C

(a)find the equation of the normal to C at P, giving your answer in the form y = mx + c where m and c are integers to be found,[4]

(b)find f(x), giving each term in simplest form.[6]

Figure 4 shows a sketch of part of the curve C₁ with equation

where x is measured in radians.

The point P shown in Figure 4 is a maximum point on C₁

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

The curve C₂ has equation

where k is a constant.

Given that the maximum value of y on C₂ is 3

(b)find the coordinates of the minimum point on C₂ which has the smallest positive x coordinate.[2]

The curve C₃ has equation

where B is a positive constant.

Given that  π 4 , A, where A is a constant, is the minimum point on C which has the

smallest positive x coordinate,

(c)find

(i)the value of A,

(ii)the smallest possible value of B.[2]