WMA11 Pure Mathematics 1 (P1): May/June 2023 Paper 01

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

Solutions relying on calculator technology are not acceptable.

Solve the inequality

[4]

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

Solutions relying entirely on calculator technology are not acceptable.

A rectangular sports pitch has length x metres and width y metres, where x > y

Given that the perimeter of the pitch is 350 m,

(a)write down an equation linking x and y[1]

Given also that the area of the pitch is 7350 m²

(b)write down a second equation linking x and y[1]

(c)hence find the value of x and the value of y[4]

(a)Express 3x²+ 12x + 13 in the form

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

(b)Hence sketch the curve with equation y = 3x²+ 12x + 13

On your sketch show clearly

• the coordinates of the y intercept

• the coordinates of the turning point of the curve[3]

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

(a)Write

where A, B, p and q are constants to be found.[4]

(b)Hence find dydx giving each coefficient in simplest form.[3]

Figure 1 shows the plan for a garden.

In the plan

• OA and CD are perpendicular to OD

• AB is an arc of the circle with centre O and radius 4 metres

• BC is parallel to OD

• OD is 6 metres, OA is 4 metres and CD is 1.5 metres

(a)Show that angle AOB is 1.186 radians to 4 significant figures.[2]

(b)Find the perimeter of the garden, giving your answer in metres to 3 significant figures.[4]

(c)Find the area of the garden, giving your answer in square metres to 3 significant figures.[4]

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

Solutions relying on calculator technology are not acceptable.

(a)Expand and simplify

[2]

(b)Express 13 + 2 in the form p + q 2 where p and q are integers.

[2]

(c)Use the results of parts (a) and (b), or otherwise, to show that

[3]

The region R₁ , shown shaded in Figure 2, is defined by the inequalities

where k is a constant.

The line x = a, where a is a constant, passes through the intersection of the lines y = 2 and y = kx

Given that the area of R₁ is 274 square units,

(a)find

(i)the value of a

(ii)the value of k[4]

(b)Define the region R₂ , also shown shaded in Figure 2, using inequalities.[2]

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

Solutions relying entirely on calculator technology are not acceptable.

(a)Find the equation of the tangent to the curve with equation

at the point P(4, 12)

Give your answer in the form ax + by + c = 0 where a, b and c are integers.[5]

The curve with equation y = f (x) also passes through the point P(4, 12)

Given that

(b)find f (x) giving the coefficients in simplest form.[5]

(i)

Figure 3 shows part of the graph of the trigonometric function with equation y = f (x)

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

On a separate diagram,

(b)sketch, for –2π < x < 2π , the graph of the curve with equation y = f  x + π 4 

Show clearly the coordinates of all the points where the curve intersects the coordinate axes.[3]

(ii)

Figure 4 shows part of the graph of the trigonometric function with equation y = g(x)

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

On a separate diagram,

(b)sketch, for –2π < x < 2π , the graph of the curve with equation y = g(x) – 2

Show clearly the coordinates of the y intercept.[2]

Figure 5 shows a sketch of the quadratic curve C with equation

The line l₁ also shown in Figure 5,

• has gradient 12

• intersects C on the negative x-axis and at the point P

(a)(i)Write down an equation for l₁[1]

(ii)Find, in terms of b, the coordinates of P[3]

Given that the line l₂ is perpendicular to l₁ and intersects C on the positive x-axis,

(b)find, in terms of b, an equation for l₂[2]

Given also that l₂ intersects C at the point P

(c)show that another equation for l₂ is

[2]

(d)Hence, or otherwise, find the value of b[2]