WMA11 Pure Mathematics 1 (P1): May/June 2022 Paper 01

Find

giving your answer in simplest form.[4]

In the triangle ABC,

(a)Use the sine rule to find the value of sin x° , giving your answer to 4 decimal places.[2]

Given also that AB is the longest side of the triangle,

(b)find the value of x, giving your answer to 2 decimal places.[3]

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

Solutions relying on calculator technology are not acceptable.

(i)Show that is an integer and find its value.

[2]

(ii)Simplify

giving your answer in the form a + b 5 where a and b are rational numbers.[3]

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

The curve has a minimum at P(−1, 0) and a maximum at Q  3 2 , 2

The line with equation y = 1 is the only asymptote to the curve.

On separate diagrams sketch the curves with equation

(i)y = f(x) − 2[3]

(ii)y = f(−x)[3]

On each sketch you must clearly state • the coordinates of the maximum and minimum points • the equation of the asymptote

The curve C has equation y = f(x)

Given that • f(x) is a quadratic expression • the maximum turning point on C has coordinates (−2, 12) • C cuts the negative x-axis at −5

(a)find f(x)[4]

The line l₁ has equation y = 45 x

Given that the line l₂ is perpendicular to l₁ and passes through (−5, 0)

(b)find an equation for l₂ , writing your answer in the form y = mx + c where m and c are constants to be found.[3]

Figure 2 shows a sketch of the curve C and the lines l₁ and l₂

(c)Define the region R, shown shaded in Figure 2, using inequalities.[2]

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

Solutions relying on calculator technology are not acceptable.

(a)Given that

[2]

(b)Hence solve the simultaneous equations

Give your answers in fully simplified surd form.[5]

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

Given that

• f′(x) = 2x + Ax₂ + 3 , where A is a constant

• f′′(x) = 0 when x = 4

(a)find the value of A.[4]

Given also that

• f(x) = 8 3 , when x = 12

(b)find f(x) , giving each term in simplest form.[5]

Figure 3 shows a sketch of the outline of the face of a ceiling fan viewed from below.

The fan consists of three identical sections congruent to OABCDO, shown in Figure 3, where • OABO is a sector of a circle with centre O and radius 9 cm • OBCDO is a sector of a circle with centre O and radius 84 cm

• angle AOD = 2π3 radians

Given that the length of the arc AB is 15 cm,

(a)show that the length of the arc CD is 35.9 cm to one decimal place.[3]

The face of the fan is modelled to be a flat surface.

Find, according to the model,

(b)the perimeter of the face of the fan, giving your answer to the nearest cm,[2]

(c)the surface area of the face of the fan.

Give your answer to 3 significant figures and make your units clear.[5]

Figure 4 shows part of the graph of the curve with equation y = sin x

Given that sin α = p , where 0 < α < 90°

(a)state, in terms of p , the value of

(i)2 sin (180° − α)

(ii)sin (α − 180°)

(iii)3 + sin (180° + α)[3]

A copy of Figure 4, labelled Diagram 1, is shown on page 27.

On Diagram 1,

(b)sketch the graph of y = sin 2x[2]

(c)Hence find, in terms of α, the x coordinates of any points in the interval 0 < x < 180° where

[3]

Figure 5 shows a sketch of the curve C with equation

where k is a constant.

(a)Find dydx[2]

The line l, shown in Figure 5, is the normal to C at the point A with x coordinate − 72

Given that l is also a tangent to C at the point B,

(b)show that the x coordinate of the point B is a solution of the equation

[4]

(c)Hence find the x coordinate of B, justifying your answer.[2]

Given that the y intercept of l is −1

(d)find the value of k.[4]