WMA11 Pure Mathematics 1 (P1): Oct/Nov 2021 Paper 01

Find

giving each term in simplest form.[5]

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

Solutions relying on calculator technology are not acceptable.

A curve has equation

The points P and Q lie on the curve.

The gradient of the curve at both point P and point Q is 2

Find the x coordinates of P and Q.[5]

(i)Solve

[3]

(ii)

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

The infinite region R, shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by C and l only.

Given that

• l has a gradient of 3

• C has equation y = 2x² – 50

• C and l intersect on the negative x‑axis

use inequalities to define the region R.[3]

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

The point Q and the point R(k, 0) lie on the curve and are shown in Figure 2.

(a)State

(i)the coordinates of Q,

(ii)the value of k.[3]

(b)Given that there are exactly two solutions to the equation

find the range of possible values for p.[2]

The line l₁ has equation 3y – 2x = 30

The line l₂ passes through the point A(24, 0) and is perpendicular to l₁

Lines l₁ and l₂ meet at the point P

(a)Find, using algebra and showing your working, the coordinates of P.[5]

Given that l₁ meets the x‑axis at the point B,

(b)find the area of triangle BPA.[3]

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

Solutions relying on calculator technology are not acceptable.

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

(a)Sketch a graph of C.

Show on your graph the coordinates of the points where C cuts or meets the coordinate axes.[3]

(b)Write f(x) in the form ax³ + bx² + cx + d, where a, b, c and d are constants to be found.[3]

(c)Hence, find the equation of the tangent to C at the point where x = 13[4]

Figure 3 shows the design for a sign at a bird sanctuary.

The design consists of a kite OABC joined to a sector OCXA of a circle centre O.

In the design

• OA = OC = 0.6 m

• AB = CB = 1.4 m

• Angle OAB = Angle OCB = 2 radians

• Angle AOC = θ radians, as shown in Figure 3

Making your method clear,

(a)show that θ = 1.64 radians to 3 significant figures,[4]

(b)find the perimeter of the sign, in metres to 2 significant figures,[2]

(c)find the area of the sign, in m² to 2 significant figures.[4]

Figure 4 shows a sketch of the curve C with equation

The point M is the maximum turning point on C.

(a)(i)Write 4 + 12x – 3x² in the form

where a, b and c are constants to be found.

(ii)Hence, or otherwise, state the coordinates of M.[5]

The line l₁ passes through O and M, as shown in Figure 4.

A line l₂ touches C and is parallel to l₁

(b)Find an equation for l₂[5]

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

Solutions relying on calculator technology are not acceptable.

Figure 5 shows a sketch of the curve with equation y = f(x) where

The point P(9, 3) lies on the curve and is shown in Figure 5.

On the next page there is a copy of Figure 5 called Diagram 1.

(a)On Diagram 1, sketch and clearly label the graphs of

Show on each graph the coordinates of the point to which P is transformed.[3]

The graph of y = f(2x) meets the graph of y = f(x) + 3 at the point Q.

(b)Show that the x coordinate of Q is the solution of

[3]

(c)Hence find, in simplest form, the coordinates of Q.[3]

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

Given that

• fʹ(x) = ax – 12x^1/3 , where a is a constant

• fʹʹ(x) = 0 when x = 27

• the curve passes through the point (1, –8)

(a)find the value of a.[3]

(b)Hence find f(x).[4]