WMA13 Pure Mathematics 3 (P3): May/June 2022 Paper 01

The curve C has equation

(a)Find dydx[2]

Given that the point P  1 3 , 1 lies on C,

(b)find the equation of the normal to C at P. Write your answer in the form ax + by + c = 0 where a, b and c are integers to be found.[4]

The functions f and g are defined by

(a)Find the value of fg(5)[2]

(b)Find f ^–1[3]

(c)Solve the equation

[4]

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

Solutions relying entirely on calculator technology are not acceptable.

Given that k is a positive constant,

(a)find

[2]

Given also that

(b)find the value of k[4]

The number of subscribers to an online video streaming service, N, is modelled by the equation

where a and b are constants and t is the number of years since monitoring began.

The line in Figure 1 shows the linear relationship between t and log₁₀ N

The line passes through the points (0, 3.08) and (5, 3.85)

Using this information,

(a)find an equation for this line.[2]

(b)Find the value of a and the value of b, giving your answers to 3 significant figures.[3]

When t = T the number of subscribers is 500 000

According to the model,

(c)find the value of T[2]

Figure 2 shows part of the graph with equation y = f(x), where

and k is a positive constant.

The graph intersects the y‑axis at the point A and has a minimum point at B as shown.

(a)(i)Find the y coordinate of A

(ii)Find, in terms of k, the x coordinate of B[2]

(b)Find, in terms of k, the range of values of x that satisfy the inequality[3]

Given that the line y = 3 – 2x intersects the graph y = f(x) at two distinct points,

(c)find the range of possible values of k[3]

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

Solutions relying entirely on calculator technology are not acceptable.

The function f is defined by

(a)Show that

where k is an integer to be found.[4]

(b)Hence, find the values of x for which f′(x) = 0[1]

Figure 3 shows a sketch of the curve C with equation y = f(x).

The curve has a local maximum at the point P

(c)Find the exact coordinates of P[2]

The function g is defined by

(d)Find the range of g[3]

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

Solutions relying entirely on calculator technology are not acceptable.

(a)Show that the equation

can be written as

[4]

(b)Hence solve, for 0 < θ < π2 , the equation

giving your answers to 3 significant figures.[4]

Figure 4 is a graph showing the velocity of a sprinter during a 100 m race.

The sprinter’s velocity during the race, v m s^–1, is modelled by the equation

where t seconds is the time after the sprinter begins to run.

According to the model,

(a)find, using calculus, the sprinter’s maximum velocity during the race.[5]

Given that the sprinter runs 100 m in T seconds, such that

(b)show that T is a solution of the equation

[4]

The iteration formula

is used to find an approximate value for T

Using this iteration formula with T₁= 10

(c)find, to 4 decimal places,

(i)the value of T₂

(ii)the time taken by the sprinter to run the race, according to the model.[3]

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

Solutions relying entirely on calculator technology are not acceptable.

Figure 5 shows the curve with equation

The point M, shown in Figure 5, is the minimum point on the curve.

(a)Show that the x coordinate of M is a solution of the equation

[4]

(b)Hence find, to 3 significant figures, the x coordinate of M.[5]