WMA13 Pure Mathematics 3 (P3): May/June 2024 Paper 01

Figure 1 shows a sketch of the graph with equation y = f (x) where

The point P, shown in Figure 1, is the vertex of the graph.

(a)State the coordinates of P[2]

(b)Use algebra to solve

[2]

(c)Find the point to which P is mapped, when the graph with equation y = f (x) is transformed to the graph with equation y = 3f (x – 2)[2]

(a)Write g (x) in the form

where A, B and C are integers to be found.[3]

(b)Hence use algebraic integration to show that

where α and β are integers to be found.[4]

(i)The variables x and y are connected by the equation

Sketch the graph of log₁₀ y against log₁₀ x

Show on your sketch the coordinates of the points of intersection of the graph with the axes.[3]

(ii)

Figure 2 shows the linear relationship between log₃ N and t.

Show that N = ab^t where a and b are constants to be found.[3]

(a)Write f (x) in the form

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

(b)Use the answer to part (a) to write f (x) in the form

where R > 0 and 0 < α < π2

Give the exact value of R and give the value of α in radians to 3 significant figures.[3]

(c)Hence, or otherwise,

(i)state the maximum value of f (x)

(ii)find the second smallest positive value of x at which a maximum value of f (x) occurs. Give your answer to 3 significant figures.[3]

The functions f and g are defined by

(a)Find f^–1(22)[2]

(b)Use differentiation to prove that g is an increasing function.[3]

(c)Find g^–1[3]

(d)Find the range of fg[2]

Solutions relying entirely on calculator technology are not acceptable.

Figure 3 shows a sketch of part of the curve with equation

The line l, shown in Figure 3, is the normal to the curve at the point P (8, 5)

(a)Use calculus to show that an equation of l is

[5]

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

(b)Use algebraic integration to find the exact area of R.[4]

Solutions relying entirely on calculator technology are not acceptable.

(a)Given that

[4]

(b)Hence or otherwise, solve, for 0  θ < 180°

[4]

Figure 4 is a graph showing the path of a golf ball after the ball has been hit until it first hits the ground.

The vertical height, h metres, of the ball above the ground has been plotted against the horizontal distance travelled, x metres, measured from where the ball was hit.

The ball travels a horizontal distance of d metres before it first hits the ground.

The ball is modelled as a particle travelling in a vertical plane above horizontal ground.

The path of the ball is modelled by the equation

(a)Find the value of d, giving your answer to 2 decimal places.

[3]

(b)Show that the maximum value of h occurs when

[4]

Using the iteration formula

(c)(i)find the value of x₂ to 2 decimal places,

(ii)find, by repeated iteration, the horizontal distance travelled by the golf ball before it reaches its maximum height. Give your answer to 2 decimal places.[3]

The curve shown in Figure 5 has equation

The point P  k, π 3  lies on the curve.

(a)Verify that k = 2[1]

(b)(i)Find dxdy in terms of y

(ii)Hence show that dydx  1x  1

[6]

The normal to the curve at P cuts the x-axis at the point N.

(c)Find the exact area of triangle OPN, where O is the origin.

Give your answer in the form aπ + bπ² where a and b are constants.[3]