WMA13 Pure Mathematics 3 (P3): January 2025 Paper 01

The equation f (x) = 0 has a single root α

(a)Show that 0.1 < α < 0.2[2]

(b)Show that α is a solution of

[1]

The iterative formula

is used to find α

(c)Starting with x₁= 0.15 and using the iterative formula,

(i)find, to 4 decimal places, the value of x₂

(ii)find, to 4 decimal places, the value of α[3]

The weed on the surface of a pond is being monitored.

The surface area of the pond covered by the weed, A m², is modelled by the equation

where t is the number of weeks after monitoring began.

Use the equation of the model to answer parts (a) and (b).

(a)Find the surface area of the pond initially covered by the weed.[1]

After T weeks, 25 m² of the pond is covered by the weed.

(b)Find the value of T, giving your answer to 2 decimal places.[2]

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

A curve has equation

Use calculus to find the range of values of x for which y is increasing.[6]

Given that

(a)(i)find the values of the constants A, B and C

(ii)show that D = 0[4]

(b)Hence, using algebraic integration, find

giving your answer in the form p + q ln 2, where p and q are integers.[5]

A hot piece of metal is cooled by dropping it into water. The temperature, H °C, of the metal, t minutes after it is dropped into the water, is modelled by the equation

(a)Find the initial temperature of the piece of metal.[1]

(b)On Diagram 1, sketch the graph of H against t. On your sketch, state the equation of the asymptote to the curve.[2]

(c)Find the value of t for which H = 144, giving your answer to 2 decimal places.

[3]

(d)Show by differentiation that

where a and b are constants to be found.[3]

The function f is defined by

(a)Find f^–1[3]

(b)Show that

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

The point P (3, 15) lies on the curve with equation y = f (x) .

(c)Find the point to which P is mapped when y = f (x) is transformed to the curve with equation y = 2f (3x) + 8[2]

Given that a and b are positive constants with a > b ,

(a)sketch, on separate diagrams, the graph with equation

(i)y = |3x – a|

(ii)y = |3x – a| – b

Show on each sketch

• the coordinates of the minimum point on the graph • the coordinates of the point at which the graph crosses the y-axis[6]

(b)Solve the equation

giving any solution for x in terms of a and b.[2]

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

Solutions relying entirely on calculator technology are not acceptable.

(i)Solve, for 0 < θ < π

giving your answers, in radians, to 3 significant figures.[5]

(ii)Solve, for 0 < x < 180°

giving your answers, in degrees, to one decimal place.[4]

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

The curve cuts the x-axis at point P

(a)State the x coordinate of P[1]

The point Q, shown in Figure 1, is the stationary point on C

(b)Use calculus to find the exact coordinates of Q[5]

(c)Hence find the range of the function g (x) where

[2]

Figure 2 shows a sketch of the curve with equation

(a)Find in terms of y. dy[2]

(b)Hence show that

where k is a constant to be found.[3]

The point P (a, b) lies on the curve and is shown in Figure 2.

Given that

• the gradient of the curve at P is – 14

• both a and b are positive

(c)find the exact values of a and b.[4]