WMA11 Pure Mathematics 1 (P1): January 2023 Paper 01

A curve C has equation

(a)Find dydx giving your answer in simplest form.[3]

(b)Hence find the exact value of the gradient of the tangent to C at the point where x = 2 giving your answer in simplest form.

[2]

The points P, Q and R have coordinates (–3, 7), (9, 11) and (12, 2) respectively.

(a)Prove that angle PQR = 90°[3]

Given that the point S is such that PQRS forms a rectangle,

(b)find the coordinates of S.[2]

Find

giving your answer in simplest form.[5]

Given that the equation

has no real roots, find the range of possible values for k.[4]

(a)By substituting p = 3^x, show that the equation

can be rewritten in the form

[3]

(b)Hence solve

[3]

Figure 1 shows the plan view for the design of a stage.

The design consists of a sector OBC of a circle, with centre O, joined to two congruent triangles OAB and ODC.

Given that

• angle BOC = 2.4 radians

• area of sector BOC = 40 m²

• AOD is a straight line of length 12.5 m

(a)find the radius of the sector, giving your answer, in m, to 2 decimal places,[2]

(b)find the size of angle AOB, in radians, to 2 decimal places.[1]

Hence find

(c)the total area of the stage, giving your answer, in m², to one decimal place,[3]

(d)the total perimeter of the stage, giving your answer, in m, to one decimal place.[4]

(a)On Diagram 1, sketch a graph of the curve C with equation

[2]

The curve C is transformed onto the curve with equation y = 6x − 2 x ≠ 2

(b)Fully describe this transformation.[2]

The curve with equation

and the line with equation

intersect at exactly two points, P and Q.

Given that the x coordinate of point P is –4

(c)find the value of k,[2]

(d)find, using algebra, the coordinates of point Q.

[4]

Figure 2 shows a sketch of the straight line l and the curve C.

Given that l cuts the y‑axis at –12 and cuts the x‑axis at 4, as shown in Figure 2,

(a)find an equation for l, writing your answer in the form y = mx + c, where m and c are constants to be found.[2]

Given that C

• has equation y = f(x) where f(x) is a quadratic expression

• has a minimum point at (7, –18)

• cuts the x‑axis at 4 and at k, where k is a constant

(b)deduce the value of k,[1]

(c)find f(x).[3]

The region R is shown shaded in Figure 2.

(d)Use inequalities to define R.[2]

Figure 3 shows a sketch of

• the curve with equation y = tan x

• the straight line l with equation y = πx

in the interval –π < x < π

(a)State the period of tan x[1]

(b)Write down the number of roots of the equation

(i)tan x = (π + 2)x in the interval –π < x < π[1]

(ii)tan x = πx in the interval –2π < x < 2π[1]

(iii)tan x = πx in the interval –100π < x < 100π[1]

Figure 4 shows a sketch of part of the curve C with equation y = f(x), where

(a)Use the given information to state the values of x for which

[2]

(b)Expand (3x + 20)(x + 6)(2x – 3), writing your answer as a polynomial in simplest form.[3]

The straight line l is the tangent to C at the point where C cuts the y‑axis.

Given that l cuts C at the point P, as shown in Figure 4,

(c)find, using algebra, the x coordinate of P

[5]

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

Given that

• f ′′(x) = 4x +

• the point P has x coordinate 4 and lies on C

• the tangent to C at P has equation y = 3x + 4

(a)find an equation of the normal to C at P[2]

(b)find f(x), writing your answer in simplest form.[6]