WMA11 Pure Mathematics 1 (P1): May/June 2024 Paper 01R

Find

writing your answer in simplest form.[4]

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

Solutions relying on calculator technology are not acceptable.

(a)Solve

[2]

(b)(i)Write

in the form (x + a)² + b where a and b are constants.

(ii)Hence solve

[4]

(c)Hence find the values of x that satisfy both

[1]

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

Solutions relying on calculator technology are not acceptable.

(a)Find dydx , giving each term in simplest form.[3]

(b)Find the solution of the equation

writing the answer in the form 2^k where k is a constant.[3]

The curve C has equation

The straight line l has equation

where k is a constant.

(a)Show that l meets C when

[2]

(b)Hence find the value of k for which l is a tangent to C[3]

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

Solutions relying on calculator technology are not acceptable.

(a)Fully factorise

[2]

(b)Hence solve

[3]

Figure 1 shows a sketch of the entrance to a tunnel.

The shape of the entrance consists of a sector BCDF, of a circle centre F, joined to two congruent (identical) triangles ABF and EDF.

Given that

(a)Show that angle AFB = 0.433 radians to 3 decimal places.[1]

(b)Find the perimeter of the entrance to the tunnel, ABCDEFA, in metres, to one decimal place.[4]

(c)Find the cross-sectional area of the entrance to the tunnel, ABCDEFA, in m², to one decimal place.[4]

The straight line l₁ shown in Figure 2 has equation 5y = 2x + 10

The points A and B lie on l₁ such that

• point A lies on the y-axis

• point B has x coordinate 10

(a)Find the distance AB writing your answer as a fully simplified surd.[3]

The straight line l₂ also shown in Figure 2

• passes through B

• is perpendicular to l₁

(b)Find an equation for l₂ writing your answer in the form ax + by + c = 0 , where a, b and c are integers.[4]

Line l₂ crosses the x-axis at the point C.

Point D is such that the points A, B, C and D form the vertices of a rectangle, shown shaded in Figure 2.

(c)Find the area of rectangle ABCD.[3]

A curve C with equation y = f (x) passes through the point R(4, 13).

Given that

(a)use integration to find f (x), giving your answer in simplest form.[5]

(b)Given that f (x) can be written in the form

find the value of the constant p and the value of the constant q.[2]

(c)Sketch the graph of y = f (2x), showing the coordinates of any points where the curve touches or crosses the coordinate axes.[4]

Figure 3 shows a sketch of

• the curve with equation y = tan  x  π 6  for 0  x  2π

• part of the straight line l with equation y = π – x

(a)State the number of solutions of the equation

(i)tan  x  π 6  = π – x in the interval 0  x  2π

(ii)tan  x  π 6  = π – x in the interval 0  x  100π

(iii)tan  x  π 6  = π + x in the interval 0  x  2π

[3]

The line with equation x = a , shown in Figure 3, is the asymptote to the curve with the smallest positive x coordinate.

(b)State the value of a[1]

The line with equation x = b , also shown in Figure 3, is the asymptote to the curve with the second smallest positive x coordinate.

The line l meets x = a at point P and meets x = b at point Q as shown in Figure 3.

(c)Find the midpoint of the line segment PQ.[4]

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

Solutions relying on calculator technology are not acceptable.

The curve C has equation

The point P, which lies on C, has coordinates (2, –8)

(a)Show that an equation of the tangent to C at P is

[5]

The point Q also lies on C.

Given that the tangent to C at Q is parallel to the tangent to C at P,

(b)find, using algebra and showing your working, the exact x coordinate of Q.[5]