WMA11 Pure Mathematics 1 (P1): Oct/Nov 2024 Paper 01

The line l₁ passes through the point A(–5, 20) and the point B(3, –4).

(a)Find an equation for l₁ giving your answer in the form y = mx + c, where m and c are constants.[3]

The line l₂ is perpendicular to l₁ and passes through the midpoint of AB

(b)Find an equation for l₂ giving your answer in the form px  qy  r  0 , where p, q and r are integers.[3]

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

Solutions relying on calculator technology are not acceptable.

(i)Simplify fully

[3]

(ii)Find the exact value of a such that

Write your answer in the form p 3 + q where p and q are fully simplified rational constants.[4]

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

(a)Find f(x) dx

[4]

(b)(i)Show that when f ′(x) = 0

(ii)Hence state the value of x for which

[5]

Figure 1 shows a sketch of part of the curves C₁ and C₂

Given that C₁

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

• cuts the x‑axis at the origin and at x = 4

• has a minimum turning point at (2, –4.8)

(a)find f(x)[3]

Given that C₂

• has equation y = g(x) where g(x) is a cubic function

• cuts the x‑axis at the origin and meets the x‑axis at x = 4

• passes through the point (6, 7.2)

(b)find g(x)[3]

The curves C₁ and C₂ meet in the first quadrant at the point P, shown in Figure 1.

(c)Use algebra to find the coordinates of P.[4]

A plot of land OAB is in the shape of a sector of a circle with centre O.

Given

• OA = OB = 5 km

• angle AOB = 1.2 radians

(a)find the perimeter of the plot of land.[2]

A point P lies on OB such that the line AP divides the plot of land into two regions R₁ and R₂ as shown in Figure 2.

Given that

(b)show that the area of R₂= 3.75 km²[3]

(c)Find the length of AP, giving your answer to the nearest 100 m.[4]

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

Solutions relying on calculator technology are not acceptable.

(a)Sketch the curve C with equation

State on your sketch

• the equation of the vertical asymptote

• the coordinates of the intersection of C with the y‑axis[3]

The straight line l has equation y = kx – 4, where k is a constant.

Given that l cuts C at least once,

(b)(i)show that

(ii)find the range of possible values for k.[6]

Figure 3 shows a plot of part of the curve C₁ with equation

where x is measured in radians.

Points P and Q lie on the curve and are shown in Figure 3.

(a)State

(i)the coordinates of P

(ii)the coordinates of Q[3]

The curve C₂ has equation y = –4 cos x + k where x is measured in radians and k is a constant.

Given that C₂ has a maximum y value of 11

(b)(i)state the value of k

(ii)state the coordinates of the minimum point on C₂ with the smallest positive x coordinate.[3]

On the opposite page there is a copy of Figure 3 labelled Diagram 1.

(c)Using Diagram 1, state the number of solutions of the equation

giving a reason for your answer.[2]

A curve C has equation y = f(x).

The point P with x coordinate 3 lies on C

Given

• f ′(x) = 4x²+ kx + 3 where k is a constant

• the normal to C at P has equation y   124 x  5

(a)show that k = –5[3]

(b)Hence find f(x).[4]

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

(a)Deduce the range of values of x for which f(x)  0[1]

(b)Find f ′(x) giving your answer in simplest form.[3]

The point R(–4, 84) lies on C.

Given that the tangent to C at the point P is parallel to the tangent to C at the point R

(c)find the x coordinate of P.[4]

(d)Find the point to which R is transformed when the curve with equation y = f(x) is transformed to the curve with equation,

(i)y = f(x – 3)

(ii)y = 4f(x)[2]