WMA11 Pure Mathematics 1 (P1): Oct/Nov 2023 Paper 01

Given that

find, in simplest form,

dy

(a)

[3]

d² y

(b)₂

[2]

Given that

express each of the following in the form kxⁿ where k and n are simplified constants.

(a)a^1/2[1]

16

(b)₃ b[1]

 ab ^4/3

(c)    2 [2]

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

Solutions relying on calculator technology are not acceptable.

(a)Write 8 2153  in the form a 3 + b 5 where a and b are integers to be found.

[3]

(b)Hence, or otherwise, solve

giving your answer in simplest form.[3]

Figure 1 shows a sketch of part of the curve C with equation y = 1x + 2

(a)State the equation of the asymptote of C that is parallel to the y-axis.[1]

(b)Factorise fully x³ + 4x² + 4x[2]

A copy of Figure 1, labelled Diagram 1, is shown on the next page.

(c)On Diagram 1, add a sketch of the curve with equation

On your sketch, state clearly the coordinates of each point where this curve cuts or meets the coordinate axes.[3]

(d)Hence state the number of real solutions of the equation

giving a reason for your answer.[1]

Diagram 1 copy of Diagram 1

Only use the copy of Diagram 1 if you need to redraw your answer to part (c).

Figure 2 shows the plan view of a frame for a flat roof.

The shape of the frame consists of triangle ABD joined to triangle BCD.

Given that

• BD = x m

• CD = (1 + x) m

• BC = 5 m

• angle BCD = θ°

(a)show that cos θ° = 13 + x5 + 5x[2]

Given also that

• x = 2 3

• angle BAC = 30°

• ADC is a straight line

(b)find the area of triangle ABC, giving your answer, in m², to one decimal place.[5]

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

Solutions relying on calculator technology are not acceptable.

The equation

where p is a constant, has two distinct real roots.

(a)Show that

[3]

(b)Hence, using algebra, find the range of possible values of p[3]

The curve C has equation y = f(x) where x > 0

Given that

4x²  10  7x^1/2 • fʹ(x) =

• the point P(4, –1) lies on C

(a)(i)find the value of the gradient of C at P

(ii)Hence find the equation of the normal to C at P, giving your answer in the form ax + by + c = 0 where a, b and c are integers to be found.[4]

(b)Find f(x).[6]

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

Solutions relying on calculator technology are not acceptable.

The curve C₁ has equation

The curve C₂ has equation

(a)Show that C₁ and C₂ meet when

[2]

Given that C₁ and C₂ meet at points P and Q

(b)find, using algebra, the exact distance PQ[5]

Figure 3 shows the plan view of the area being used for a ball-throwing competition.

Competitors must stand within the circle C and throw a ball as far as possible into the target area, PQRS, shown shaded in Figure 3.

Given that

• circle C has centre O

• P and S are points on C

• OPQRSO is a sector of a circle with centre O

• the length of arc PS is 0.72 m

• the size of angle POS is 0.6 radians

(a)show that OP = 1.2 m[1]

Given also that

• the target area, PQRS, is 90 m²

• length PQ = x metres

(b)show that

[3]

(c)Hence calculate the total perimeter of the target area, PQRS, giving your answer to the nearest metre.[3]

Figure 4 shows a sketch of part of the curve C₁ with equation

where n is a constant.

The curve C₁ cuts the positive x-axis for the first time at point P(270, 0), as shown in Figure 4.

(a)(i)State the value of n

(ii)State the period of C₁[2]

The point Q, shown in Figure 4, is a minimum point of C₁

(b)State the coordinates of Q.[2]

The curve C₂ has equation y  2sin x  k, where k is a constant.

The point R  a, 12 5  and the point S  a,  3 5 , both lie on C

Given that a is a constant less than 90

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

Figure 5 shows part of the curve C with equation y = f (x) where

(a)Write 2x²  12x  14 in the form

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

Given that C has a minimum at the point P

(b)state the coordinates of P[1]

The line l intersects C at (–1, 28) and at P as shown in Figure 5.

(c)Find the equation of l giving your answer in the form y = mx + c where m and c are constants to be found.[3]

The finite region R, shown shaded in Figure 5, is bounded by the x-axis, l, the y-axis, and C.

(d)Use inequalities to define the region R.[3]