WMA11 Pure Mathematics 1 (P1): January 2025 Paper 01

Find

giving 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.

Given that

• the point A has coordinates 2 3, 5

• the point B has coordinates 7 3, 8

• the straight line l₁ passes through A and B

(a)show that the gradient of l₁ is p 3, where p is a rational constant to be found. You must show each step of your working.[2]

The straight line l₂ is perpendicular to l₁ and passes through A.

(b)Find the equation of l₂, giving your answer in the form y = mx + c, where m and c are constants.[3]

The population of a town was monitored.

Exactly 5 years after monitoring began, the population was 58 000

Exactly 10 years after monitoring began, the population was 65 000

Given that the population of the town, P thousand, t years after monitoring began can be modelled by the equation

where a and b are constants,

(a)find the value of a and the value of b.[3]

According to the model, exactly T years after monitoring began, the population was 85 000

Making your method clear,

(b)find the value of T, giving your answer to one decimal place.

[2]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

(i)Given that

express, in simplest form, in terms of y and a

(a)a^3x+1[1]

5

(b)

[3]

(ii)(a)Use the substitution p = 9^t to show that the equation

can be rewritten as

[2]

(b)Hence solve

[3]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

The curve C has equation

(a)Find dydx , giving your answer in simplest form.[2]

Given that

• the point P lies on C

• the line with equation y = k – 5x, where k is a constant, is the tangent to C at P

(b)show that the x coordinate of P satisfies the equation

[2]

(c)Hence find the value of k.[4]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

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

Given that

• the point P(4, –5) lies on C

2x² + ax + b • f ′(x) = , where a and b are constants 4 x

• the gradient of the tangent to C at P is 7

(a)show that

[2]

Given also that a + b = –9

(b)find, in simplest form, f(x)[7]

Curve C is transformed to the curve with equation y = f(x – 3)

Given that point P is transformed to the point Q,

(c)state the coordinates of Q.[1]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The curve C has equation

where k is a positive constant.

(a)Sketch the graph of C.

Show on your sketch

• the coordinates of any points of intersection of C with the coordinate axes

• the equation of the horizontal asymptote to C

stating each in terms of k.[3]

The line l has equation y = –kx – 6

Given that l intersects C at 2 distinct points,

(b)find the range of possible values of k.[5]

Figure 1 shows a sketch of a design for a badge. The design consists of a triangle OAB joined to a sector OBC of a circle with centre O In the design

• OB = 3.4 cm

• AB = 1.9 cm

• angle AOB = π6 radians

• angle OAB > π2 radians

Making your method clear,

(a)find the size of angle OAB, giving your answer in radians to 4 significant figures,[3]

(b)find the area of triangle OAB, in cm², giving your answer to 3 significant figures.[2]

Given that the ratio of the area of sector OBC to the area of triangle OAB is 3 : 2

(c)show that angle BOC is 0.462 radians to 3 significant figures.[3]

(d)Hence find the perimeter of the badge, in cm, to the nearest integer.[5]

(a)Express 6x − 274 − x² in the form a + b(x + c)² where a, b and c are constants to

be found.[3]

Figure 2 shows part of a sketch of curve C₁ with equation

Given that the point P is the maximum point on C₁

(b)state the coordinates of P[2]

Figure 2 also shows part of a sketch of curve C₂ with equation

where k is a constant and x is measured in radians.

Given that C₁ and C₂ intersect on the x‑axis at point A and at point B, as shown in Figure 2,

(c)(i)state the x coordinate of B

(ii)state the value of k

(iii)state the period of C₂[3]

The line segment L joins P and B. The region R, shown shaded in Figure 2, is bounded by L, C₁ and C₂

(d)Use inequalities to define R.[5]