WMA11 Pure Mathematics 1 (P1): January 2024 Paper 01

Find

writing your answer in simplest form.[5]

The triangle ABC is such that

• AB = 15 cm

• AC = 25 cm

• angle BAC = θ°

• area triangle ABC = 100 cm²

(a)Find the value of sin θ°[2]

Given that θ > 90

(b)find the length of BC, in cm, to 3 significant figures.[3]

The curve C has equation

(a)Find dydx writing your answer in simplest form.[4]

The point P (2, 4) lies on C.

(b)Find an equation for the tangent to C at P writing your answer in the form ax + by + c = 0, where a, b and c are integers.[3]

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

(a)By substituting p = 2^x, show that the equation

can be written in the form

[3]

(b)Hence solve

[3]

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

The straight line l₁ , shown in Figure 1, passes through the points P(–2, 9) and Q(10, 6).

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

The straight line l₂ passes through the origin O and is perpendicular to l₁

The lines l₁ and l₂ meet at the point R as shown in Figure 1.

(b)Find the coordinates of R[4]

(c)Find the exact area of triangle OPQ.[3]

Figure 2 shows a plot of part of the curve C₁ with equation

with x being measured in degrees.

The point P, shown in Figure 2, is a minimum point on C₁

(a)State the coordinates of P[2]

The point Q lies on a different curve C₂

Given that point Q

• is a maximum point on the curve

• is the maximum point with the smallest x coordinate, x > 0

(b)find the coordinates of Q when

(i)C₂ has equation y = 5 cos x – 2

(ii)C₂ has equation y = –5 cos x[4]

(a)Sketch the graph of the curve C with equation

where k is a positive constant.

Show on your sketch

• the coordinates of any points where C cuts the coordinate axes

• the equation of the vertical asymptote to C[4]

Given that the straight line with equation y = 9 – x does not cross or touch C

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

Figure 3 shows a sketch of the plan view of a platform.

The plan view of the platform consists of a sector DOC of a circle centre O joined to a sector AOBEA of a different circle, also with centre O.

Given that

• angle AOB = 0.8 radians

• arc length CD = 9 m

• DA : AO = 3 : 5

(a)show that AO = 7.03 m to 3 significant figures.[3]

(b)Find the perimeter of the platform, in m, to 3 significant figures.[3]

(c)Find the total area of the platform, giving your answer in m² to the nearest whole number.[3]

The curve C₁ has equation y = f (x).

Given that

• f (x) is a quadratic expression

• C₁ has a maximum turning point at (2, 20)

• C₁ passes through the origin

(a)sketch a graph of C₁ showing the coordinates of any points where C₁ cuts the coordinate axes,[2]

(b)find an expression for f (x).[3]

The curve C₂ has equation y = x(x²– 4)

Curve C₁ and C₂ meet at the origin, and at the points P and Q

Given that the x coordinate of the point P is negative,

(c)using algebra and showing all stages of your working, find the coordinates of P[5]

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

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

Given that

• the point P 2, 8 2 lies on C

• f'(x)  4 x³  kx₂ where k is a constant

• f''(x) = 0 at P

(a)find the exact value of k,[4]

(b)find f (x), giving your answer in simplest form.[4]