WMA11 Pure Mathematics 1 (P1): Oct/Nov 2022 Paper 01

The curve C has equation

(a)Find dydx , giving your answer in simplest form.[3]

The point R  2 , 132  lies on C.  

(b)Find the equation of the tangent to C at the point R. Write your answer in the form ax + by + c = 0, where a, b and c are integers to be found.[3]

Given that

where a and b are constants,

(a)find the value of a and the value of b.[2]

(b)Hence find

writing each term in simplest form.[5]

The share price of a company is monitored.

Exactly 3 years after monitoring began, the share price was £1.05

Exactly 5 years after monitoring began, the share price was £1.65

The share price, £V, of the company is modelled by the equation

where t is the number of years after monitoring began and p and q are constants.

(a)Find the value of p and the value of q.[3]

Exactly T years after monitoring began, the share price was £2.50

(b)Find the value of T, according to the model, giving your answer to one decimal place.[2]

(a)Solve

[4]

(b)Hence solve

giving your answer in simplified surd form.[2]

Solutions relying entirely on calculator technology are not acceptable.

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

Given that

12 x • f ′(x) = + – 4

• the point P(9, 8) lies on C

(a)find, in simplest form, f(x)[5]

The line l is the normal to C at P

(b)Find the coordinates of the point at which l crosses the y‑axis.[4]

(a)Given that k is a positive constant such that 0 < k < 4 sketch, on separate axes, the graphs of

(i)y = (2x – k)(x + 4)²

(ii)y = kx₂

showing the coordinates of any points where the graphs cross or meet the coordinate axes, leaving coordinates in terms of k, where appropriate.[5]

(b)State, with a reason, the number of roots of the equation

[1]

Figure 1 shows the curve with equation y = f(x).

The points P(–4, 6), Q(–1, 6), R(2, 6) and S(3, 6) lie on the curve.

(a)Using Figure 1, find the range of values of x for which

[3]

(b)State the largest solution of the equation

[1]

(c)(i)Sketch the curve with equation y = f(–x).

On your sketch, state the coordinates of the points to which P, Q, R and S are transformed.

(ii)Hence find the set of values of x for which

[4]

Figure 2 shows the plan view of a design for a pond.

The design consists of a sector AOBX of a circle centre O joined to a quadrilateral AOBC.

• BC = 8 m

• OA = OB = 3 m

• angle AOB is 2π3 radians

• angle BCA is π6 radians

(a)Calculate (i) the exact area of the sector AOBX,

(ii)the exact perimeter of the sector AOBX.[5]

(b)Calculate the exact area of the triangle AOB.[2]

(c)Show that the length AB is 3 3 m.[2]

(d)Find the total surface area of the pond. Give your answer in m² correct to 2 significant figures.[5]

Figure 3 shows a sketch of the curve C with equation

(a)Write 12 x²– 10x + 22 in the form

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

The point M is the minimum turning point of C, as shown in Figure 3.

(b)Deduce the coordinates of M[2]

The line l is the normal to C at the point P, as shown in Figure 3.

Given that l has equation y = k – 18 x, where k is a constant,

(c)(i)find the coordinates of P

(ii)find the value of k[6]

Figure 4 is a copy of Figure 3. The finite region R, shown shaded in Figure 4, is bounded by l, C and the line through M parallel to the y‑axis.

(d)Identify the inequalities that define R.[3]