WMA11 Pure Mathematics 1 (P1): January 2021 Paper 01

A curve has equation

(a)Find, in simplest form,

[3]

The point P lies on the curve and has x coordinate 12

(b)Find an equation of the normal to the curve at P, writing your answer in the form ax + by + c = 0, where a, b and c are integers to be found.[5]

A tree was planted.

Exactly 3 years after it was planted, the height of the tree was 2 m.

Exactly 5 years after it was planted, the height of the tree was 2.4 m.

Given that the height, H metres, of the tree, t years after it was planted, can be modelled by the equation

where p and q are constants,

(a)find, to 3 significant figures where necessary, the value of p and the value of q.[4]

Exactly T years after the tree was planted, its height was 5 m.

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

Figure 1 shows a sketch of part of the curve C₁ with equation y = 4 cos x°

The point P and the point Q lie on C₁ and are shown in Figure 1.

(a)State

(i)the coordinates of P,

(ii)the coordinates of Q.[3]

The curve C₂ has equation y = 4 cos x° + k, where k is a constant.

Curve C₂ has a minimum y value of –1

The point R is the maximum point on C₂ with the smallest positive x coordinate.

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

The points P and Q, as shown in Figure 2, have coordinates (–2,13) and (4, –5) respectively.

The straight line l passes through P and Q.

(a)Find an equation for l, writing your answer in the form y = mx + c, where m and c are integers to be found.[3]

The quadratic curve C passes through P and has a minimum point at Q.

(b)Find an equation for C.[3]

The region R, shown shaded in Figure 2, lies in the second quadrant and is bounded by C and l only.

(c)Use inequalities to define region R.[2]

Figure 3 shows the plan view of a viewing platform at a tourist site.

The shape of the viewing platform consists of a sector ABCOA of a circle, centre O, joined to a triangle AOD.

Given that

● OA = OC = 6 m ● AD = 14 m ● angle ADC = 0.43 radians ● angle AOD is an obtuse angle ● OCD is a straight line

find

(a)the size of angle AOD, in radians, to 3 decimal places,[3]

(b)the length of arc ABC, in metres, to one decimal place,[2]

(c)the total area of the viewing platform, in m², to one decimal place.[4]

(a)Sketch the curve with equation

[2]

(b)On a separate diagram, sketch the curve with equation

stating the coordinates of the point of intersection with the x-axis and, in terms of k, the equation of the horizontal asymptote.[3]

(c)Find the range of possible values of k for which the curve with equation

does not touch or intersect the line with equation y = 3x + 4[5]

Solutions relying on calculator technology are not acceptable.

(a)Solve the equation

[4]

(b)Solve the equation

[5]

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

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

(b)Expand f (x) to the form

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

The line l, also shown in Figure 4, passes through the y intercept of C and is parallel to the x-axis.

The line l cuts C again at points P and Q, also shown in Figure 4.

(c)Using algebra and showing your working, find the length of line PQ. Write your answer in the form k 3, where k is a constant to be found.

[5]

(i)Find

giving your answer in simplest form.[5]

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

Given

● fʹ(x) = x²+ ax + b where a and b are constants

● the y intercept of C is –8

● the point P(3, –2) lies on C

● the gradient of C at P is 2

find, in simplest form, f(x).[6]