Q1.

(a) Find the remainder when f(x) is divided by (x −1).

(2)
(b) Use the factor theorem to show that (x+1) is a factor of f(x).
(2)
(c) Factorise f(x) completely.
(4)
(Total 8 marks)

Q2.

(a) Use the factor theorem to show that (x + 2) is a factor of f(x).

(2)
(b) Factorise f(x) completely.
(4)
(Total 6 marks)

Q3.

f (x) = x4 + 5x3 + ax + b,
where a and b are constants.

The remainder when f(x) is divided by (x − 2) is equal to the remainder when f(x) is divided by (x + 1).

(a) Find the value of a.

(5)
Given that (x + 3) is a factor of f(x),

(b) find the value of b.

(3)
(Total 8 marks)

Q4.

where k is a constant.

(a) Write down the value of f(k).

 (1)
When f(x) is divided by (x − 2) the remainder is 4

(b) Find the value of k.

(2)
(c) Factorise f(x) completely.
(3)
(Total 6 marks)

Q5.

(a) Find the remainder when f(x) is divided by (x − 3).

(2)
Given that (x − 5) is a factor of f(x),

(b) find all the solutions of f(x) = 0.

(5)
(Total 7 marks)

Q6.

f(x) = ax3 − 11x2 + bx + 4, where a and b are constants.

When f(x) is divided by (x − 3) the remainder is 55

When f(x) is divided by (x + 1) the remainder is −9

(a) Find the value of a and the value of b.

(5)
Given that (3x + 2) is a factor of f(x),

(b) factorise f(x) completely.

(4)
(Total 9 marks)

Q7.

f(x) = 6x3 + 3x2 + Ax + B, where A and B are constants.

Given that when f(x) is divided by (x + 1) the remainder is 45,

(a) show that B – A = 48

(2)
Given also that (2x + 1) is a factor of f(x),
(b) find the value of A and the value of B.
(4)
(c) Factorise f(x) fully.
(3)

(Total for question = 9 marks)

Q8.

(a) Find the remainder when

x3 − 2x2 − 4x + 8
is divided by
(i) x − 3,
(ii) x + 2.
(3)
(b) Hence, or otherwise, find all the solutions to the equation

x3 − 2x2 − 4x + 8 = 0.
(4)

(Total 7 marks)

Q9.

Given that k is a negative constant and that the function f(x) is defined by

(a) show that

(3)
(b) Hence find f' (x) , giving your answer in its simplest form.
(3)
(c) State, with a reason, whether f(x) is an increasing or a decreasing function.

Justify your answer.

(2)

(Total for question = 9 marks)

Q10.

(a) Show that

(7)
(b) Hence, or otherwise, find f ′(x) in its simplest form.
(3)

(Total 10 marks)

Q11.

Figure 1

Figure 1 shows a sketch of the curve with equation y = f(x) where

f(x) = (x + 3)2 (x − 1), x

The curve crosses the x-axis at (1, 0), touches it at (−3, 0) and crosses the y-axis at (0, −9)

(a) In the space below, sketch the curve C with equation y = f(x + 2) and state the coordinates of the
points where the curve C meets the x-axis.
(3)
(b) Write down an equation of the curve C.
(1)
(c) Use your answer to part (b) to find the coordinates of the point where the curve C meets the y-axis.
(2)
(Total 6 marks)
Q12.

Figure 1

Figure 1 shows a sketch of the curve C with equation y = f(x).

The curve C passes through the point (−1, 0) and touches the x-axis at the point (2, 0).

The curve C has a maximum at the point (0, 4).

(a) The equation of the curve C can be written in the form

y = x3 + ax2 + bx + c

where a, b and c are integers.

Calculate the values of a, b and c.

(5)

(b) Sketch the curve with equation y = f( ) in the space provided on page 24

Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes.
(3)
(Total 8 marks)
Q13.

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

The curve passes through the origin O and the points A(5, 4) and B(–5, –4).

In separate diagrams, sketch the graph with equation

(a) y = | f (x) |,
(3)
(b) y = f( | x | ) ,
(3)
(c) y = 2f (x +1) .
(4)
On each sketch, show the coordinates of the points corresponding to A and B.

(Total 10 marks)

Q14.

(a) Express

as a single fraction in its simplest form.

(4)
Given that

(b) show that

(2)
(c) Hence differentiate f (x) and find f '(2)
(3)

(Total 9 marks)

Q15.

f(x) = 2x3 − 5x2 + ax + 18

where a is a constant.

Given that (x − 3) is a factor of f(x),

(a) show that a = − 9

(2)
(b) factorise f(x) completely.
(4)
Given that

g(y) = 2(33y) − 5(32y) − 9(3y) + 18

(c) find the values of y that satisfy g(y) = 0, giving your answers to 2 decimal places where appropriate.
(3)
(Total 7 marks)
Q16.

The functions f and g are defined by

(a) State the range of f.

(1)
(b) Find fg(x), giving your answer in its simplest form.
(2)
(c) Find the exact value of x for which f (2x + 3) = 6
(4)
−1
(d) Find f , the inverse function of f, stating its domain.
(3)
(e) On the same axes sketch the curves with equation y = f (x) and y = f−1(x), giving the coordinates of all
the points where the curves cross the axes.
(4)
(Total 14 marks)

Q17.

The function f is defined by

(a) Find f −1(x).

(3)
(b) Find the domain of f −1.
(1)
The function g is defined by

(c) Find fg (x), giving your answer in its simplest form.

(3)
(d) Find the range of fg.
(1)

(Total 8 marks)
Q18.

The function f is defined by

(a) Sketch the graph with equation y = f(x), showing the coordinates of the points where the graph cuts or
meets the axes.
(2)
(b) Solve f(x) = 15 + x.
(3)
The function g is defined by

(c) Find fg(2).

(2)
(d) Find the range of g.
(3)
(Total 10 marks)

Q19.

Figure 2 shows a sketch of part of the curve with equation The curve meets the
coordinate axes at the points where k is a constant and k > 1, as shown in
Figure 2.

On separate diagrams, sketch the curve with equation

(a)
(3)

(b)
(2)
Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the
axes.

Given that

(c) state the range of f,

(1)
(d) find f −1(x),
(3)
−1
(e) write down the domain of f .
(1)
(Total 10 marks)

Q20.

(a) By writing
(2)
(b) Given that

show that

where g(x) is a function of lnx.

(5)

(Total for question = 7 marks)

Q21.

The curve C has equation

(a) Find as a single fraction, simplifying your answer.

(3)
(b) Hence find the exact coordinates of the stationary points of C.
(6)

(Total for question = 9 marks)

Q22.

(i)

(a) Find writing your answer as a single fraction in simplest form.

(4)

b) Hence find the set of values of x for which

(2)
(ii) Given

show that , where C is a constant to be determined.

(You may assume the double angle formulae.)

(4)

(Total for question = 10 marks)

Q23.

Figure 3

The curve shown in Figure 3 has parametric equations

x = t − 4 sin t, y = 1 − 2 cos t, − 2π⁄3 ≤ t ≤ 2π⁄3

The point A, with coordinates (k, 1), lies on the curve.

Given that k > 0

(a) find the exact value of k,

(2)
(b) find the gradient of the curve at the point A.
(4)
There is one point on the curve where the gradient is equal to − ⁄2 1

(c) Find the value of t at this point, showing each step in your working and giving your answer to 4
decimal places.

[Solutions based entirely on graphical or numerical methods are not acceptable.]

(6)

(Total 12 marks)
Q24.

Figure 2 shows a sketch of the curve C with parametric equations

The point P lies on C and has coordinates .

(a) Find the exact value of at the point P.

Give your answer as a simplified surd.
(4)

The point Q lies on the curve C, where =0

(b) Find the exact coordinates of the point Q.

(2)

(Total for question = 6 marks)

Q25. The curve C has equation

y= x3 − 9x + + 30, x>0

(a) Find .
(4)
(b) Show that the point P(4,−8) lies on C.
(2)
(c) Find an equation of the normal to C at the point P, giving your answer in the form ax + by + c = 0 ,
where a, b and c are integers.
(6)

(Total 12 marks)

Q26.

Figure 2

Figure 2 shows a sketch of the curve C with parametric equations

(a) Find an expression for in terms of t.

(3)

Find the coordinates of all the points on C where =0

(5)

(Total 8 marks)
Q27.

A curve C has parametric equations

(a) Find in terms of t.

(4)

The tangent to C at the point where cuts the x-axis at the point P.

(b) Find the x-coordinate of P.

(6)

(Total 10 marks)

Q28.

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

Given that the point P(4, 5) lies on C, find

(a) f(x),
(5)
(b) an equation of the tangent to C at the point P, giving your answer in the form ax + by + c = 0, where a,
b and c are integers.
(4)

(Total 9 marks)
Q29.

Figure 3 shows part of the curve C with parametric equations

The point P lies on C and has coordinates

(a) Find the value of θ at the point P.

(2)
The line l is a normal to C at P. The normal cuts the x-axis at the point Q.

(b) Show that Q has coordinates (k √3, 0), giving the value of the constant k.
(6)
The finite shaded region S shown in Figure 3 is bounded by the curve C, the line x = √3 and the x-axis.
This shaded region is rotated through 2π radians about the x-axis to form a solid of revolution.

(c) Find the volume of the solid of revolution, giving your answer in the form where p and
q are constants.
(7)
(Total 15 marks)
Q30.

The curve C has equation

(a) Find in its simplest form.

(4)
(b) Find an equation of the tangent to C at the point where x = 2
(4)

(Total 8 marks)

Q31.

Given that , find in their simplest form

(a)
(3)

(b)
(3)

(Total for question = 6 marks)

Q32.

Differentiate with respect to x, giving your answer in its simplest form,

(a) x2ln(3x)
(4)

(b)
(5)

(Total 9 marks)
Q33.

The point P is the point on the curve x = 2tan with y-coordinate .

Find an equation of the normal to the curve at P.

(7)

(Total 7 marks)

Q34.

(a) By writing sec x as , show that = sec x tan x.

(3)
Given that y = e2x sec 3x,

(b) find .
(4)

The curve with equation y = e2x sec 3x, − <x< , has a minimum turning point at (a, b).

(c) Find the values of the constants a and b, giving your answers to 3 significant figures.
(4)

(Total 11 marks)

Q35.

(i) Differentiate with respect to x

(a) x2 cos3x
(3)

(b)
(4)
(ii) A curve C has the equation

The point P on the curve has x-coordinate 2. Find an equation of the tangent to C at P in the form ax +
by + c = 0, where a, b and c are integers.
(6)

(Total 13 marks)
Q36.

(i) (a) Show that 2 tan x − cot x = 5 cosec x may be written in the form

a cos2x + b cos x + c = 0
stating the values of the constants a, b and c.
(4)
(b) Hence solve, for 0 ≤ x < 2π, the equation
2 tan x − cot x = 5 cosec x
giving your answers to 3 significant figures.
(4)
(ii) Show that

tan θ + cot θ ≡ λ cosec 2θ, θ± ,

stating the value of the constant λ.
(4)

(Total 12 marks)

Q37.

Water is being heated in an electric kettle. The temperature, θ °C, of the water t seconds
after the kettle is switched on, is modelled by the equation

(a) State the value of θ when t = 0

(1)
Given that the temperature of the water in the kettle is 70°C when t = 40,

(b) find the exact value of λ, giving your answer in the form , where a and b are
integers.
(4)
When t = T, the temperature of the water reaches 100°C and the kettle switches off.

(c) Calculate the value of T to the nearest whole number.

(2)

(Total for question = 7 marks)

Q38.

(a) Differentiate with respect to x,

(i) e3x(sin x + 2 cos x),
(3)
(ii) x3 ln (5x + 2).
(3)

Given that y =

(b) show that

(5)

(c) Hence find and the real values of x for which

(3)

(Total 14 marks)

Q39.

Given that y = 3x2 ,

(a) show that log3y = 1 + 2log3 x

(3)
(b) Hence, or otherwise, solve the equation

1 + 2log3x = log3(28x − 9)
(3)

(Total 6 marks)
Q40.

Find the values of x such that

2log3x − log3(x − 2) = 2
(5)
(Total 5 marks)
Mark Scheme
Q1.

Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Q8.
Q9.
Q10.
Q11.
Q12.
Q13.
Q14.

Q15.
Q16.
Q17.

Q18.
Q19.

Q20.
Q21.
Q22.
Q23.
Q24.
Q25.
Q26.
Q27.

Q28.
Q29.
Q30.
Q31.
Q32.
Q33.
Q34.

Q35.
Q36.
Q37.
Q38.
Q39.
Q40.