Functions [140 marks]
1. [Maximum mark: 5]
The following diagram shows the graph of y = f (x). The graph has
a horizontal asymptote at y = −1. The graph crosses the x-axis at
x = −1 and x = 1, and the y-axis at y = 2.
2
On the following set of axes, sketch the graph of y = [f (x)] + 1
, clearly showing any asymptotes with their equations and the
coordinates of any local maxima or minima.
[5]
2. [Maximum mark: 18]
(a) Express −3 + √3i in the form re iθ , where r > 0 and
−π < θ ⩽ π. [5]
Let the roots of the equation z 3 = −3 + √ 3i be u, v and w.
(b) Find u, v and w expressing your answers in the form re iθ ,
where r > 0 and −π < θ ⩽ π. [5]
On an Argand diagram, u, v and w are represented by the points U, V and W
respectively.
(c) Find the area of triangle UVW. [4]
(d) By considering the sum of the roots u, v and w, show that
= 0.
5π 7π 17π
cos + cos + cos
18 18 18
[4]
3. [Maximum mark: 8]
The function f is defined by f (x) = e
2x
− 6e
x
+ 5, x ∈ R, x ⩽ a. The
graph of y = f (x) is shown in the following diagram.
(a) Find the largest value of a such that f has an inverse function. [3]
(b) For this value of a, find an expression for f −1 (x), stating its
domain. [5]
4. [Maximum mark: 21]
A function f is defined by f (x) =
3
2
x +2
, x ∈ R.
(a) Sketch the curve y = f (x), clearly indicating any asymptotes
with their equations and stating the coordinates of any points
of intersection with the axes. [4]
The region R is bounded by the curve y = f (x), the x-axis and the lines
x = 0 and x = √ 6. Let A be the area of R.
(b) √ 2π
Show that A =
2
. [4]
The line x = k divides R into two regions of equal area.
(c) Find the value of k. [4]
Let m be the gradient of a tangent to the curve y = f (x).
(d) Show that m = −
6x
.
2
(x +2)
2
[2]
(e)
Show that the maximum value of m is .
27 2
√
32 3 [7]
5. [Maximum mark: 20]
(a) Use the binomial theorem to expand (cos θ + i sin θ)
4
. Give your answer in the form a + bi where a and b are
expressed in terms of sin θ and cos θ. [3]
(b) Use de Moivre’s theorem and the result from part (a) to show
4 2
cot θ−6 cot θ+1
that cot 4θ = 3
. [5]
4 cot θ−4 cot θ
(c) Use the identity from part (b) to show that the quadratic
π 3π
equation x 2 − 6x + 1 = 0 has roots cot
2
8
and cot 2 8
. [5]
(d) Hence find the exact value of cot 2
3π
. [4]
8
(e) Deduce a quadratic equation with integer coefficients, having
π 3π
roots cosec 2 8
and cosec 2 8
. [3]
6. [Maximum mark: 9]
2
Let f (x) , x ∈ R, x ≠ −2.
2x −5x−12
=
x+2
(a) Find all the intercepts of the graph of f (x) with both the x
and y axes. [4]
(b) Write down the equation of the vertical asymptote. [1]
(c) As x → ±∞ the graph of f (x) approaches an oblique
straight line asymptote.
Divide 2x 2 − 5x − 12 by x + 2 to find the equation of this
asymptote. [4]
7. [Maximum mark: 9]
2
Let f (x) , x ∈ R, x ≠ −1.
x −10x+5
=
x+1
(a) Find the co-ordinates of all stationary points. [4]
(b) Write down the equation of the vertical asymptote. [1]
(c) With justification, state if each stationary point is a minimum,
maximum or horizontal point of inflection. [4]
8. [Maximum mark: 8]
Let f (x) = 2
2x+6
x +6x+10
, x ∈ R.
(a) Show that f (x) has no vertical asymptotes. [3]
(b) Find the equation of the horizontal asymptote. [2]
(c) 1
Find the exact value of ∫ f (x) dx, giving the answer in the
0
form ln q, q ∈ Q. [3]
9. [Maximum mark: 5]
(a) Solve 2x 2 − 15x + 18 < 0. [3]
(b) The function f is defined by f (x) = √ 2x − 15x + 18,
2
where x ∈ R , x ≤ k.
Find the greatest value of k for which f −1 exists, justifying
your answer. [2]
10. [Maximum mark: 7]
π π
Consider the function f (x) = sec (x −
4
), for 0 ≤ x ≤
2
.
(a) Determine the range of f . [3]
The region bounded by the graph of y = f (x), the x-axis and the lines
π
x = 0 and x =
2
is rotated 2π radians about the x-axis.
(b) Find the volume of revolution generated. [4]
11. [Maximum mark: 8]
Points A and B lie on the circumference of a circle of radius r cm with centre
at O.
The sector OAB is shown on the following diagram. The angle AÔB is
denoted as θ and is measured in radians.
The perimeter of the sector is 10 cm and the area of the sector is 6. 25 cm
2
.
(a) Show that 4r 2 − 20r + 25 = 0. [4]
(b) Hence, or otherwise, find the value of r and the value of θ. [4]
12. [Maximum mark: 6]
The graph of y = f (|x|) for −6 ≤ x ≤ 6 is shown in the following
diagram.
(a) On the following axes, sketch the graph of y = |f (|x|)| for
−6 ≤ x ≤ 6.
[2]
It is given that f is an odd function.
(b) On the following axes, sketch the graph of y = f (x) for
−6 ≤ x ≤ 6.
[2]
4
It is also given that∫ 0 f (|x|) d x = 1. 6.
(c) Write down the value of
(c.i) ∫
0
f (x) d x; [1]
−4
(c.ii) ∫
4
(f (|x|) + f (x)) d x. [1]
−4
13. [Maximum mark: 16]
Consider the function f (x) , x ≠ 2.
4x+2
=
x−2
(a) Sketch the graph of y = f (x). On your sketch, indicate the
values of any axis intercepts and label any asymptotes with
their equations. [5]
(b) Write down the range of f . [1]
Consider the function g(x) = x
2
+ bx + c. The graph of g has an axis of
symmetry at x = 2.
1
The two roots of g(x) = 0 are −
2
and p, where p ∈ Q.
(c) Show that p =
9
. [1]
2
(d) Find the value of b and the value of c. [3]
(e) Find the y-coordinate of the vertex of the graph of y = g(x). [2]
(f ) Find the product of the solutions of the equation
f (x) = g(x). [4]
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