Topic 2 Part 1 [373 marks]

The functions f and g are defined by f(x) = a x2 + bx + c, x ∈ R and g(x) = p sin x + qx + r, x ∈ R where
a, b, c, p, q, r are real constants.

1a. Given that f is an even function, show that b

= 0. [2 marks]

1b. Given that g is an odd function, find the value of r.

[2 marks]

The function h is both odd and even, with domain R. [2 marks]

1c.
Find h(x).

3 −2 1
R
 3x−2 1
A function f is defined by f(x) = 2x−1
, x ∈ R, x ≠ 2
.

2a. Find an expression for f −1 (x). [4 marks]

B
2b. Given that f(x) can be written in the form f(x) = A + 2x−1
, find the values of the constants A and B. [2 marks]

3x−2
2c. Hence, write down ∫ 2x−1 dx. [1 mark]

5 4 3 2 R
 Let p(x) = 2x5 + x4 − 26x3 − 13x2 + 72x + 36, x ∈ R.

3a.
For the polynomial equation p(x) = 0, state [3 marks]

(i) the sum of the roots;

(ii) the product of the roots.

3b.
A new polynomial is defined by q(x) = p(x + 4). [2 marks]

Find the sum of the roots of the equation q(x) = 0.

The functions f and g are defined by f(x) = 2x + π5 , x ∈ R and g(x) = 3 sin x + 4, x ∈ R.

4a. Show that g ∘ f(x) = 3 sin(2x + π5 ) + 4. [1 mark]

∘
4b.
Find the range of g ∘ f. [2 marks]

4c. Given that g ∘ f ( 3π

20
) = 7, find the next value of x, greater than 3π
20
, for which g ∘ f(x) = 7. [2 marks]

The graph of y = g ∘ f(x) can be obtained by applying four transformations to the graph of y = sin x. State [4 marks]
4d.
what the four transformations represent geometrically and give the order in which they are applied.

3 R
 Let y(x) = xe3x , x ∈ R.

dy [2 marks]
5a. Find dx .

dn y [7 marks]
5b. Prove by induction that
dxn
= n3n−1 e3x + x3n e3x for n ∈ Z+ .

Find the coordinates of any local maximum and minimum points on the graph of y(x). [5 marks]
5c.
Justify whether any such point is a maximum or a minimum.

( )
 Find the coordinates of any points of inflexion on the graph of y(x). Justify whether any such point is a point of [5 marks]
5d.
inflexion.

6a. State the set of values of a for which the function x ↦ loga x exists, for all x ∈ R+ . [2 marks]

6b. Given that [6 marks]

logx y = 4logy x, find all the possible expressions of y as a function of x.

3
R
 3x
The function f is defined by f(x) = x−2
, x ∈ R, x ≠ 2.

7a. Sketch the graph of y = f(x), indicating clearly any asymptotes and points of intersection with the x and y axes. [4 marks]

7b. Find an expression for f −1 (x). [4 marks]

7c. Find all values of x for which f(x) = f −1 (x). [3 marks]

3
 3
7d. Solve the inequality |f(x)| < . [4 marks]
2

3
7e. Solve the inequality f (|x|) < 2
. [2 marks]

Consider the functions f(x) = tan x, 0 ≤ x < π

2
and g(x) = x+1
x−1
, x ∈ R, x ≠ 1.

8a.
Find an expression for g ∘ f(x), stating its domain. [2 marks]

sin +cos
 sin x+cos x
8b. Hence show that g ∘ f(x) = sin x−cos x
. [2 marks]

8c.
Let y = g ∘ f(x), find an exact value for [6 marks]
dy –
dx
at the point on the graph of y = g ∘ f(x) where x = π
6
, expressing your answer in the form a + b√3, a, b ∈ Z.

π
8d.
Show that the area bounded by the graph of y = g ∘ f(x), the x-axis and the lines x = 0 and x = 6
is [6 marks]
–
ln(1 + √3).

3 2
 The cubic equation x3 + px2 + qx + c = 0, has roots
α, β, γ. By expanding (x − α)(x − β)(x − γ) show that

9a.
(i) p = −(α + β + γ); [3 marks]

(ii) q = αβ + βγ + γα;
(iii) c = −αβγ.

9b.
It is now given that p = −6 and q = 18 for parts (b) and (c) below. [5 marks]

(i) In the case that the three roots α, β, γ form an arithmetic sequence, show that one of the roots is 2.
(ii) Hence determine the value of c.

In another case the three roots α, β, γ form a geometric sequence. Determine the value of [6 marks]
9c.
c.

1 −−−−−
 1
10a. Show that √n+√n+1 = √−
n−−
+−−
1 − √−
n where n ≥ 0, n ∈ Z. [2 marks]

– 1
10b. Hence show that √2 − 1 < . [2 marks]
√2

r=n
> √−
1 [9 marks]
10c. Prove, by mathematical induction, that ∑ n for n ≥ 2, n ∈ Z.
√r
r=1

3 R ′
11. A function f is defined by f(x) = x3 + ex + 1, x ∈ R. By considering f ′ (x) determine whether f is a one-to- [4 marks]
one or a many-to-one function.

12a.
A function f is defined by f(x) = (x + 1)(x − 1)(x − 5), x ∈ R. [3 marks]

Find the values of x for which f(x) < |f(x)|.

12b. A function g is defined by g(x) = x2 + x − 6, x ∈ R. [7 marks]

1
Find the values of x for which g(x) < g(x)
.

[5 marks]
13. When the polynomial
3x3 + ax + b is divided by
(x − 2), the remainder is 2, and when divided by
(x + 1), it is 5. Find the value of a and the value of b.
 [6 marks]
14. The equation
5x3 + 48x2 + 100x + 2 = a has roots
r1 ,
r2 and
r3 .
Given that
r1 + r2 + r3 + r1 r2 r3 = 0, find the value of a.

[4 marks]
15. One root of the equation
x2 + ax + b = 0 is
2 + 3i where
a, b ∈ R . Find the value of
a and the value of
b.

Let
f(x) = x(x + 2)6 .

[5 marks]
16a. Solve the inequality
f(x) > x.

[5 marks]
16b. Find
∫ f(x)dx.

Let
e2x +1
f(x) = ex −2
.

[4 marks]
17a. Find the equations of the horizontal and vertical asymptotes of the curve
y = f(x).

[8 marks]
17b. (i) Find
f ′ (x).
(ii) Show that the curve has exactly one point where its tangent is horizontal.
(iii) Find the coordinates of this point.

[4 marks]
17c. Find the equation of
L1, the normal to the curve at the point where it crosses the y-axis.

The line
L2 is parallel to
L1 and tangent to the curve
y = f(x).

[5 marks]
17d. Find the equation of the line
L2.
 [18 marks]
18. Let
f(x) = |x| − 1.
(a) The graph of
y = g(x) is drawn below.

(i) Find the value of

(f ∘ g)(1).
(ii) Find the value of
(f ∘ g ∘ g)(1).
(iii) Sketch the graph of
y = (f ∘ g)(x).
(b) (i) Sketch the graph of
y = f(x).
(ii) State the zeros of f.
(c) (i) Sketch the graph of
y = (f ∘ f)(x).
(ii) State the zeros of
f ∘ f.
(d) Given that we can denote


f ∘ f ∘ f ∘ … ∘ f as
n times

fn,
(i) find the zeros of
f 3;
(ii) find the zeros of
f 4;
(iii) deduce the zeros of
f 8.
(e) The zeros of
f 2n are
a1 , a2 , a3 , … , aN .
(i) State the relation between n and N;
(ii) Find, and simplify, an expression for
N
∑ |ar | in terms of n.
r=1
 [6 marks]
19. The roots of a quadratic equation
2x2 + 4x − 1 = 0 are
α and
β.
Without solving the equation,
(a) find the value of
α2 + β2 ;
(b) find a quadratic equation with roots
α2 and
β 2.

[2 marks]
20a. Sketch the graph of
y= ∣∣cos( x4 )∣∣ for
0 ⩽ x ⩽ 8π.

[3 marks]
20b. Solve
∣∣cos( x )∣∣ = 1
4 2
for
0 ⩽ x ⩽ 8π.

The function f is defined by

f(x) = {
1 − 2x, x ≤ 2
3
4
(x − 2)2 − 3, x > 2

[2 marks]
21a. Determine whether or not
fis continuous.

[4 marks]
21b. The graph of the function
g is obtained by applying the following transformations to the graph of
f:
a reflection in the
y–axis followed by a translation by the vector
2
( ).
0
Find
g(x).

Consider the following functions:

h(x) = arctan(x), x ∈ R

g(x) = x1 ,
x ∈ R,
x≠0

[2 marks]
22a. Sketch the graph of
y = h(x).

[2 marks]
22b. Find an expression for the composite function
h ∘ g(x) and state its domain.
 [7 marks]
22c. Given that
f(x) = h(x) + h ∘ g(x),
(i) find
f ′ (x) in simplified form;
(ii) show that
π
f(x) = 2
for
x > 0.

[3 marks]
22d. Nigel states that
f is an odd function and Tom argues that
f is an even function.
(i) State who is correct and justify your answer.
(ii) Hence find the value of
f(x) for
x < 0.

The graphs of
y = x2 e−x and
y = 1 − 2 sin x for
2 ⩽ x ⩽ 7 intersect at points A and B.
The x-coordinates of A and B are
xA and
xB.

[2 marks]
23a. Find the value of
xA and the value of
xB.

[3 marks]
23b. Find the area enclosed between the two graphs for
xA ⩽ x ⩽ xB .

1
The function f is defined by f(x) = x
, x ≠ 0.
The graph of the function y = g(x) is obtained by applying the following transformations to
the graph of y = f(x) :
−3 0
a translation by the vector ( ) ; a translation by the vector ( ) ;
0 1

Find an expression for g(x). [2 marks]

24a.
 State the equations of the asymptotes of the graph of g . [2 marks]
24b.

The quadratic equation 2x2 − 8x + 1 = 0 has roots α and β.

Without solving the equation, find the value of [2 marks]

25a.
(i) α + β;
(ii) αβ.

2 2
25b. Another quadratic equation x
2 + px + q = 0, p, q ∈ Z has roots α
and .
β
[4 marks]

Find the value of p and the value of q.

R
 The function f is defined as f(x) = e3x+1 , x ∈ R.

26a. (i) Find f −1 (x). [4 marks]

(ii) State the domain of f −1 .

26b. The function g is defined as g(x) = ln x, x ∈ R+ . [5 marks]

The graph of y = g(x) and the graph of y = f −1 (x) intersect at the point P .
Find the coordinates of P .

26c.
The graph of y = g(x) intersects the x-axis at the point Q. [3 marks]

Show that the equation of the tangent T to the graph of y = g(x) at the point Q is y = x − 1.

= ( )
26d.
A region R is bounded by the graphs of y = g(x), the tangent T and the line x = e. [5 marks]

Find the area of the region R .

26e.
A region R is bounded by the graphs of y = g(x), the tangent T and the line x = e. [6 marks]

(i) Show that g(x) ≤ x − 1, x ∈ R+.

1
(ii) By replacing x with
x
in part (e)(i), show that
x−1
x
≤ g(x), x ∈ R+ .

27. Consider p(x) = 3x3 + ax + 5a, a ∈ R. [6 marks]

The polynomial p(x) leaves a remainder of −7 when divided by (x − a).

Show that only one value of a satisfies the above condition and state its value.
 The seventh, third and first terms of an arithmetic sequence form the first three terms of a geometric sequence.

The arithmetic sequence has first term

a and non-zero common difference
d.

a
28a.
Show that d = 2
. [3 marks]

The seventh term of the arithmetic sequence is 3. The sum of the first n terms in the arithmetic sequence [6 marks]
28b.
exceeds the sum of the first
n terms in the geometric sequence by at least 200.
Find the least value of
n for which this occurs.
 Compactness is a measure of how compact an enclosed region is.

The compactness,
C , of an enclosed region can be defined by C = 4A2 , where
πd
A is the area of the region and
d is the maximum distance between any two points in the region.
For a circular region, C = 1.
Consider a regular polygon of
n sides constructed such that its vertices lie on the circumference of a circle of diameter
x units.

29a. If n > 2 and even, show that C = n

2π
sin 2π
n
. [3 marks]

2π
n sin [4 marks]
29b. If n > 1 and odd, it can be shown that C = n
.
π(1+cos πn )

Find the regular polygon with the least number of sides for which the compactness is more than 0.99.
 The vertical cross-section of a container is shown in the following diagram.

The curved sides of the cross-section are given by the equation y = 0.25x2 − 16. The horizontal cross-sections are
circular. The depth of the container is 48 cm.

30a. If the container is filled with water to a depth of h cm, show that the volume, V cm3 , of the water is given by [3 marks]

4π ( h2
2
V = + 16h).

30b. Once empty, water is pumped back into the container at a rate of 8.5 cm3 s−1 . At the same time, water [3 marks]
250 √h
continues leaking from the container at a rate of cm3 s−1 .
π(h+16)

Using an appropriate sketch graph, determine the depth at which the water ultimately stabilizes in the container.
 A random variable X has probability density function
⎧0
⎪
⎪
x<0
⎪
⎪1 0≤x<1
f(x) = ⎨ 2
⎪
⎪
⎪
1
1≤x<3
⎩4
⎪
0 x≥3

31a.
Sketch the graph of y = f(x). [1 mark]

Find the cumulative distribution function for X. [5 marks]

31b.

Find the interquartile range for X. [3 marks]

31c.
 The probability density function of a continuous random variable X is given by

⎧
⎪
⎪
0, x < 0
⎪
⎪ sin x
f(x) = ⎨ 4
, 0≤x≤π
.
⎪
⎪
⎪
⎩
⎪
a(x − π), π < x ≤ 2π
0, 2π < x

32a.
Sketch the graph y = f(x). [2 marks]

Find P(X ≤ π). [2 marks]

32b.

1
32c. Show that a = π2
. [3 marks]
 Write down the median of X. [1 mark]
32d.

Calculate the mean of X. [3 marks]

32e.

Calculate the variance of X. [3 marks]

32f.

π 3π
32g. Find P ( 2 ≤X≤ ). [2 marks]
2
 π 3π
32h. Given that 2 ≤X≤ 2
find the probability that π ≤ X ≤ 2π. [4 marks]

33a. (i) Use the binomial theorem to expand (cos θ + i sin θ)5 . [6 marks]

(ii) Hence use De Moivre’s theorem to prove

sin 5θ = 5cos4 θ sin θ − 10cos2 θsin3 θ + sin5 θ.

(iii) State a similar expression for cos 5θ in terms of cos θ and sin θ.

Let z = r(cos α + i sin α), where α is measured in degrees, be the solution of z 5 − 1 = 0 which has the smallest
positive argument.

Find the value of r and the value of α . [4 marks]

33b.

4 2
 4
33c. Using (a) (ii) and your answer from (b) show that 16sin α − 20sin2 α + 5 = 0. [4 marks]

33d. Hence express [5 marks]

√a+b √c
sin 72∘ in the form d
where a, b, c, d ∈ Z.

34a. Sketch the graph of y = (x − 5)2 − 2 |x − 5| − 9, for 0 ≤ x ≤ 10. [3 marks]

(x − 5)2 − 2 |x − 5| − 9 = 0
 The graph of y = ln(5x + 10) is obtained from the graph of y = ln x by a translation of a units in the direction of the
x-axis followed by a translation of b units in the direction of the y-axis.

35a. Find the value of a and the value of b. [4 marks]

35b. The region bounded by the graph of y = ln(5x + 10), the x-axis and the lines x = e and x = 2e, is rotated [2 marks]
through 2π radians about the x-axis. Find the volume generated.

Farmer Bill owns a rectangular field, 10 m by 4 m. Bill attaches a rope to a wooden post at one corner of his field, and
attaches the other end to his goat Gruff.

Given that the rope is 5 m long, calculate the percentage of Bill’s field that Gruff is able to graze. Give your [4 marks]
36a.
answer correct to the nearest integer.
36b. Bill replaces Gruff’s rope with another, this time of length a, 4 < a < 10, so that Gruff can now graze exactly [4 marks]
one half of Bill’s field.

Show that a satisfies the equation

4 −−−−−−
a2 arcsin( ) + 4√a2 − 16 = 40.
a

36c. Find the value of a . [2 marks]

A particle moves in a straight line, its velocity v ms−1 at time t seconds is given by v = 9t − 3t2 , 0 ≤ t ≤ 5.
At time t = 0, the displacement s of the particle from an origin O is 3 m.

37a. Find the displacement of the particle when t = 4. [3 marks]

0≤ ≤5
 Sketch a displacement/time graph for the particle, 0 ≤ t ≤ 5, showing clearly where the curve meets the axes [5 marks]
37b.
and the coordinates of the points where the displacement takes greatest and least values.

37c. For [3 marks]

2πt
t > 5, the displacement of the particle is given by s = a + b cos 5
such that
s is continuous for all
t ≥ 0.
Given further that s = 16.5 when t = 7.5, find the values of a and b.

37d. For
[4 marks]
2πt
t > 5, the displacement of the particle is given by s = a + b cos 5
such that
s is continuous for all
t ≥ 0.
Find the times t1 and t2 (0 < t1 < t2 < 8) when the particle returns to its starting point.

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