SUBJECT
GRADE
EXAMINER
Advanced Programme
Mathematics Paper 1
12
Mrs MH Povall
NAME
DATE
9 July 2015
MARKS
200
MODERATORS Mrs Serafino
DURATION
2 hour
TEACHER
QUESTION NO
1
2
3
4
5
6
7
8
DESCRIPTION
Proof by Induction
Algebra including logs, partial fractions,
complex numbers and complex roots
All graphs including the graph of the
absolute value
Discontinuity and differentiability
Differentiation and applications of
differentiation
Graphs of polynomials and rational
functions
MAXIMUM
MARK
13
32
15
16
39
40
Integration and trigonometry
33
Problems involving sectors and arc
lengths
12
TOTAL
200
INSTRUCTIONS:
1. Write your name and your Mathematics teacher’s name on this test.
2. Answer all questions in the answer booklet provided.
3. Show all working out , as answers only will not guarantee you full marks.
ACTUAL MARK
QUESTION 1
13 MARKS
Use mathematical induction to prove that:
72𝑛−1 + 1 is divisible by 8 for all 𝑛 ∈ ℕ.
QUESTION 2
32 MARKS
2.1. Solve for x: 2 log 3 𝑥 + log 𝑥 27 = log 3 243
2.2.
(7)
Given:𝑓(𝑥) = 2𝑙𝑜𝑔3 (𝑥 − 1)
a) Determine the equation of f 1 ( x) , the inverse of f ( x) in the form f 1 ( x)  ....
(4)
1
b) Sketch the graphs of f ( x) and f ( x) on the same axes, clearly labelling intercepts
with the axes and asymptotes.
(8)
2.3. If g ( x) 
2x
, decompose g ( x ) into partial fractions.
( x  3)2
(6)
2.4. Find an equation, with a leading coefficient of 1 and with the lowest possible
degree, which has the following roots: 𝑥 = 2 − 𝑖 and 𝑥 = 5
Grade 12
2 of 7
(7)
Advanced Programme Mathematics Paper 1
QUESTION 3
15 MARKS
The sketch shows the graph of g ( x)   x 2  2 x  2 and f ( x)  x , which intersect
at A (-2; 2) and B.
4
y
3
A
2
1
B
x
−6
−4
−2
2
4
6
−1
−2
−3
−4
3.1
Determine the x - value of the point B (leave you answer in surd form)
(4)
3.2
Draw a sketch graph of h( x)  g ( x)
(3)
3.3
Find all the solutions for 𝑥 if h( x)  f ( x) correct to 2 decimal places.
(8)
QUESTION 4
16 MARKS
Consider the graph of g(x) alongside:

 x 2  6 x  5 if x  3

g ( x)   2 x  4
if - 3  x  3
 2
 x  4
if x  3
 3
-3
3
It is given that g(x) is continuous for x  R; x  3
4.1. Without proof, name the discontinuity at x = -3.
(2)
4.2 Discuss the differentiability of 𝑔(𝑥) at 𝑥 = −3 and 𝑥 = 3. Provide reasons for your
answers.
(4)
4.3 Draw a sketch graph of g ' ( x ) . Indicate all critical points on the axes clearly.
Grade 12
3 of 7
(10)
Advanced Programme Mathematics Paper 1
QUESTION 5
39 MARKS
5.1. Find the following limit, if it exists:
lim
x 1
x2  8  3
x2 1
(6)
5.2. Find the derivatives of the following functions (you do not need to fully simplify
your answers)
(a) 𝑓(𝑥) = 𝑠𝑖𝑛2 (3𝑥)
(4)
(b) g ( x)  ( x 2  2) 2  4 x 2
(8)
3
(b) ℎ(𝑥) =
√1−2𝑥
𝑥3
(6)
5.3 Given 𝑓(𝑥) = (1 + 7𝑥)−1
Find a formula for the nth derivative of f (x) .
(5)
5.4. Below is the graph of a lemniscate which has the equation
2(𝑥 2 + 𝑦 2 )2 = 25(𝑥 2 − 𝑦 2 ) use implicit differentiation to find the gradient of
the tangent at the point A(3; 1).
(10)
4
y
3
2
A (3; 1)
1
x
−6
−4
−2
2
4
6
−1
−2
−3
−4
Grade 12
4 of 7
Advanced Programme Mathematics Paper 1
QUESTION 6
Given :
40 MARKS
g ( x) 
x3  x 2  x  3
( x  1) 2
6.1 a) Determine the y-intercept of g.
(1)
b) Show that g has an x-intercept on x  [2;3] .
(2)
c) Use Newton’s method with x0 = 2 to find this x-intercept to an accuracy of 4
decimal places.
(8)
6.2 Show, using polynomial long division only, that g can be written as
4
.
g ( x)  x  1 
( x  1) 2
(4)
6.3 Hence write down the equations of the oblique and vertical asymptotes.
(2)
6.4 a) Use your result in 6.2 to find a simple expression for g ' ( x) and g ' ' ( x) .
(5)
b) Hence determine the only turning point of g (x) and use g' ' (x) to classify it.
(6)
c) Use g ' ( x ) to determine the values of x for which g is increasing.
(6)
6.5 Draw a neat graph of 𝑔(𝑥).
Grade 12
(6)
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Advanced Programme Mathematics Paper 1
QUESTION 7
33 MARKS
7.1. Determine the following integrals: (you do not have to simplify your answers)
(a) ∫ 𝑠𝑖𝑛2𝑥. 𝑠𝑖𝑛 5𝑥𝑑𝑥
(b)
7.2.
(6)
𝑥
∫ √4−𝑥 2 𝑑𝑥
(7)
Use integration by parts to calculate the value of:
∫ 𝑥𝑐𝑜𝑠2𝑥𝑑𝑥
7.3.
(7)


a) Show that Dx cos 4 x  sin 4 x  2 sin 2 x .
(6)
b) Hence show that this implies that:
cos 4 x  sin 4 x  cos 2 x  C
(4)
c) Now prove that in fact cos 4 x  sin 4 x  cos 2 x .
(3)
Grade 12
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Advanced Programme Mathematics Paper 1
QUESTION 8
12 MARKS
The diagram shows the cross-section of a wooden log, radius 50 cm, floating in water.
10cm of the log is above the water (diagram is not drawn to scale).
10cm
θ
8.1. Show that 𝜃 = 1.287 radians
(4)
8.2. What is the area of the cross-section that is below the water?
(8)
Grade 12
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Advanced Programme Mathematics Paper 1