ST. JAGO HIGH SCHOOL
INTERNAL ASSESSMENT – MODULE 3
LOWER SIXTH FORM
MAY 2021
PURE MATHEMATICS UNIT 1
1 hour 10 minutes
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1. This examination paper consists of THREE questions for a total of 60 marks.
2. Answer ALL questions.
3. Write your solutions, with full working, on the answer sheets provided.
4. Unless otherwise stated in the question, any numerical answer that is not exact
MUST be written correct to THREE significant figures.
Examination Materials Permitted
Mathematical instruments
Silent, non-programmable, electronic calculator
.
DO NOT OPEN THIS EXAMINATION PAPER UNTIL INSTRUCTED TO DO SO.
1.
Evaluate the following
a)
b)
2𝑥 2 −7𝑥−15
lim (
𝑥→5
𝑥 2 −4𝑥−5
sin 7𝑥
lim (
2𝑥
𝑥→0
)
(5 marks)
) where 𝑥 is in radians.
(4 marks)
[9 marks]
2.
a)
b)
Find
𝑑𝑦
𝑑𝑥
for each of the following.
3
i)
𝑦 = 𝑒 2𝑥+5 √3𝑥 2 − 5
(5 marks)
ii)
𝑦 = sin4 (3𝑥 − 5)
(4 marks)
The equation of a curve is y =
2𝑥
1+𝑥 3
. Find the equation of the normal to
the curve at the point where 𝑥 = 2.
c)
3.
a)
b)
(5 marks)
An open metal container, with a square base of side 𝑥 m and a height of
𝑦 m, is to be made from 96 m2 of a thin sheet of metal. The volume of the
container is 𝑣 m3.
1
i)
Show that 𝑣 = 24𝑥 − 4 𝑥 3 .
(3 marks)
ii)
Given that 𝑥 can vary, show that the maximum volume is 64√2 m3.
(5 marks)
[22 marks]
i)
Integrate 5sec 2 (3𝑥 + 1) with respect to 𝑥.
(3 marks)
ii)
Use the substitution 𝑢 = 𝑥 5 − 3 to determine ∫(𝑥 5 − 3)8 (5𝑥 9 ) 𝑑𝑥.
(6 marks)
For the graph shown below, find the area bounded by the curve, the 𝑥-axis
and the lines 𝑥 = 0.5 and 𝑥 = 1.
𝑦
1
𝑦=
0
c)
1
(3𝑥 + 1)2
1 𝑥
(5 marks)
A model for the height, y metres, of a mango tree at time t years after being
planted assumes that, while the mango tree is growing, the rate of increase
1
of the height is proportional to (9 − 𝑦)3 . It is given that when 𝑡 = 0, 𝑦 = 1
𝑑𝑦
and = 0.2.
𝑑𝑡
i)
Form a differential equation connecting 𝑦 and 𝑡.
(3 marks)
ii)
Solve the differential equation, expressing y in terms of t.
(8 marks)
iii)
Calculate the time taken for the tree to reach half its maximum height.
(4 marks)
[29 marks]
END OF TEST