3U Test 6 (60 marks)
1. (3 marks) Solve: |4x − 1| > 2(x(1 − x)
7. (6 marks) Sketch the functions
2. (2 marks) Find the Range of 𝑓(𝑥) without
using calculus
cos 𝑥 + sin 𝑥
𝑓(𝑥) =
cos 𝑥 + sin 𝑥 + 2
3. (4 marks) Simplify and find the exact value:
a) cos42𝑡𝑎𝑛!" (−3)9
"
b) 𝑡𝑎𝑛!" 𝑥 + 𝑡𝑎𝑛!" #
4. (4 marks) Sketch
$
a) 𝑦 = % − 𝑐𝑜𝑠 !" >1 −
b) 𝑦 = 𝑐𝑜𝑠 !" (cos 𝑥)
a)
y = f !" (|𝑥|)
b)
𝑦 = 𝑒 '(#!")
c)
𝑦 = ln(𝑓(𝑥))
8. (6 marks) Given
&#
?
%
1
𝑥
Find the stationary points and
𝑓(𝑥) = 𝑥 +
a)
determine their nature.
5. (2 marks) All the letters of the word
𝐸𝑁𝐶𝑈𝑀𝐵𝑅𝐴𝑁𝐶𝐸 are arranged in a line.
b)
Sketch 𝑦 = 𝑓(𝑥)
Find the total number of arrangements
c)
What is the largest domain
which contain all of the vowels in
containing 𝑥 = ±3 for which 𝑓(𝑥)
alphabetical order but separated by at least
has an inverse function?
one consonant.
6. (3 marks) A cylinder, open at both ends, is
inscribed in a sphere of constant radius R.
d)
Hence, sketch 𝑦 = 𝑓 !" (𝑥)
e)
Find the equation of the inverse
function.
The radius of the cylinder is r and the height
is h. Find the maximum surface area of the
cylinder in terms of R.
9. (3 marks) The cost of running a ship at a
*!
constant speed of v km/hr is 160 + "+++
dollars per hour. If the ship’s maximum
speed is 16km/hr in a 1000km journey, find
what the minimum cost would be.
10. (5 marks) In the diagram, ABCD is a unit
square. ∠𝐴𝐸𝐺 = ∠𝐹𝐻𝐶 = 𝛼. 𝐴𝐸 = 𝑝
and 𝐹𝐶 = 𝑞
13. (3 marks) 3 numbers are chosen at random
from the numbers 1 to 100. Calculate the
probability of the three numbers is divisible
by 3, 6 and 12 in any order.
14. (8 marks) A convex quadrilateral 𝑃𝑄𝑅𝑆 has
side lengths 𝑎, 𝑏, 𝑐, 𝑑 and an area of √𝑎𝑏𝑐𝑑
units2. The quadrilateral has an inscribed
circle as shown in the diagram below. Let 𝑘
be the length of the diagonal 𝑆𝑄 and
a) Prove that 𝑝 + 𝑞 + 𝑝𝑞 = 1
∠𝑆𝑃𝑄 = 𝛼 and ∠𝑆𝑅𝑄 = 𝛽.
b) Find the max area of EBFD (in exact
value)
11. (3 marks) A circle centre O with radius r
inscribed in ∆𝐴𝐵𝐶. Prove that
𝑎
𝐴
𝐵
𝐶
= cos 𝑐𝑜𝑠𝑒𝑐 𝑐𝑜𝑠𝑒𝑐
𝑟
2
2
2
a) Prove that
(𝑎 − 𝑏)% = (𝑐 − 𝑑)%
12. (8 marks) Given
𝑥
𝑓(𝑥) = ± (4 − 𝑥 %
2
,
a) Find 𝑓 (𝑥)
b) Hence, show that
b) Find stationary points and determine
their nature.
c) Hence, by using the area of the
c) Find the points which have vertical
tangents.
d) Find the gradient at 𝑥 = 0
e) Sketch the graph. (show all important
points)
𝑎𝑏(1 − cos 𝛼) = 𝑐𝑑(1 − cos 𝛽)
quadrilateral 𝑃𝑄𝑅𝑆, show that the
quadrilateral 𝑃𝑄𝑅𝑆 is a cyclic
quadrilateral