INSTITUTE OF BASIC AND BIOMEDICAL SIENCES
DEPARTMENT OF MATHEMATICS
MAT 101−FOUNDATION MATHEMATICS TEST 2
INSTRUCTIONS:
6TH MARCH, 2020.
1. Answer all questions
2. Show all essential working to obtain full marks.
3. Calculators are not allowed in this test
TIME ALLOWED:
Two (2) hours
1. (a) Work out the following
(i)
7!
8
9
8
(ii) (5) + (5) + (2)
4!3!
[1] [2]
(b) Express in factorial notation
(i) 10 × 9 × 8
(ii)
2𝑛(2𝑛−1)
2×1
[1] [2]
(c) The sum of three terms of an arithmetic series is 12. If the 20th term is −32, find the first
term and the common difference
2. (a) Find the sixth term in the binomial expansion of (1 − 3𝑥)9
[4]
[4]
(b) Find the angle subtended at the center of the circle of radius 10cm and arc length 200cm,
giving your answer in degrees.
[3]
(c) Use mathematical induction to prove the sum formulas below for all positive integers
𝑆𝑛 =
3(3𝑛 −1)
2
, 𝑓𝑜𝑟 𝑎𝑛 = 3𝑛
[4]
3. (a) If 𝑡𝑎𝑛𝜃 =
1
and 𝜃 is in the third quadrant, find the values of the other five
√5
trigonometric ratios.
[3]
(b) Using trigonometric identities and fundamental trigonometric function values, find each of
the following
(i) 2𝑠𝑖𝑛15° cos 15°
[2]
(ii) 𝐶𝑜𝑠15°
[2]
(c) Prove the following identities
1
(i) 𝐶𝑜𝑠𝑒𝑐𝐴 + 𝑐𝑜𝑡𝐴 = 𝑐𝑜𝑡 2 𝐴
(ii) Prove that
(
1+𝑠𝑖𝑛𝜃 2
𝑐𝑜𝑠𝜃
[3]
1−𝑠𝑖𝑛𝜃 2
) +(
𝑐𝑜𝑠𝜃
)
= 2 + 4𝑡𝑎𝑛2 𝜃
[4]
4. (a) The function 𝑓 is defined, for 0° ≤ 𝑥 ≤ 360°, by 𝑓(𝑥) = 1 + 3𝑐𝑜𝑠2𝑥.
(i) State the (a) amplitude of 𝑓
[1]
(b) period of 𝑓
[1]
(c) vertical shift
[1]
(d) phase shift
[1]
(ii) Sketch the graph of 𝑦 = 𝑓(𝑥)
[3]
(b) ) Solve the trigonometric equation below in the interval 0 ≤ 𝑥 ≤ 2𝜋
1 − 𝑐𝑜𝑠𝑥 = √3 𝑠𝑖𝑛𝑥
[4]
3
1
4
2
(c) (i) If 𝑡𝑎𝑛𝑥 = , find the value of 𝑐𝑜𝑠 𝑥
1
[2]
1
(ii) Express 𝑐𝑜𝑡𝑥𝑐𝑜𝑡 2 𝑥 in terms of 𝑡 where 𝑡 = 𝑡𝑎𝑛 2 𝑥
[2]