MSIN0180 - Quantitative Methods for Business 2020/21
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MSIN0180 Quantitative Methods for Business
Main Summer Assessment
2020/21
Examination length: TWENTY-FOUR (24) hours
There is ONE (1) section to the examination paper. It consists of FIVE (5) questions from which you must answer all FIVE (5).
This examination paper is designed to be completed in TWO (2) hours.
All questions carry equal marks.
1. This is a question on the solution of linear equations and vector spaces
a. Consider the system of linear equations
x + y + az = 1 、
x + ay + z = b .
x _ y + 3z = 1 .
where a; b are real numbers. Giving your reasoning, Önd the values of a and b such that the system has
i. exactly one solution,
ii. more than one solution,
iii. no solutions.
[10 Marks]
b. Consider the set of 4 vectors in R4 ; v1 = (1; 1; 1; 0); v2 = (1; _1; 0; _1); v3 = (1; 1; 0; 3); v4 = (1; _1; 1; k) : For what value of k does this form a linearly dependent set?
[10 Marks]
2. This is a question on di§erentiation and applications
a. Find the equation of the normal to the tangent line to the graph of 3x2 _ 7y2 + 4xy _ 8x = 0
at the point P (_1; 1) ; giving the equation in the form ay + bx + c = 0: [5 Marks]
b. Express the function
2x2
f (x) =
in partial fractions. Hence, Önd the value when x = 1: [6 Marks]
c. Sketch the graph of
y = x3 + 2x2 _ x _ 2;
showing clearly turning points and intercepts on both axes. [9 Marks]
3. This question is on di§erential equations
a. Find the general solution of the following second order di§erential equation y\\ + 3y + 2y\ = 4e字x + 2sin x:
[10 Marks]
b. Find the general solution of the di§erential equation
dy 1
dx 2
Hint: You may use the following result in your integration working
sin(A + B) + sin(A _ B)
2
[10 Marks]
4. DeÖne the deÖnite integrals In (x) for n e Zf by
In (x) 三
1
xn ^1 _ xdx:
d
Use integration by parts to derive the recursion formula
In (x) = aIn-1 (x) n > 1:
where a should be stated. [16 Marks]
Hence evaluate I4 (x) : [4 Marks]
5. This question is on limits
a. Using the expansion of sin(kx) when x is small, show that
x-(li)m二d ╱ 、 = ╱a3 _ 83、
[10 Marks]
Use the fact that if
lim log f (x) _二 l;
x-二a
then
lim el。g f {x3 _二 el
x-二a
lim (sinx)tan x
x-二T/3
[10 Marks]
2023-04-12