Question $1$

$[3]$

Solve for $x$ and $y$ in the following system:

$2x+4y=8$

$x^2+y=4$

Question $2$

$[5]$

Solve for $x$ in the following inequalities and graph your solution

$2\mathrm{.}1|1-4x|-230$

$[3]$

Question $3$

$[8]$

Solve for $x$ in the following equations:

$3\mathrm{.}1,\frac{5^{3x-1}}{5^{x+1}}=10$

$3\mathrm{.}2,lo{g}_5(lo{g}_3(x^2-4))=4$

$[2]$

$3\mathrm{.}3,\frac{1}{\sqrt[\phantom{2}]{2x^2-1}}=\frac{3}{x+2}$

Question $4$

$[2]$

On the provided Venn Diagrams, shade the region that represents:

$4\mathrm{.}1,(SM)(PM)$

$[1]$

Introductory Mathematical Analysis A $-$ MAA$00$A$1$ & MAT$00$A$1-$ Supplementary Exam $-2019$

$\mathrm{.}4.$

$4\mathrm{.}2(SMP{)}^{\text{'}}$

$[1]$

Question $5$

$[9]$

A standard deck of cards contains $52$ cards, evenly split between two colours $(26$ cards each$).$ Each colour has two suites, for four suites in total. Hearts, clubs, diamonds and spades. There are $13$ cards in each suite. If you draw three cards from a deck one at a time without replacement, what is the probability that:

$5\mathrm{.}1$ All three cards are diamonds?

$[2]$

$5\mathrm{.}2$ You don't draw any spades?

$[2]$

$5\mathrm{.}3$ If you note only the colour each time you draw a card, construct a tree diagram detailing all of the possible outcomes with their probabilities.

$[3]$

$5\mathrm{.}4$ Now determine the probability of drawing three red cards or three black cards in a row

$[2]$

Introductory Mathematical Analysis A $-$ MAA$00$A$1$ & MAT$00$A$1-$ Supplementary Exam $-2019$

$5-$

Question $6$

$[6]$

Given that:

$P(E)=20\%,P(F)=40\%,$ and $P(E|F)=20\%$

$6\mathrm{.}1$ Determine whether or not events E and F are independent events? Show all calculations and provide a reason for your answer.

$[2]$

$6\mathrm{.}2$ Find $P(EF).$

$[2]$

$6\mathrm{.}3$ Find $P(EF).$

$[2]$

Question $7$

$[7]$

Given the following constraints:

$f(x)=\{\begin{array}{c}2x+2y120\\ 9x+9y720\\ 0x50\\ 0y40\end{array}$

And the objective function

$Z=300x+150y$

$7\mathrm{.}1$ Graph all the inequalities on the same set of axes, clearly indicating all intercepts and the feasible region.

$[4]$

Introductory Mathematical Anaysis A $-$ MAA$00$A$1$ & MAT$00$A$1-$ Supplementary Exam $-2019$

$-6-$

$7\mathrm{.}2$ Determine the corner points of the feasible region.

$[2]$

$7\mathrm{.}3$ Maximise the Objective Function.

$[1]$

Question $8$

$[5]$

Differentiate each of the following functions. Do not simplify your answer.

$8\mathrm{.}1f(t)=e^{t^2}(4t^3-7t^2+9t).$

$[3]$

$8\mathrm{.}2g(x)=(3x^2-9x-1{)}^{50}+e^{}+3x^2-ln(x^2)$

$[2]$

Question $9$

$[5]$

Lesotho's consumption function may be defined as:

$C(I)=\frac{4I^3-7\text{I}}{5I^2+2}$

Where I, C and S are in billions of dollars.

$9\mathrm{.}1$ Determine the Marginal Propensity to Consume when $I=100.$

$[3]$

$9\mathrm{.}2$ Determine the Marginal Propensity to Save when $l=100.$

$[1]$

$9\mathrm{.}3$ From this data, are consumers more likely to save or to consume goods and services when $I=100?$

$[1]$