MAA$00$A$1$ PAPER A $2022$

$\frac{7}{12}$

$3\mathrm{.}2$ Simplify the following expression:

$(2)$

$\frac{\frac{1}{2}lo{g}_2(x^2)-(lo{g}_2(x))(lo{g}_2(e))}{lo{g}_2(e)}+ln(2x)-ln2$

$3\mathrm{.}3$ Differentiate $y$ with respect to $x$:$y=e^{2x}+x^{2}ln(x^2-x+1)$

$(3)$

$3\mathrm{.}4$ Differentiate $y$ with respect to $x$ :

$(2)$

$y=\frac{2x^2-ln(3x)}{x{e}^x}$

Question $4[8$ marks$]$

$4\mathrm{.}1$ An investor has a choice of investing a sum of money at $8\%$ compounded semiannually or at $7\mathrm{.}8\%$ compounded monthly. Which is the better of the two rates?

$(2)$

$4\mathrm{.}2$ A debt of R$3500$ due in four years and R$5000$ due in six years is to be repaid by a single payment of R$1500$ now and two equal payments that are due each consecutive year from now. Assume that the interest rate is $7\%$ compounded semiannually.

$($a$)$ Draw a detailed timeline for the above.

$1\mathrm{.}5$

$($b$)$ How much are each of the equal payments? Write the equation of value first.

$2\mathrm{.}5$

MAA$00$A$1$ PAPER A $2022$

$9/12$

$4\mathrm{.}3$ If interest is compounded continuously at an annual rate of $0\mathrm{.}07,$ how many years would it take for a principal P to triple? Give your answer to the nearest year.

$(2)$

Question $5[8$ marks$]$

$5\mathrm{.}1$ If $P(E)=\frac{1}{3},P(EF)=\frac{7}{12},$ and $,P(EF)=\frac{1}{12},$ find $,P(E|F).$

$(2)$

$5\mathrm{.}2$ From a group of three women and four men, two people are selected at random to form a committee. Find the probability that the committee consists of women only.

$(2)$

MAA$00$A$1$ PAPER A $2022$

$10/12$

$5\mathrm{.}3$ A new test was developed for detecting a communicable disease, which is believed to affect $3\%$ of the population. Results of extensive testing indicate that $86\%$ of people who have this disease will have a positive reaction to the test, whereas $7\%$ of those who do not have the disease will also have a positive reaction.

$($a$)$ What is the probability that a randomly selected person who has a positive reaction will actually have this disease?

$(2)$

$10$

$($b$)$ What is the probability that a randomly selected person who has a negative reaction will actually have this disease?

$(2)$

Question $5[8$ marks$]$

$5\mathrm{.}1$ If $P(E)=\frac{1}{3},P(EF)=\frac{7}{12},$ and $,P(EF)=\frac{1}{12},$ find $,P(E|F).$

$(2)$

$5\mathrm{.}2$ From a group of three women and four men, two people are selected at random to form a committee. Find the probability that the committee consists of women only.

$(2)$

MAA$00$A$1$ PAPER A $2022$

$10/12$

$5\mathrm{.}3$ A new test was developed for detecting a communicable disease, which is believed to affect $3\%$ of the population. Results of extensive testing indicate that $86\%$ of people who have this disease will have a positive reaction to the test, whereas $7\%$ of those who do not have the disease will also have a positive reaction.

$($a$)$ What is the probability that a randomly selected person who has a positive reaction will actually have this disease?

$(2)$

$($b$)$ What is the probability that a randomly selected person who has a negative reaction will actually have this disease?

$2$

Question $6[16$ marks$]$

$6\mathrm{.}1$ Find the derivative of the following functions using the rule mentioned next to it$.$

$($a$)h(x)=(5x^7+\frac{2}{x^6}+2x)(\sqrt[5]{x}-\sqrt[\phantom{2}]{x}+e^{ln(x^2)}).[$Hint: Use the product rule.$]3$

$($b$)$ Let $E(x)=\frac{f(x)}{g(x)}$ and suppose we are given that $f^{\text{'}}(2)=10,f(2)=-8,g^{\text{'}}(2)=\frac{f^{\text{'}}(2)}{2}$ and $g(2)=2f(2).$ Find $E^{\text{'}}(2).[$Hint: Use the quotient rule.$]$

$(2)$

$6\mathrm{.}2$ Let $y=\frac{x^2}{x-1}.$

$($a$)$ Find the relative extremum values of $y$ using the first derivative test.

$(5)$

MAA$00$A$1$ PAPER A $2022$

$\frac{12}{12}$

$($b$)$ Determine the open intervals on which the function $y$ is increasing and decreasing.

$2$

$($c$)$ Show that $y^{\text{'}\text{'}}=\frac{2}{(x-1{)}^3}.$

$($b$)$ Let $E(x)=\frac{f(x)}{g(x)}$ and suppose we are given that $f^{\text{'}}(2)=10,f(2)=-8,g^{\text{'}}(2)=\frac{f^{\text{'}}(2)}{2}$ and $g(2)=2f(2).$ Find $E^{\text{'}}(2).[$Hint: Use the quotient rule.$]$

$2$

$6\mathrm{.}2$ Let $y=\frac{x^2}{x-1}.$

$($a$)$ Find the relative extremum values of $y$ using the first derivative test.

MAA$00$A$1$ PAPER A $2022$

$\frac{12}{12}$

$($b$)$ Determine the open intervals on which the function