2019 Nov P2 Q10.2

Written on
November 2019 - Paper 2

Question 1

The table below shows the monthly income (in rands) of

6

different people and the amount (in rands) that each person spends on the monthly repayment of a motor vehicle.

$\quad$ MONTHLY INCOME (IN RANDS)	$9\,000$	$13\,500$	$15\,000$	$16\,500$	$17\,000$	$20\,000$
$\quad$ MONTHLY REPAYMENT (IN RANDS)	$2\,000$	$3\,000$	$3\,500$	$5\,200$	$5\,500$	$6\,000$

1.1

Determine the equation of the least squares regression line for the data.

(3)

1.2

If a person earns

R14\,000

per month, predict the monthly repayment that the person could make towards a motor vehicle.

(2)

1.3

Determine the correlation coefficient between the monthly income and the monthly repayment of a motor vehicle.

(1)

1.4

A person who earns

R18\,000

per month has to decide whether to spend

R9\,000

as a monthly repayment of a motor vehicle, or not. If the above information is a true representation of the population data, which of the following would the person most likely decide on:

\quad A

Spend

R9\,000

per month because there is a very strong positive correlation between the amount earned and the monthly repayment.

\quad B

NOT to spend

R9\,000

per month because there is a very weak positive correlation between the amount earned and the monthly repayment.

\quad C

Spend

R9\,000

per month because the point

(18\,000\;;\,9\,000)

lies very near to the least squares regression line.

\quad D

NOT to spend

R9\,000

per month because the point

(18\,000\;;\,9\,000)

lies very far from the least squares regression line.

(2)

\textbf{[8]}

Written on
November 2019 - Paper 2

Question 2

A survey was conducted among

100

people about the amount that they paid on a monthly basis for their cellphone contracts. The person carrying out the survey calculated the estimated mean to be

R\,309

per month. Unfortunately, he lost some of the data thereafter. The partial results of the survey are shown in the frequency table below:

$\quad$ AMOUNT PAID (in Rands)	$\quad$ FREQUENCY
$\quad 0<x \leq 100$	$\quad 7$
$\quad 100<x \leq 200$	$\quad 12$
$\quad 200<x \leq 300$	$\quad a$
$\quad 300<x \leq 400$	$\quad 35$
$\quad 400<x \leq 500$	$\quad b$
$\quad 500<x \leq 600$	$\quad 6$

2.1

How many people paid

R\,200

or less on their monthly cellphone contracts?

(1)

2.2

Use the information above to show that

a=24

and

b=16

.

(5)

2.3

Write down the modal class for the data.

(1)

2.4

On the grid provided in the ANSWER BOOK, draw and ogive (cumulative frequency graph) to represent the data.

(4)

2.5

Determine how many people paid more than

R\,420

per month for their cellphone contracts.

(2)

\textbf{[13]}

Written on
November 2019 - Paper 2

Question 3

In the diagram, $P$ , $R(3\,;\,5)$ , $S(-3\,;\,-7)$ and $T(-5\,;\,k)$ are vertices of a trapezium $PRST$ and $PT \parallel RS$ . $RS$ and $PR$ cut the $y$ -axis at $D$ and $C(0\,;\,5)$ respectively. $PT$ and $RS$ cut the $x$ -axis at $E$ and $F$ respectively. $P\hat EF=\theta$ .

$3.1$	Write down the equation of $PR$ .	$(1)$
$3.2$	Calculate the:
$\quad 3.2.1$	Gradient of $RS$	$(2)$
$\quad 3.2.2$	Size of $\theta$	$(3)$
$\quad 3.2.3$	Coordinates of $D$	$(3)$
$3.3$	If it is given that $TS=2\sqrt{5}$ , calculate the value of $k$ .	$(4)$
$3.4$	Parallelogram $TDNS$ , with $N$ in the $4^{th}$ quadrant, is drawn. Calculate the coordinates of $N$ .	$(3)$
$3.5$	$\Delta PRD$ is reflected about the $y$ -axis to form $\Delta P^\prime R^\prime D^\prime$ . Calculate the size of $R\hat DR^\prime$ .	$(3)$
		$\textbf{[19]}$

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November 2019 - Paper 2

Question 4

In the diagram, a circle having centre $M$ touches the $x$ -axis at $A(-1\,;\,0)$ and the $y$ -axis at $B(0\,;\,1)$ . A smaller circle, centred at $N(-\frac{\displaystyle 1}{\displaystyle 2}\,;\,\frac{\displaystyle 3}{\displaystyle 2})$ , passes through $M$ and cuts the larger circle at $B$ and $C$ . $BNC$ is a diameter of the smaller circle. A tangent drawn to the smaller circle at $C$ , cuts the $x$ -axis at $D$ .

$4.1$	Determine the equation of the circle centred at $M$ in the form $(x-a)^2+(y-b)^2=r^2$	$(3)$
$4.2$	Calculate the coordinates of $C$ .	$(2)$
$4.3$	Show that the equation of the tangent $CD$ is $y-x=3$ .	$(4)$
$4.4$	Determine the values of $t$ for which the line $y=x+t$ will NOT touch or cut the smaller circle.	$(3)$
$4.5$	The smaller circle centred at $N$ is transformed such that point $C$ is translated along the tangent to $D$ . Calculate the coordinates of $E$ , the new centre of the smaller circle.	$(3)$
$4.6$	If it is given that the area of quadrilateral $OBCD$ is $2a^2$ square units and $a>0$ , show that $a=\frac{\displaystyle \sqrt{7}}{\displaystyle 2}$ units.	$(5)$
		$\textbf{[20]}$

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November 2019 - Paper 2

Question 5

$5.1$	Simplify the following expression to ONE trigonometric term: $\frac{\displaystyle\sin{x}}{\displaystyle\cos{x}.\tan{x}}+\sin(180^{\circ}+x)\cos(90^{\circ}-x)$	$(5)$
$5.2$	$\textbf{Without using a calculator}$ , determine the value of: $\frac{\displaystyle \sin^2 35^{\circ}-\cos^2 35^{\circ}}{\displaystyle 4\sin 10^{\circ}\cos 10^{\circ}}$	$(4)$
$5.3$	Given: $\cos 26^{\circ}=m$ $\textbf{Without using a calculator}$ , determine $2\sin^2 77^{\circ}$ in terms of $m$ .	$(4)$
$5.4$	Consider: $f(x)=\sin(x+25^{\circ})\cos 15^{\circ}-\cos(x+25^{\circ})\sin 15^{\circ}$
$\quad 5.4.1$	Determine the general solution of $f(x)=\tan 165^{\circ}$	$(6)$
$\quad 5.4.2$	Determine the value(s) of $x$ in the interval $x\in[0^{\circ}\,;\,360^{\circ}]$ for which $f(x)$ will have a minimum value.	$(3)$
		$\textbf{[22]}$

Written on
November 2019 - Paper 2

Question 6

In the diagram, the graphs of $f(x)=\sin x-1$ and $g(x)=\cos 2x$ are drawn for the interval $x\in[-90^{\circ}\,;\,360^{\circ}]$ . Graphs $f$ and $g$ intersect at $A$ . $B(360^{\circ}\,;\,-1)$ is a point on $f$ .

$6.1$	Write down the range of $f$ .	$(2)$
$6.2$	Write down the values of $x$ in the interval $x\in[-90^{\circ}\,;\,360^{\circ}]$ for which graph $f$ is decreasing.	$(2)$
$6.3$	$P$ and $Q$ are points on the graphs $g$ and $f$ respectively such that $PQ$ is parallel to the $y$ -axis. If $PQ$ lies between $A$ and $B$ , determine the value(s) of $x$ for which $PQ$ will be a maximum.	$(6)$
		$\textbf{[10]}$

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November 2019 - Paper 2

Question 7

The diagram below shows a solar panel, $ABCD$ , which is fixed to a flat piece of concrete slab $EFCD$ . $ABCD$ and $EFCD$ are two identical rhombuses. $K$ is a point on $DC$ such that $DK=KC$ and $AK\perp DC$ . $AF$ and $KF$ are drawn. $A\hat DC=C\hat DE=60^{\circ}$ and $AD=x$ units.

$7.1$	Determine $AK$ in terms of $x$ .	$(2)$
$7.2$	Write down the size of $K\hat CF$ .	$(1)$
$7.3$	It is further given that $A\hat KF$ , the angle between the solar panel and the concrete slab, is $y$ . Determine the area of $\Delta AKF$ in terms of $x$ and $y$ .	$(7)$
		$\textbf{[10]}$

Written on
November 2019 - Paper 2

Question 8

$8.1$	In the diagram, $PQRS$ is a cyclic quadrilateral. Chord $RS$ is produced to $T$ . $K$ is a point on $RS$ and $W$ is a point on the circle such that $QRKW$ is a parallelogram. $PS$ and $QW$ intersect at $U$ . $P\hat ST=136^{\circ}$ and $\hat Q_1=100^{\circ}$ .

Determine, with reasons, the size of:
$\quad 8.1.1$	$\hat R$	$(2)$
$\quad 8.1.2$	$\hat P$	$(2)$
$\quad 8.1.3$	$P\hat QW$	$(3)$
$\quad 8.1.4$	$\hat U_2$	$(2)$
		$\textbf{[9]}$

Written on
November 2019 - Paper 2

Question 8.2

$8.2$	In the diagram, the diagonals of quadrilateral $CDEF$ intersect at $T$ . $EF=9$ units, $DC=18$ units, $ET=7$ units, $TC=10$ units, $FT=5$ units and $TD=14$ units.

Prove, with reasons, that:
$\quad 8.2.1$	$E\hat FD=E\hat CD$	$(4)$
$\quad 8.2.2$	$D\hat FC=D\hat EC$	$(3)$
		$\textbf{[7]}$

Written on
November 2019 - Paper 2

Question 9

In the diagram, $O$ is the centre of the circle. $ST$ is a tangent to the circle at $T$ . $M$ and $P$ are points on the circle such that $TM=MP$ . $OT$ , $OP$ and $TP$ are drawn. Let $\hat O_1=x$ .

Prove, with reasons, that $S\hat TM=\frac{\displaystyle 1}{\displaystyle 4}x$ .		$\textbf{[7]}$

Written on
November 2019 - Paper 2

Question 10.1

$10.1$	In the diagram, $\Delta ABC$ is drawn. $D$ is a point on $AB$ and $E$ is a point on $AC$ such that $DE\parallel BC$ . $BE$ and $DC$ are drawn.

	Use the diagram to prove the theorem which states that a line drawn parallel to one side of a triangle divides the other two sides proportionally, in other words prove that $\frac{AD}{DB}=\frac{AE}{EC}$	$\textbf{[6]}$

Written on
November 2019 - Paper 2

Question 10.2

$10.2$	In the diagram, $ST$ and $VT$ are tangents to the circle at $S$ and $V$ respectively. $R$ is a point on the circle and $W$ is a point on chord $RS$ such that $WT$ is parallel to $RV$ . $SV$ and $WT$ are drawn. $WT$ intersects $SV$ at $K$ . Let $\hat S_2=x$ .

$10.2.1$	Write down, with reasons, THREE other angles EACH equal to $x$ .	$(6)$
$10.2.2$	Prove, with reasons, that:
$\quad (a)$	$WSTV$ is a cyclic quadrilateral	$(2)$
$\quad (b)$	$\Delta WRV$ is isosceles	$(4)$
$\quad (c)$	$\Delta WRV\,\|\|\|\,\Delta TSV$	$(3)$
$\quad (d)$	$\frac{RV}{SR}=\frac{KV}{TS}$	$(4)$
		$\textbf{[19]}$

2019-nov-p2

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 8.2

Question 9

Question 10.1

Question 10.2