CSEC Mathematics Paper 2 May/June 1989 - Free CXC Past Paper with Answers

Given that $x = \frac{y-2}{y-3}$, express $y$ in terms of $x$.

Factorize completely $x^2 - y^2 - 4x + 4y$.

Solve $\frac{3(x-2)}{2} - \frac{x-3}{4} = 4$.

Solve $6x^2 + 17x - 14 = 0$.

A survey on a sample of persons who read at least one of the magazines $P$, $Q$ and $R$ yielded the following data:

• 72 persons read $P$.
• 53 persons read $Q$.
• 29 persons read $R$.
• 14 persons read only $P$ and $R$.
• 9 persons read only $Q$ and .

Use $x$ to represent the number of persons who read all three magazines. Draw a carefully labelled Venn diagram to represent the data.

Determine the value of $x$.

Calculate the number of persons in the sample.

The figure above, not drawn to scale, represents a fish tank in the shape of a cuboid of height 30 cm.

Calculate the volume of the tank.

If there are 40 litres of water in the tank, calculate the height of the water in the tank.

Construct a triangle $DAB$ such that $AB = 8$ cm, $AD = 6.5$ cm and angle $DAB = 70^\circ$. Through $D$, construct $DC$ parallel to $AB$. Construct also the line $BC$ perpendicular to $AB$. Measure and state the length of $DC$. (Show all construction lines clearly.)

State THREE properties which define a rhombus, with respect to its sides, angles and diagonals.

$ABCD$ is a rhombus (not drawn to scale) with $AO = 4.8$ cm and $BO = 3.6$ cm. Calculate

the length of $AB$

the size of angle $BAD$ to the nearest degree

the area of $ABCD$.

Given that $R$ is an anticlockwise rotation of $90^\circ$ about the origin and $M$ is a reflection on the $x$-axis, draw accurate diagrams to show

the image $A''B''C''$ of $ABC$ under $R$

the image $A'''B'''C'''$ of $A'B'C'$ under .

If the transformation $Q = MR$, obtain a single matrix to represent $Q$.

If the transformation $Q = MR$, obtain a single matrix to represent $Q$.

Describe clearly $Q$ as a single geometrical transforma-tion.

Given that $f(x) = 5x$ and $g(x) = x-2$,

calculate $f(2)$ and $g(2)$

determine $x$ when $fg(x) = 0$

prove that $g f^{-1}(23) = 5$.

The figure above represents the distance-time graph of two persons, Andy and Bob, journeying between two offices, $M$ and $N$. Andy leaves $M$ at the same time that Bob leaves $N$. Use the graph to

determine the distance $MN$

determine how much time Andy spent in walking between $M$ and $N$

determine how far from $M$ Andy and Bob met

calculate Andy's average speed for the whole distance

explain fully the segment $PQ$ on Bob's graph.

The cumulative frequency distribution of the daily wages, in dollars, for a group of workers is shown in the table below:

Using 2 cm to represent $5 and 1 cm to represent 5 workers, construct on graph paper, a cumulative frequency curve for the data.

Using your graph, estimate

the semi-interquartile range for the distribution

the number of workers who received more than $47.50 in daily wages

the probability that a worker selected at random receives a wage which is more than $45.50 but less than $55.00.

Two places, $P$ and $Q$, are on the same parallel of latitude $48^\circ$ N but on longitude $57^\circ$ E and $123^\circ$ W respectively.

Calculate, to the nearest hundred kilometre, the distance $PQ$, travelling

• due east along their parallel of latitude
• along the great circle over the North Pole.

Two aircraft, $X$ and $Y$, travelling at the same ground speed left from $P$ for $Q$ at the same time. If $X$ used the great circle route over the north pole while $Y$ travelled due east, calculate the longitude of the point at which aircraft was, at the time when aircraft reached . [Take the radius of the earth to be 6400 km and to be 3.14]

Prove that $\frac{1}{\tan A} + \tan A = \frac{1}{\sin A \cos A}$.

Three stations, $P$, $Q$ and $R$, are on level ground such that $P$ is due south of $Q$ and $R$ is on a bearing of from . The bearing of from is . A vertical tower is situated at . Given that m and is 84 m, calculate

the distance $PR$ to the nearest metre

the angle of elevation of $T$ from $P$.

The table above shows the cost of tuition fees per week for a private school. The conditions under which the school operates are as follows:

1. The maximum number of places at the school is 75.
2. There must be a minimum of 20 full-time pupils.
3. The number of part-time pupils must be at least half the number of full-time pupils.
4. The minimum weekly income must be $8000.

Let $x$ represent the number of full-time pupils and $y$ represent the number of part-time pupils.

Write FOUR inequalities not including $x > 0$, $y > 0$, to represent the conditions stated above.

Using a scale of 1 cm to 5 units on both axes draw the graphs of the inequalities.

If the profit on each full-time pupil is $40 and that on each part-time pupil $30, determine from your graph

the number of full-time and part-time pupils for the maximum profit to be obtained

the sum representing the maximum profit.

Copy and complete the table below for $y = 3 \sin 2x$.

Using a scale of 2 cm to represent $\frac{\pi}{8}$ radians on the $x$-axis, and 2 cm to 1 unit on the $y$-axis, draw the graph of the function above for $0 \le x \le \pi$.

Using your graph

state the minimum value of the function

state the values of $x$ within the domain $0 < x < \pi$, for which the function is negative

determine the solutions of $3 \sin 2x = 1.5$.

Given $X = \begin{pmatrix} p & q \\ r & s \end{pmatrix}$ and $Y = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}$,

evaluate $XY$ and $X+Y$.

if $XY = X+Y$, determine the values of $p$, $q$ and $r$.

Given that $A$ and $B$ are two points whose position vectors with respect to the origin are $\underline{a}$ and $\underline{b}$ respectively,

draw on a diagram the point $C$ whose position vector is $\frac{2\underline{a} + \underline{b}}{3}$

prove that $ABC$ is a straight line, using a vector method

determine the ratio $AB:BC$.

express each of the position vectors $\vec{OA}$ and $\vec{OB}$ in the form $\begin{pmatrix} x \\ y \end{pmatrix}$. If $\vec{OA} + \vec{OB} = \vec{c}$, show that $|\vec{c}| = \sqrt{34}$.

With $O$ as origin, $\triangle OMN$ is transformed under a translation ($T$) to $\triangle PQR$ such that $N(3, 2)$ moves to . (i) Determine the matrix of the transformation. The point is mapped to the point under an enlargement (), centre , and scale factor . (ii) State the value of . (iii) Hence determine the co-ordinates of the mapping of the origin under the combined transformation represented by .