CSEC Mathematics Paper 2 May/June 1993 - Free CXC Past Paper with Answers

There are 68 students in Form V. 15 students study Mathematics only. 12 students study Physics only. 8 students study Physics and Chemistry only. 2 students study Physics and Mathematics only. 3 students study Mathematics, Physics and Chemistry. 4 students do not study any of these subjects.

Draw a carefully labelled Venn diagram to represent the information given above.

Given that $a$ and $b$ are unit vectors as shown in the diagram above, (i) write the position vectors $\vec{OP}$ and $\vec{OQ}$ in terms of $a$ and $b$ (ii) determine the length of $\vec{OP}$.

Solve $\frac{2P}{5} + \frac{P}{3} = 3$.

Given that $y$ varies inversely as $x^2$ and that $y = 3$ when $x = 2$, calculate the value of when .

The cost of four chairs and a small table is $684. The cost of six chairs and a large table is $1196. The cost of the large table is TWICE the cost of the small table.

Given that $c$ is the cost, in dollars, of a chair and $b$ is the cost, in dollars, of a small table, (i) write a pair of simultaneous equations to represent the information given (ii) calculate the cost of the large table.

Using ruler and compasses only, construct a triangle ABC with $AB = 9.5$ cm, $AC = 7.5$ cm and angle $BAC = 60^\circ$.

Locate the point D such that $DB$ is perpendicular to $AB$ and $CD$ is parallel to $AB$.

Measure and state the length of $BD$ in centimetres.

Triangle PQR with P(1,1), Q(11,2), and R(1,9) is mapped onto triangle P'Q'R' with P'(3,3), Q'(8,3.5), and R'(3,7).

(i) Using a scale of 2 cm to 1 unit on both axes, draw on graph paper triangles PQR and P'Q'R'. Note: Draw the x-axis on the longer side of the graph paper. (ii) Hence, describe fully the transformation which maps triangle PQR onto triangle P'Q'R'.

An aeroplane travels a distance of 3700 km in $7\frac{1}{2}$ hours. Calculate its average speed.

An aeroplane left Kingston at 21:00 hours local time and travelled for 9 hours arriving at Los Angeles airport at 01:30 hrs local time on the following day. Calculate the difference in time between Kingston and Los Angeles.

(i) The angle of elevation from a point $P$ on the ground to the top of a tower 20 m tall, is $65^\circ$. Calculate the distance of $P$ from the foot of the tower. (ii) A point $T$ is 3.2 m nearer than $P$ to the foot of the tower. Calculate the angle of depression of $T$ from the top of the tower.

A survey was taken among 100 customers to find out the time spent waiting in lines for service at the bank. The following table shows the result of the survey.

Waiting time (in minutes) | No. of customers

1 - 5 | 5 6 - 10 | 12 11 - 15 | 15 16 - 20 | 19 21 - 25 | 21 26 - 30 | 25 31 - 35 | 3

Construct a cumulative frequency table to represent the data above.

Using a scale of 2 cm to 5 minutes on the horizontal axis and 2 cm to 10 customers on the vertical axis, draw a cumulative frequency curve to illustrate the information.

Estimate the proportion of customers who waited more than 16 minutes.

Calculate the probability that a customer chosen at random would have waited for more than 27 minutes.

Copy and complete the table below for the function $y = 5 + x - 2x^2$.

x | -3 | -2 | -1 | 0 | $\frac{1}{2}$ | 1 | 2 | 3

y | -16 | 2 | | 4 | | | -1 | -10

Using a scale of 2 cm to 1 unit on the x-axis and 1 cm to 1 unit on the y-axis, draw the graph of $y = 5 + x - 2x^2$ for $-3 \le x \le 3$.

Using your graph or otherwise, determine the range of values of $x$ for which $x - 2x^2 > -3$.

Notes for this question: Take $\pi = \frac{22}{7}$ Volume of cone $= \frac{1}{3}\pi r^2 h$

The diagram above, not drawn to scale, shows the MAJOR sector, AOB, of a circle of radius 6 cm. It represents the net of a cone.

Show by calculation that the circumference of the base of the cone is 22 cm.

Calculate (i) the radius of the base of the cone (ii) the height of the cone, giving your answer correct to one decimal place.

Calculate to two significant figures the volume of liquid, in litres, that the cone holds when filled.

Given that $f(x) = x + 3 - 2x^2$, (i) derive $f(x)$ in the form $f(x) = c + a(x + b)^2$, where $a$, $b$ and $c$ are constants (ii) determine the value of $x$ at which the maximum value of $f(x)$ occurs (iii) state the maximum value of $f(x)$.

A composite function $k$ is defined as $k(x) = (2x - 1)^2$.

(i) Express $k(x)$ as , where and are two simpler functions. (ii) Show that .

A boy has $280. He wants to buy $x$ records at $35 each and $y$ tapes at $40 each. He must buy more than one tape but not more than four tapes. He must also buy at least three records.

Write THREE inequalities in $x$ and $y$ to represent the above information.

Using a scale of 2 cm to represent 1 unit on EACH axis, draw the graphs of the inequalities.

SHADE the region that satisfies the THREE inequalities.

Determine the maximum amount spent and state the $(x, y)$ value that gives this amount.

In the figure above, not drawn to scale, the quadrilateral PQRS is inscribed in the circle, centre O. PR passes through O. The tangents TP and TS are drawn to the circle from T. Angle RSV = $20^\circ$.

Calculate, giving reasons, (i) angle PQR (ii) angle SPR (iii) angle PST (iv) angle PTS.

In triangle ABC above, not drawn to scale, $AB = 4$ cm, $AC = 7$ cm and angle $BAC = \alpha$.

Given that $\sin^2 \alpha = 0.64$, determine (i) the exact value of $\cos^2 \alpha$ (ii) the value of $\alpha$, if (iii) the length of BC, correct to one decimal place.

A point $K$ is on a bearing of $025^\circ$ from another point $M$. The boat leaves $K$ to go to $M$. The engine speed of the boat is 55 kmh$^{-1}$. A wind is blowing from the East at a speed of 25 kmh$^{-1}$.

(i) Using a scale of 1 cm to represent 5 kmh$^{-1}$, find by accurate drawing the course of the boat the resultant speed of the boat.

If $K$ is 12 km away from $M$, and the boat leaves $K$ at 07:00 hrs, calculate the time at which the boat reaches $M$.

Using a scale of 2 cm to represent 1 unit on EACH axis, draw $\triangle PQR$ with $P(1,4)$, $Q(3,1)$ and $R(4,2)$.

$\triangle PQR$ is transformed by the matrix $\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$. (i) Determine the coordinates of the image, . (ii) Draw triangle and describe the transformation fully.

Determine the $2 \times 2$ matrix that will transform $\triangle P'Q'R'$ onto .

$P$ is the point $(6,4)$ and $Q$ is the point $(8,2)$. $M$ and $N$ are the mid points of $OP$ and $OQ$ respectively, where $O$ is the origin. (i) Determine the vector $\vec{PQ}$. (ii) Determine the vector $\vec{MN}$. (iii) State the relationship between $\vec{MN}$ and $\vec{PQ}$.

Given the equations $x - y = -5$ $3x + 2y = -5$. (i) write the equations in matrix form (ii) determine the inverse of the matrix (iii) hence, solve the equations.