CSEC Mathematics Paper 2 May/June 1986 - Free CXC Past Paper with Answers

The diagram above (not drawn to scale) shows a right triangular prism with $AB = 15$cm, $AD = 10$cm and $ED = 12$cm. Calculate the volume of the prism.

$A$, $B$ and $C$ are three sets such that $C$ is a subset of $B$ and $n(U)=n(A\cup B\cup C)=58$, , , , and . represents the universal set. Draw a Venn diagram to represent the above information and hence determine .

Solve the simultaneous equations $3x - \frac{y}{2} = 1$ $x - 3y = 10$

If $\frac{1}{R} = \frac{1}{V} - \frac{2}{T}$, express in terms of and .

The number of calls a taxi-driver received during a period of 75 days is given in the table below:

Draw a histogram to illustrate the data.

Calculate, correct to the nearest whole number, the mean number of calls received per day during the period of 75 days.

Calculate the number of days the taxi-driver received (i) less than 3 calls (ii) 6 or more calls.

Calculate the probability that on any day chosen at random from the sample the taxi-driver received no calls.

[A graph is provided for this question]. The graph provided shows the fall in temperature of a body after it was put into a special container to cool.

Using the graph, estimate (i) the temperature of the body four minutes after it was put into the container (ii) the average rate of cooling of the body, in degrees per minute, for the first twelve minutes, after being put into the container (iii) the rate of cooling of the body when its temperature was $64^\circ$.

Assuming that after 20 minutes, the temperature of the body continues to fall at a constant rate, estimate the temperature of the body 1 hour after being put into the container.

The figure $OAB$ above (not drawn to scale) represents a flower bed in the shape of a sector of a circle centre $O$, radius 10 metres. The triangular region $OAB$ is planted in roses and the segment bounded by the chord $AB$ and the arc $AB$ is planted in marigolds. Given that angle $AOB = 36^\circ$, calculate the area of the segment assigned to the marigolds. [Take $\pi$ to be 3.14]

$A(4, 2)$, $B(8, 1)$, $C(2, 5)$ are the vertices of a triangle. $P$ and are the midpoints of the sides and respectively. (i) Determine the position vectors of and . (ii) Determine the components of the vectors and . (iii) Describe the relation between and .

The points $A(3, 2)$, $B(7, 2)$ and $C(3, 5)$ are vertices of triangle $ABC$. Triangle $ABC$ is mapped onto triangle $A'B'C'$ by a glide reflection consisting of a reflection in the line $y = -x$ and a translation by the vector $\begin{pmatrix} -1 \\ 1 \end{pmatrix}$. Draw on graph paper triangles $ABC$ and $A'B'C'$.

$A''B''C''$ is the image of triangle $A'B'C'$ after reflection in the -axis. (i) Construct the centre of rotation that maps triangle onto triangle . (ii) State the coordinates of the centre of rotation. (iii) Describe fully the single transformation that maps triangle onto triangle .

A farmer wishes to buy some goats and cows. He has pasture for only 50 animals. Goats cost $300 each and cows $600 each. He can expect to make a profit of $250 on each goat and $350 on each cow. If he has $24 \ 000$ to spend and he buys $g$ goats and $c$ cows, obtain TWO inequalities connecting $g$ and $c$.

Illustrate these inequalities on the same graph.

Determine how many of EACH type of animal he should buy to obtain a maximum profit.

Express $9x^2 - 30x + 21$ in the form $a(x - b)^2 - c$, where $a$, $b$ and $c$ are integers. (ii) Hence, or otherwise determine the solution set of $9x^2 - 30x + 21 > 0$.

Sketch the curve of the function $f(x) = 9x^2 - 30x + 21$ for $\frac{2}{3} \le x \le \frac{8}{3}$. Show clearly on the diagram (i) the points where the curve cuts the -axis and give their co-ordinates (ii) the minimum value of .

In this question assume that the earth is a sphere of radius 6 377 km. Take $\pi$ to be $\frac{22}{7}$. The positions of two places $P$ and $Q$ on the earth's surface are $(80^\circ N, 15^\circ E)$ and $(80^\circ N, 165^\circ W)$ respectively. Describe the shortest route from $P$ to $Q$.

Calculate the distance $PQ$ measured along the parallel of latitude, $80^\circ$, giving your answer in km correct to two significant figures.

Calculate the nearest kilometre, the distance $PQ$ measured along the meridian and by way of the South Pole.

$A$, $B$ and $C$ are 3 points on the banks of a river with $B$ and $C$ on one side of the river and $A$ on the opposite bank. Assume that the banks are parallel, $BC = 20$ metres, angle $ABC = 30^\circ$, angle $ACB = 45^\circ$ and the width $AD$ of the river is $w$ metres. (i) Express the lengths of $DC$ and $BD$ in terms of $w$. (ii) Hence, determine the width of the river correct to 2 significant figures.

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

A man tried to swim straight across a river 120 metres wide to the point directly opposite his starting point. However, he landed at a point 60 metres downstream on the opposite bank after swimming for 90 seconds. Calculate (i) the speed of the current (ii) the man's speed in still water.

By drawing, or calculation, determine (i) the direction, relative to the current, in which the man should swim so as to land directly opposite his starting point on the other bank (ii) the time it takes him to swim across the river to this point. [Assume the river flows between parallel banks.]

The table below shows the frequency distributions of the marks obtained by students of Schools A and B on the same Mathematics test.

Calculate to one decimal place estimates of (i) the mean of the students' marks in School A (ii) the standard deviation of the students' marks in school B

Compare the means and standard deviations of the two sets of marks and comment on the spread of marks obtained by EACH school.

In a spelling competition, two contestants Alfred and Brian are given different words to spell. The probability that Alfred spells correctly any word given to him is $\frac{2}{3}$ and the probability that Brian spells any of his words correctly is $\frac{4}{5}$. Calculate the probability that both Alfred and Brian spell their word correctly

either Alfred or Brian, but not both, spells his word correctly

when given three words, Alfred spells (i) only the first two words correctly (ii) any two words correctly.

Copy and complete the truth table above for $p \Rightarrow q$ and $q \Rightarrow p$. Is $p \Rightarrow q$ logically equivalent to $q \Rightarrow p$? Justify your answer.

Given $P$: A quadrilateral is a square $Q$: A quadrilateral is a rhombus write a verbal statement which corresponds to $P \Rightarrow Q$.

Using $\sim P$ and $\sim Q$ write (i) a symbolic statement (ii) a verbal statement each of which is logically equivalent to $P \Rightarrow Q$.

Show that the symbolic statement you gave in part c(i) above is logically equivalent to $P \Rightarrow Q$.

Prove that angle $DAB = 90^\circ$.

Express the area of triangle $PAD$ in terms of $x$ and $y$ and hence show that the area of $PQRS$ is $(x^2 + 2xy) \text{cm}^2$. (ii) Hence, prove the Pythagorean result $DC^2 = DS^2 + SC^2$.