CSEC Mathematics Paper 2 May/June 1981 - Free CXC Past Paper with Answers

Using ruler and compasses only, construct a triangle $ABC$ with $\angle A = 60^\circ$, $\angle B = 45^\circ$ and $AB = 10 \text{ cm}$. Measure and state the length of $BC$ in centimetres.

Find a point $D$ on $AB$ such that $CD = DB$. Measure and state the length of $AD$ in centimetres.

Solve for $y$ the equation $3(y^2 + 3) = 28y$

If 5 is added both to the numerator and denominator of a fraction, the result is equivalent to $\frac{2}{3}$. If 3 is subtracted from both the numerator and denominator of the original fraction, the new result is equivalent to $\frac{1}{2}$. Find the original fraction.

If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, and $B$ is the inverse of $A$, find $B$.

If $X = \begin{pmatrix} 2 & 3 \\ c & \frac{1}{2a} \end{pmatrix}$, and , find in terms of .

A woman wishes to invest $1 500. She can buy savings bonds which pay simple interest at the rate of 8% per annum or she can start a savings account which pays compound interest at the same rate. Calculate, to the nearest cent, the difference in the amounts of the two investments at the end of 3 years.

Copy the table and fill in the missing values of $x$, the midpoints of the intervals.

Use these values to calculate the mean number of children.

Fathers in a certain village are asked to indicate how many children they have. The table below shows the data collected in one year, where $f$ is the frequency.

(a) Copy the table and fill in the missing values of $x$, the midpoints of the intervals. (b) Use these values to calculate the mean number of children.

that their sum is 9

that their sum is 9 or 15

If two digits are chosen at random from the set of ten digits $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, calculate the probability (a) that their sum is 9 (b) that their sum is 9 or 15

Copy and complete the table below for the function $f: x \to 2^{x-1}$.

Using scales of 2 cm to represent 1 unit on the x-axis, and 4 cm to represent 1 unit on the $f(x)$-axis, plot the graph of $f(x) = 2^{x-1}$, for $-2 \le x \le 3$. ()

Using the same axes and the same scales, draw a suitable straight line to solve the equation $\frac{5x}{4} = 2^{x-1}$

Show that the equation in (iii) is equivalent to the equation $5x = 2^{x+1}$.

The diagram shows a straight stretch of river flowing between parallel banks. $A$, $B$ and $C$ are three points on the banks of the river. If $\angle CBA = 30^\circ$, $\angle BCA = 55^\circ$ and $BC = 20 \text{ m}$, calculate the width $AD$ of the river.

A rectangular field is shown on the map. Its dimensions on the map can be measured only to the nearest $0.1 \text{ mm}$. You read the measurements on the map as $8.6 \text{ mm}$ and $5.2 \text{ mm}$. Calculate the largest possible area of the field (on the ground).

Determine the percentage error made by using the calculated area if the area of the field is actually $18 \ 000 \text{ m}^2$, giving the answer correct to one decimal place.

A map is drawn to a scale of $1:20 \ 000$.

(a) A rectangular field is shown on the map. Its dimensions on the map can be measured only to the nearest $0.1 \text{ mm}$. You read the measurements on the map as $8.6 \text{ mm}$ and $5.2 \text{ mm}$. Calculate the largest possible area of the field (on the ground). (b) Determine the percentage error made by using the calculated area if the area of the field is actually , giving the answer correct to one decimal place.

Copy and complete the table below for $y = \sin x + \cos x$, where $x$ is measured in radians.

Using a scale of 1 cm to represent $\frac{\pi}{10}$ radians on the horizontal axis and 5 cm to represent 1 unit on the vertical axis, draw on graph paper the graph of $y = \sin x + \cos x$, for .

Use your graph to find, in the range $0 \le x \le \pi$, (a) the solutions of $\sin x + \cos x = 1.2$ (b) the maximum value of $\sin x + \cos x$.

Express $g$ as the subject of the formula $T = 2\pi \sqrt{\frac{k}{g}}$

Find the ranges of values of $N$ for which $6 < 6N^2 - 5N \le 34$

The table below shows the distance, $d$ metres, a car travels in $t$ seconds along a straight road.

Using scales of 2 cm to represent 1 second on the horizontal axis and 2 cm to represent 5 metres on the vertical axis, plot on graph paper a graph to illustrate the relation between the distance, $d$ and the time, $t$.

From your graph determine the speed of the car at $t = 2, 3, 4$.

Using the values obtained in (ii), plot a straight line graph. From this graph determine the acceleration of the car.

$A$ and $B$ have coordinates $(1, 2)$ and $(-5, -6)$ respectively. The lines $AC$ and have gradients 3 and respectively.

$S$ is the midpoint of $AC$ and $T$ is the midpoint of $BC$. Calculate (a) the gradient of $ST$; (b) the length of $ST$.

A bag contains 2 white marbles and 3 blue marbles. (a) A marble is selected at random from the bag. Calculate the probability that the marble is white. (b) If the white marble is NOT replaced in the bag, calculate the probability that the second marble selected from the bag will be blue.

In a certain Caribbean island 52% of the births are female. A recently married couple plans to have 3 children. Calculate the probability that (a) all three children will be boys (b) at least one of the children will be a girl

Given that $p, q$ and $r$ are statements such that $p \implies q \implies r \implies p$, prove that $p$ and $q$ are logically equivalent.

(a) Complete the following statement: "The square of the difference of two numbers is identically equal to the sum of their squares ........." (b) Write the completed statement in symbols. (c) Prove that for any two numbers, $m$ and $n$ $m^2 + n^2 \ge 2mn$ (d) When is ?