CSEC Mathematics Paper 2 May/June 1987 - Free CXC Past Paper with Answers

Factorize $a^2 - ab - ac + bc$.

Solve the equation $2^{4x} = 64$.

Given that $\begin{pmatrix} x & 2 \\ -1 & y \end{pmatrix} \begin{pmatrix} 2 & -1 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} 8 & a \\ b & 5 \end{pmatrix}$ calculate the values of , , and .

Without using your protractor, construct a triangle $OAB$ in which angle $OAB = 60^\circ$ and $OA = AB = 4$ cm. Hence, construct the rhombus $OFAB$.

Determine the area of the rhombus $OFAB$ to 2 significant figures.

On the same figure, draw a circle with centre $O$ passing through $B$, $A$ and $F$. Hence, calculate the area between the chord $AF$ and the minor arc $AF$. Give your answer to 2 significant figures. (Use )

In the parallelogram $ABCD$ above (not drawn to scale), $AB = 9$ cm, $BD = 10$ cm and angle $BCD = 44.4^\circ$.

(i) Calculate, to the nearest whole number, the size of the angle $BDA$. (ii) Using the cosine rule, calculate the length of $AD$.

Given that $f(x) = x^2 - 2x - 3$ and $x$ is an integer, write down the range of values of $f(x)$ for the following: (i) $1 < x < 4$ (ii) $-2 \le x \le 6$

The function $g(x) = \frac{x+1}{x-1}$. (i) Calculate $g(\frac{1}{2})$. (ii) Given that and and so on, show that , and hence, deduce the value of .

All the pupils who neither walk nor ride to school have their lunch at school. Ten of the pupils who walk and seven of the pupils who ride have their lunch at school. Draw a Venn diagram to represent this information. Indicate clearly the number of elements of each set and subset.

The diagram above (not drawn to scale) shows two line segments $PQ$ and $MN$ intersecting at $O$. The points $A$, $B$ and $C$ are equidistant from $O$. Angle $AOC = 40^\circ$.

$A$ is mapped onto $C$ by means of a reflection in the line $k$ (not shown). Determine the acute angle which $k$ makes with $ON$. Give a reason for your answer.

A mirror line $k$ (not shown) passes through $O$. A reflection in $k$ maps $A$ onto $B$. Calculate the acute angle which $k$ makes with .

Describe fully a transformation which maps the line $k$ onto the line $L$.

Given that $OA = 8.5$ cm, calculate the length of (i) $BC$ (ii) $AC$ (to the nearest integer).

The frequency distribution of the marks awarded to 120 candidates in an examination is as follows:

Construct a cumulative frequency table to represent the information above.

Estimate the upper and lower quartiles of the distribution.

Using an assumed mean of 23, calculate an estimate of the mean mark to the nearest whole number.

A candidate is selected at random. Calculate the probability that his mark is less than 16.

Show that $(x+2)^2 + (x-3)(x+3) - 2 = 2x^2 + 4x - 7$.

Given that $2x^2 + 4x - 7 = A(x+h)^2 + K$, calculate the values of , and .

Determine the minimum value of $2x^2 + 4x - 7$.

Solve, correct to 2 decimal places, the equation $2x^2 + 4x - 7 = 0$.

A racing car starts from rest at a point $A$ and travels with uniform acceleration of $12.5 \text{ m s}^{-2}$ for 8 seconds. For the next 20 seconds it travels at a constant speed. It then brakes uniformly to a stop in 15 seconds.

Draw a speed-time graph to represent the information above, using a scale of 1 cm to represent a speed of $10 \text{ m s}^{-1}$ on the vertical axis and 1 cm to represent 2 seconds on the horizontal axis.

Hence or otherwise, determine (i) the distance travelled during the first 8 seconds (ii) the average speed (in $\text{m s}^{-1}$) for the entire journey.

The diagram above (not drawn to scale) shows the angles of elevation of $T$, the top of a vertical mast, from the points $P$ and $Q$ on the same side of $R$ on a horizontal plane. $P, Q$ and $R$ lie on a straight line. $PQ = 25$ metres. Angles $TPQ$ and $TQR$ are $53.1^\circ$ and $64.6^\circ$ respectively.

(i) Show that the length of $QT$ is 100 metres to the nearest metre. (ii) Hence or otherwise, calculate the height of the mast.

Prove that $\tan^2 x - \sin^2 x = \tan^2 x \sin^2 x$.

In this question, use $\pi$ as $\frac{22}{7}$ and the radius of the earth to be 6 370 km.

An aeroplane flies due east from a point $X$ in latitude $42^\circ N$, longitude $15^\circ W$ to a point $Y$ in the latitude $42^\circ N$, longitude $30^\circ E$. Calculate the distance $XY$ to the nearest km.

The aeroplane then flies due North from $Y$ for a distance of 1 557 km to a position $Z$. Calculate the latitude of the point $Z$ to the nearest degree.

Calculate, to the nearest km, the distance of $Z$ from the north pole.

Given that the matrix $L = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and the matrix $M = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ (i) show that $L$ represents a reflection in the line $y = x$ (ii) calculate the matrix product $LM$ and show that the combination of transformations represented by $L$ and $M$ is NOT commutative.

The position vectors of the points $A$, $B$ and $S$ are as follows: $\vec{OA} = \begin{pmatrix} 4 \\ 0 \end{pmatrix}$, and where is the origin. and are transformed into the position vectors and respectively by . (i) Calculate and . (ii) Describe a combination of two transformations that maps triangle onto triangle , and find the determinant of the matrix that carries out this trans-formation.

In $\triangle PQR$, $PS = a$, and $\vec{PT} = b$. $S$ and $T$ are the midpoints of $PQ$ and $PR$ respectively.

Show that: (i) $ST$ is parallel to $QR$ (ii) $ST = \frac{1}{2}QR$.

(i) $N$ is the singular matrix $\begin{pmatrix} 4 & 2 \\ 3 & k \end{pmatrix}$. Find the value of . (ii) The matrix is . Find its inverse and hence determine the point of intersection of the lines and .