CSEC Mathematics Paper 2 May/June 1988 - Free CXC Past Paper with Answers

Draw a carefully labelled Venn diagram to illustrate this data.

Given that $n(U) = 40 \ 000$, $n(S) = 25 \ 000$, $n(C) = 10 \ 000$ and $n(P) = 8000$, Determine $n(S \cup C)$ and $n(C \cap P)'$.

Calculate (i) the length of $AB$ (ii) the gradient of $AB$ (iii) the co-ordinates of $X$.

Determine the equation of the perpendicular bisector of $AB$ and state the co-ordinates of the point at which the perpendicular bisector meets the $y$-axis.

Simplify (i) $2 \ (5x - y) - 3 \ (3x - y)$ (ii) $81^{\frac{-1}{2}} \times 27^{\frac{2}{3}}$

Calculate correct to one decimal place the values of $x$ for which $2x^2 + 2x - 8 = 3x - 6$

Given that $A = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$, $B = \begin{pmatrix} p & q \\ 0 & r \end{pmatrix}$ and that $AB = A + 2B$, calculate the values of $p, q$ and $r$.

In the diagram above, not drawn to scale, $PQ$ and $LM$ represent parallel edges of an east-west river bank. Angle $LPR = 55^\circ$ and angle $RQM = 25^\circ$.

Copy in your answer booklet the table above and complete the cumulative frequency column.

Draw the cumulative frequency curve using 2 cm to represent each class interval and 2 cm to represent 10 applicants.

Using your curve, answer the following: (i) A score of 45 was considered as acceptable for the programme. Estimate the number of applicants who qualified for entry. (ii) Assuming there were places for only 15 applicants, estimate the lowest score that would be used to select them. (iii) Calculate the probability that an applicant chosen at random obtained a score of at least 45.

evaluate $f(-2)$ and $g(-2)$

determine $f^{-1}(x)$

calculate the value of $x$, if $f(x) = 12$

calculate the values of $x$, for which (i) $f(x) = 0$ (ii) $f(x)$ is undefined.

Use ruler and compasses only for this question. All construc-tion lines and arcs must be clearly shown and be of sufficient length and clarity to permit assessment. (i) Construct a triangle $ABC$ in which $AB = 8$ cm, $\hat{A} = 45^\circ$ and $\hat{B} = 60^\circ$. (ii) Construct also the perpendicular bisector of $AC$ to meet $AB$ at $X$. (iii) Measure accurately and state the lengths of $AC$ and $AX$.

A rectangular steel pyramid of height 6 cm and base dimensions 11 cm by 18 cm, is melted down and rolled into a cylinder of height 7 cm. Calculate (i) the radius of the cylinder in cm. (ii) the mass of the cylinder in kg, if the density of the steel is 5 g/cm$^3$.

Notes: Volume of pyramid $= \frac{1}{3}Ah$ Volume of cylinder Take to be

Copy and complete the table below for the functions $x \to \cos x$ and $x \to \tan x$, where $x$ is measured in degrees.

Draw the graph of $x \to \cos x$ for $0^\circ \le x \le 50^\circ$, using a horizontal scale of 2 cm to represent and a vertical scale of 10 cm to represent 1 unit.

Using the same scales and on the same axes, draw the graph of $x \to \tan x$, for $0^\circ < x \le 50^\circ$.

Using your graphs, (i) solve the equation $\tan x = \cos x$ in the range $0^\circ \le x \le 50^\circ$ (ii) derive the value for which in the range . (iii) deduce the corresponding values of , for each value of , in the range .

Find the range of values of $x$ for which (i) $x(x + 1) > 0$ (ii) $x(x + 1) \le 6$

The diagram above shows a shaded region $ABC$ bounded by the line $AB$ represented by the equation $x + 3y = 6$, the line $AC$ represented by the equation and the broken line drawn parallel to . (i) State THREE inequalities in and to describe the shaded region . (ii) Given that , find the maximum value of for any point in the region .

Given that $\tan \alpha = \frac{2}{K}$ calculate the value of (i) $\cos \alpha$ (ii) $\sin (90^\circ - \alpha)$ (iii) $\cos (90^\circ - \alpha)$.

$P$ and $Q$ are two points on level ground. $P$ is 10 metres due south of a vertical tower and the angle of elevation of the top of the tower from $P$ is $75^\circ$. lies 30 metres away on a bearing of from . (i) Illustrate this information by means of a sketch, taking care to indicate clearly the appropriate distances and angles. (ii) Calculate, in metres, the height of the tower the distance of from the foot of the tower.

A model aeroplane, flying on a course due north with an air speed of 10 kilometres per hour, encounters a wind blowing from the west which causes the aeroplane to go $30^\circ$ off course. Use a scale of 1 cm to represent 1 km h$^{-1}$, determine by drawing, (i) the wind speed (ii) the ground speed of the aeroplane.

A helicopter is situated at a point $P(25.8^\circ \text{ N}, 2^\circ \text{ W})$ on the earth's surface. It has a range of 600 km. Calculate the latitude and longitude of the furthest places which the helicopter can reach by flying (i) due east (ii) due north. [Note: Consider that the earth is a sphere with circumference 40 000 km.]

In the figure above, not drawn to scale, the transformation $R$ is an anti-clockwise rotation about the origin $O$ through an angle of $60^\circ$.

The images of the points $L, M$ and $N$ under $R$ are respectively $L', M'$ and $N'$. $L$ is the point $(2, 0)$ and $M$ is the point $(0, 6)$. $N'$ is the point $(\sqrt{3}, -1)$, not shown. (i) Calculate the co-ordinates of $L'$ and $M'$. (ii) Hence, determine the image of the vectors $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ under the transformation $R$ and write down the matrix for $R$.

The transformation $T$ is represented by the matrix $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$. Describe fully the transformation .

Calculate the co-ordinates of the image of $N$ under the combina-tion of transformations denoted by $TR$.

Determine the $2 \times 2$ matrix which represents the combination of transformations denoted by $RT$.

(i) Calculate the determinant of the matrix $\begin{pmatrix} 4 & 3 \\ -1 & \frac{1}{2} \end{pmatrix}$

(ii) Find the inverse of the matrix $\begin{pmatrix} 4 & 3 \\ -1 & \frac{1}{2} \end{pmatrix}$.

(iii) Hence, use a matrix method to solve simultaneously the pair of equations $8x + 6y = 28$ $-2x + y = 8$

Given that $\vec{a}$ and $\vec{b}$ are two vectors in the same plane,