CSEC Mathematics Paper 2 May/June 1984 - Free CXC Past Paper with Answers

Draw a Venn diagram to represent these sets and show on the diagram in the appropriate regions the members of $P$, $Q$, $R$ and $P \cap Q$.

In the figure above (not drawn to scale) $PS$ is an arc of a circle of radius 5 cm and $Q$ is the centre of the circle. $RQ = 6$ cm, $PR = 7$ cm.

Calculate (a) the size of angle $Q$ (b) the area of the shaded portion bounded by the arc $PS$ and the line segments $PR$ and $RS$. (use $\pi = 3.14$)

The points $A(0, 9)$ and $B(0, 4)$ are mapped by a rotation with centre $C$ on to the points $A'(8, 7)$ and $B'(4, 4)$. (i) Using a scale of 2 cm to 1 unit on both axes, plot the points $A, B, A'$ and $B'$. (ii) State (a) What is the relation of $A$ and $A'$ to $C$? (b) the size of the angle $BMC$, where $M$ is the midpoint of $B B'$. (iii) By suitable constructions find the coordinates of $C$. Measure and state the size of the angle of rotation to the nearest degree.

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

(ii) Using a scale of 2 cm to represent 1 unit on both axes, draw on graph paper the graph of the function for $-3 \le x \le 2$. (iii) Draw the line $y = x + 3$ on the same graph and write down the coordinates where $y = x + 3$ cuts the curve. (iv) Hence, solve the equation $x^2 + 2x - 2 = x + 3$.

Simplify the expression $4a^{-\frac{1}{2}}(a^{\frac{3}{2}} - a^{-\frac{1}{2}})$ State your answer using positive indices.

The table below shows the distribution the scores obtained by 100 candidates in an examination.

(i) Complete the column for the cumulative frequency on the answer sheet provided. (ii) Using 1 cm to represent 10 units on both axes draw the cumulative frequency curve on graph paper. (iii) From the curve estimate (a) the median (b) the probability that a student chosen at random obtained a score less than or equal to 56.

An object moves from rest and travels for 70 seconds. The speed changes uniformly over each given time interval shown in the table below. The speed shown is the value at the end of the time interval.

(i) Using a suitable scale, draw the speed-time graph on graph paper. (ii) Calculate the acceleration during the last twenty seconds. (iii) Calculate the total distance travelled in the seventy seconds.

A sliding door is to be put in a wall of area 81 m$^2$. The area of the door must not be greater than $\frac{1}{4}$ of the total wall space. The width of the door is 5 m more than the height.

Calculate the greatest possible height of the door.

In this question assume that the earth is a sphere with circumference 40 000 km at the equator. (i) Given that $P$ ($0^{\circ}$N, $75^{\circ}$W) and $Q$ ($0^{\circ}$N, $x^{\circ}$W) are points on the earth with $PQ$ 1 800 km west of $P$, calculate $x$. (ii) Calculate (a) the shortest distance between $A$ ($30^{\circ}$N, $42^{\circ}$W) and $B$ ($24^{\circ}$S, $42^{\circ}$W) (b) the distance between $R$ ($45^{\circ}$N, $50^{\circ}$W) and $S$ ($45^{\circ}$N, $22^{\circ}$E) measured along the parallel of latitude in an eastward direction from $R$.

In triangle $ABC$ above, angle $C$ is obtuse and $CD$ is perpendicular to $AB$. Prove that $\frac{a}{\sin A} = \frac{b}{\sin B}$. (ii) Prove that for any acute angle $A$ $\frac{1}{\tan A} + \tan A = \frac{1}{\sin A \cos A}$

The image of the point $P(x, y)$ under the transformation $T = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ is $P'(x', y')$. (i) Express $x', y'$ in terms of $x$ and $y$. (ii) Calculate the coordinates of the images of the points $A$ (0, 1), $B$ (2, 1), $C$ (2, 4) and $D$ (0, 4), under $T$. (iii) Plot on graph paper the figures $ABCD$ and $A'B'C'D'$. (iv) Describe the transformation geometrically.

The table below shows the frequency distribution of the marks obtained by students of Form A and B on a mathematics test.

(i) Calculate an estimate of the mean and standard deviation of the marks of students in Form A. (ii) COMPARE the means and standard deviations of the two marks of the two forms and COMMENT on these statistics.

Reasoning and Logic

It is given that $1 + 3 = 4 = 2^2$ $1 + 3 + 5 = 9 = 3^2$ $1 + 3 + 5 + 7 = 16 = 4^2$ $1 + 3 + 5 + \dots + 17 + 19 = 100 = 10^2$