CSEC Mathematics Paper 2 May/June 1996 - Free CXC Past Paper with Answers

Calculate the value of $\frac{2\frac{1}{3} - 1\frac{1}{2}}{1\frac{5}{6}}$ , expressing your answer as a fraction in lowest terms.

Evaluate $\sqrt{1.073}$ , giving your answer to three significant figures.

The Peters' family consists of Mr Peters, his wife and two children. Mr Peters' monthly salary is $2,800.

Table showing income tax allowances for a year are calculated as follows:

	Husband	Wife	Each child
	$1000.	$600.	$200.

Table showing Tax Rates for a year

Chargeable Income	Rate of tax
For every dollar of the first $12 000.	5 cents
" " " " next $8 000.	15 cents
" " " " next $20 000.	35 cents

Calculate for Mr Peters, (i) his salary for the year (ii) his total allowances (iii) his chargeable income (iv) the amount he has to pay as income tax (v) the percent of his salary paid as income tax.

his salary for the year

his total allowances

his chargeable income

the amount he has to pay as income tax

the percent of his salary paid as income tax.

Given that $l = -2$ , $m = 3$ and $n = 7$ , calculate the value of $lm(m - n)$ .

If $P = \frac{q}{5} + r$ , express $s$ in terms of $p$ , and .

Factorize completely: (i) $8h^2 - 4h$ (ii) $4d^2 - 1$

Solve the equation $\frac{4x + 5}{4} - \frac{9 + 2x}{3} = 0$ .

The Venn Diagram (Figure 1) shows two sets, $X$ and $Y$ , which are subsets of the universal set $\varepsilon$ . Copy the diagram and shade the region which represents $(X \cup Y) \cap X'$ .

Figure 2 is a Venn Diagram showing the universal set $\varepsilon$ and three subsets $A$ , $B$ and $C$ . The lower case letters written in the diagram are members of the various subsets. (i) List the members of $(A \cup B \cup C)'$ . (ii) List the members of . (iii) State the value of .

List the members of $(A \cup B \cup C)'$ .

List the members of $B \cap C$ .

State the value of $n(B \cup C)$ .

The diagram above shows vector $b$ and vector $c$ . Express in the form $\begin{pmatrix} x \\ y \end{pmatrix}$ (i) $b$ (ii) $c$ (iii) $b - c$ . Calculate (iv) $|b - c|$ (v) the size of the angle between $b - c$ and the x-axis.

the size of the angle between $b - c$ and the x-axis.

(i) Construct triangle $LMN$ with $MN = 10$ cm, $LM = 6.5$ cm and angle $LMN = 50^\circ$ . (ii) Measure and state the length of . (iii) Measure and state the size of angle . (iv) Using ruler and compasses ONLY, construct so that is perpendicular to and meets in .

Construct triangle $LMN$ with $MN = 10$ cm, $LM = 6.5$ cm and angle $LMN = 50^\circ$ .

Measure and state the length of $LN$ .

Measure and state the size of angle $LNM$ .

Using ruler and compasses ONLY, construct $LX$ so that $LX$ is perpendicular to $MN$ and meets $MN$ in $X$ .

If $f(x) = 2x - 1$ and $g(x) = \frac{1}{2}(x + 2)$ , calculate (i) $f(3)$ (ii) $g^{-1}(x)$ (iii) $gf(3)$

Write your answers to this question on the answer sheet provided. The diagram drawn on the answer sheet shows triangle $ABC$ . $T$ is a translation vector represented by $\begin{pmatrix} -6 \\ 0 \end{pmatrix}$ . is rotation about the point . (i) Write down the coordinates of , and . (ii) Draw and label triangle which is the image of triangle under . (iii) Draw and label triangle which is the image of triangle under .

Write down the coordinates of $A$ , $B$ and $C$ .

Draw and label triangle $A'B'C''$ which is the image of triangle $ABC$ under .

Draw and label triangle $A''B''C''$ which is the image of triangle $ABC$ under .

The graph above shows a straight line $EF$ intersecting the x-axis at $F$ . (i) State the coordinates of $F$ . (ii) Calculate the gradient of $EF$ . (iii) Determine the equation of $EF$ .

State the coordinates of $F$ .

Calculate the gradient of $EF$ .

Determine the equation of $EF$ .

The velocity-time graph shown above represents the journey of a motor cyclist, between two points. Using the graph, determine, by stating the units used, (i) the length of time during which the cyclist accelerated (ii) the acceleration (iii) the length of time during which the velocity was constant (iv) the deceleration.

the length of time during which the cyclist accelerated

the length of time during which the velocity was constant

The table above shows the cost of a plot of land over a period of time.

On graph paper, using a scale of 2 cm to represent 1 year on the horizontal axis and 1 cm to represent $10,000 on the vertical axis, draw a line graph to represent the data.

From your graph: Determine the period during which the cost of the plot of land showed the greatest rate of increase.

State the year during which the cost of the plot of land decreased.

Estimate the cost of the plot of land in the year 1996.

State one reason for your estimate in (iv) above.

A firm manufacturing computers found that on the average 65 out of 80 of its computers passed the required standard set for quality control. Calculate

the probability that a computer selected at random would be defective

the number of computers the firm would expect to be defective, if it manufactured 336 computers.

The diagram $VXY$ above, not drawn to scale, represents a right open-ended paper cone. The slant height $VX$ is 14 cm, the base diameter $XY$ is 21 cm and $VO$ is the height of the cone. [Take $\pi$ to be $\frac{22}{7}$ ].

Calculate, in cm, the circumference of the base of the cone.

Show that the height of the cone is approximately 9.3 cm.

Calculate the volume, in $\text{cm}^3$ , of the cone.

The cone is cut along a straight line drawn from $V$ to $X$ . The paper is flattened out to form a plane figure.

Draw a diagram of the plane figure, showing the position of $V$ .

Write the length of each side on your diagram.

Calculate the size of the angle, in degrees, at $V$ .

Given that $f(x) = x^3$ , copy and complete the table below:

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$-27$			$0$		$8$	$27$

Using 2 cm to represent 1 unit on the x-axis and 1 cm to represent 5 units on the y-axis, draw the graph of $f(x) = x^3$ for $-3 \le x \le 3$ .

On the graph of $f(x) = x^3$ :

Draw a tangent to the curve at the point $x = 2$ .

Estimate the gradient of $f(x)$ at $x = 2$ .

Draw the graph of $g(x) = -6x - 20$ .

Using the graphs, obtain the value of $x$ for which $f(x) = g(x)$ .

A florist uses $x$ lilies and $y$ roses to make bouquets. The total number of flowers in a bouquet must be less than or equal to 45.

Write an inequality in $x$ and $y$ to represent this condition.

There must be at least 10 lilies in the bouquet. Write an inequality to represent this condition.

The number of lilies is less than or equal to twice the number of roses. Write an inequality in $x$ and $y$ to represent this condition.

Using a scale of 1 cm to 5 units on each axis, graph the inequalities in (i), (ii) and (iii) above.

Shade ONLY the region which satisfies the three inequalities in (i), (ii) and (iii) above.

The profit on a lily is $1, and the profit on a rose is $2. Write an expression for the total profit made on a bouquet.

Use the graph to determine the maximum profit that could be made on a bouquet.

Two ships leave a Port $O$ . One ship travels on a bearing of $340^\circ$ to a point $P$ which is 50 km from $O$ . The other ship travels on a bearing of $060^\circ$ to a point $Q$ , 85 km from $O$ .

(i) Draw a diagram to represent the position of the port and the two ships. On your diagram, carefully label North, the given angles and the distances travelled.

Calculate the distance $PQ$ in km.

Determine the bearing of $P$ from $Q$ .

Given that $\tan \theta = \frac{2}{3}$ and $\theta$ is an acute angle, show that $\sin θ \cos θ = \frac{2 \sqrt{3}}{}$ .

Convert (i) $160^\circ$ to radians

$\frac{2\pi}{3}$ radians to degrees.

The diagram above, not drawn to scale, shows a circle of radius 10 cm. $O$ is the centre of the circle. $J, K$ and $L$ are points on the circumference of the circle. Angle $JOK = 150^\circ$ . Take $\pi$ to be 3.14.

Show that the area of the minor sector $OJK$ is $\frac{125\pi}{3} \text{ cm}^2$ .

If the area of the triangle $LOK$ is 43.3 $\text{cm}^2$ , calculate, in degrees, the size of the angle $LOK$ .

The diagram above, not drawn to scale, shows $TP$ and $TQ$ as tangents to a circle from a point $T$ . The circle has centre $O$ and the angle $OMQ = 70^\circ$ . Calculate, giving reasons for your answer, the measure of (i) angle

In the triangle $OXY$ , $\vec{OX} = x$ and $\vec{OY} = y$ . $M$ is the mid-point of $XY$ . $\vec{OL} = \frac{2}{3}\vec{OX}$ and $\vec{ON} = \frac{3}{2}\vec{OY}$ .

(i) Copy the triangle $OXY$ and indicate on the triangle, the positions of $L, M$ and $N$ .

Express in terms of $x$ and $y$ $XY, LM$ and $MN$ .

Using a vector method, Prove $L, M$ and $N$ are collinear.

State the value of the ratio $LM : MN$ .

A plane is defined by two perpendicular coordinate axes, $Ox$ and $Oy$.

(i) Write down a $2 \times 2$ matrix which represents $R$ , a reflection in the line $y = x$ .

Write down a $2 \times 2$ matrix which represents $S$ , an anticlockwise rotation through $180^\circ$ about the origin.

Describe, in words, the transformation $Q$ which is represented by the vector $\begin{pmatrix} -2 \\ 1 \end{pmatrix}$ .

Determine the image of the point $(3, 4)$ under the following transformation $RS$ .

Describe, in words, $RS$ as a single geometrical transformation.

Calculate the values of $a$ and $b$ such that the point $(a, b)$ is an invariant point under the transformation $SQ$ .

CSEC Mathematics Paper 2 May/June 1996 Questions

Question 1 (12 marks)

Question 2 (10 marks)

Question 3 (4 marks)

Question 4 (15 marks)

Question 5 (11 marks)

Question 6 (12 marks)

Question 7 (12 marks)

Question 8 (10 marks)

Question 9 (15 marks)

Question 10 (15 marks)

Question 11 (15 marks)

Question 12 (17 marks)

Question 13 (13 marks)

Question 14 (12 marks)