CSEC Mathematics Paper 2 May/June 1997 - Free CXC Past Paper with Answers

Answer ALL the questions in this section.

Calculate the exact value of (i) $\frac{2.8 + 1.36}{4 - 2.7}$ (ii) $(\frac{27}{8})^{\frac{1}{3}}$

Calculate $9.72 \times 12.05$ and write your answer (i) exactly (ii) correct to 2 decimal places (iii) correct to 2 significant figures (iv) in standard form.

If $\frac{3x+1}{3} - \frac{x-2}{2} = 2 + \frac{2x-3}{3}$ , find the value of x.

A rectangular block of wood is 6 cm high. The cross-section of the block is a square of side 16 cm. A cylindrical container is carved out of the block. The cylinder is 5 cm deep and the diameter of the cross-section is 14 cm. The figure above, not drawn to scale, shows the top surface of the container.

Calculate (a) the volume of the rectangular block of wood (b) the area of the wooden top surface of the container (c) the capacity, in litres, of the container (d) the volume of wood in the container. (Use $\pi = \frac{22}{7}$ )

The charges for electricity are given in the table below:

ITEM	RATE
Rental	$2.00 per month
Energy charge	15¢ per KWH
Fuel charge	2.05¢ per KWH

The meter readings for the KWH used for December and January were:

Previous Reading 30 November	Present Reading 31 January
05783	06593

A government tax of 15% of the total charges is added to the bill.

Calculate (i) the number of KWH units used (ii) the total amount to be paid.

The charges for electricity are given in the table below:

ITEM	RATE
Rental	$2.00 per month
Energy charge	15¢ per KWH
Fuel charge	2.05¢ per KWH

The meter readings for the KWH used for December and January were:

Previous Reading 30 November	Present Reading 31 January
05783	06593

A government tax of 15% of the total charges is added to the bill.

Calculate (i) the number of KWH units used (ii) the total amount to be paid.

$f$ and $g$ are functions defined as follows: $f : x \rightarrow 3x - 5$ $g : x$

(i) Using ruler and compasses only, construct the trapezium $KLMN$ with $KL$ parallel to $NM$ , $KL = 4$ cm, cm, and cm. (ii) Show that . (iii) Show, by calculation, that the distance between and is cm.

In a competition, 3 points are awarded for winning a match, 0 points if the match was lost, and 1 point for a draw.

The results of seven matches played by three teams, $J$ , $K$ and $L$ are:

Team $J$ won 3, lost 1, and drew 3; Team $K$ won 2, lost 4, and drew 1; Team won 3, lost 2, and drew 2.

The figure above, not drawn to scale, is a sketch of a asymmetrical jar. The jar is in the shape of two cylinders joined by a frustum of a cone. The diameter of the bottom cylinder is 8 cm and of the top cylinder is 2 cm. The height of each cylinder is 2 cm. The total height of the jar is 8 cm. Construct full-size drawings of (i) the plan (ii) an elevation.

On the answer sheet provided, use a scale of 2 cm to represent 10 cm on the x-axis, and 2 cm to represent 10 pieces of ribbon on the y-axis. Draw a cumulative frequency curve to represent the information given in the table.

Using your graph, estimate (i) the lower quartile (ii) the upper quartile.

SECTION II Answer TWO questions in this section. RELATIONS AND FUNCTIONS An answer sheet is provided for part (b) of this question.

Sketch the graph of the function $F = x^2 + x - 6$ for $-4 \le x \le 3$ .

An answer sheet is provided for this question.

The diagram on the graph sheet represents the distance-time graphs of a bus and a car.

The bus leaves a town $X$ at 08:30 hrs and travels to another town $Y$ . Its distance-time graph is $DFGH$ . The car leaves $Y$ and travels to $X$ . Its distance-time graph is . There is a check-point between and where all vehicles MUST stop.

TRIGONOMETRY AND GEOMETRY

In the figure above, not drawn to scale, $SR$ is a tangent to the circle. The chords $SN$ and $MP$ intersect at $Q$ . The chord $MOP$ produced meets the tangent at . Angle , angle and angle .

In this question, use $\pi = 3.14$ .

A wheel is turning at the rate of 33 revolutions per minute. Express this speed in radians per second, correct to 2 significant figures.

An airport is situated at $G (40^\circ \text{N}, 50^\circ \text{W})$ . Another airport at lies east of . The airports at and are apart, measured along the circle of latitude. (i) Calculate the length of the circle of latitude , expressing the answer in standard form. (ii) Calculate the longitude of . (iii) Another airport is situated at . Calculate the shortest distance between and , expressing the answer in standard form. [Use radius of the earth = ]

$KLMN$ is a quadrilateral. The position vector of $K$ is $\begin{pmatrix} -2 \\ 1 \end{pmatrix}$ , of is and of is .

An answer sheet is provided for part (a) of this question.

The position vector of $H$ is $\begin{pmatrix} -1 \\ 1 \end{pmatrix}$ , of is , and of is . $H, K$ and are the vertices of . The triangle is reflected in the line . (i) State the position vectors of the vertices of the image. (ii) Using 2 cm to represent 1 unit on each axis, draw on graph paper the triangle and its image. (iii) Name the axes of symmetry of the shape made by the object and the image together.

Factorize completely: (i) $15x^2y - 20xy^2$ (ii) $3 - 12h^2$

Given that $m = -3$ , $n = 2$ , $p = -1$ , find the value of $\frac{m(p-n)^2}{3p + m}$ .

the volume of the rectangular block of wood

the area of the wooden top surface of the container

the capacity, in litres, of the container

the volume of wood in the container. (Use $\pi = \frac{22}{7}$ )

The Venn diagram above represents information about the 40 members of a youth club. $F = \text{ {members who play football} }$ $C = \text{ {members who play cricket} }$ $T = \text{ {members who play tennis} }$

The same number, x, play football only and tennis only. (i) Calculate the number who play football. (ii) State the information represented by the shaded region of the Venn diagram. (iii) State the relationship between the members of C and F, and between C and T.

Calculate the number who play football.

State the information represented by the shaded region of the Venn diagram.

State the relationship between the members of C and F, and between C and T.

Calculate the value of $f(-3)$ .

Write expressions for (i) $f^{-1}(x)$ (ii) $g^{-1}(x)$ .

Hence, or otherwise, write an expression for $(gf)^{-1}$ .

Using ruler and compasses only, construct the trapezium $KLMN$ with $KL$ parallel to $NM$ , $KL = 4$ cm, $NM = 7$ cm, $\angle KLM = 120^\circ$ and $LM = 6$ cm.

Show that $\angle NKL = \angle MNK = 90^\circ$ .

Show, by calculation, that the distance between $KL$ and $MN$ is $3\sqrt{3}$ cm.

A plane takes off at an angle of elevation of $17^\circ$ to the ground. After 25 seconds the plane has travelled a horizontal distance of 2400 metres. (i) Draw a sketch to represent the given information. (ii) Calculate to 2 significant figures the height of the plane above the ground after 25 seconds.

Draw a sketch to represent the given information.

Calculate to 2 significant figures the height of the plane above the ground after 25 seconds.

In a competition, 3 points are awarded for winning a match, 0 points if the match was lost, and 1 point for a draw.

The results of seven matches played by three teams, $J$ , $K$ and $L$ are:

Team $J$ won 3, lost 1, and drew 3; Team $K$ won 2, lost 4, and drew 1; Team $L$ won 3, lost 2, and drew 2.

(i) Write a column matrix to represent the points awarded for win, lose and draw. Name the matrix $P$ . (ii) Write a matrix to represent the results of the matches played by the three teams. Name the matrix $R$ . (iii) Construct a matrix product which may be used to find the number of points each team was awarded at the end of the competition. (iv) Using your matrix product, calculate the total points awarded to each team.

A piece of ribbon is chosen at random. Calculate the probability that it is NOT longer than 50 cm.

From the graph, estimate for the domain the range of $F$

the interval of the domain for which $F(x) < 0$ .

The figure on the graph sheet shows the graph of the curve $y = 2x^2 - 3x - 2$ .

Using the graph, find the gradient of the curve at the point where $x = 2$ .

On the same axes, draw the graph of $5y = 18x + 36$ .

Using your graphs, find the solution of the simultaneous equations $y + 2 = 2x^2 - 3x$ and $5y = 18x + 36$ .

the distance between $X$ and $Y$

the position of $M$ .

What information about the vehicles is given by

At what time did the car

For the journey between $Y$ and $M$ , calculate the velocity in kmh $^{-1}$ of

Calculate the average speed of the car for the whole journey to the nearest kmh $^{-1}$ .

Calculate, giving reasons

Given that $SR = 12$ cm, show that the area of $\triangle MRS = 39$ cm $^2$ , correct to 2 significant figures.

Find (i) $\vec{LM}$ (ii) $\vec{KM}$ .

Write the equation of the line (i) $LM$ (ii) $KM$ .

$P$ is the midpoint of $KM$ . Find the position vector of $P$ .

$P$ is also the midpoint of $LN$ . Show that the position vector of $N$ is $\begin{pmatrix} -2 \\ -2 \end{pmatrix}$ .

Show that $KLMN$ is a parallelogram, giving reasons.

The matrix $N = \begin{pmatrix} -1 & 1 \\ 1 & -2 \end{pmatrix}$ describes a transformation on $\triangle GDF$ . The coordinates of the image are $G'(3, 2)$ , $D'(5, 2)$ and $F'(4, 3)$ . Find (i) the matrix which would transform $\triangle G'D'F'$ onto $\triangle GDF$ (ii) the coordinates of $G, D$ and $F$ .

CSEC Mathematics Paper 2 May/June 1997 Questions

Question 1 (10 marks)

Question 2 (13 marks)

Question 3 (10 marks)

Question 4 (19 marks)

Question 5 (10 marks)

Question 6 (13 marks)

Question 7 (5 marks)

Question 8 (15 marks)

Question 9 (45 marks)

Question 10 (18 marks)

Question 11 (15 marks)

Question 12 (15 marks)

Question 13 (15 marks)

Question 14 (15 marks)