CSEC Mathematics Paper 2 May/June 1994 - Free CXC Past Paper with Answers

Evaluate $3.7 \times 10^2 + 2.4 \times 10^3$, giving your answer in standard form.

Janet's gross salary is $2400 per month. Her tax-free allowances are shown in Table A below.

TABLE A: Tax-Free Allowances | National Insurance | 5% of gross salary | | Personal Allowance | $3000 per year |

Calculate (i) her gross yearly salary (ii) her total tax-free allowances for the year (iii) her taxable yearly income.

(iv) A 10% tax is charged on the first $20 000 of taxable income. A 20% tax is charged on the portion of taxable income above $20 000.

Calculate the amount of income tax Janet pays for a year.

Given the following information: $U = \{3, 6, 9, 12, \dots, 27\}$ $E = \{\text{even numbers}\}$ $G = \{\text{numbers greater than } 20\}$ $E$ and $G$ are subsets of $U$.

(i) List the members of $E$ and of $G$. (ii) Draw a Venn diagram to represent the above data. (iii) State $n(E \cup G)'$.

The width of a rectangular field is $w$ metres. The length is 6 metres more than twice the width. Write, in terms of $w$, algebraic expressions for

• the length of the field
• the area of the field.

(ii) The area of the field is $360 \text{ m}^2$.

• Write an algebraic equation for the area of the field.
• Determine the value of $w$.

(i) The scale used for a map is $1:250 \ 000$. The distance $MN$ on the map is $4.4 \text{ cm}$. Calculate, in kilometres, the actual distance of $M$ from $N$.

(ii) A car leaves $M$ at $09:50 \text{ hrs}$ and arrives at $N$ at $10:04 \text{ hrs}$ the same day. Calculate

• the time, in minutes, taken for the journey
• the average speed of the car in kilometres per hour, giving your answer to the nearest whole number.

In the figure above, not drawn to scale, $TF$ is perpendicular to $FY$, $FX = 40 \text{ m}$, $XY = 20 \text{ m}$ and angle . Calculate, correct to 2 significant figures, (i) the length of (ii) angle .

Using ruler and protractor, construct a quadrilateral $VWXY$ in which $YX = 8 \text{ cm}$, angle $XYV = 80^\circ$, $VY = 6 \text{ cm}$, $XW = 7 \text{ cm}$ and angle $XYW = 35^\circ$. Measure and state the length of $VW$ correct to one decimal place.

In the figure above, not drawn to scale, $C$ is the centre of the circle of radius $5 \text{ cm}$. $AB$ is a chord of length $6 \text{ cm}$. $J$ is a point on the circumference so that $$. (i) Calculate the perpendicular distance of from . (ii) Show that approximately.

The points $P(2, 1)$, $Q(4, 2)$ and $R(3, -5)$ are the vertices of $\triangle PQR$. $\triangle PQR$ is mapped onto $\triangle P'Q'R'$ by a translation $\begin{pmatrix} -1 \\ 2 \end{pmatrix}$.

(i) On graph paper, using $1 \text{ cm}$ to represent $1 \text{ unit}$ on each axis, plot the points $P, Q$ and $R$. (ii) Write the position vector of $P$ in the form $\begin{pmatrix} a \\ b \end{pmatrix}$. (iii) Calculate the vector , using the sum of two vectors. (iv) Calculate the coordinates of $P', Q'$ and .

$\triangle PQR$ is mapped by a reflection onto $\triangle P''Q''R''$ where is and is .

Given that $\begin{pmatrix} 2 & -3 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -2 \\ 3 \end{pmatrix} + \begin{pmatrix} 1 \\ 1 \end{pmatrix}$. calculate the values of $x$ and $y$.

A straight line $HK$ cuts the $y$-axis at $H(0, -1)$. The gradient of $HK$ is $\frac{2}{3}$. (i) Show that the equation of the line is .

The results of a test for defective bottles are shown in the table below, where $x$ is the number of defective bottles per carton, and $n$ is the number of cartons.

Illustrate the information given above on a bar chart.

Each carton contains a total of 48 bottles. Calculate, for the sample (i) the total number of bottles (ii) the total number of defective bottles.

The company produces 250 cartons each day but only ships cartons which contain no defective bottles. Using the results of the above sample, (i) calculate the probability of randomly choosing a carton which contains no defective bottles (ii) estimate the number of cartons that is likely to be shipped in a five-day week.

(i) Indicate, on your answer sheet, the value of $x$ for which $2^x = 12$. (ii) Indicate also, the value of $y$ for which $y = 2^{1.4}$. (iii) Determine the gradient of the curve $y = 2^x$ at $x = 2$, giving your answer correct to 2 significant figures.

The figure above shows the velocity-time graph $BRSE$ of a moving body. Calculate, clearly stating the units, (i) the acceleration of the body during the first two minutes (ii) the acceleration of the body during the last minute (iii) the total distance moved during the first ten minutes.

Given that $f(x) = 2x^2 + x - 3$, copy and complete the table below.

Using $2 \text{ cm}$ to represent $1 \text{ unit}$ on the $x$-axis, and $1 \text{ cm}$ to represent $1 \text{ unit}$ on the $$-axis, draw the graph of for .

Using the graph, (i) determine the value of $x$ for the minimum value of $f(x)$ (ii) determine the minimum value of $f(x)$ (iii) state the values of $x$ for $f(x) = 0$.

Using the same scale and the same axes as in part (b), (i) draw the graph of $g(x) = 3x - 1$ (ii) shade the area which represents $f(x) \le g(x)$.

In the diagram above, not drawn to scale, $O$ is the centre of the circle and $AOB$ is a diameter. $D$ is a point on the circumference and $F$ is the midpoint of the chord $BE$. Angle $ABE = 70^\circ$. (i) Calculate angle $BDE$. (ii) Show that $\triangle OFB$ and $\triangle AEB$ are similar. (iii) Calculate angle $AOF$.

(Note: State reasons and show necessary working.)

$VMNPQ$ is a pyramid on a square base $MNPQ$ of side $40 \text{ cm}$. (i) Draw a diagram to represent the pyramid. Clearly label the vertices. (ii) Draw a plan of the pyramid, viewed from above. State the scale used. (iii) The height of the pyramid is $20 \text{ cm}$. Show that the length of the sloping edge $VM$ is .

(In this question take the radius of the earth to be $6400 \text{ km}$ and $\pi$ to be $3.142$.) (i) A ship $X$ travelled due South from $P$ ($15^\circ \text{ W}$, $30^\circ \text{ N}$) to $Q$. Calculate the circumference of the circle that passes through $P$ and $Q$. Given that the distance $PQ$ is $3114 \text{ km}$, calculate the position of $Q$. (ii) Another ship $Y$ travelled due East from $P$ ($15^\circ \text{ W}$, $30^\circ \text{ N}$) to $T$ ($30^\circ \text{ E}$, $30^\circ \text{ N}$). Show that the radius of the circle that passes through $P$ and $T$ is $6400 \cos 30^\circ$. Calculate the distance $PT$.

In triangle $VXR$, $VX = 5 \text{ cm}$, $VR = 4 \text{ cm}$ and angle $XVR = 55^\circ$. Calculate to 3 significant figures (i) the area of triangle (ii) the perpendicular distance of from (iii) the magnitude of angle .

Sketch a diagram to show the position vectors of $F$ and of $G$.

Determine the vector $\vec{FG}$.

Show that the position vector (i) of $H$ is $\begin{pmatrix} -1 \\ 3 \end{pmatrix}$ (ii) of $M$ is $$.

of $H$ is $\begin{pmatrix} -1 \\ 3 \end{pmatrix}$

of $M$ is $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$.

Use vectors to show that (i) the diagonals of the parallelogram $OFGH$ bisect one another (ii) $OFGH$ is a square.

the diagonals of the parallelogram $OFGH$ bisect one another

$OFGH$ is a square.

Show, with the aid of diagrams, that (i) the transformation represented by the matrix $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ is NOT a rotation. (ii) the transformation represented by the matrix is NOT a reflection.

the transformation represented by the matrix $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ is NOT a rotation.

the transformation represented by the matrix $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ is NOT a reflection.

Determine the matrix which represents a reflection in the line $y = -x$.

The matrix $A = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ and the matrix . (i) Calculate . (ii) Identify the coordinates of the image of the point under the combined transformation represented by . (iii) Describe completely the combination of the two transformations represented by .

Calculate $AK$.

Identify the coordinates of the image of the point $(5, 3)$ under the combined transformation represented by $AK$.

Describe completely the combination of the two transformations represented by $AK$.