CSEC Mathematics Paper 2 May/June 2000 - Free CXC Past Paper with Answers

Factorize completely: (i) $3xy - x$ (ii) $4a^2 - 9$ (iii) $(x-y)^2 - x + y$

The floor of a room is in the shape of a rectangle. The floor is $c$ metres long. The width of the floor is 2 metres less than its length. (i) State, in terms of $c$ (a) the width of the floor (b) the area of the floor. (ii) If the area of the floor is $15 \text{ m}^2$, write down an equation in $c$ to show this information. (iii) Use the equation to determine the width of the floor.

The Venn diagram below shows the number of students doing Mathematics ($M$), Accounts ($A$) and French ($F$) in a class of 50.

(i) Write down an expression, in terms of $x$, for the number of students who do Accounts. (ii) Write down an equation, in terms of $x$, which shows the information in the Venn diagram. (iii) Determine the number of students who do Mathematics only. (iv) Determine the number of students who do French.

Solve the simultaneous equations $3a - 2b = 12$ $2a + b = 1$

Using ruler and compasses only: (i) Construct $\triangle ABD$ such that $BD = 7 \text{ cm}$, angle $ABD = \text{angle } ADB = 60^\circ$. (ii) Construct $AX$ which is perpendicular to $BD$ and which intersects $BD$ at $X$. (iii) Complete the quadrilateral $ABCD$ such that $AC = 2AX$. (iv) Measure and write down the length of $AC$. (v) Name the type of quadrilateral you have drawn. [All construction lines must be clearly shown.]

Write your answer to this question on the answer sheet provided. The diagram on the answer sheet shows triangle $PRS$. $T$ is a translation given by the vector $\begin{pmatrix} -2 \\ 3 \end{pmatrix}$. is a rotation of about the origin . (i) Draw and label triangle , the image of triangle under . (ii) Under the rotation , triangle is mapped onto triangle . Draw and label triangle .

Given that $f:x \to 3-x$ $g:x \to \frac{x+2}{x-5}$ Calculate $g(2)$.

State the value of $x$ for which $g(x)$ is not defined.

Derive an expression for $gf(x)$.

Calculate the value of $f^{-1}(4)$.

An aircraft left Jamaica at 13:55 hrs and travelled to Barbados via Antigua. The average speed of the aircraft was 420 km per hour. It arrived in Antigua at 16:45 hrs local time.

Given that Antigua is ONE hour AHEAD of Jamaica, calculate the distance between Jamaica and Antigua.

[Take $\pi = 3.14$] In the diagram above, not drawn to scale, $OLN$ is a sector of a circle, centre $O$ and $ON$ is produced to $D$. Angle $DLO = 90^\circ$, angle $DOL = 30^\circ$ and $DO = 8.7 \text{ cm}$. Calculate (i) the length, in cm, of $DL$ (ii) the radius of the sector $OLN$ (iii) the area of the sector $OLN$.

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

The table below gives the marks of 100 students in an examination. Copy and complete the table in your answer booklet.

Plot a cumulative frequency curve to show this information using a scale of 1 cm to represent 5 marks on the horizontal axis and 1 cm to represent 10 students on the vertical axis.

Show on your graph how to estimate (i) the lower quartile (ii) the upper quartile.

Use your estimates to calculate the interquartile range.

Given that 20% of the students passed the examination, determine the pass mark.

Solve $8x^2 + 3y^2 = 50$ $2x + y = 5$

Write the expression $9x^2 - 9x + 1$ in the form $a(x+b)^2 + c$, where , and are real numbers. (ii) Hence, state whether the function has a maximum or minimum value. (iii) State the value of at which this maximum or minimum value occurs.

Write the expression $9x^2 - 9x + 1$ in the form $a(x+b)^2 + c$, where , and are real numbers.

Hence, state whether the function $y = 9x^2 - 9x + 1$ has a maximum or minimum value.

State the value of $x$ at which this maximum or minimum value occurs.

Copy and complete the table below for the function $f(x) = -x^2 + 3x - 2$.

Using 2 cm to represent 1 unit on the $x$-axis and 1 cm to represent 10 students on the $y$-axis, draw the graph of $f(x) = -x^2 + 3x - 2$ for .

Using the graph, determine: (i) the maximum value of $f(x)$ (ii) The value of $x$ for which $f(x)$ is maximum (iii) The values of $x$ for which $f(x) > 0$.

the maximum value of $f(x)$

The value of $x$ for which $f(x)$ is maximum

The values of $x$ for which $f(x) > 0$.

Using the same scale and the same axes, draw the line $g(x) = -3$. (ii) Use the graphs of $f(x) = -x^2 + 3x - 2$ and to find the roots of the equation .

Using the same scale and the same axes, draw the line $g(x) = -3$.

Use the graphs of $f(x) = -x^2 + 3x - 2$ and $g(x) = -3$ to find the roots of the equation .

Draw a diagram to show the information above, labelling all given points, distances and angles.

Calculate, to 3 significant figures (i) the length $RG$ (ii) the height $RN$, of the antenna.

the length $RG$

the height $RN$, of the antenna.

A third point, $C$, lies on the ground 5.4 m from $N$. The angle of depression of $C$ from $R$ is $x$ degrees.

Calculate, to the nearest degree, the value of $x$.

Copy and complete the diagram and show the positions of Town $I$ and Town $J$. (ii) Calculate the radius of the circle of latitude on which Towns $I$ and $J$ are situated. (iii) Calculate the shortest distance, measured along the earth's surface, between the two towns.

Copy and complete the diagram and show the positions of Town $I$ and Town $J$.

Calculate the radius of the circle of latitude on which Towns $I$ and $J$ are situated.

Calculate the shortest distance, measured along the earth's surface, between the two towns.

In the figure above, not drawn to scale, $O$ is the centre of the circle $ABCDE$ and $TEF$ is a tangent to the circle at $E$.

Given that $DEF = 30^\circ$, calculate, giving reasons to support your answer, the sizes of the angle (i) (ii) (iii) (iv) .

$ACD$

$EAD$

$EOD$

$BCD$.

The position vectors of the points $P$, $Q$ and $R$ relative to an origin $O$ are $\vec{OP} = \begin{pmatrix} 7 \\ 9 \end{pmatrix}$, $\vec{OQ} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}$ and $\vec{OR} = \begin{pmatrix} -3 \\ 2 \end{pmatrix}$. (i) Express, in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, the vectors $\vec{PQ}$ and $\vec{RQ}$. (ii) Show, by calculation, that $PQ$ makes an angle of $45^\circ$ with the $x$-axis.

Express, in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, the vectors $\vec{PQ}$ and .

Show, by calculation, that $PQ$ makes an angle of $45^\circ$ with the $x$-axis.