CSEC Mathematics Paper 2 May/June 2006 - Free CXC Past Paper with Answers

Simplify $\frac{x-3}{3} - \frac{x-2}{5}$.

Factorise (a) $x^2 - 5x$ (b) $x^2 - 81$ (ii) Simplify $\frac{a^2+4a}{{}^{}}$.

Two cassettes and three CD's cost $175$ while four cassettes and one CD cost $125$. (i) Given that one cassette costs $x$ and one CD costs $y$, write two equations in $x$ and $y$ to represent the information. (ii) Calculate the cost of one cassette.

In the quadrilateral $KLMN$, not drawn to scale, $LM = LN = LK$, $\angle KLM = 140^\circ$, and $\angle LKN = 40^\circ$.

Giving the reason for each step of your answer, calculate the size of (i) $\angle LNK$ (ii) $\angle NLM$ (iii) $\angle KNM$.

Using a ruler, a pencil and a pair of compasses, construct the triangle $ABC$ in which $AB = 8$ cm, $\angle BAC = 60^\circ$, and $AC = 5$ cm. (Credit will be given for a neat, clear diagram)

Measure and state the length of $BC$.

Find the perimeter of $\triangle ABC$.

Draw on your diagram the line $CD$ which is perpendicular to $AB$ and meets $AB$ at $D$.

Determine the length of $CD$.

Calculate the area of $\triangle ABC$ giving your answer to 1 decimal point.

The owner of a parking lot wishes to park $x$ vans and $y$ cars for persons attending a function. The lot provides parking space for no more than $60$ vehicles.

Identify, by shading, the region which satisfies all three inequalities.

Write an inequality to represent this information.

To get a good bargain, he must provide parking space for at least $10$ cars. Write an inequality to represent this information.

The number of cars parked must be fewer than or equal to twice the number of vans parked. Write an inequality to represent this information.

Using a scale of $2$ cm to represent $10$ vans on the $x$-axis and $2$ cm to represent $10$ cars on the $y$-axis, draw the graphs of the lines associated with the inequalities at (i), (ii) and (iii) above.

The parking fee for a van is $6$ and for a car is $5$. Write an expression in $x$ and $y$ for the total fees charged for parking $x$ vans and $y$ cars.

Using your graph write down the coordinates of the vertices of the shaded region.

Calculate the maximum fees charged.

The diagram below, not drawn to scale, shows a vertical tower, $FT$, and a vertical antenna, $TW$, mounted on the top of the tower.

A point $P$ is on the same horizontal ground as $F$, such that $PF = 28$ m, and the angles of elevation of $T$ and $W$ from $P$ are $40^\circ$ and $54^\circ$ respectively.

Copy and label the diagram clearly showing a) the distance $28$ m b) the angles of $40^\circ$ and $54^\circ$ c) any right angles.

Calculate the length of the antenna $TW$.

The diagram below, not drawn to scale, shows a circle, centre $O$. The lines $BD$ and $DCE$ are tangents to the circle, and angle $BCD = 70^\circ$.

$\angle OCE$

$\angle BAC$

$\angle BOC$

$\angle BDC$.

The diagram below, not drawn to scale, shows parallelogram $EFGH$ in which $EF = 6$ cm, $EH = 4.2$ cm, and angle $FEH = 70^\circ$. Calculate (i) the length of the diagonal $HF$ (ii) the area of the parallelogram $EFGH$.

the length of the diagonal $HF$

the area of the parallelogram $EFGH$.

In this question, use $\pi = 3.14$ and assume the earth to be a sphere of radius 6370 km. The diagram below, not drawn to scale, shows a sketch of the earth with the North and South poles labelled $N$ and $S$ respectively. The circle of latitude $41^\circ N$ is shown. Arcs representing circles of longitude and are drawn but not labelled.

Copy the sketch above, and draw and label two arcs to represent a) the Equator b) the Greenwich Meridian.

Two points, $Y$ and $M$, on the surface of the earth have coordinates $Y(41^\circ N, 74^\circ W)$ and . a) Insert the points and on your diagram. b) Calculate, correct to the nearest kilometre, the circumference of the circle of latitude . c) Calculate the shortest distance between and measured along the circle of latitude .

Copy and complete the diagram to show (i) the point $B$ such that $OABC$ is a parallelogram (ii) the vector $\vec{u}$ where $\vec{u} = \vec{OA} + \vec{OC}$.

the point $B$ such that $OABC$ is a parallelogram

the vector $\vec{u}$ where $\vec{u} = \vec{OA} + \vec{OC}$.

Write as a column vector, in the form $\begin{pmatrix} x \\ y \end{pmatrix}$ the vector (i) $\vec{OA}$ (ii) (iii) .

$\vec{OA}$

$\vec{OC}$

$\vec{AC}$.

Given that $G$ is the midpoint of $OB$, use a vector method to (i) determine the coordinates of $G$ (ii) prove, using a vector method, that $A$, $G$, and $C$ lie on a straight line.

determine the coordinates of $G$

prove, using a vector method, that $A$, $G$, and $C$ lie on a straight line.

The value of the determinant of $M = \begin{pmatrix} 2 & 3 \\ -1 & x \end{pmatrix}$ is 9. (i) Calculate the value of $x$. (ii) For this value of $x$, find $M^{-1}$. (iii) Show that $M^T M = I$.

Calculate the value of $x$.

For this value of $x$, find $M^{-1}$.

Show that $M^T M = I$.

The graph below shows the line segment $AC$ and its image $A'C'$ after a transformation by the matrix $\left(\begin{array}{cc}p& q\\ r& s\end{array}\right)$.

Write in the form of a single $2 \times 2$ matrix, the coordinates of a) $A$ and $C$ b) $A'$ and $$.

Using matrices only, write an equation to represent the transformation of $AC$ onto $A'C'$.

Determine the values of $p, q, r$ and $s$.