CSEC Mathematics Paper 2 May/June 2009 - Free CXC Past Paper with Answers

Factorise completely:

$2ax + 3ay - 2bx - 3by$

$5x^2 - 20$

$3x^2 + 4x - 15$

One packet of biscuits costs $x$ and one cup of ice cream costs $y$. One packet of biscuits and two cups of ice cream cost $8.00, while three packets of biscuits and one cup of ice cream cost $9.00.

Write a pair of simultaneous equations in $x$ and $y$ to represent the given information above.

Solve the equations obtained in (b) (i) above to find the cost of one packet of biscuits and the cost of one cup of ice cream.

In a survey of 50 students,

• 23 owned cellular phones
• 18 owned digital cameras
• $x$ owned cellular phones and digital cameras
• $2x$ owned neither.

Let $C$ represent the set of students in the survey who owned cellular phones, and $D$ the set of students who owned digital cameras.

Copy and complete the Venn diagram below to represent the information obtained from the survey.

Write an expression in $x$ for the TOTAL number of students in the survey.

Calculate the value of $x$.

The diagram below, not drawn to scale, shows a rhombus, $PQRS$, with the diagonal $PR = 6$ cm, and the angle $RPQ = 60^\circ$.

Using a ruler, a pencil, and a pair of compasses, construct the rhombus $PQRS$ accurately.

Join $QS$. Measure and state, in centimetres, the length of $QS$.

The table below shows two readings taken from an aircraft's flight record.

For the period of time between the two readings, calculate

the distance travelled in kilometres

the average speed of the aircraft in km/h.

The map shown below is drawn to a scale of 1:50 000.

Measure and state, in centimetres, the distance on the map from $L$ to $M$ along a straight line.

Calculate the actual distance, in kilometres, from $L$ to $M$.

The actual distance between two points is 4.5 km. Calculate the number of centimetres that should be used to represent this distance on the map.

Given that $f(x) = 2x - 5$ and $g(x) = x^2 - 31$, calculate the value of

$f(-2)$

$gf(1)$

$f^{-1}(3)$.

Given that $y = x^2 + 2x - 3$

Copy and complete the table below.

Using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis, draw the graph of $y = x^2 + 2x - 3$ for $-4 \le x \le 2$.

Describe, FULLY, the single transformation which maps triangle $A$ onto

triangle $D$.

triangle $B$.

State the coordinates of the vertices of triangle $C$, the image of triangle $A$ after a reflection in the line $y = x$.

Copy and complete the table, showing the cumulative frequency.

On the answer sheet provided, use the values from your table to complete the cumulative frequency curve.

Use your graph from (b) above to estimate

the median for the data

the number of students who waited for no more than 29 minutes

the probability that a student, chosen at random from the group, waited for no more than 17 minutes.

Draw Diagram 4 in the sequence.

Complete the table by inserting the appropriate values at the rows marked (i), (ii) and (iii).

Solve the pair of simultaneous equations $y = 4 - 2x$ $y = 2x^2 - 3x + 1$.

Express $2x^2 - 3x + 1$ in the form $a(x + h)^2 + k$, where , and are real numbers.

Using your answer from (b) above, or otherwise, calculate

the minimum value of $2x^2 - 3x + 1$

the value of $x$ for which the minimum occurs.

Sketch the graph of $y = 2x^2 - 3x + 1$, clearly showing

• the coordinates of the minimum point
• the value of the $y$-intercept
• the values of $x$ where the graph cuts the -axis.

Sketch on your graph of $y = 2x^2 - 3x + 1$, the line which intersects the curve at the values of $x$ and $y$ calculated in (a) above.

The owner of a shop wishes to buy $x$ guitars and $y$ violins. To satisfy the demands of his customers, the number of violins must be less than or equal to the number of guitars.

Write an inequality to represent this information.

The cost of one guitar is $150 and the cost of one violin is $300. He has $4 500 to spend on the purchase of these instruments.

Write an inequality to represent this information.

To get a good bargain, the owner of the shop must buy at least 5 violins.

Write an inequality to represent this information.

Using a scale of 2 cm on the horizontal axis to represent 5 guitars, and 2 cm on the vertical axis to represent 5 violins, draw the graphs of the lines associated with the THREE inequalities written in (a) (i), (ii) and (iii) above.

Shade the region on your graph that satisfies all THREE inequalities.

State the coordinates of the vertices of the shaded region.

The owner of the shop sells the instruments to make a profit of $60 on each guitar and $100 on each violin.

Express the TOTAL profit in terms of $x$ and $y$.

Calculate the maximum profit.

The diagram below, not drawn to scale, shows a circle, centre $O$. The line $DCE$ is a tangent to the circle. Angle $ACE = 48^\circ$ and angle $OCB = 26^\circ$.

Calculate:

$\angle ABC$

$\angle AOC$

$\angle BCD$

$\angle BAC$

$\angle OAC$

$\angle OAB$

The diagram below, not drawn to scale, shows the positions of two hurricane tracking stations, $P$ and $Q$, relative to a point $O$. $P$ is on a bearing of $025^\circ$ from , and km. is on a bearing of from and km.

Copy the diagram above. On your diagram label the angles that show the bearings of $025^\circ$ and $080^\circ$.

the length, to the nearest kilometre, of $OQ$

Copy the sketch and

label the arc which represents: a) $60^\circ N$ b) $35^\circ E$

insert the points: a) $J(60^\circ N, 35^\circ E)$ b) $K(60^\circ N, 15^\circ W)$

Calculate, to the nearest kilometre, the SHORTEST distance from

$J$ to the North Pole measured along the common circle of longitude

$J$ to $K$ measured along the common circle of latitude.

A point $H$ is located 2002 km due south of $K$ along the common circle of longitude. Calculate the latitude of $H$.

The points $A$ and $B$ have position vectors $\vec{OA} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ where $O$ is the origin $(0, 0)$. The point $G$ lies on the line $AB$ such that $\vec{AG} = \frac{1}{3}\vec{AB}$.

Express in the form $\begin{pmatrix} x \\ y \end{pmatrix}$

the vectors $\vec{AB}$ and $\vec{AG}$

the position vector $\vec{OG}$.

In the diagram below, not drawn to scale, $B$ is the midpoint of $OX$. $C$ is the midpoint of $AB$, and $D$ is such that $\overrightarrow{OD}=2$. The vectors and are such that and .

Write in terms of $\mathbf{a}$ and $\mathbf{b}$: a) $\vec{AB}$ b) $\stackrel{}{}$ c) d)

State TWO geometrical relationships between $DX$ and $DC$.

State ONE geometrical relationship between the points $D$, $C$, and $X$.

The value of the determinant of $M = \begin{pmatrix} x & 4 \\ 3 & x \end{pmatrix}$ is 13.

Calculate the values of $x$.

The transformation $R$ is represented by the matrix $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$.

Write a single matrix, in the form $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ to represent the combined transformation $$ followed by .

Calculate the image of the point $(5, -2)$ under the combined transformation in (b) (i) above.

The matrix $N = \begin{pmatrix} 3 & -1 \\ 2 & 5 \end{pmatrix}$.

Determine the inverse matrix of $N$.

Hence, calculate the value of $x$ and the value of $y$ for which $3x - y = 5$ $2x + 5y = 9$.