CSEC Mathematics Paper 2 May/June 2012 - Free CXC Past Paper with Answers

Factorise completely: (i) $2x^2y + 6xy^2$ (ii) $9x^2 - 4$ (iii) $4x^2 + 8xy - xy - 2y^2$

$2x^2y + 6xy^2$

$9x^2 - 4$

$4x^2 + 8xy - xy - 2y^2$

Solve for $x$: $\frac{2x - 3}{3} + \frac{5 - x}{2} = 3$

Solve the simultaneous equations: $3x - 2y = 10$ $2x + 5y = 13$

In a survey of 36 students, it was found that

• 30 play tennis,
• $x$ play volleyball ONLY,
• 9 play BOTH tennis and volleyball,
• 4 play neither tennis nor volleyball.

Given that: $V = \{\text{students who play volleyball}\}$ $T = \{\text{students who play tennis}\}$

Copy and complete the Venn diagram below to show the number of students in the subsets marked $y$ and $z$.

Copy and complete the Venn diagram below to show the number of students in the subsets marked $y$ and $z$.

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

Write an equation in $x$ to represent the total number of students in the survey and hence solve for $x$.

The diagram below, not drawn to scale, shows the journey of a ship which started at port $P$, sailed 15 km due south to port $Q$, and then a further 20 km due west to port $R$.

(i) Copy the diagram and label it to show the points $Q$ and $R$, and the distances 20 km and 15 km. (ii) Calculate $PR$, the shortest distance of the ship from the port where the journey started. (iii) Calculate the measure of angle , giving your answer to the nearest degree.

Copy the diagram and label it to show the points $Q$ and $R$, and the distances 20 km and 15 km.

Calculate $PR$, the shortest distance of the ship from the port where the journey started.

Calculate the measure of angle $QPR$, giving your answer to the nearest degree.

Calculate (i) the length of the arc $ABC$ (ii) the perimeter of the sector $OABC$ (iii) the area of the sector $OABC$.

the length of the arc $ABC$

the perimeter of the sector $OABC$

the area of the sector $OABC$.

The prism is 20 cm long and is a solid made of tin. Calculate (i) the volume of the prism (ii) the mass of the prism, to the nearest kg, given that 1 cm$^3$ of tin has a mass of 7.3 kg.

the volume of the prism

the mass of the prism, to the nearest kg, given that 1 cm$^3$ of tin has a mass of 7.3 kg.

Using a ruler, a pencil and a pair of compasses, construct triangle $PQR$ with $PQ = 8$ cm, $\angle PQR = 60^\circ$ and $\angle QPR = 45^\circ$.

Measure and state the length of $RQ$.

The line $\ell$ passes through the points $S(6, 6)$ and $T(0, -2)$.

Determine

the gradient of the line, $\ell$

the equation of the line, $\ell$

the midpoint of the line segment, $TS$

the length of the line segment, $TS$.

Locate the centre of enlargement, showing your method clearly.

State the scale factor and the coordinates of the centre of the enlargment.

Determine the value of $\frac{\text{Area of } PQR}{\text{Area of } LMN}$.

Draw and label triangle $ABC$ with coordinates $(-4, 4)$, $(-1, 4)$ and $(-1, 2)$ respectively.

Describe fully the single transformation which maps triangle $LMN$ on to triangle $ABC$.

Copy and complete the table to show the cumulative frequency.

Using a scale of 2 cm to represent 10 years on the x-axis and 1 cm to represent 5 persons on the y-axis, draw the cumulative frequency curve for the data.

Use your graph drawn at (b) above to estimate

the median age for the data

the probability that a person who visited the clinic was 75 years or younger.

On the answer sheet provided, draw the fourth figure (Figure 4) in the sequence.

Study the patterns in the table below, and on your answer sheet, complete the rows numbered (i), (ii), (iii) and (iv). The breaks in the columns are to indicate that the rows do not follow one after the other.

4

100

20

n

Solve the pair of simultaneous equations: $y = 8 - x$ $2x^2 + xy = -16$

State, giving the reason for your answer, whether the line $y = 8 - x$ is a tangent to the curve $2x^2 + xy = -16$.

An answer sheet is provided for this question.

A florist makes bouquets of flowers, each consisting of $x$ roses and $y$ orchids. For each bouquet, she applies the following constraints:

• the number of orchids must be at least half the number of roses
• there must be no more than 12 flowers
• there must be no more than 12 flowers

Write THREE inequalities for the constraints given.

On the answer sheet provided, shade the region of the graph which represents the solution set for the inequalities in (b) (i).

State the coordinates of the points which represent the vertices of the region showing the solution set.

The florist sells a bouquet of flowers to make a profit of $3 on each rose and $4 on each orchid. Determine the MAXIMUM possible profit on the sale of a bouquet.

The diagram below, not drawn to scale, shows a quadrilateral $QRST$ in which $QS = 7$ cm, $ST = 10$ cm, $QT = 8$ cm, $\angle SRQ = 60^\circ$ and $\angle RQS = 48^\circ$.

Calculate

the length of $RS$

the measure of $\angle QTS$.

The diagram below, not drawn to scale, shows a circle, centre $O$. The line $UVW$ is a tangent to the circle, $ZOXW$ is a straight line and angle $UOX = 70^\circ$.

Calculate, showing working where necessary, the measure of angle a) $OUZ$ b) $UVY$ c) $UWO$.

Name the triangle in the diagram which is congruent to triangle a) $ZOU$ b) $YXU$.

The points $A$, $B$ and $C$ have position vectors $\vec{OA} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}$, $\vec{OB} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ and $\vec{OC} = \begin{pmatrix} 12 \\ -2 \end{pmatrix}$.

Express in the form $\begin{pmatrix} x \\ y \end{pmatrix}$ the vector a) $\vec{BA}$ b) .

State ONE geometric relationship between $\vec{BA}$ and $\vec{BC}$.

Draw a sketch to show the relative positions of $A$, $B$ and $C$.

Calculate the values of $a$ and $b$ such that $\begin{pmatrix} a & -4 \\ 1 & b \end{pmatrix} \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$.

Calculate the values of $a$ and $b$ such that $\begin{pmatrix} a & -4 \\ 1 & b \end{pmatrix} \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$.

Hence, or otherwise, write down the inverse of $\begin{pmatrix} 2 & -4 \\ 1 & -3 \end{pmatrix}$.

Use the inverse of $\begin{pmatrix} 2 & -4 \\ 1 & -3 \end{pmatrix}$ to solve for $$ and in the matrix equation .