CSEC Mathematics Paper 2 January 2018 - Free CXC Past Paper with Answers

Given that $a * b$ means $\sqrt{a+4b}$, where the positive root is taken, determine

the value of $1 * 2$

whether the operation denoted by $*$ is commutative. Justify your answer.

Solve the inequality $3 - 2x > 5$.

Represent your answer in (b) (i) on the number line shown below.

Statement one: Two adult tickets and three children tickets cost $43.00. Statement two: One adult ticket and one ticket for a child cost $18.50.

Let $x$ represent the cost of an adult ticket and $y$ the cost of a ticket for a child. Write TWO equations in $x$ and $y$ to represent the information above.

Solve the equations to determine the cost of an adult ticket.

State the value of $n(P \cap R)$.

List the members of a) $M \cap P$ b) $M \cup R'$.

Using a ruler, a pencil and a pair of compasses, construct triangle $PQR$ with $PQ = 8$ cm, angle $PQR = 120^\circ$ and $$ cm.

Using a ruler, a pencil and a pair of compasses, construct triangle $PQR$ with $PQ = 8$ cm, angle $PQR = 120^\circ$ and $$ cm.

Measure and state the length of the side $PR$.

On your diagram in (b) (i), construct the point $S$, such that $PQRS$ forms a parallelogram.

Write the equation of the line, $L$, in the form $y = mx + c$.

Hence, determine the gradient of the line, $L$.

The point $P$ with coordinates $(r, 2)$ lies on the line $L$. Determine the value of $r$.

Find the equation of the straight line passing through the point $(6, 0)$ which is perpendicular to $L$.

Draw the straight lines $x + y = 10$ and $y = x$ on the grid below.

Draw the straight lines $x + y = 10$ and $y = x$ on the grid below.

On the same grid, shade the region which satisfies the FOUR inequalities $x \ge 0$ $y \ge 0$ $x + y \le 10$ $$.

The regular polygon $EFGHIJ$, shown below, has centre $O$. Triangle $OEF$ is equilateral and $EF = 5$ cm.

What is the name of the polygon shown above?

Calculate the perimeter of the polygon $EFGHIJ$.

Determine the size of each interior angle of the polygon.

Show, by calculation, that the area of the polygon, to the nearest whole number, is $65$ cm$^2$.

A tank has a cross section with dimensions identical to the polygon $EFGHIJ$ in 5 (a). Water is poured into the tank at a rate of $75$ cm$^3$ per second. After 52 seconds the tank is $\frac{2}{5}$ full. Determine the capacity of the tank, in litres.

Calculate the height, $h$, in metres, of the tank.

State the coordinates of $R$.

On the diagram above, draw (i) $\Delta P'Q'R'$, a reflection of $\Delta PQR$ in the line $y = 1$ (ii) $\Delta P''Q''R''$, a reflection of $\Delta P'Q'R'$ in the line $x = 0$.

$\Delta P'Q'R'$, a reflection of $\Delta PQR$ in the line

$\Delta P''Q''R''$, a reflection of $\mathrm{\Delta }{P}^{\mathrm{\prime }}{Q}^{\mathrm{\prime }}{R}^{\mathrm{\prime }}$ in the line .

Describe, fully, the single transformation that maps $\Delta P''Q''R''$ onto $\Delta PQR$.

Triangle $PQR$ undergoes an enlargement of scale factor 2. Calculate the area of its image.

The marks obtained by 10 students in a test, scored out of 60, are shown below.

29 38 26 42 38 45 35 37 38 31

For the data above, determine

the range

the median

the interquartile range

the probability that a student chosen at random scores less than half the total marks in the test.

The frequency distribution below shows the masses, in kg, of 50 adults prior to the start of a fitness programme.

On the grid on page 23, using a scale of 2 cm to represent 5 units on the x-axis and 1 cm to represent 1 unit on the y-axis, draw a frequency polygon to represent the information in the table.

Draw Figure 4 of the sequence.

Study the patterns of numbers in each row of the table below. Each row relates to one of the figures in the sequence of figures above. Some rows have not been included in the table.

Complete the rows numbered (i), (ii), (iii) and (iv).

Complete the row for Figure (i).

Complete the row for Figure (ii).

Complete the row for Figure (iii).

Complete the row for Figure (iv).

(i) Show, by calculation, that the EXACT roots of the quadratic equation $x^2 + 2x - 5 = 0$ are $-1 \pm \sqrt{6}$.

Show, by calculation, that the EXACT roots of the quadratic equation $x^2 + 2x - 5 = 0$ are $-1 \pm \sqrt{6}$.

Hence, or otherwise, solve the simultaneous equations $2 + x = y$ $xy = 5$

The incomplete table below shows values of $x$ and $y$ for the function $y = 2^x$ for integer values of $x$ from $$ to .

Complete the table for the function $y = 2^x$.

On the grid provided on page 27, draw the graph of $y = 2^x$, 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.

Drawing appropriate lines on your graph, determine the value of $x$ for which $2^x = 11$.

MEASUREMENT, GEOMETRY AND TRIGONOMETRY

10. (a) The diagram below, not drawn to scale, shows a circle with centre $O$. The points $A, B, C$ and $D$ are on the circumference of the circle. $EAF$ and $EDG$ are tangents to the circle at $A$ and $D$ respectively. $\angle AOD = 114^\circ$ and $\angle CDG = 18^\circ$.

Calculate, giving reasons for EACH step of your answer, the measure of

$\angle ACD$

$\angle AED$

$\angle OAC$

$\angle ABC$.

The diagram below shows a cuboid.

Give your answer correct to one decimal place.

A straight adjustable wire connects $R$ to $P$ along the top of the cuboid. Calculate the length of the wire $RP$.

The connection at $P$ is now adjusted and moved to $T$.

Calculate the length of the wire $RT$.

Calculate the angle $TRV$.

Complete the following statements:

The size of the angle through which the wire moves from $RP$ to $RT$ is $\underline{\hspace{5cm}}$. An angle which is the same in size as $RTV$ is .

VECTORS AND MATRICES Given the vectors $\vec{OP} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$, $\vec{PQ} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}$ and $\vec{RS} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}$.

determine the vector $\vec{OQ}$.

show that $\vec{OQ}$ is parallel to $\vec{RS}$, giving a reason for your answer.

$XYZ$ is a triangle and $M$ is the midpoint of $XZ$. $\vec{XY} = \mathbf{a}$ and .

$\vec{XZ}$

$\vec{MY}$

The matrices $A$ and $B$ are given as $A = \begin{pmatrix} -1 & 0 \\ 3 & 2 \end{pmatrix}$ and .

Determine $A^{-1}$, the inverse of $A$.

Show that $A^{-1}A = I$, the identity matrix.

Determine the matrix $A^2$.

Explain why the matrix product $AB$ is NOT possible.

Without calculating, state the order of the matrix product $BA$.