CSEC Mathematics Paper 2 May/June 2013 - Free CXC Past Paper with Answers

Factorize completely: (i) $2x^2 - 8x$ (ii) $3x^2 - 5x - 2$

$2x^2 - 8x$

$3x^2 - 5x - 2$

(i) Make $C$ the subject of the formula $F = \frac{9}{5}C + 32$. (ii) Given that $$, calculate the value of .

Make $C$ the subject of the formula $F = \frac{9}{5}C + 32$.

Given that $F = 113$, calculate the value of $C$.

500 tickets were sold for a concert. Of these $x$ tickets were sold at $6 and the remainder at $10 each. (i) Write an expression, in terms of $x$, for a) the number of tickets sold at $10 each b) the TOTAL amount of money collected for the sale of the 500 tickets. (ii) The sum of $4108 was collected for the sale of the 500 tickets. Calculate the number of tickets sold at $6 each.

Write an expression, in terms of $x$, for a) the number of tickets sold at $10 each b) the TOTAL amount of money collected for the sale of the 500 tickets.

The sum of $4108 was collected for the sale of the 500 tickets. Calculate the number of tickets sold at $6 each.

A survey of the 30 students in Form 5 showed that some students used cameras ($C$) or mobile phones ($M$) to take photographs.

• 20 students used mobile phones
• 4x students used ONLY cameras
• x students used BOTH mobile phones and cameras
• 2 students did not use either cameras or phones.

(i) Copy the Venn diagram below and complete it to show, in terms of $x$, the number of students in each region. (ii) Write an expression, in terms of $x$, which represents the TOTAL number of students in the survey. (iii) Determine the number of students in Form 5 who used ONLY cameras.

Copy the Venn diagram below and complete it to show, in terms of $x$, the number of students in each region.

Write an expression, in terms of $x$, which represents the TOTAL number of students in the survey.

Determine the number of students in Form 5 who used ONLY cameras.

In the diagram below, not drawn to scale, $AEC$ and $ADB$ are straight lines. $\angle ABC = \angle ADE = 90^\circ$. m, m and m.

Calculate the length of $BC$.

Explain why triangles $ABC$ and $ADE$ are similar.

Determine the length of $DE$.

The diagram below shows an isosceles triangle $CDE$. $G$ is the midpoint of $CD$.

(i) Measure and state, in centimetres, the length of $DE$. (ii) Measure and state, in degrees, the size of $\angle ECD$. (iii) Determine the perimeter of the triangle $CDE$. (iv) Calculate the area of the triangle $CDE$.

Measure and state, in centimetres, the length of $DE$.

Measure and state, in degrees, the size of $\angle ECD$.

Determine the perimeter of the triangle $CDE$.

Calculate the area of the triangle $CDE$.

$A (-1, 4)$ and $B (3, 2)$ are the end points of a line segment $AB$. Determine (i) the gradient of $AB$ (ii) the coordinates of the midpoint of (iii) the equation of the perpendicular bisector of .

the gradient of $AB$

the coordinates of the midpoint of $AB$

the equation of the perpendicular bisector of $AB$.

The incomplete table below shows one pair of values for $A$ and $R$ where $A$ is directly proportional to the square of $R$.

Express $A$ in terms of $R$ and a constant $k$.

Calculate the value of the constant $k$.

Copy and complete the table.

Given that $f(x) = \frac{2x+1}{3}$ and $g(x) = 4x+5$, determine the values of:

$fg(2)$

$f^{-1}(3)$

A car, travelling along a straight road at a constant speed of 54 km/h, takes 20 seconds to travel the distance between two sign posts.

Calculate

the speed of the car in m/s

the distance, in metres, between the two sign posts.

An answer sheet is provided for this question.

The graph below shows triangle $LMN$ and its image $L'M'N'$ after undergoing a single transformation.

Describe fully the transformation that maps $\triangle LMN$ onto $\triangle L'M'N'$.

On the answer sheet provided, draw triangle $L''M''N''$ the image of triangle $LMN$, after a translation by the vector .

Name and describe a combination of TWO transformations which may be used to map $\triangle L''M''N''$ onto $\mathrm{△}{L}^{\mathrm{\prime }}{M}^{\mathrm{\prime }}{}^{}$.

Copy and complete the table to show the cumulative frequency.

Using a scale of 1 cm to represent $5 on the horizontal axis and 1 cm to represent 5 students on the vertical axis, draw the cumulative frequency graph for the data.

(Marks will be awarded for axes appropriately labelled, points correctly plotted, and a smooth curve carefully drawn.)

Use your graph to estimate

the median amount of money spent

the probability that a student chosen at random spent less than $23 during the week.

Show on your graph, using broken lines, how these estimates were determined.

Draw a sketch of Diagram 4, the fourth diagram in the sequence.

Complete the table by inserting the missing values at the rows marked (i) and (ii).

Row (i) in the table.

Row (ii) in the table.

Write the rules which may be used to find the values of $W$ and of $B$ where $N$ is known.

$W = \text{______}

$B = \text{______}

Trish wishes to buy $x$ oranges and $y$ mangoes which she intends to carry in her bag. Her bag has space for only 6 fruits.

More information about the number of oranges and mangoes associated with the good bargain is represented by

$y \le 2x$.

Write an inequality to represent this information.

To get a good bargain, she must buy AT LEAST 2 mangoes.

Write an inequality to represent this information.

Write the information represented by this inequality as a sentence in your own words.

On the answer sheet provided, draw the lines associated with the two inequalities obtained in (i) and (ii) above.

Shade on your graph the region which represents the solution set for the three inequalities.

Write $3x^2 - 12x + 8$ in the form $a(x + h)^2 + k$ where $a, h$ and are constants.

Sketch the graph of $y = 3x^2 - 12x + 8$, showing on your sketch a) the intercept on the $y$-axis b) the coordinates of the minimum point.

The diagram below, not drawn to scale, shows a circle with centre $O$. $EBC$ is a tangent to the circle. $\angle OBA = 40^\circ$ and $\angle OBF = 35^\circ$.

Calculate, giving reasons for your answer, the measure of

$\angle EBF$

$\angle BOA$

$\angle AFB$

$\angle OAF$

The diagram below, not drawn to scale, shows three points $R, S$ and $F$ on the horizontal ground. $FT$ is a vertical tower of height 25 m. The angle of elevation of the top of the tower, $T$, from $R$ is $27^\circ$. is due east of and is due south of . m.

Sketch separate diagrams of the triangles $RFT$, $TFS$ and $SFR$. Mark on EACH diagram the given measures of sides and angles.

Show, by calculation, that $RF = 49.1$ m.

Calculate the length of $SR$ correct to 1 decimal place.

Calculate the angle of elevation of the top of the tower, $T$, from $S$.

Express in terms of $\mathbf{a}$ and $\mathbf{b}$ the vectors

$\vec{AB}$

$\vec{PQ}$.

State TWO geometrical relationships that exist between $OB$ and $PQ$. Give reasons for your answers.

Given that $M = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}$

Evaluate $M^{-1}$, the inverse of $M$.

Show that $M^{-1}M = I$.

Use a matrix method to solve for $r, s, t$ and $u$ in the equation $\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} \begin{pmatrix} r & s \\ t & u \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 4 & -1 \end{pmatrix}$.