CSEC Mathematics Paper 2 May/June 2010 - Free CXC Past Paper with Answers

Given that $a = -1$, $b = 2$ and $c = -3$, find the value of: (i) $a + b + c$ (ii) $b^2 - c^2$

$a + b + c$

$b^2 - c^2$

Write the following phrases as algebraic expressions: (i) seven times the sum of $x$ and $y$ (ii) the product of TWO consecutive numbers when the smaller number is $y$

seven times the sum of $x$ and $y$

the product of TWO consecutive numbers when the smaller number is $y$

Solve the pair of simultaneous equations: $2x + y = 7$ $x - 2y = 1$

Factorise completely: (i) $4y^2 - z^2$ (ii) $2ax - 2ay - bx + by$ (iii)

$4y^2 - z^2$

$2ax - 2ay - bx + by$

$3x^2 + 10x - 8$

A survey was conducted among 40 tourists. The results were:

• 28 visited Antigua (A)
• 30 visited Barbados (B)
• 3 visited neither Antigua nor Barbados

(i) Copy and complete the Venn Diagram below to represent the given information above. (ii) Write an expression, in $x$, to represent the TOTAL number of tourists in the survey. (iii) Calculate the value of $x$.

Copy and complete the Venn Diagram below to represent the given information above.

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

Calculate the value of $x$.

The diagram below, not drawn to scale, shows a wooden toy in the shape of a prism, with cross section $ABCDE$. $F$ is the midpoint of $EC$, and $\angle BAE = \angle CBA = 90^\circ$.

the length of $EF$

the length of $DF$

the area of the face $ABCDE$.

When $y$ varies directly as the square of $x$, the variation equation is written $y = kx^2$, where $k$ is a constant. (i) Given that $y = 50$ when $x = 10$, find the value of $k$. (ii) Calculate the value of $y$ when $x = 30$.

Given that $y = 50$ when $x = 10$, find the value of $k$.

Calculate the value of $y$ when $x = 30$.

Using a ruler, a pencil and a pair of compasses, construct triangle $EFG$ with $EG = 6$ cm, $\angle FEG = 50^\circ$ and $\mathrm{\angle }$.

Measure and state a) the length of $EF$ b) the size of $\angle EFG$.

The functions $f$ and $g$ are defined as $f(x) = 2x - 5$ and $g(x) = x^2 + 3$. (i) Calculate the value of a) $f(4)$ b) $g(4)$. (ii) Find $f^{-1}(x)$.

Calculate the value of a) $f(4)$ b) $g(4)$.

Find $f^{-1}(x)$.

The diagram below shows the graph of $y = x^2 + 2x - 3$ for the domain $-4 \le x \le 2$.

the scale used on the x-axis

the value of $y$ for which $x = -1.5$

the values of $x$ for which $y = 0$

the range of values of $y$, giving your answer in the form $a \le y \le b$, where $a$ and $b$ are real numbers.

The diagram below, not drawn to scale, shows two straight lines, $PQ$ and $RS$, intersecting a pair of parallel lines, $TU$ and $VW$.

Determine, giving a reason for EACH of your answers, the value of

$x$.

$y$.

The diagram below shows triangle $LMN$, and its image, triangle $L'M'N'$, after undergoing a rotation.

Describe the rotation FULLY by stating a) the centre b) the angle c) the direction.

State TWO geometric relationships between triangle $LMN$ and its image, triangle $L'M'N'$.

Triangle $LMN$ is translated by the vector $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$. Determine the coordinates of the image of the point $L$ under this transformation.

A class of 24 students threw the cricket ball at sports. The distance thrown by each student was measured to the nearest metre. The results are shown below. 22 50 35 52 47 30 48 34 45 23 43 40 55 29 46 56 43 59 36 63 54 32 49 60

Copy and complete the frequency table for the data shown above.

State the lower boundary for the class interval $20 - 29$.

Using a scale of 1 cm on the $x$-axis to represent 5 students, and a scale of 1 cm on the $y$-axis to represent 1 student, draw a histogram to illustrate the data.

Determine

the number of students who threw the ball a distance recorded as 50 metres or more

the probability that a student, chosen at random, threw the ball a distance recorded as 50 metres or more.

The diagram below shows the first three figures in a sequence of figures. Each figure is made up of squares of side 1 cm.

On your answer sheet, draw the FOURTH figure (Fig. 4) in the sequence.

Study the patterns in the table shown below, and on the answer sheet provided, complete the rows numbered (i), (ii), (iii) and (iv).

4

5

15

n

The diagram below shows the speed-time graph of the motion of an athlete during a race.

(i) Using the graph, determine a) the MAXIMUM speed b) the number of seconds for which the speed was constant c) the TOTAL distance covered by the athlete during the race. (ii) During which time-period of the race was a) the speed of the athlete increasing b) the speed of the athlete decreasing c) the acceleration of the athlete zero?

Using the graph, determine a) the MAXIMUM speed b) the number of seconds for which the speed was constant c) the TOTAL distance covered by the athlete during the race.

During which time-period of the race was a) the speed of the athlete increasing b) the speed of the athlete decreasing c) the acceleration of the athlete zero?

A farmer supplies his neighbours with $x$ pumpkins and $y$ melons daily, using the following conditions: First condition : $y \ge 3$ Second condition : $y \le x$ Third condition : the total number of pumpkins and melons must not exceed 12. (i) Write an inequality to represent the THIRD condition. (ii) Using a scale of 1 cm to represent one pumpkin on the x-axis and 1 cm to represent one melon on the y-axis, draw the graphs of the THREE lines associated with the THREE inequalities. (iii) Identify, by shading, the region which satisfies the THREE inequalities. (iv) Determine, from your graph, the minimum values of and which satisfy the conditions.

Write an inequality to represent the THIRD condition.

Using a scale of 1 cm to represent one pumpkin on the x-axis and 1 cm to represent one melon on the y-axis, draw the graphs of the THREE lines associated with the THREE inequalities.

Identify, by shading, the region which satisfies the THREE inequalities.

Determine, from your graph, the minimum values of $x$ and $y$ which satisfy the conditions.

In the diagram below, not drawn to scale, $PQ$ is a tangent to the circle $PTSR$, so that $\angle RQP = 46^\circ$, $\angle ROP = 90^\circ$ and is a straight line.

$\angle PTR$

$\angle TPR$

$\angle TSR$.

The diagram below, not drawn to scale, shows a vertical flagpole, $FT$, with its foot, $F$, on the horizontal plane $EFG$. $ET$ and $GT$ are wires which support the flagpole in its position. The angle of elevation of $T$ from is , m, m and .

the height, $FT$, of the flagpole

the length of $EG$

the angle of elevation of $T$ from $E$.

$A$ and $B$ are two $2 \times 2$ matrices such that $A = \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & -2 \\ -2 & 1 \end{pmatrix}$.

(i) Find $AB$. (ii) Determine $B^{-1}$, the inverse of $B$. (iii) Given that $\left(\begin{array}{cc}5& -2\\ -2& 1\end{array}\right)(\begin{array}{}\end{array}$, write as the product of TWO matrices. (iv) Hence, calculate the values of and .

Find $AB$.

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

Given that $\begin{pmatrix} 5 & -2 \\ -2 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$, write as the product of TWO matrices.

Hence, calculate the values of $x$ and $y$.

The diagram below, not drawn to scale, shows triangle $JKL$.

$M$ and $N$ are points on $JK$ and $JL$ respectively, such that $$ and .

Copy the diagram in your answer booklet and show the points $M$ and $N$.

Given that $\vec{JM} = \mathbf{u}$ and $\vec{JN} = \mathbf{v}$, write in terms of and , an expression for a) b) c) .

Using your findings in (b) (ii), deduce TWO geometrical relationships between $KL$ and $MN$.