CSEC Mathematics Paper 2 May/June 2002 - Free CXC Past Paper with Answers

Simplify

$3m - 2(m+1)$

$\frac{3}{y} - \frac{2}{y-2}$

Solve the equation $2(x - 1) = \frac{5}{2}$.

Calculate the range of values of $v$ when $5 - v \le 2v - 1$.

The diagram above, not drawn to scale, shows a square $KLMN$, where $KM = 8$ cm and $KN = x$ cm. Show that $x^2 = 32$.

Using ruler and compasses only (i) draw the diagonal $KM = 8$ cm (ii) construct the perpendicular bisector of $KM$. Hence, draw the square $KLMN$.

draw the diagonal $KM = 8$ cm

construct the perpendicular bisector of $KM$. Hence, draw the square $KLMN$.

In the diagram above, $OC = OC'$, $BC = B'C'$ and all angles are right angles. can be mapped onto by a transformation, , followed by another transformation, . Describe fully the transformations (i) (ii) .

$J$

$K$.

In the Venn diagram above, $U = \{ \text{whole numbers less than 10} \}$, and $A$ and $B$ are subsets of $U$. (i) Describe $A$ and $B$ in words. (ii) List the members of $A \cap B$ and describe the set, in words, in relation to $A$ and $B$. (iii) Determine $n(A \cup B)$.

Describe $A$ and $B$ in words.

List the members of $A \cap B$ and describe the set, in words, in relation to $A$ and $B$.

Determine $n(A \cup B)$.

The diagram above, not drawn to scale, shows $ABCDEF$, a vertical cross-section of a container with $ED$ being the top edge. $DC$ and $EF$ are vertical edges. $BC$ and are arcs of a circle of radius 7 cm and . cm; cm; cm. (i) Taking , show that the area of is 459 cm. (ii) Water is poured into the container until the water level is 4 cm from the top. If the container is 40 cm long and has uniform cross-section, calculate, to the nearest litre, the volume of water in the container.

Taking $\pi = \frac{22}{7}$, show that the area of $ABCDEF$ is 459 cm$^2$.

Water is poured into the container until the water level is 4 cm from the top. If the container is 40 cm long and has uniform cross-section, calculate, to the nearest litre, the volume of water in the container.

Given that $\vec{PR} = \begin{pmatrix} b \\ -2b \end{pmatrix}$ and $\vec{PS} = \begin{pmatrix} 3b \\ b+1 \end{pmatrix}$. (i) express EACH of the vectors $\vec{RP}$ and $\vec{RS}$ in the simplest form (ii) determine the values of $b$ if $| \vec{PR} | = \sqrt{20}$ units.

express EACH of the vectors $\vec{RP}$ and $\vec{RS}$ in the simplest form

determine the values of $b$ if $| \vec{PR} | = \sqrt{20}$ units.

A man travelled a total distance of 8 km in 54 minutes by running and walking. He ran $x$ km at 10 km h$^{-1}$ and walked the remaining distance at 5 km h$^{-1}$. (i) Write an expression in $x$ for the time, in hours, that a) he ran b) he walked. (ii) a) Form an equation in for the total time, in hours, spent travelling. b) Calculate the value of . c) Hence, calculate the distance the man walked.

Write an expression in $x$ for the time, in hours, that a) he ran b) he walked.

a) Form an equation in $x$ for the total time, in hours, spent travelling. b) Calculate the value of $x$. c) Hence, calculate the distance the man walked.

The functions, $f$ and $g$, are defined by $f(x) = \frac{x}{3} + 1$ and $g(x) = 2x - 1$. Calculate $g(-3)$.

Find, in its simplest form, (i) $f^{-1}(x)$ (ii) $g^{-1}(x)$ (iii) $fg(x)$ (iv) .

$f^{-1}(x)$

$g^{-1}(x)$

$fg(x)$

$(fg)^{-1}(x)$.

Show that $(fg)^{-1}(x) = g^{-1}f^{-1}(x)$.

The diagram above, not drawn to scale, shows a circle, centre $O$. $BA$ is parallel to $CD$ and $\angle CBD = 45^\circ$. (i) Calculate, giving reasons, the values of $x$ and $y$. (ii) Show that $ABCD$ is a square, giving the reasons for your answer.

Calculate, giving reasons, the values of $x$ and $y$.

Show that $ABCD$ is a square, giving the reasons for your answer.

Points $O, P$ and $Q$ are in the same horizontal plane. $P$ is 15 m away from $O$ on a bearing of $040^\circ$ from $O$. is on a bearing of from , and m. (i) Sketch a diagram to show the positions of $O, P$ and . Clearly indicate North on your diagram. (ii) Calculate the distance .

Sketch a diagram to show the positions of $O, P$ and $Q$. Clearly indicate North on your diagram.

Calculate the distance $OQ$.

Calculate the number of hours spent on sports.

Calculate the angles in a pie chart that would be used to represent the hours spent on (i) Mathematics (ii) Languages.

Draw a pie chart to represent the distribution of hours in the week, which the student spends on the activities indicated in the table above.

One hour in the 30 hours is chosen at random. Calculate the probability that the student is (i) playing sports (ii) studying Mathematics or Languages.

SECTION II RELATIONS, FUNCTIONS AND GRAPHS

Given that $y = \frac{1}{2}x^2$, copy and complete the table below.

Using scales 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 the function $y$ for $-2 \le x \le 3$.

Using the graph (i) solve the equation, $\frac{1}{2}x^2 = 4$ (ii) determine the values of $x$ for which $$.

Using the same axes and scales (i) draw the graph of $y = 2$ (ii) write down the equation in $x$ whose root is given by the intersection of the graphs, $y = 2$ and $y = \frac{1}{2}x^2$.

The diagram above shows the graph of the function $y = px^2 + qx + r$. (i) Determine the values of $p$, $q$, and $r$. (ii) State TWO ways in which the graphs of the functions $y = px^2$ and $y = px^2 + qx + r$ are similar. (iii) State ONE way in which the graphs of the two functions is different.

In the diagram above, not drawn to scale, $AB$ is the straight line joining $A(-1, 9)$ and $B(3, 1)$. (i) Calculate the gradient of the line, $AB$. (ii) Determine the equation of the line, . (iii) Write the coordinates of , the point of intersection of and the -axis. (iv) Write the equation of the line through , the origin, that is perpendicular to . (v) Write the equation of the line through that is parallel to .

GEOMETRY AND TRIGONOMETRY

In $\triangle JKL$, the coordinates of the vertices are $J(0, 1)$, $K(5, -4)$ and $L(7, 2)$. (i) Draw $\triangle JKL$. (ii) Determine the coordinates of $M$, the midpoint of $KL$. (iii) Show by calculation, that $JK = JL$.

In the diagram above, not drawn to scale, $ABCT$ is a circle. $AC$ and $BT$ are diameters. $TP$, the tangent at $T$, meets $BA$ produced at , so that . Calculate, giving reasons for all statements, the size of (i) (ii) (iii) (iv) .

In the diagram above, not drawn to scale, $ST = 5$ cm, $TW = 9$ cm and $STW = 52^\circ$. Calculate (i) the length of $SW$ (ii) the area of $\triangle STW$.

The diagram above, not drawn to scale, shows $ABCD$, a bucket of height 18 cm. The bucket is made by removing a cone $VAB$, of height $h$ cm, from a larger cone $VCD$. $AMB$, the diameter of the circular base of the bucket is cm. , the diameter of the open top, is 28 cm. (i) Show, giving reasons, that . (ii) Determine the value of .

Given that $M = \begin{pmatrix} n+1 & 2 \\ 2n & n \end{pmatrix}$ (i) write down an expression for the determinant of $M$ (ii) find $M^{-1}$.

The coordinates of the vertices of $\triangle PQS$ are $P(1, 5)$, $Q(4, -1)$ and $S(6, 0)$. (i) Write down the position vectors, and . (ii) Determine the position vectors, and , given that and are the midpoints of and respectively. (iii) Determine the vectors and . (iv) Hence, state TWO geometrical relationships between and .

Write down the matrix (i) $M_y$ that represents reflection in the y-axis (ii) $R_t$ that represents a rotation of $180^\circ$ about the origin.

Determine the single matrix, $U$, that represents a transformation, $M_y$, followed by another transformation, $R_t$.

Describe geometrically the transformation represented by (i) $R_p = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ (ii) .

On the answer sheet provided, using a scale of 1 cm to represent 1 unit on each axis, draw the pentagon $ABCDE$ with vertices $A(1, 2)$, $B(4, 2)$, $C(4, 5)$, and .

Draw the image of $ABCDE$ under the transformation represented by (i) $R_p$, and label that image $A'B'C'D'E'$. (ii) , and label that image .