CSEC Mathematics Paper 2 May/June 1998 - Free CXC Past Paper with Answers

Factorize completely: (i) $2y^2 + 3y$ (ii) $81 - m^2$ (iii) $2x^2 - x - 15$

If $S$ varies directly as $(r + 1)$, and $S = 8$ when $r = 3$, calculate the value for when .

Ninety tickets were sold for a concert. $x$ tickets were sold for $3.00 each and the rest of the tickets were sold for $4.00 each. Write an expression in $x$ to represent the number of tickets sold at (i) $3.00 each (ii) $4.00 each. (iii) If the total sales on all tickets amounted to $300.00, how many of the tickets costing $3.00 were sold?

The Universal set, $\xi$, contains the sets $J, K$ and $L$ such that $J \subset K$, $J \cap K \neq \phi$ and $L \subset (J \cup K)^c$. Draw a Venn diagram to show the relation amongst the sets $J, K$ and $L$.

The Venn diagram below shows sets $C$ and $D$, where $P$ and $Q$ are subsets of $C$ and $D$ respectively.

Given that $n(C) = 33$, $n(D) = 41$, $n(C \cap D) = x$, write down, in terms of , the values of and .

If $n(C \cup D) = 51$, calculate the value of $x$.

The matrix $H = \begin{pmatrix} h & 2 \\ -h & 1 \end{pmatrix}$. (i) Determine . (ii) Evaluate , if .

All construction lines must be clearly shown. (i) Draw a line $PR$ which measures 7.5 cm. Construct the perpendicular bisector, $QS$, of $PR$ to meet $PR$ at $O$, such that $OQ$ and $OS$ both measure 5 cm. Complete the parallelogram $PQRS$. (ii) State the name of this type of parallelogram $PQRS$. (iii) Measure and state, in cm, the length of $PQ$. (iv) Measure and state the size of angle $QPS$.

On the graph paper provided for this question, the point $P(x, y)$ has been marked in. (i) Write down the coordinates of $P$. (ii) Through $P$, draw a straight line whose $y$-intercept is 4. (iii) Calculate the gradient of the straight line. (iv) Determine the equation of the straight line.

$\vec{AB} = \begin{pmatrix} -3 \\ 2 \end{pmatrix}$ and $\vec{EF} = \begin{pmatrix} 5 \\ c \end{pmatrix}$. (i) Calculate $|\vec{AB}|$, the length of $\vec{AB}$. (ii) If $|\vec{EF}| = 3|\vec{AB}|$, show that the value of $c = 10\sqrt{2}$.

The transformation, $M$, denotes a reflection in the line $x = 2$. The transformation, $T$, denotes a translation represented by $\begin{pmatrix} 2 \\ 0 \end{pmatrix}$. (i) On graph paper, using a scale of 2 cm to 1 unit on both the and axes, plot the points and . Join . On your graph: (ii) Draw the image of under the transformation, . (iii) Draw the image of under the transformation, . (iv) Describe, geometrically, a single transformation which will map onto .

Given $f(x) = x^2$ and $g(x) = 5x + 3$, calculate: (i) $f(-2)$ (ii) $g^2(-2)$ (iii) $g^{-1}(x)$

The diagram above shows a rectangular sheet of metal, $ABCD$, supported by a vertical wall (shaded), which is at right angles to the level ground $OX$. $AB$ measures 3 metres and $AD$ measures 10 metres. $A$ is 4 metres above . (i) Calculate the size of the angle . (ii) Hence, calculate the size of the angle . (iii) If represents the height in metres of above the ground, calculate .

Using the graph construct a frequency distribution table to represent the data for the children of the 30 families.

Determine the mean and the median of the distribution.

Calculate the probability that a child, chosen at random, comes from a family with at least 4 children.

If the sample size were increased to 42 families, how many families would be expected to have at least 4 children?

Sketch a diagram of the cylinder, and clearly mark in 100 cm as a dimension of the cylinder.

Calculate the radius, in cm, of the base of the cylinder.

Calculate the area of the base of the cylinder.

If the cylinder is closed at one end, calculate, in cm$^3$, the capacity of the cylinder.

If 50 litres of liquid is poured into the cylinder, calculate, in cm, the depth of the liquid in the cylinder.

Calculate the set of values of $x$ for which $x^2 + 6x - 5 > 2$.

Given that $f(x) = 2x^2 + 2x - 1$, copy and complete the table below.

Using 1 cm to represent 1 unit on the x-axis and 1 cm to represent 5 units on the f(x) axis, draw the graph of $f(x) = 2x^2 + 2x - 1$ for $-3 \le x \le 2$.

On the graph for $f(x) = 2x^2 + 2x - 1$, draw in the axis of symmetry for $f(x)$, and state the value of where this occurs.

From your graph, obtain estimates for:

• the values of $x$ for which $f(x) = 0$
• the set of values of $x$ for which $f(x) < 0$

A car starts from rest and accelerates for 5 seconds to a point, $P$, reaching a velocity of 30 ms$^{-1}$. It maintains this velocity for 10 seconds.

Using 1 cm to represent 2 seconds on the time axis and 1 cm to represent 5 metres/sec. on the velocity axis, draw the velocity-time graph for this part of the car's journey.

Using your graph, calculate:

• the velocity of the car after 2 seconds
• the car's acceleration in the first 5 seconds
• the distance travelled in the 15 seconds
• the average speed for the 15 seconds.

The velocity-time graph, not drawn to scale, shows the movement of a particle along a straight line. The particle passes a point, $A$, when $t = 0$. It maintains a constant speed, $V$ ms$^{-1}$, for $T$ seconds, and then decelerates uniformly before coming to rest at $B$. (i) If the average speed for the whole movement is 40 ms$^{-1}$, calculate the value of $V$. (ii) Given that the deceleration between $T$ seconds and $4T$ seconds is 5 ms$^{-2}$, calculate the value of $T$.

GEOMETRY AND TRIGONOMETRY In the figure above, not drawn to scale, $PQR$ is a circle with centre, $O$. The angle $PQO = 53^\circ$. $PO$ is parallel to $QR$. Calculate, giving reasons to support your answers, the size, in degrees, of the angles: (i) $POQ$ (ii) $OQR$ (iii) $PRQ$ (iv) $QPR$

The diagram represents a right circular cone with a base radius of 9 cm and a slant height of 15 cm. The cone is cut along OC, and unrolled to form a sector, OCC', from a circle of centre, O. [Take $\pi$ to be 3.14] Calculate:

The length, in cm, of the arc, CC'

The size, in radians, of the angle COC' subtended by the arc CC'

The area, in $\text{cm}^2$, of the curved surface of the cone

Draw a diagram to represent the triangular course carefully labelling:

• The points K, L and M
• The distances 4 km and 5 km
• The north direction
• The bearing $300^\circ$

Calculate:

The distance KM in km

The bearing of M from K

VECTORS AND MATRICES

Write down the inverse of the matrix $\begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix}$.

Hence or otherwise determine the value for a, b, c and d in the matrix equation $\begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 12 \\ 5 \end{pmatrix}$.

The coordinates of the points E, D and F, are $(1, 3)$, $(2, -1)$ and $(m, n)$ respectively. $\vec{OE}$, and are position vectors relative to the origin, O.

Express $\vec{OE}$, $\vec{OD}$ and in the form .

Given $\vec{OF} = 3\vec{OE} - \vec{OD}$, determine the values of m and n.

If G is the midpoint of OD, use a vector method to prove

• EF : GE = 2 : 1
• G, E and F are collinear.

If $\begin{pmatrix} 5 \\ m-1 \end{pmatrix} \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ is a singular matrix, calculate the value of m.

The matrix $R = \begin{pmatrix} \text{Cos } \theta & -\text{Sin } \theta \\ \text{Sin } \theta & \text{Cos } \theta \end{pmatrix}$. Determine the coordinates of the image of