CSEC Mathematics Paper 2 May/June 1990 - Free CXC Past Paper with Answers

Consider the following three statements:

1. Some students play basketball.
2. Tall students are over 2 metres in height.
3. All basketball players are tall students.

Represent the statements in a suitable Venn diagram, showing and stating an appropriate universal set.

Show on your Venn diagram that (i) Nina is 1.5 m tall (ii) Robert, who is 2.2 m tall, does not play basketball.

Solve the following equations for $x$: $3(x+2)^2 = 7(x+2)$

Solve the following equations for $x$: $9^x = \frac{1}{27}$

Solve the following equations for $x$: $3x = y$, given than $\sin y = \frac{\sqrt{3}}{2}$ where .

Using a scale of 1 cm to 1 unit on each axis, draw on graph paper triangle ABC whose vertices are A (2, 2), B (5, 2) and C (2, 4). (i) $\Delta A'B'C'$ is the image of $\Delta ABC$ under an enlargement, centre (0, 0) and scale factor $k = 2$. Draw $\Delta A'B'C'$, and state the coordinates of the vertices. (ii) $\Delta A''B''C''$ is the image of $\Delta ABC$ under a transformation $T$. The vertices are $A'' (-2, -4)$, $B'' (-5, -4)$ and $C'' (-2, -2)$. Give a full geometrical description of the transformation $T$.

Given that $f(x) = 2 \sin \frac{x}{2}$, copy and complete the following table:

Using a scale of 2 cm to represent $\frac{\pi}{12}$ on the $x$-axis and 10 cm to represent 1 unit on the $f(x)$ axis, draw the graph of the function $$ for the domain .

By drawing a suitable tangent, determine the gradient of the curve at $x = \frac{\pi}{2}$.

Given that $A = \begin{pmatrix} 3 & 5 \\ 1 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 6 & 4 \\ -1 & -2 \end{pmatrix}$. (i) Calculate the matrix $AB$. (ii) If $C = \begin{pmatrix} x & -8 \\ 4 & y \end{pmatrix}$ and $2A + C = AB$, calculate the values of $x$ and $y$.

The ratio of the prices of two different sheets of glass is $2:5$. The total bill for 20 sheets of the cheaper glass and 10 sheets of the more expensive glass is J$1080. If J dollars represent the cost of one sheet of the cheaper glass, determine (i) an expression in $d$ for the cost of ONE sheet of the more expensive glass (ii) the value of $d$ (iii) the cost of ONE sheet of the more expensive glass.

The frequency distribution of the lengths of 100 rods measured to the nearest mm is given in the following table.

Using an assumed mean of 222 mm, calculate the mean length of all the rods.

A two-digit number in base ten is formed by choosing the tens digit from the set $\{4, 5, 6, 7, 8\}$ and the units digit from the set $\{1, 2, 3, 9\}$. (i) Determine the total number of different two-digit numbers that may be formed. (ii) Calculate the probability that the resulting number is even. (iii) Hence, deduce the probability that the resulting number is odd.

LMNOPQ is a hexagon (not drawn to scale) with $\hat{Q} = 130^\circ$, $\hat{P} = 110^\circ$ and $\hat{O} = 90^\circ$. $\hat{L} = \hat{M} = \hat{N}$. (i) Calculate the value of $\hat{L}$. (ii) Given that $PO = 4$ cm and the area of $\Delta NOP$ is 12 cm$^2$, calculate the length of $PV$ in cm, giving your answer correct to one decimal place.

Copy and complete the following table for the function $f: x \to \frac{1}{x^2}$ for real values of $x$ in the domain $0.5 \le x < 3.5$.

Using a scale of 4 cm to represent 1 unit on the $x$-axis and 2 cm to represent 0.1 unit on the $f(x)$ axis, draw the graph of the function $f: x \to \frac{1}{x^2}$ for real values of in the domain .

(i) Using the same scale and axes, draw the graph of the function $g: x \to \frac{5}{10}$ for the same domain. (ii) Use your graph to solve the equation $\frac{1}{x^2} = \frac{x}{10}$ for real values of such that . (iii) Hence, deduce one solution of the equation for real values of .

At a factory, two types of motor cycles, $M_1$ and $M_2$, are assembled. The factory can produce a maximum of 80 motor cycles daily. For each $M_1$ motor cycle made, there must be at least two $M_2$ motor cycles produced. It takes one workman to produce each $M_1$ motor cycle but three workmen to produce each $M_2$ motor cycle. Factory policy requires that at least 90 workmen be employed.

Let $x$ represent the number of $M_1$ motor cycles and $y$ the number of $M_2$ motor cycles produced daily.

Write down THREE inequalities, not including $x \ge 0$, $y \ge 0$, which represent the above conditions.

Using a scale of 2 cm to 10 motor cycles on EACH axis, shade, on graph paper, the region which satisfies the inequalities.

Given that a profit of $250 is made on each $M_1$ motor cycle and $200 on each $M_2$ motor cycle, use your graph to determine the values of $x$ and $$ which give the maximum profit.

The foot, $F$, of a hill and the base $B$, of a vertical tower $TB$, 21 metres on the same horizontal level. From the top, $T$, of the tower, the angle of depression of $F$ is $36.9^\circ$. $P$ is a point on the hill $24.5$ m away from $F$ along the line of greatest slope. $T, B, F$ and $P$ all lie in the same vertical plane. The angle of depression of $P$ from $T$ is $19.4^\circ$.

Draw a sketch to represent the information given above.

Show that (i) $TF$ is 35 m approximately (ii) $\sin \hat{TPF} = \frac{2}{3}$ approximately.

Calculate (i) the gradient of the hill (ii) the height, in metres, of $P$ above $F$, giving your answer correct to one decimal place.

A helicopter is set on a course of $095^\circ$. A wind is blowing from a direction $50^\circ$ east of north at a speed of 40 kmh$^{-1}$. Using a scale of 1 cm to 20 kmh$^{-1}$, determine by drawing, (i) the track and ground speed of the helicopter when it has an air speed of 100 kmh$^{-1}$. (ii) the course which must be set if the helicopter is to fly on a track of $095^\circ$.

In an experiment, the distance $d$ of a moving object from its starting point is given by the equation $d = \sqrt{3} - 2 \cos \theta$, where the angle is measured in radians. (i) For , determine all the values of for which . (ii) State the value of when , giving your answer in surd form. (iii) State the largest possible value of , giving your answer in surd form.

The diagram above, not drawn to scale, shows a regular 8-sided polygon $ABCDEFGH$, centre $O$. $OE = 6$. Show that $\angle COE = 90^\circ$.

Calculate (i) the angle $CDE$ (ii) the length of $CE$ in terms of $l$.

Given that $P$ is the midpoint of $CE$, show that $OP = PE$.

Given that $\vec{x}$ represents the unit vector in the direction $\vec{OD}$, and is the unit vector in direction , express, in terms of and (i) (ii) .

Four points on a plane have the following coordinates: $O(0, 0)$, $P(0, 12)$, $Q(6, 0)$, $R(0, -3)$. The triangle $QOR$ is mapped onto triangle $QOP$ by a combination of two transformations – a reflection and a stretch. (i) Describe fully the transformations and determine the matrix for (a) the reflection (b) the stretch (c) the combination of the two transformations. (ii) Determine the matrix which maps triangle $QOP$ onto triangle $QOR$.

The vertices of triangle $ABC$ have coordinates $A(2, 3)$; $B(4, 6)$; $C(6, 3)$ and the triangle has vertices ; ; . is the image of under a shear with a positive scale factor. Determine, for the shear, (i) the equation of the invariant line (ii) its scale factor.