CSEC Mathematics Paper 2 May/June 1991 - Free CXC Past Paper with Answers

Factorize completely:

$1 - (a + b)^2$

$(2x^2 + xy - y^2) + 2x - y$.

Solve: $\frac{x}{5} - 5 = \frac{x}{15} + 4$

The scores obtained by a class of ten students in a test were: $2, 3, 4, 4, 5, 6, 6, 7, 11, 12$. Calculate

the median mark

the mean mark

the standard deviation of the marks; showing clearly all steps in your calculation.

Determine the probability that a pupil selected at random from the class obtained

exactly 6 marks

at LEAST 6 marks.

Solve the simultaneous equations: $x + y = 5$ $xy = 6$

The figure $ABCDEF$ above, not drawn to scale, represents a wedge with measurements as shown. $BC$ is perpendicular to the plane $FEDC$. Calculate

the length, in cm, of $BD$

the surface area, in cm$^2$, of the wedge

the volume, in cm$^3$, of the wedge

the size of angle $BDC$.

Using ruler and compasses only, construct a parallelogram $ABCD$, such that $AB = 6.5$ cm, $AD = 5.7$ cm and the angle $DAB = 60^\circ$. Measure and state the length of in centimetres. [Note: All construction lines must be clearly shown.]

There are 50 students in Form VI. All students study Mathematics. 17 study Biology. 18 study Chemistry. 24 study Physics. 5 study Physics, Chemistry and Mathematics. 7 study Physics, Biology and Mathematics. 6 study Chemistry, Biology and Mathematics. 2 study all four subjects. Draw a carefully labelled Venn diagram to represent the data, using the universal set as the set of students who study Mathematics.

Determine the number of students who study at LEAST TWO subjects.

Calculate the number of students who study Mathematics only.

The functions $f$ and $g$ are defined by: $f: x \to 5 + x$ $g: x \to x^2$ Determine expressions for the functions:

$fg$

$g^{-1}$

The distance $S$, in metres, moved by a particle from its starting point, at time $t$ seconds, is given by $S = 3t + t^2$. Copy and complete the table below for $S=3$.

Using a scale of 1 cm to represent 1 second and 1 cm to represent 10 m, draw the graph of the function $S = 3t + t^2$ for the range $0 \le t \le 8$.

Using the graph, estimate the distance moved in 4.5 seconds.

Draw a tangent to the curve at $t = 6$. Estimate the value of the tangent at this point. Give an interpretation of this value.

In the graph above, $A$ and $B$ are points such that $\vec{OA} = \mathbf{a}$ and $\vec{OB} = \mathbf{b}$. The point $P$ (not shown) is such that $\vec{OP} = \frac{1}{2}\mathbf{a} + \mathbf{b}$.

Write $\vec{OP}$ in the form $\begin{pmatrix} x \\ y \end{pmatrix}$.

Determine the length of $OP$.

The rectangle $WXYZ$, with coordinates $(2,2)$, $(5,2)$, $(5,4)$ and respectively, is mapped onto parallelogram with coordinates , , and respectively under a shear $(S)$.

Plot the rectangle $WXYZ$ and its image $W'X'Y'Z'$.

Determine the equation of the invariant line.

Calculate the area of $W'X'Y'Z'$.

State the coordinates of the image of the point $(2,3)$ under the shear $S$.

In the diagram above, not drawn to scale, the points $X, Q$ and $R$ are on a straight line in the same horizontal plane. The angle of depression of a point $Q$ from the top of a tower $PX$, 10 m high, is $70^\circ$. The angle of depression of $R$ from the top of the tower is $40^\circ$. Calculate the length $QR$ to one decimal place.

In the diagram above, not drawn to scale, $PQ, XY$ and $SR$ are parallel lines. $QY = 10$ cm, $YR = 5$ cm and $XY = 3$ cm.

Prove that triangles $PQX$ and $RSX$ are similar.

Calculate the lengths of $PQ$ and $RS$.

Calculate the ratio of the areas of triangles $PQX$ and $RSX$.

Prove that $\frac{1}{1 + \sin^2 \theta} + \frac{1}{\cos^2 \theta} = \frac{1}{\sin^2 \theta \cos^2 \theta}$.

A ship sails 4 nautical miles from port $P$ on a course $069^\circ$ to a point $Q$, changes course to $295^\circ$ and sails a further 5 nautical miles to a port .

Draw a carefully labelled diagram of the entire route taken. Show the north direction.

Calculate the distance $PR$ to 2 significant figures.

Determine the bearing of $P$ from $R$.

If $\sin A = \frac{3}{5}$, $\cos B = \frac{15}{17}$ and $A$ and $B$ are acute angles, determine the exact value of $\tan A \tan B$.

Take the radius of the earth to be 6400 km and $\pi$ to be 3.142.

The distance between Town $P$ ($80^\circ N, 72^\circ N$) and Town $R$ ($$) is 800 km. If Town is east of Town , determine 's E, the longitude of . Give your answer to the nearest degree.

The shortest distance between Town $P$ ($80^\circ E, 72^\circ N$) and Town $Q$ ($$) is 10054.4 km. Determine , the latitude of .

Use the method of completing the square to determine the value of $x$ for which the expression $4x^2 + 4x + 3$ is a minimum.

Hence deduce the minimum value of $4x^2 + 4x + 3$.

A motorist starting a car from rest, accelerates uniformly to a speed of 40 km h$^{-1}$ in 2 minutes. He maintains this speed for another 3 minutes. He then applies the brakes and decelerates uniformly to rest in 1 minute.

Draw a diagram of the velocity-time graph to show the different segments of the journey. Determine

the acceleration, in km h$^{-2}$, during the first two minutes

the retardation, in km h$^{-2}$, during the last minute

the total length of the journey

the average speed for the whole journey in km h$^{-1}$.

Copy and complete the table below for $y = \frac{5}{x}$.

Using a scale of 2 cm to represent 1 unit on both $x$ and $y$ axes, draw the graph of the function above for $1 \le x \le 6$.

Using the same axes and scale given above, draw the graph of the function $x + y = 6$ for $0 \le x \le 6$.

Shade on your graph the region which satisfies the inequalities $y \le \frac{5}{x}$, $x + y \le 6$, , .

Indicate a point $Z$ in the region which satisfies the inequalities $y \le \frac{5}{x}$ and $x + y > 6$.

Identify, by writing the letter $T$, the region bounded by the curve and the straight line.

State the inequalities which describe the region bounded by the curve and the straight line.

If $A = \begin{pmatrix} 3 & 2 \\ -2 & 3 \end{pmatrix}$.

If $A = \begin{pmatrix} 3 & 2 \\ -2 & 3 \end{pmatrix}$. evaluate the determinant of A.

If $A = \begin{pmatrix} 3 & 2 \\ -2 & 3 \end{pmatrix}$. determine .

If $A = \begin{pmatrix} 3 & 2 \\ -2 & 3 \end{pmatrix}$. using solve the simultaneous equations:

S is the transformation represented by $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 3 \\ 0 \end{pmatrix}$.

S is the transformation represented by $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 3 \\ 0 \end{pmatrix}$. Perform the transformation S on a square with vertices as A(0,0), B(0,1), C(1,1), D(1,0) and write, in coordinate form, the images of A, B, C and D.

S is the transformation represented by $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 3 \\ 0 \end{pmatrix}$. Describe, in words, the single transformation represented by transformation S.

T is the matrix $\begin{pmatrix} p & q \\ r & s \end{pmatrix}$.

T is the matrix $\begin{pmatrix} p & q \\ r & s \end{pmatrix}$. Determine the elements of T which map A(2,1) onto A'(4,3) and B(3,2) onto B'(5,6).

T is the matrix $\begin{pmatrix} p & q \\ r & s \end{pmatrix}$. Derive the matrix S, such that TS represents the transformation matrix for a reflection in the x-axis.

The figure ABC is a triangle with X and Y the mid-points of AB and AC respectively. Using a vector method prove that

The figure ABC is a triangle with X and Y the mid-points of AB and AC respectively. Using a vector method prove that $\vec{BC}$ is parallel to $\vec{XY}$.

The figure ABC is a triangle with X and Y the mid-points of AB and AC respectively. Using a vector method prove that $\vec{BC} = 2\vec{XY}$.