CSEC Mathematics Paper 2 May/June 2005 - Free CXC Past Paper with Answers

Factorise (i) $5a^2b + ab^2$ (2 marks) (ii) $9k^2 - 1$ (2 marks) (iii) $2y^2 - 5y + 2$ (2 marks)

Expand and simplify $(2x + 5)(3x - 4)$

Adam, Imran and Shakeel were playing a card game. Adam scored $x$ points Imran scored 5 points fewer than Adam Shakeel scored twice as many points as Imran Together they scored 39 points.

(i) Write down, in terms of $x$, an expression for the number of points scored by Shakeel. (2 marks) (ii) Write an equation which may be used to find the value of $x$. (2 marks)

In the diagram shown above, the Universal set, $(U)$, represents all the students in a class. The set $M$ represents the students who take Music. The set $D$ represents the students who take Drama. If 24 students take Music, calculate

(i) the number of students who take BOTH Music and Drama (ii) the number of students who take Drama ONLY. (4 marks)

A straight line passes through the point $P(-3, 5)$ and has a gradient of $\frac{2}{3}$.

(i) Write down the equation of this line in the form $y = mx + c$. (5 marks) (ii) Show that this line is parallel to the line $2x - 3y = 0$. (2 marks)

Is a medium pizza twice as large as a small pizza? Use calculations to support your answer. (5 marks)

A medium pizza is cut into 3 equal parts, and each part is sold for $15.95. A small pizza is sold for $12.95. Which is the better buy? Use calculations to support your answer. (5 marks)

On graph paper, draw the $x$-axis and the $y$-axis. Using a scale of 1 cm to represent 1 unit on both axes, draw the triangle $DEF$ with vertices $D(1, 1)$, $E(3, 1)$ and $F(1, 4)$. (3 marks)

(i) Draw the image of $\triangle DEF$ under reflection in the line $x = 4$. Name the image $\triangle D'E'F'$. (ii) Draw the image of under the translation . Name the image . (iii) Name the type of transformation that maps onto . (5 marks)

A vertical stick of height 1.8 m casts a shadow of length 2 m on the horizontal as shown in the diagram below, not drawn to scale.

Calculate, to the NEAREST degree, the angle of elevation of the sun. (4 marks)

In the diagram shown below, $ABCDE$ is a pentagon. $\angle BAE = 108^\circ$, $\angle ABC = 90^\circ$, $\angle AED = 80^\circ$, $\angle ADC = 57^\circ$ and $AE$ is parallel to $CD$.

Calculate the size of the angle marked (i) $x^\circ$ (ii) $y^\circ$ Show all steps in your calculations and give reasons for your answers. (4 marks)

The functions $f$ and $g$ are defined by $f(x) = \frac{1}{2}x + 5$, .

Using a horizontal scale of 2 cm to represent a height of 5 cm and a vertical scale of 2 cm to represent 50 applicants, draw a cumulative frequency curve of the heights. Start your horizontal scale at 150 cm.

Use your graph to estimate (i) the number of applicants whose heights are less than 170 cm. (ii) the median height of applicants. (iii) the height that 25% of the applicants are less than (iv) the probability that an applicant selected at random has a height that is no more than 162 cm. Credit will be given for drawing appropriate lines on your graph to show how the estimates were obtained.

Study the number pattern in the table below and complete lines (i), (ii) and (iii) in your answer booklet.

Show that $(a - b)^2 (a + b) + ab(a + b) = a^3 + b^3$.

Write $5x^2 + 2x - 7$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are real numbers.

Hence, or otherwise, determine (i) the minimum value of the function $y = 5x^2 + 2x - 7$ (ii) the value of x at which the minimum occurs

Find the values of x for which $5x^2 + 2x - 7 = 0$.

Sketch the graph of $y = 5x^2 + 2x - 7$, clearly showing (i) the coordinates of the minimum point (ii) the value of the y-intercept (iii) the points where the graph cuts the x-axis.

The speed-time graph below shows the movement of a cyclist. Using the graph, calculate (i) the acceleration of the cyclist during the first 15 seconds (ii) the distance traveled by the cyclist between the period $t = 15$ and $t = 35$ seconds.

The graph below represents the 5-hour journey of an athlete.

(i) What was the average speed during the first 2 hours? (ii) What did the athlete do between 2 and 3 hours after the start of the journey? (iii) What was the average speed on the return journey?

In the diagram above, not drawn to scale, $P$ and $Q$ are midpoints of the sides $XY$ and $XZ$ of triangle $XYZ$. Given that $XP = 7.5$ cm, $XQ = 4.5$ cm and the area of triangle $XPQ = 13.5 \text{ cm}^2$, calculate (i) the size of angle $PXQ$, expressing your answer correct to the nearest degree. (ii) the area of triangle $XYZ$.

The figure $SJKM$ above, not drawn to scale, is a trapezium with $SJ$ parallel to $MK$. angle $MJK = 124^\circ$, angle , and metres.

Draw a sketch of the earth showing the location of Antigua and of Belize, their associated circles of latitude and longitude, the equator, and the Greenwich Meridian.

Calculate the shortest distance between Antigua and Belize measured along their `common circle of latitude.

A town, Bahia Blanka, situated in South America, lies on a meridian $62^\circ$W and has a latitude of $38^\circ$S. Calculate the shortest distance between Antigua and Bahia Blanka measured along the common circle of longitude.

Express in terms of $\mathbf{x}$ and $\mathbf{y}$: (i) $\vec{AB}$ (ii) $\stackrel{}{}$ (iii)

Show that $\vec{AP} = \mathbf{x} - 2\mathbf{y}$.

Given that $E$ is the mid-point of $DC$, prove that $A$, $P$ and $E$ are collinear.

Given that $\mathbf{x} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$ and $\mathbf{y} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$, use a vector method to prove that triangle is isosceles.

Given that $M = \begin{pmatrix} 2 & 5 \\ 7 & 15 \end{pmatrix}$. (i) Show that $M$ is a non-singular matrix. (ii) Write down the inverse of $M$. (iii) Write down the $2 \times 2$ matrix which is equal to the product $M \times M^{-1}$. (iv) Pre-multiply both sides of the following matrix equation by $M^{-1}$. $\begin{pmatrix} 2 & 5 \\ 7 & 15 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -3 \\ 17 \end{pmatrix}$ Hence solve for $x$ and $y$.

(i) Write down the $2 \times 2$ matrix, $R$, which represents a reflection in the $y$-axis. (ii) Write down the $2 \times 2$ matrix, $N$, which represents a clockwise rotation of about the origin. (iii) Write down the matrix, which represents a translation of units parallel to the -axis and units parallel to the -axis. (iv) The point undergoes the following combined transformations such that maps onto maps onto Determine the coordinates of and .