CSEC Mathematics Paper 2 May/June 1995 - Free CXC Past Paper with Answers

Factorize completely: (i) $9 - 25m^2$ (ii) $2x^2 - x - 15$ (iii) $x + y - ax - ay$

Solve $\frac{p - 1}{2} - \frac{p}{3} = 1$.

If $a * b = \sqrt{ab}$, where the positive root is taken, calculate $(2 * 18) * 24$.

Given that $a = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $b = \begin{pmatrix} 4 \\ 1 \end{pmatrix}$. (i) sketch vector $a + b$. (ii) express $a + b$ as a column vector (iii) determine the magnitude of $a + b$.

A survey of 156 visitors to the Caribbean found that:

• 118 persons visited Barbados.
• 98 persons visited Antigua.
• 110 persons visited Tobago.
• 25 persons visited Barbados and Antigua only.
• 35 persons visited Barbados and Tobago only.
• 30 persons visited Tobago and Antigua only.
• $x$ visitors had visited all three countries.

(i) Draw a carefully labelled Venn diagram to represent the information above. (ii) Write an algebraic expression in $x$ to represent the number of travellers who visited Barbados only. (iii) Write an equation in $x$ to show the total number of visitors in the survey. (iv) Calculate the number of travellers who visited all three countries.

A straight line is drawn through the points $A(-5, 3)$ and $B(1, 2)$. (i) Determine the gradient of $AB$. (ii) Write the equation of the line $AB$.

Using rulers and compasses only, construct

• the triangle $CAB$ with angle $CAB = 60^\circ$, with $AB = 8$ cm and cm

In the trapezium $DEFG$ above, not drawn to scale, $DE = 10$ cm, $DG = 13$ cm and $GX = 5$ cm. Angle $EFX$ and angle $DXF$ are right angles. Calculate (i) the length of $DX$ (ii) the area of trapezium $DEFG$.

In this problem, take $\pi$ to be $\frac{22}{7}$. A piece of wire, formed into a circle, encloses an area of $1386 \text{ cm}^2$. (i) Calculate the radius of the circle. (ii) Calculate the length of the wire used to form the circle. (iii) The wire is then bent to form a square. Calculate, in , the area of the square.

In the diagram above, not drawn to scale, $PR = 7$ cm, $RO = 5$ cm, $QO = 10$ cm and angles at $R$ and $S$ are right angles. Calculate (i) the size of angle $POR$. (ii) the length of $RS$.

In the diagram above, the transformation $M$ followed by another transformation $N$, maps triangle $XUV$ onto triangle $TUW$. Describe fully transformations $M$ and .

The table below shows the heights of a sample of seedlings measured to the nearest centimetre.

Calculate (i) the number of seedlings in the sample. (ii) the mean height of the seedlings in the sample.

Draw a histogram to display the data.

Calculate the probability that a seedling chosen at random will measure 10 cm or more in height.

Given $f(x) = \frac{1}{2}x$ and $g(x) = x - 2$, calculate (i) $g(-2)$ (ii) $fg(4)$ (iii) $f^{-1}(4)$

Copy on graph paper the diagram above, and show by shading ONLY, the region which satisfies both the inequalities $x > 5$ $x \le y$.

Given $\pi$ radians $= 180^\circ$, express (i) $\frac{\pi}{6}$ radians in degrees (ii) $210^\circ$ in radians.

Use the answer sheet provided to answer this question. A function $f$ is defined on the set of real numbers by $f(x) = \sin x$. (i) Complete the table for the given values of $x$ where $x$ is stated in radians and the value of $$ is given to two places of decimals.

By drawing an appropriate straight line on the graph of $f(x)$, estimate the range of values of $x$ (in degrees) for which $\sin x \ge 0.8$.

Solve the following equations: $x^2 + 9y^2 = 37$ $x - 2y = -3$.

If $x = 1$ is one of the root of the equation $(x - c)^2 = 4(x + c + 2)$, calculate to 2 decimal places, the possible values of the constant .

In this question, take the radius of the earth to be 6400 km and use $\pi = 3.14$. (i) the latitude of $Q$, if $Q$ lies 1600 km due north of $A$ (ii) the distance in km between $Q$ and $M$, if $M$ is due east of $Q$ and on longitude $20^\circ E$ (iii) the longitude of $L$, if $L$ lies 1200 km due west of $R$.

Using the diagram above, not drawn to scale, state the coordinates of: (i) Point $A$ (ii) Point $R$ (iii) Point $T$

Prove the identity $\sin^2 \theta = (1 + \cos \theta)(1 - \cos \theta)$

In the triangle above, not drawn to scale, the right angle and the angle $\theta$ are indicated. Given that $\cos \theta = \frac{3}{5}$, calculate the value of $\sin \theta + \tan \theta$.

In the diagram above, not drawn to scale, $O$ is the centre of the circle $LMNT$ and $PTQ$ is a tangent to the circle at $T$. Given that $\angle LTQ = 65^\circ$, calculate, stating your reasons, the size of: (i) (ii) (iii) (iv)

State $\vec{PQ}$ and $\vec{XY}$ in terms of and .

Given that $\vec{XR} = \frac{1}{2}\vec{XQ}$, write and in terms of and .

Prove by a vector method that the points $Y, R$ and $P$ are collinear.

If $T = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$ and $G = \begin{pmatrix} -2 \\ 3 \end{pmatrix}$, evaluate $TG$.

The equation of a straight line $PQ$ is $y = 3x + 1$ and $T$ is a transformation represented by the matrix $(\begin{array}{cc}3& 0\\ 0& \end{array}$. Determine (i) the value of , if is the point . (ii) the coordinates of the image of under the transformation . (iii) the coordinates of the point if is invariant under the transformation .