WAEC · 2014 · Nov/Dec · General Maths · Paper 2

1. (a)

   Factorize completely: $m^2 - 2mn + n^2 - 9r^2$.

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   $(m - n + 3r)(m - n - 3r)$

2. (b)

   Solve simultaneously the equations $5x - 4y = 6$ and $3^{3(y - x)} = \frac{1}{27}$.

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   Separate values with commas, e.g. 3, −2

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1. (a)

   A man has a wife and 6 children and his total income in a year was GH¢ 850.00. He was given the following tax-free allowances: personal GH¢ 120.00; wife GH¢ 30.00; children GH¢ 25.00 per child, for a maximum of 4 children; medical GH¢ 40.00. The rest was taxed as follows: first GH¢ 200.00 at $10\%$; next GH¢ 200.00 at $15\%$; next GH¢ 200.00 at $20\%$; remainder at $25\%$. Calculate his: (i) taxable income; (ii) monthly tax.

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   Separate values with commas, e.g. 3, −2

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1. (a)

   In the diagram, $QSP$ and $QTR$ are straight lines, $|PS| = 6\text{ cm}$, $|QS| = 4\text{ cm}$, $|QT| = 5\text{ cm}$ and $\angle QTS = \angle RPQ$. Calculate $|TR|$.

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2. (b)

   If $\sin x = \frac{5}{13}$, $0^\circ \le x \le 90^\circ$, evaluate, without tables or a calculator, $\dfrac{\cos x - 2\sin x}{2\tan x}$.

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1. (a)

   Without using tables or a calculator, evaluate $\log_{10}\left(\frac{75}{10}\right) - 2\log_{10}\left(\frac59\right) + \log_{10}\left(\frac{100}{243}\right)$.

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2. (b)

   Given that $X = \{x : 10 \le x < 15\}$ and $Y = \{\text{even numbers} < 18\}$ are subsets of $U = \{10, 11, 12, \ldots, 20\}$, find: (i) $X \cap Y$; (ii) $n(X' \cap Y)$.

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1. (a)

   height of the pyramid;

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2. (b)

   value of $\angle OXZ$, correct to the nearest degree;

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3. (c)

   volume of the pyramid.

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1. (a)

   In a class of 52 students, 16 are Science students. If $\frac13$ of the boys and $\frac14$ of the girls are Science students, how many boys are in the class?

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2. (b)

   The sum of the first and third terms of a Geometric Progression (G.P.) is 40 while the fourth and sixth terms are in the ratio $1 : 4$. Find the: (i) common ratio; (ii) fifth term.

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   Separate values with commas, e.g. 3, −2

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1. (a)

   A spherical tank of diameter $3\text{ m}$ is filled with water from a pipe of radius $30\text{ cm}$ at $0.2\text{ m}$ per second. Calculate, correct to 3 significant figures, the time, in minutes, it takes to fill the tank. $\left[\text{Take }\pi = \frac{22}{7}\right]$

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2. (b)

   When $k$ is added to the expression $y^2 - 12y$, the expression becomes $(y + p)^2$. Find the values of $p$ and $k$.

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   Separate values with commas, e.g. 3, −2

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1. (a)

   Using a ruler and a pair of compasses only, construct: (i) a parallelogram $PQRS$ with $RS$ as the base such that $|PQ| = 7.8\text{ cm}$, $|QR| = 5.6\text{ cm}$ and $\angle QRS = 120^\circ$; (ii) a rectangle $ABRS$ equal in area to the parallelogram $PQRS$.

   Model answer

   Use only a ruler and a pair of compasses: leave every construction arc visible, because the examiner looks for them. Draw the base $RS = 7.8$ cm. Construct $120^\circ$ at $R$ and mark $Q$ with $RQ = 5.6$ cm. Through $Q$ draw the line parallel to $RS$ and mark $P$ with $QP = 7.8$ cm. For the rectangle, draw perpendiculars to $RS$ at $R$ and $S$ to meet the line $QP$ (produced) at $B$ and $A$. It has the same base and lies between the same parallels, so it has the same area. Measured: $|AP| = 2.8$ cm and $|AS| \approx 4.85$ cm.

2. (b)

   Measure: (i) $|AP|$; (ii) $|AS|$.

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   Separate values with commas, e.g. 3, −2

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1. (a)

   An object was thrown vertically upwards from the top of a cliff and its height, $y$ metres, above sea level after $t$ seconds is given by $y = -16t^2 + 64t + 5$. Copy and complete the table of values for $0 \le t \le 4.0$.

   $t$	0.0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0
   $y$	5			65			53

   Model answer

   $t$	0.0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0
   $y$	5	33	53	65	69	65	53	33	5

   For example $t = 0.5$: $-4 + 32 + 5 = 33$, and $t = 2.0$: $-64 + 128 + 5 = 69$. The values are symmetrical about $t = 2$.

2. (b)

   Using scales of 2 cm to 0.5 seconds on the $t$-axis and 2 cm to 10 m on the $y$-axis, draw the graph of $y = -16t^2 + 64t + 5$ for $0 \le t \le 4.0$.

   Model answer

   Plot every point from the table, then join them with one smooth curve (not straight lines between points). Scale: 2 cm to 0.5 s across, 2 cm to 10 m up. The path is symmetrical about $t = 2$.

   For (c): (i) at $t = 1.75$ s the height is about 68 m; (ii) the line $y = 50$ meets the curve at $t \approx 0.9$ s and $3.1$ s; (iii) the maximum height is 69 m.

3. (c)

   Use the graph to find the: (i) height reached when $t = 1.75$ seconds; (ii) times the object was at a height of $50\text{ m}$; (iii) maximum height reached.

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   Separate values with commas, e.g. 3, −2

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1. (a)

   (i) Solve the inequality $\frac12x - \frac56(x + 2) \le 1 + x$. (ii) Illustrate the solution on a number line.

   Model answer

   (i) $x \ge -2$. (ii) A solid dot at $-2$ (it is included) with an arrow to the right.

2. (b)

   From a point $P$ on level ground and directly west of a pole, the angle of elevation of the top of the pole is $45^\circ$, and from a point $Q$ east of the pole, the angle of elevation of the top of the pole is $58^\circ$. If $|PQ| = 10\text{ m}$, calculate, correct to 2 significant figures, the: (i) distance from $P$ to the pole; (ii) height of the pole.

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   Separate values with commas, e.g. 3, −2

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1. (a)

   The probabilities that Manful, John and Ernest will pass an examination are $\frac23$, $\frac58$ and $\frac34$ respectively. Find the probability that all three will pass the examination.

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2. (b)

   The scores of students in a test were recorded as follows:

   4 2 1 6 5 3 5 6 1 2 1 5 5 6 3 4 3 5 1 5

   (i) Construct a frequency distribution table. (ii) Represent the information in a bar chart. (iii) Calculate the interquartile range.

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1. (a)

   In the diagram, $\overline{TS}$ is a tangent to the circle $PQRS$ at $S$, and $O$ is the centre of the circle ($QOS$ is a diameter). If $\angle TSP = 21^\circ$ and $\angle RQP = 100^\circ$, find, with reasons: (i) $\angle SPR$; (ii) $\angle QSR$.

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   Separate values with commas, e.g. 3, −2
2. (b)

   Given the relation $T = \sqrt{\dfrac{U}{\frac1f + \frac1g}}$: (i) make $g$ the subject of the relation; (ii) find $g$ when $T = 3$, $f = 4$ and $U = 5$.

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1. (a)

   If $\mathbf x = \begin{pmatrix} -2 \\ 4 \end{pmatrix}$ and $\mathbf y = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$, find, correct to 1 decimal place, $|\mathbf x + \mathbf y|$.

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2. (b)

   $P(6, 4)$, $Q(-2, -2)$ and $R(4, -6)$ are the vertices of triangle $PQR$. (i) Determine the coordinates of $M$ and $S$, the midpoints of $\overline{PQ}$ and $\overline{PR}$ respectively. (ii) Find $\overrightarrow{QR}$ and $\overrightarrow{MS}$. (iii) State the relationship between $\overrightarrow{QR}$ and $\overrightarrow{MS}$. (iv) Find the equation of $\overline{MS}$.

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   (i) $M(2, 1)$, $S(5, -1)$; (ii) $\overrightarrow{QR} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}$, $\overrightarrow{MS} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$; (iii) $\overrightarrow{QR} = 2\overrightarrow{MS}$; (iv) $2x + 3y = 7$

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