WAEC · 2012 · May/June · General Maths · Paper 2

1. (a)

   Simplify: $\dfrac{1\frac14 + \frac79}{1\frac49 - 2\frac23 \times \frac{9}{64}}$.

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2. (b)

   Given that $\sin x = \frac23$, evaluate, leaving your answer in surd form and without using tables or a calculator, $\tan x - \cos x$.

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1. (a)

   Sonny is twice as old as Wale. Four years ago, he was four times as old as Wale. When will the sum of their ages be 66?

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1. (a)

   In the diagram, $\overline{TU}$ is a tangent to the circle. $\angle RVU = 100^\circ$ and $\angle URS = 36^\circ$. Calculate the value of angle $STU$.

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2. (b)

   In triangle $XYZ$, $|XY| = 5\text{ cm}$, $|YZ| = 8\text{ cm}$ and $|XZ| = 6\text{ cm}$. $P$ is a point on the side $XY$ such that $|XP| = 2\text{ cm}$ and the line through $P$, parallel to $YZ$, meets $XZ$ at $Q$. Calculate $|QZ|$.

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1. (a)

   A box contains 40 identical discs which are either red or white. If the probability of picking a red disc is $\frac14$, calculate the number of: (i) white discs; (ii) red discs that should be added such that the probability of picking a red disc will be $\frac13$.

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

   A salesman bought some plates at ₦50.00 each. If he sold all of them for ₦600 and made a profit of $20\%$ on the transaction, how many plates did he buy?

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1. (a)

   angle which $XY$ subtends at the centre $O$;

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2. (b)

   area of the shaded minor segment.

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1. (a)

   A boy had $M$ dalasis (D). He spent D15 and shared the remainder equally with his sister. If the sister's share was equal to $\frac13$ of $M$, find the value of $M$.

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2. (b)

   A number of tourists were interviewed on their choice of means of travel. Two-thirds said that they travelled by road, $\frac{13}{30}$ by air and $\frac{4}{15}$ by both air and road. If 20 tourists did not travel by either air or road, (i) represent the information on a Venn diagram; (ii) how many tourists: (A) were interviewed; (B) travelled by air only?

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1. (a)

   (i) Using a scale of 2 cm to 1 unit on both axes, draw on the same graph sheet the graphs of $y - \frac{3x}{4} = 3$ and $y + 2x = 6$. (ii) From your graph, find the coordinates of the point of intersection of the two graphs. (iii) Show, on the graph sheet, the region satisfied by the inequality $y - \frac34x \ge 3$.

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

   Given that $x^2 + bx + 18$ is factorised as $(x + 2)(x + c)$, find the values of $c$ and $b$.

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1. (a)

   $|PT|$;

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2. (b)

   the distance between $H$ and the top of the tower;

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3. (c)

   how far $H$ must be from the foot of the tower if the angle of depression of $H$ from the top of the tower is to be $40^\circ$.

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1. (a)

   Three towns $X$, $Y$ and $Z$ are such that $Y$ is 20 km from $X$ and 22 km from $Z$. Town $X$ is 18 km from $Z$. A Health Centre is to be built to serve the three towns, located such that patients from $X$ and $Y$ always travel equal distances to it, while patients from $Z$ travel exactly 10 km. Using a scale of 1 cm to 2 km, find by construction, using a pair of compasses and ruler only, the possible positions of the Health Centre.

   Model answer

   Use only a ruler and a pair of compasses: leave every construction arc visible, because the examiner looks for them. At 1 cm to 2 km the sides are $XY = 10$ cm, $YZ = 11$ cm and $XZ = 9$ cm: draw $XY$, then arcs of 9 cm from $X$ and 11 cm from $Y$ to fix $Z$. "Equal distances from $X$ and $Y$" is the perpendicular bisector of $XY$; "exactly 10 km from $Z$" is the circle centre $Z$, radius 5 cm. They meet at two points, $H_1$ and $H_2$: about 28 km and 12.7 km from $X$. The nearer one, $H_2$, is more convenient for all three towns.

2. (b)

   (i) In how many possible locations can the Health Centre be built? (ii) Measure and record the distances of the locations from town $X$. (iii) Which of these locations would be convenient for all the three towns?

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1. (a)

   mean;

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2. (b)

   standard deviation of the distribution.

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1. (a)

   In the diagram, $ABCD$ is a rectangular garden $(3n - 1)\text{ m}$ long and $(2n + 1)\text{ m}$ wide. A wire mesh $135\text{ m}$ long is used to mark its boundary and to divide it into 8 equal plots (3 lines along the length and 5 across). Find the value of $n$.

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2. (b)

   A cylinder with base radius $14\text{ cm}$ has the same volume as a cube of side $22\text{ cm}$. Calculate the ratio of the total surface area of the cylinder to that of the cube. $\left[\text{Take }\pi = \frac{22}{7}\right]$

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   $73 : 77$ (about $0.95 : 1$)

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1. (a)

   Copy and complete the table of values for $y = 1 - 4\cos x$.

   $x$	$0^\circ$	$30^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$150^\circ$	$180^\circ$	$210^\circ$	$240^\circ$	$270^\circ$	$300^\circ$
   $y$	$-3.0$			$1.0$				$4.5$			$-1.0$

   Model answer

   $x$	$0^\circ$	$30^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$150^\circ$	$180^\circ$	$210^\circ$	$240^\circ$	$270^\circ$	$300^\circ$
   $y$	$-3.0$	$-2.5$	$-1.0$	$1.0$	$3.0$	$4.5$	$5.0$	$4.5$	$3.0$	$1.0$	$-1.0$

   Work to one decimal place in degree mode; for example $x = 60^\circ$: $1 - 4(0.5) = -1.0$.

2. (b)

   Using a scale of 2 cm to $30^\circ$ on the $x$-axis and 2 cm to 1 unit on the $y$-axis, draw the graph of $y = 1 - 4\cos x$ for $0^\circ \le x \le 300^\circ$.

   Model answer

   Plot every point from the table, then join them with one smooth curve (not straight lines between points). Scale: 2 cm to $30^\circ$, 2 cm to 1 unit. The curve rises from $-3$ at $0^\circ$ to its highest point, 5, at $180^\circ$.

   For (c): (i) it crosses the $x$-axis at $x \approx 76^\circ$ and $284^\circ$; (ii) at $x = 105^\circ$, $y \approx 2.0$; (iii) the line $y = 1.5$ meets it at $x \approx 97^\circ$ and $263^\circ$.

3. (c)

   Use the graph to: (i) solve the equation $1 - 4\cos x = 0$; (ii) find the value of $y$ when $x = 105^\circ$; (iii) find $x$ when $y = 1.5$.

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

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