WAEC · 2009 · Nov/Dec · General Maths · Paper 2

1. (a)

   Simplify, without using tables or a calculator, $3\frac13 \div 1\frac14 - \frac49$.

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2. (b)

   Simplify, without using tables or a calculator, $2 + \sqrt{96} - 4(\sqrt6 - 1)^2$, and express your answer in the form $m + n\sqrt6$, where $m$ and $n$ are real numbers.

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1. (a)

   Solve the equation $\dfrac{4x - 1}{3} - \dfrac{3x - 1}{2} = \dfrac{5 - 2x}{4}$.

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2. (b)

   From a shop, Kofi bought 2 singlets and 3 shirts for GH¢31.00, while Kwasi bought 3 singlets and 2 shirts for GH¢29.00. How much will Yaw pay for one singlet and one shirt bought from the same shop?

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1. (a)

   What is the probability that either of the two patients survives?

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2. (b)

   What is the probability that neither of the two patients survives?

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3. (c)

   What is the probability that at least one of the two patients survives?

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1. (a)

   Calculate the base radius of the cone, correct to two decimal places.

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2. (b)

   Calculate the greatest volume of liquid which the cone can hold, correct to the nearest $\text{cm}^3$.

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1. (a)

   In the diagram, the two circles intersect at $X$ and $Y$. The centre $O$ of the smaller circle is on the circumference of the bigger circle. $A$ and $B$ are any two points on the major arcs, one on each circle. Find an equation connecting $a$ and $b$.

   Model answer

   Join $OX$ and $OY$. In the small circle, $\angle XOY = 2b^\circ$ (angle at the centre). $AXOY$ is a cyclic quadrilateral of the big circle, so $a + 2b = 180$.

2. (b)

   In the diagram, $\angle QPR = \angle PTR = 90^\circ$, $|PR| = 8$ cm and $|QP| = 6$ cm. Find $|TR|$.

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1. (a)

   By how much is $11000_2$ greater than or less than $111_2 \times 11_2$? Give your answer in base two.

   Model answer

   $111_2 \times 11_2 = 10101_2$ and $11000_2 - 10101_2 = 11_2$, so $11000_2$ is greater by $11_2$ (3 in base ten).

2. (b)

   A shopkeeper has 20 television sets in stock. He sells 18 of them at a profit of 15% and the remaining two at a loss of 5%. Find his percentage profit on the 20 sets.

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1. (a)

   Using class intervals 21 – 30, 31 – 40, …, prepare a frequency distribution table.

   Model answer

   Marks	21–30	31–40	41–50	51–60	61–70	71–80
   Class boundaries	20.5–30.5	30.5–40.5	40.5–50.5	50.5–60.5	60.5–70.5	70.5–80.5
   Frequency	2	10	12	15	8	3

2. (b)

   Draw a histogram to represent the distribution.

   Model answer

   Use the class boundaries 20.5, 30.5, …, 80.5 on the horizontal axis and draw bars with no gaps, of heights 2, 10, 12, 15, 8 and 3.

3. (c)

   Use your histogram to estimate the modal mark.

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4. (d)

   If a student is selected at random, find the probability that he or she obtains a mark greater than 63.

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1. (a)

   Calculate the dimensions of the field.

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

   Calculate the cost of clearing the field at ₦6.50 per square metre, leaving a margin 2 m wide along the longer sides.

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3. (c)

   Calculate the percentage of the field not cleared.

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1. (a)(i)

   $A$ (lat $43^\circ$N, long $77^\circ$E), $B$ (lat $43^\circ$N, long $103^\circ$W) and $C$ (lat $57^\circ$S, long $77^\circ$E) are three points on the surface of the earth. Find the distance from $A$ to $B$ along latitude $43^\circ$N. [Take $\pi = 3.142$ and $R = 6400$ km]

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2. (a)(ii)

   Find the distance from $A$ to $C$ along the great circle joining the two points.

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3. (b)

   In triangle $XYZ$, $|XY| = 9$ cm, $|XZ| = 10$ cm and $\angle YXZ = 75^\circ$. Find $|YZ|$.

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1. (a)

   Using a ruler and a pair of compasses only, construct a triangle $ABC$ such that $|AB| = 7.1$ cm, $|AC| = 7$ cm and $\angle BAC = 105^\circ$. Construct the bisector of $\angle BAC$ to meet $BC$ at $X$, and the perpendicular bisector of $AC$ to meet $AX$ produced at $Y$.

   Model answer

   Construct $105^\circ$ at $A$ as $90^\circ + 15^\circ$ (bisect the $30^\circ$ between $90^\circ$ and $120^\circ$). Mark $B$ and $C$, join $BC$. Bisect angle $BAC$ and extend the bisector beyond $BC$. Construct the perpendicular bisector of $AC$; where it meets $AX$ produced is $Y$. Leave all arcs visible.

2. (b)

   Measure: (i) $|XY|$; (ii) $|BC|$.

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

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1. (a)

   Copy and complete the table of values for $y = x^2 - 2$ for $-3 \le x \le 4$.

   $x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
   $y$		$2$		$-2$		$2$

   Model answer

   $x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
   $y$	$7$	$2$	$-1$	$-2$	$-1$	$2$	$7$	$14$

   The row is symmetric about $x = 0$: $x = 3$ and $x = -3$ both give 7.

2. (b)

   Using a scale of 2 cm to 1 unit on the $x$-axis and 2 cm to 2 units on the $y$-axis, draw the graph of $y = x^2 - 2$.

   Model answer

   Plot the eight points and join them with a smooth U-shaped curve through $(0, -2)$. For (c): it cuts the $x$-axis at $x \approx \pm 1.4$; the line $y = 1$ meets it at $x \approx \pm 1.7$; the tangent at $x = -1$ has gradient $-2$.

3. (c)(i)

   Use your graph to find the roots of the equation $x^2 - 2 = 0$.

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   Separate values with commas, e.g. 3, −2
4. (c)(ii)

   Use your graph to find the values of $x$ for which $x^2 - 3 = 0$.

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   Separate values with commas, e.g. 3, −2
5. (c)(iii)

   Use your graph to find the gradient of the curve at the point where $x = -1$.

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1. (a)

   Three positive numbers are in arithmetic progression (A.P.). The sum of the squares of the three numbers is 155, while the sum of the numbers is 21. If the common difference is positive, find the numbers.

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

   If the total surface area of a sphere is $154\text{ cm}^2$, find its radius. [Take $\pi = \frac{22}{7}$]

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1. (b)

   The pilot of an aircraft 2000 metres above sea level observes at an instant that the angles of depression of two boats, which are in a direct straight line with the aircraft, are $58^\circ$ and $72^\circ$. Find, correct to the nearest metre, the distance between the two boats.

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