Theory paper · 13 questions

WAEC · 2019 · Private, 2nd series · General Maths · Paper 2

Topics include Modular arithmetic, Binary operations, Solid mensuration, Indices & standard form, Surds, Probability.

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Answer every question in order, timed if you like (suggested 3 h 15 min). You're marked when you hand in, then you see where to focus and the working for each question.

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Question 1

(a)
Draw a table for multiplication $\otimes$ in modulo 8 on the set $T = \{2, 3, 5, 7\}$ .
Show the answer

$\otimes$ 2 3 5 7

2 4 6 2 6

3 6 1 7 5

5 2 7 1 3

7 6 5 3 1
(b)
Use the table to find the solution set of: (i) $3 \otimes n = 5$ ; (ii) $n \otimes n = 1$ .
Show the answer
(i) $\{7\}$ ; (ii) $\{3, 5, 7\}$

$\otimes$	2	3	5	7
2	4	6	2	6
3	6	1	7	5
5	2	7	1	3
7	6	5	3	1

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Question 2

(a)
The slant height of a cone is $18.7\text{ cm}$ and the diameter is $24\text{ cm}$ . Calculate, correct to three significant figures, the curved surface area of the cone. $\left[\text{Take }\pi = \frac{22}{7}\right]$
(b)
Solve: $\dfrac{128^x \times 2}{16^{(1 - x)}} = 8^{\frac23 x}$ .

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Question 3

(a)
Without using mathematical tables or calculators, simplify $3\sqrt{\frac32} - 4\sqrt{\frac23} - \sqrt{24}$ .
(b)
The probabilities of two candidates, $M$ and $N$ , passing an examination are $\frac23$ and $\frac45$ respectively. Find the probability that: (i) only one candidate will pass; (ii) at least one candidate will pass.
Separate values with commas, e.g. 3, −2

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Question 4

(a)
If $\cos\theta = \frac{15}{17}$ , find the value of $\dfrac{\tan\theta}{1 + 2\tan\theta}$ .
(b)
Find the value of $y$ if $\dfrac{\log_{10} y}{\log_{10} 64} = \frac12$ .

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Question 5 ✱✱

(a)
In the diagram, $MNPR$ is a circle with centre $O$ . The reflex angle at $O$ is $196^\circ$ and $\angle NMO = 52^\circ$ . Find the value of $m$ ( $= \angle OPN$ ).
(b)
A farmer uses $\frac25$ of his land to grow cassava, $\frac13$ of the remainder for plantain and the rest for yam. Find the part of the land used for yam.

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Question 6

(a)
Given that $2^m \times \left(\frac18\right)^n = 128$ and $4^m \div 2^{-4n} = \frac{1}{16}$ , find the value of $(m - n)$ .
(b)
Find the equation of the line joining the points $\left(-2, \frac12\right)$ and $\left(1, -\frac23\right)$ .
Show the answer
$18y + 7x + 5 = 0$

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Question 7

The following are the scores of 48 students in a promotion test.

52 56 25 56 68 73 66 64 56 48 15 88 20 39 9 50 98 54 54 40 50 96 53 16 36 44 18 97 65 21 60 44 54 32 84 52 92 49 37 94 72 88 89 35 59 34 72 60

(a)
Construct a frequency distribution table using the class intervals $0$ – $9$ , $10$ – $19$ , $20$ – $29$ , …
Model answer

Class interval Frequency

0–9 1

10–19 3

20–29 3

30–39 6

40–49 5

50–59 12

60–69 6

70–79 3

80–89 4

90–99 5

Total 48

Tally each score into its class, then count; the frequencies must add up to 48.
(b)
Draw a histogram for the distribution.
Model answer
Draw bars on the class boundaries, not the class limits (−0.5, 9.5, …, 99.5), with no gaps between them. The height of each bar is the frequency, and each axis is labelled.

To estimate the mode, take the tallest bar. Join its top-left corner to the top-left corner of the bar on its right, and its top-right corner to the top-right corner of the bar on its left. Read down from where the two lines cross: the mode is about 54.9.
(c)
(i) Use the histogram to estimate the modal score. (ii) If the pass mark for promotion is 30, find, correct to one decimal place, the percentage of students who will be promoted.
Separate values with commas, e.g. 3, −2

Class interval	Frequency
0–9	1
10–19	3
20–29	3
30–39	6
40–49	5
50–59	12
60–69	6
70–79	3
80–89	4
90–99	5
Total	48

Try it on a graph

Histogram bars (class boundaries 9.5, 19.5, …) with the crossed lines that locate the mode.

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Question 8

(a)
The fifth term of an Arithmetic Progression (A.P.) is 11 and the eighth term is 20. Find the: (i) 12th term; (ii) sum of the first 12 terms.
Separate values with commas, e.g. 3, −2
(b)
A trader collected ₦36,000.00 in cash after allowing a discount of $15\%$ . How much was the discount?

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Question 9

(a)
A selection interview was conducted for 110 students into various subjects in a Science department. 25 were selected for Physics, 45 for Biology and 48 for Mathematics. 10 were selected for Physics and Mathematics, 8 for Biology and Mathematics and 6 for Physics and Biology, while 5 were selected for all the three subjects. (i) Illustrate the information on a Venn diagram. (ii) How many students were selected for Biology but neither Physics nor Mathematics? (iii) How many students were not selected for any of the three subjects?
Separate values with commas, e.g. 3, −2
(b)
The mean of $22$ , $18$ , $(2x + 1)$ , $10$ and $20$ is $15$ . Find the median.

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Question 10

Commercial arithmetic

A man bought a house at $350,000.00. He paid $20\%$ of the cost from his own resources and the rest with a loan he took from the bank at $7\%$ simple interest per annum for 8 years. Calculate the:

(a)
total cost of the house to the man;
(b)
percentage increase in the cost of the house as a result of the loan;
(c)
percentage loss, correct to two decimal places, if after paying the loan, he renovates the house at a cost of $10,000.00 and then sells it for $460,000.00.

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Question 11 ✱✱

Quadratics & their graphs

(a)
Copy and complete the table of values for $y = x^2 + \frac12$ for $-4 \le x \le 4$ .

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

$y$ $9.5$ $0.5$

Model answer

$x$ −4 −3 −2 −1 0 1 2 3 4

$y$ 16.5 9.5 4.5 1.5 0.5 1.5 4.5 9.5 16.5

For example, at $x = -4$ : $y = 16 + 0.5 = 16.5$ .
(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 $y = x^2 + \frac12$ .
Model answer
Plot every point from the table, then join them with one smooth curve (not straight lines between points). The curve is symmetrical about the $y$ -axis, with its lowest point at $(0, 0.5)$ .

For (c): (i) $x^2 = 5$ is $x^2 + \frac12 = 5.5$ , so draw $y = 5.5$ : $x \approx \pm 2.2$ . (ii) $y$ decreases as $x$ increases on the left half, $-4 \le x < 0$ .
(c)
Use the graph to: (i) solve $x^2 = 5$ ; (ii) find the range of values of $x$ for which $y$ decreases as $x$ increases.
Separate values with commas, e.g. 3, −2

$x$	−4	−3	−2	−1	0	1	2	3	4
$y$	16.5	9.5	4.5	1.5	0.5	1.5	4.5	9.5	16.5

Try it on a graph

y = x² + ½ with the line y = 5.5.

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Question 12

A town $J$ is $20\text{ km}$ from a lorry station, $K$ , on a bearing $065^\circ$ . Another town, $T$ , is $8\text{ km}$ from $K$ on a bearing $155^\circ$ . Calculate:

(a)(i)
to the nearest kilometre, the distance of $T$ from $J$ ;
(a)(ii)
to the nearest degree, the bearing of $T$ from $J$ .

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K is at the origin; north is up. The dashed line is JT.

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Question 13

(a)
In the diagram, $A$ , $B$ , $C$ , $D$ are points on the circle with centre $O$ . $\angle AOD = 130^\circ$ and $\angle BAO = 26^\circ$ . Find: (i) $\angle ODB$ ; (ii) $\angle BOD$ .
Separate values with commas, e.g. 3, −2
(b)
A number is selected at random from each of the sets $\{1, 2, 6\}$ and $\{3, 4, 5\}$ . Find the probability that the sum of the numbers selected is greater than seven.

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