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Mathematics
UTME / JAMB 2022
Geometry and Trigonometry
Solution
Mr Adu spends his annual salary on food(f), rent(r), car maintenance, gifts(g), savings(s) and some miscellaneous (m) as indicate in the table below:
F
R
C
G
S
M
28%
15%
20%
14%
10%
13%
If the above information is represented on a pie chart. What angle represents his spending on food?
A
108.5
B
100.8
C
98.8
D
120.5
E
120.5
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{28 * 360}{100}\)
= 100.8º
Source: Myschool Past Questions Archive (Record #64229)
Mathematics
UTME / JAMB 2022
Algebra
Solution
If A = \(\begin{pmatrix} 2 & 1 \\ 2 & 3 \\ 1 & 2 \end{pmatrix}\) and B = \(\begin{pmatrix} 3 & 2 \\ 4 & 2 \end{pmatrix}\). Find AB
A
\(\begin{pmatrix} 18 & 6 \\ 12 & 10 \\ 10 & 6 \end{pmatrix}\)
B
\(\begin{pmatrix} 10 & 6 \\ 13 & 10 \\ 12 & 6 \end{pmatrix}\)
C
\(\begin{pmatrix} 10 & 6 \\ 12 & 10 \\ 11 & 6 \end{pmatrix}\)
D
\(\begin{pmatrix} 10 & 6 \\ 18 & 10 \\ 11 & 6 \end{pmatrix}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Given A = \(\begin{pmatrix} 2 & 1 \\ 2 & 3 \\ 1 & 2 \end{pmatrix}\) and B = \(\begin{pmatrix} 3 & 2 \\ 4 & 2 \end{pmatrix}\).
We can multiply these matrices since the number of colums in A = number of rows in B
AB = \(\begin{pmatrix} (2*3)+(1*4) & (2*2)+(1*2) \\ (2*3)+(3*4) & (2*2)+(3*2) \\ (1*3)+(2*4) & (1*2)+(2*2) \end{pmatrix}\)
AB = \(\begin{pmatrix} (6+4) & (4+2) \\ (6+12) & (4+6) \\ (3+8) & (2+4) \end{pmatrix}\)
= \(\begin{pmatrix} 10 & 6 \\ 18 & 10 \\ 11 & 6 \end{pmatrix}\)
Source: Myschool Past Questions Archive (Record #64273)
Mathematics
UTME / JAMB 2022
Number and Numeration
Solution
If y varies inversely as x and x = 3 when y =4. Find the value of x when y = 12
A
4
B
1
C
2
D
3
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
y = 1/x; y = \(\frac{k}{x}\)
k =xy → 3 * 4 = 12
x = \(\frac{k}{y}\)
x = \(\frac{12}{12}\) = 1
Source: Myschool Past Questions Archive (Record #64392)
Mathematics
UTME / JAMB 2022
Algebra
Diagram
Solution
The number line represented by the inequality
Attached Figure / Diagram
A
x≤2
B
x<2
C
x≥2
D
x>2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
x \ge 2 is represented by an arrow with an filled-in dot above the 2.
The arrow points in the direction of all the numbers that are greater than 2.
Source: Myschool Past Questions Archive (Record #64410)
Mathematics
WAEC / WASSCE 2022
Number and Numeration
Solution
Solve \(\frac{y+2}{4}\) - \(\frac{y-1}{3}\) > 1
A
y < -10
B
y < -2
C
y < 2
D
y < 10
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
multiply both sides by the LCM
12 x \(\frac{y+2}{4}\) - 12 x \(\frac{y-1}{3}\) >12 x 1
3[y+2] - 4[y-1] > 12
3y + 6 - 4y + 4 > 12
-y + 2 > 12
-y > 12 - 10 = 2
y < -2
Source: Myschool Past Questions Archive (Record #64856)
Mathematics
WAEC / WASSCE 2022
Number and Numeration
Solution
(a) Given that (7 -2x), 9, (5x + 17) are consecutive terms of a Geometric Progression (G. P) with common ratio, r>0, find the values of x.
(b) Two positive numbers are in the ratio 3:4. The sum of thrice the first number and twice the second is 68. Find the smaller number.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) r =\(\frac{9}{7-2x}\)
r = \(\frac{5x+1}{7-2x}\)
\(\frac{9}{7-2x}\) = \(\frac{5x+1}{7-2x}\)
9x9=(7-2x) (5x +17)
81 = 35x + 119 - 10x\(^2\) - 34x
10x\(^2\) - x - 38
(x -2)(10x+19) = 0
x = 2, x = \(\frac{19}{10}\)
x = 2
(b) Let the integers be x and y
x:y= 3:4
\(\frac{x}{y}\) = \(\frac{3}{4}\)
x = \(\frac{3y}{4}\)
3x + 2y=68
3( \(\frac{3}{4}\)y)+ 2y = 68
\(\frac{9}{4}\)y + 2y = 68
9y + 8y = 272
17y = 272
y= 16
x = \(\frac{3}{4}\) * 16 →12
Source: Myschool Past Questions Archive (Record #65854)
Mathematics
WAEC / WASSCE 2022
Number and Numeration
Solution
Given that y =(\(\frac{pr}{m} - p^2r\))\(^{\frac{-3}{2}}\)
(a) make r the subject;
(b) find the value of r when y = -8, m = 1 and p = 3.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) y =(\(\frac{pr}{m} - p^2r\))\(^{\frac{-3}{2}}\)
y = (\(\frac{pr - mrp^2}{m})^\frac{-3}{2}\)
y = [\(\frac{m}{pr - mrp^2}]^\frac{3}{2}\)
\(y^\frac{2}{3} = \frac{m}{r(p - mp^2)}\)
r = \(\frac{m}{y^\frac{2}{3}[p - mp^2]}\)
b) Given y = -8, m = 1 p = 3
r = \(\frac{m}{y^\frac{2}{3}[p - mp^2]}\)
r = \(\frac{1}{(-8)^\frac{2}{3}[ 3 - (1)3^2]}\)
r = \(\frac{1}{(-2)^2[3 - 9]}\)
r = \(\frac{1}{[4][-6]}\) = \(\frac{-1}{24}\)
Source: Myschool Past Questions Archive (Record #65855)
Mathematics
WAEC / WASSCE 2022
Geometry and Trigonometry
Diagram
Solution
In the diagram, BCDE is a circle with centre
A. ∠BCD = (2x + 40)^\circ, ∠BAD = (5x - 35)º, ∠BED = (2y+ 10)^\circ and ∠ADC=40". Find:
(a) the values of x and y;
Attached Figure / Diagram
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) 2x + 40 + 2y + 10 = 180^\circ...(1)
2x + 2y = 130
x + y = 65º
5x - 35 = 2(2y+ 10) ... (2)
5x - 35 = 4y +20
5x - 4y = 55
5x - 4(5-x) = 55
5x - 260 + 4x =55
9x = 315
x = 35^\circ
y= 65 - 35
y=30^\circ
∠ABC + [2 x º35 + 40] + [5(35)-350)] + 40^\circ =360º
∠ABC + 110 + 140^\circ+ 40º = 360
∠ABC + 290^\circ = 360^\circ
∠ABC = 70^\circ
Source: Myschool Past Questions Archive (Record #65857)
Mathematics
WAEC / WASSCE 2022
Geometry and Trigonometry
Solution
a) Given that m = tan 30º and n = tan 45º, simplify, without using calculator, \(\frac{ m - n}{mn}\), leaving the answer in the form p + \(\sqrt{q}\)
b) There are 20 women in a bus. 15 of them wear glasses and 10 wear wristwatches. If a woman is chosen at random from the bus, find the probability that she wears both glasses and wristwatches.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
a) m = tan 30º = \(\frac{\sqrt{3}, n = tan 45º = 1
\(\frac{ m - n}{mn}\), = \(\frac{\frac{\sqrt{3}}{3} - 1}{\frac{\sqrt{3}}{3}(1)}\)
= \(\frac{\frac{\sqrt{3} - 3}{3}}{\frac{\sqrt{3}}{3}}\)
= \(\frac{\sqrt{3} - 3}{3}\) X \(\frac{3}{\sqrt{3}}\) = \(\frac{\sqrt{3} - 3}{\sqrt{3}}\)
= \(\frac{\sqrt{3} - 3}{\sqrt{3}}\) x \(\frac{\sqrt{3}}{\sqrt{3}}\)
= \(\frac{3 - 3\sqrt{3}}{3}\) = 1 - \(\sqrt{3}\)
b) see the diagram above..
Let the number of women wearing both wristwatches and glasses be x
15 - x + x + 10 - x = 20
25 - x = 20
x = 5
Therefore, the number of women wearing wrist watches only = 10 - x = 10 - 5 = 5
and number of women wearing glasses only = 15 - x = 15 - 5 = 10
Pr(both) = \(\frac{5}{20}\) = \(\frac{1}{4}\) = 0.25
Source: Myschool Past Questions Archive (Record #65858)
Mathematics
WAEC / WASSCE 2022
Geometry and Trigonometry
Diagram
Solution
The graph shows the relation of the form y = mx\(^2\) + nx + r, where m, n and r are constants.
Using the graph:
(a) state the scale used on both axes; (b) find the values of m, n and r; (c) find the gradient of the line through P and Q; (d) state the range of values of x for which y > Q.
Attached Figure / Diagram
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) x-axis: 2 cm to 2 units or 1 cm to 1 unit
y-axis: 2 cm to 10 units or 1 cm to 5 units
(b) x = -2, x = 4
(x+2)(x-4) = 0
x\(^2\) -4x + 2x -8 = 0
x\(^2\) -2x - 8 = 0
y = 8 + 2x - x(curve has a max. point)
m = -1,n = 2, r =8
(c) P(-5,-27),Q(3,5)
Gradient = \(\frac{5+27}{3+5}\)
= \(\frac{32}{8}\)
= 4
(d) Range: {x:-2 < x < 4)
Source: Myschool Past Questions Archive (Record #65860)
Mathematics
WAEC / WASSCE 2022
Number and Numeration
Solution
(a) A man purchased 180 copies of a book at N250.00 each. He sold y copies at N300.00 each and the rest at a discount of 5 kobo in the Naira of the cost price.
If he made a profit of N7,125.00, find the value of y.
(b) A trader bought x bags of rice at a cost C = 24x + 103 and sold them at a price, S = \(\frac{x^2}{20} - 33x\).
Find the expression for the profit (i) If 20 bags of rice were sold,
(ii) calculate the percentage profit.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) Total cost 180 x 250
=N45,000.00
Total selling price = 300y + (180-y) x 250 x \(\frac{95}{100}\)
300y + 42750 -237.5y
= 62.5y + 42750
Profit = 62.5y + 42750 -4500
7125 = 62.5y - 2250
9375 = 62.5y
y = 150
(b)) Profit = \(\frac{x^2}{20} - 33x\) (24x+ 103)
9x - \(\frac{x^2}{20}\) - 103
(ii) Cost = 24(20) +103 = 583.00
Selling Price =33(20) - \(\frac{20^2}{20}\) = 640
Percentage profit = \(\frac{640-583}{583}\) x 100%
= 9.78%
Source: Myschool Past Questions Archive (Record #65864)
Mathematics
WAEC / WASSCE 2022
Number and Numeration
Solution
Item
food & drinks
fuel
rent
building project
education
savings
Percentage%
35
7.5
1.0
15
17.5
x
The table shows the monthly expenditure (in percentages) of Mr. Okafor's salary.
(a) Calculate the percentage of Mr. Okafor's salary that was put into savings.
(b) Illustrate the information on a pie chart.
(c) If Mr. Okafor's annual gross salary is $28,800.00 and he pays tax of 12%.
Calculate; (i) his monthly tax; (ii) the amount saved each month.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) 35 + 7.5 + 10 + 15 + 17.5 + x = 100
85 + x = 100.
x = 100 - 85
x = 15
items
%
degree
food & drinks
35
\(\frac{35}{100} * 360\) = 126º
fuel
7.5
\(\frac{7.5}{100} * 360\) = 27º
rent
10
\(\frac{10}{100} * 360\) = 36º
building project
15
\(\frac{15}{100} * 360\) = 54º
education
17.5
\(\frac{17.5}{100} * 360\) = 63º
savings
15
\(\frac{15}{100} * 360\) = 54º
(c) income tax = \(\frac{12}{100} * 28,800\) = $3,456.00
Monthly tax = \(\frac{3456}{12}\) = $288.00
(ii) Monthly net salary = \(\frac{1}{12} (288 - 3456)\) = $2,112.00
Amount saved each month = \(\frac{15}{100} * 2112\) = $316.80
Source: Myschool Past Questions Archive (Record #65866)
Mathematics
WAEC / WASSCE 2022
Geometry and Trigonometry
Solution
(a) Copy and complete the table of values for y = 3Sinx + 7Cosx for 0^\circ
xº
0
20
40
60
80
100
120
140
160
180
y
7.0
4.2
-0.9
(b) Using a scale of 2cm to 20^\circ on the x-axis and 2cm to 2 units on the y-axis, draw the graph of y = 3Sinx + 7Cosx for 0^\circ
(c) Using the graph, find the;
(i) value of y when x= 150^\circ,
(i) range of values of x for which y > 0.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
xº
0
20
40
60
80
100
120
140
160
180
y
7.0
7.6
7.3
6.1
4.2
1.7
-0.9
-3.4
-5.6
7.0
(b)
(C)(i) From the graph, When x is 150^\circ y = -4.6 \pm 0.2
(ii) Range of value of x for which y > is x: 0º \le x \le 113
Source: Myschool Past Questions Archive (Record #65867)
Mathematics
WAEC / WASSCE 2022
Geometry and Trigonometry
Solution
Age
3
4
5
6
7
8
9
10
No. of children
2
6
5
x
6
9
8
5
The table shows the distribution of ages of a number of children in a school. If the mean of the distribution is 7, find the;
(a) value of x, (b) standard deviation of their ages.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
Mean = \(\frac{\sum fx}{ \sum f}\) = (3x2) +(4x6) + (5x5) + (6x) + (7x6) + (8x9) + (9x8) + (10x5) \div (2 + 6 + 5 + x + 6 + 9 + 8 + 5)
7 = (291 + 6x) \div (41+x)
7(41+x) = 291 + 6x
287 + 7x = 291 + 6x
x = 4
(b)
Age(x in yrs)
No.of child
Fx\(^2\)
3
2
18
4
6
96
5
5
125
6
4
144
7
6
294
8
9
576
9
8
648
10
5
500
Σf = 45
Σfx\(^2\) = 2401
Standard deviation = \(\sqrt \frac{196}{45} = \sqrt 4.3555\)
= 2.087
Source: Myschool Past Questions Archive (Record #65868)
Mathematics
WAEC / WASSCE 2022
Geometry and Trigonometry
Diagram
Solution
(a) In the diagram, O is the centre of the circle XYZ. angle ZXO=34 ^\circ and angle XOY=146 ^\circ . Find angle OYZ.
(b) The exterior angles of a polygon are 42, 38,57, x, (x+ y)^\circ. (2x- 15)^\circ and (3x -y)º? If x is 7 less than y, find the values of x and y.
Attached Figure / Diagram
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) ∠XYZ = \(\frac{1}{2}\) (146) = 73º
∠OZX = 34º
∠OZY = 73 - 34
∠OYZ = ∠OZY = 39
(b) x = y - 7
42 + 38 + 57 + x + x + y + 2x - 15 + 3x - y = 360
122^\circ+7x = 360^\circ
7x = 360º - 122 = 238^\circ
7x = 238º
x = 34
: 34 = y - 7
34 + 7 = y
41 = y
Source: Myschool Past Questions Archive (Record #65869)