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Mathematics
WAEC / WASSCE 2017
Geometry and Trigonometry
Solution
(a) It takes 8 students two- thirds of an hour to fill 12 tanks with water. How many tanks of water will 4 students fill in one- third of an hour at the same rate?
(b) A chord, 20 cm long, is 12 cm from the centre of the circle. Calculate, correct to one decimal place, the :
(i) angle subtended by the chord at the centre of the circle;
(ii) perimeter of the minor segment cut off by the chord. [Take \(\pi = 3.142\)].
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) \(Time = \frac{2}{3} hour = \frac{2}{3} \times 60 = 40 mins\)
4 students will take \(\frac{8 \times 40}{4} = 80 minutes\)
\(Time = \frac{1}{3} hour = \frac{1}{3} \times 60 = 20 minutes\)
It will take 20 minutes to fill \(\frac{12 \times 20}{80} = 3 tanks\)
(b)
\(\tan (\frac{\theta}{2}) = \frac{10}{12}\)
\(\frac{\theta}{2} = \tan^{-1} (0.8337) = 39.8^\circ\)
\(\theta = 39.8^\circ \times 2 = 79.6^\circ\)
(ii) Perimeter of minor segment = Length of chord + length of arc
But \(r^{2} = 12^{2} + 10^{2}\)
= \(144 + 100 = 244\)
\(r = \sqrt{244} = 15.62 cm\)
Length of arc = \(\frac{79.6}{360} \times 2 \times 3.142 \times 15.62 = 21.71cm\)
Perimeter of minor segment = 20 + 21.71 = 41.71cm
Source: Myschool Past Questions Archive (Record #52385)
Mathematics
WAEC / WASSCE 2017
Algebra
Solution
(a) Using completing the square method, solve, correct to 2 decimal places, the equation \(3y^{2} - 5y + 2 = 0\).
(b) Given that \(M = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix}, N = \begin{pmatrix} m & x \\ n & y \end{pmatrix}\) and \(MN = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}\), find the matrix N.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) \(3y^{2} - 5y + 2 = 0\)
\(\frac{3y^{2}}{3} - \frac{5y}{3} = \frac{-2}{3}\)
\(y^{2} - \frac{5}{3}y = \frac{-2}{3}\)
\(\frac{1}{2} of \frac{-5}{3} = \frac{-5}{6}\)
\(y^{2} - \frac{5}{3}y + (-\frac{5}{6})^{2} = \frac{-2}{3} + (\frac{-5}{6})^{2}\)
\((y - (\frac{5}{6}))^{2} = \frac{1}{36}\)
Taking square root of both sides,
\(\sqrt{(y - (\frac{5}{6}))^{2}} = \sqrt{\frac{1}{36}}\)
\(y - \frac{5}{6} =\pm {\frac{1}{6}}\)
\(y = \frac{5}{6} \pm \frac{1}{6}\)
\(y = \frac{5}{6} + \frac{1}{6} = 1.00\) or \(y = \frac{5}{6} - \frac{1}{6} = \frac{2}{3} = 0.67\).
(b) \(M = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix}, N = \begin{pmatrix} m & x \\ n & y \end{pmatrix}\)
\(MN = \begin{pmatrix} m + 2n & x + 2y \\ 4m + 2n & 4x + 2y \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}\)
\(m + 2n = 2 ... (1)\)
\(4m + 2n = 3 ... (2)\)
Multiplying (1) by 4, we have
\(4m + 8n = 8 ... (3)\)
(3) - (1) : \(8n - 2n = 8 - 2 = 6\)
\(6n = 6 \implies n = 1\)
\(m + 2n = m + 2(1) = m + 2 = 2\)
\(m = 2 - 2 = 0\)
\(x + 2y = 1 ... (1)\)
\(4x + y = 4 ... (2)\)
Multiply (1) by 4,
\(4x + 8y = 4 ... (3)\)
(3) - (2) : \(8y - y = 4 - 4 = 0\)
\(\implies y = 0\)
\(x + 2y = x + 2(0) = x + 0 = 1\)
\(x = 1 - 0 = 1\)
\(\therefore N = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\).
Source: Myschool Past Questions Archive (Record #52406)
Mathematics
WAEC / WASSCE 2017
Number and Numeration
Solution
(a) The operation (*) is defined on the set of real numbers, R, by \(x * y = \frac{x + y}{2}, x, y \in R\).
(i) Evaluate \(3 * \frac{2}{5}\).
(ii) If \(8 * y = 8\frac{1}{4}\), find the value of y.
(b) In \(\Delta ABC, \overline{AB} = \begin{pmatrix} -4 \\ 6 \end{pmatrix}\) and \(\overline{AC} = \begin{pmatrix} 3 \\ -8 \end{pmatrix}\). If P is the midpoint of \(\overline{AB}\), express \(\overline{CP}\) as a column vector.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) (i) \(3 * \frac{2}{5} = \frac{3 + \frac{2}{5}}{2}\)
= \(\frac{\frac{17}{5}}{2}\)
= \(\frac{17}{10}\)
(ii) \(8 * y = 8\frac{1}{4}\)
\(\frac{8 + y}{2} = \frac{33}{4}\)
\(66 = 32 + 4y \implies 4y = 66 - 32 = 34\)
\(y = \frac{34}{4} = 8.5\)
(b) \(\overrightarrow{AP} = \frac{1}{2}(\overrightarrow{AB})\)
= \(\frac{1}{2} \begin{pmatrix} -4 \\ 6 \end{pmatrix}\)
= \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\)
Midpoint = \(\overrightarrow{AP} + \overrightarrow{PC} = \overrightarrow{AC}\)
\(\overrightarrow{PC} = \overrightarrow{AC} - \overrightarrow{AP}\)
= \(\begin{pmatrix} 3 \\ -8 \end{pmatrix} - \begin{pmatrix} -2 \\ 3 \end{pmatrix}\)
= \(\begin{pmatrix} 5 \\ -11 \end{pmatrix}\)
\(\overrightarrow{CP} = -1 \times \begin{pmatrix} 5 \\ -11 \end{pmatrix}\)
= \(\begin{pmatrix} -5 \\ 11 \end{pmatrix}\)
Source: Myschool Past Questions Archive (Record #52423)
Mathematics
UTME / JAMB 2016
Number and Numeration
Diagram
Solution
The venn diagram shows a class of 50 students with the games they play. How many students play only two games?
Attached Figure / Diagram
A
15
B
16
C
20
D
18
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Number of students that plays only two games = 4 + 8 + 3 = 15
Source: WAEC / JAMB / NECO Classroom Archive (Record #44756)
Mathematics
UTME / JAMB 2016
Statistics and Probability
Diagram
Solution
mark
2
3
4
Frequency
4
4
y
The table above shows the frequency distribution of marks obtained by a group of students. If the total mark is 48, find the value of y.
Attached Figure / Diagram
A
6
B
8
C
7
D
5
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
(2 x 4) + (3 x 4) + (4 x y) = 48
8 + 12 + 4y = 48
4y = 48 - 20
\(\frac{4y}{4}\) = \(\frac{28}{4}\)
y = 7
Source: WAEC / JAMB / NECO Classroom Archive (Record #44776)
Mathematics
UTME / JAMB 2016
Calculus
Solution
Integrate \(\frac{2x^3 + 2x}{x}\) with respect to x
A
\(\frac{2x^3}{3}\) - 2x + k
B
x3 + 2x + k
C
\(\frac{2x^3}{3}\) + 2x + k
D
x3 - 2x + k
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\int\) \(\frac{2x^3 + 2x}{x}\) = \(\int\) \(\frac{2x^3}{x}\) + \(\frac{2x}{x}\) = \(\int\) (2x2 + 2) dx
\(\frac{2x^2+1}{2 + 1}\) + 2x + k
= \(\frac{2x^3}{3}\)+ 2x + k
Source: Myschool Past Questions Archive (Record #44744)
Mathematics
UTME / JAMB 2016
Statistics and Probability
Solution
If the mean of 4, y, 8 and 10 is 7. Find Y?
A
6
B
10
C
7
D
9
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{4 + y + 8 + 10}{4}\) = 7
22 + y = 4 x 7
22 + y = 28
y = 28 - 22 = 6
Source: Myschool Past Questions Archive (Record #44745)
Mathematics
UTME / JAMB 2016
Calculus
Solution
Find the derivative of y = ( \(\frac{1}{3}\)X + 6)\(^2\)
A
2( \(\frac{1}{3}\)X + 6
B
\(\frac{2}{3}\) ( \(\frac{1}{3}\)X + 6)
C
\(\frac{2}{3}\) ( \(\frac{1}{3}\)X + 6)2
D
\(\frac{1}{3}\) ( \(\frac{1}{3}\)X + 6)2
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Y = ( \(\frac{1}{3}\)X + 6)\(^2\)
\(\frac{dy}{dx}\) = 2( \(\frac{1}{3}\)X + 6) \(\frac{1}{3}\)
= \(\frac{2}{3}\) ( \(\frac{1}{3}\)X + 6)
Source: Myschool Past Questions Archive (Record #44746)
Mathematics
UTME / JAMB 2016
Number and Numeration
Solution
In a school of 150 students, 80 offer French while 60 offer Arabic and 20 offer neither. How many students offer both subjects?
A
45
B
10
C
35
D
30
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
80 - x + x + 60 - x + 20 = 150
160 - x = 150
x = 160 - 150 = 10
Source: Myschool Past Questions Archive (Record #44747)
Mathematics
UTME / JAMB 2016
Number and Numeration
Solution
If the 2nd term of a G.P is \(\frac{8}{9}\) and the 6th term is 4\(\frac{1}{2}\). Find the common ratio.
A
2
B
\(\frac{3}{2}\)
C
\(\frac{2}{3}\)
D
3
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
T2 = \(\frac{8}{9}\), T6 = 4\(\frac{1}{2}\)
ar = \(\frac{8}{9}\)
ar5 = \(\frac{9}{2}\)
\(\frac{ar}{ar^5}\) = \(\frac{8}{9}\) x \(\frac{2}{9}\)
\(\frac{1}{r^4}\) = \(\frac{16}{81}\)
r = \(\sqrt[4]{\frac{18}{16}}\) = \(\frac{3}{2}\)
Source: Myschool Past Questions Archive (Record #44748)
Mathematics
UTME / JAMB 2016
Number and Numeration
Diagram
Solution
From the diagram above, find the value of < OTQ
Attached Figure / Diagram
A
230o
B
55o
C
115o
D
65o
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
o
Source: Myschool Past Questions Archive (Record #44749)
Mathematics
UTME / JAMB 2016
Geometry and Trigonometry
Solution
The sum of the interior angles of a polygon is a given as 1080o. Find the number of the sides of the polygon.
A
5
B
7
C
6
D
8
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{(n - 2) 180}{180}\) = \(\frac{1080}{180}\)
(n - 2) = 6
n = 6 + 2 = 8
Source: Myschool Past Questions Archive (Record #44750)
Mathematics
UTME / JAMB 2016
Geometry and Trigonometry
Diagram
Solution
In the diagram above, l1 is parallel to l2, Find the value of < PMT
Attached Figure / Diagram
A
82o
B
36o
C
72o
D
118o
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
o - 118o = 62o
o ( Alternative angle)
y + 2y + 10o + 62o = 180o (Angles on a straight line)
3y = 180 - 72
\(\frac{3y}{3}\) = \(\frac{108}{3}\)
y = 36o
o
Source: Myschool Past Questions Archive (Record #44751)
Mathematics
UTME / JAMB 2016
Geometry and Trigonometry
Diagram
Solution
From the diagram above, Find the value of < ROP
Attached Figure / Diagram
A
110o
B
70o
C
95o
D
85o
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
o (Exterior angle of cyclic quadrilateral)
Source: Myschool Past Questions Archive (Record #44755)
Mathematics
UTME / JAMB 2016
Number and Numeration
Solution
If line p = 5x + 3 is parallel to line p = wx + 5. Find the value of w.
A
7
B
3
C
6
D
5
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
P = 5x + 3, P = wx + 5
M1 = 5, M2 = w
M1 = M2 (parallel lines)
w = 5
Source: Myschool Past Questions Archive (Record #44768)