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Mathematics
UTME / JAMB 2009
Geometry and Trigonometry
Solution
Evaluate ∫sec\(^2\)\theta d\theta?
A
sec θ tan θ + k
B
tan θ + k
C
2sec θ + k
D
sec θ + k
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
∫sec2\theta d\theta = ∫ 1/cos2 d\theta
∫(cos)-2 d\theta,
let u = cos \theta
∴∫u-2 = 1/u + c
∫cos \theta = sin \theta + c
∫sec-2\theta = 1/u sin \theta + c
= (sin\theta / cos\theta) + c
= Tan \theta + c
Source: Myschool Past Questions Archive (Record #9363)
Mathematics
UTME / JAMB 2009
Geometry and Trigonometry
Diagram
Solution
The pie chart above represents 400 fruits on display in a grocery store. How many apples are in the store?
Attached Figure / Diagram
A
45
B
50
C
60
D
75
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Sectorial angle of apple
100 + 100 + 115 + x = 30o
315 + x = 360
x = 360 - 315
x = 45o
360o represents 400
1o represents 400 / 360
45o represents (400/360) x (45/1)
= 50 apples
Source: Myschool Past Questions Archive (Record #9365)
Mathematics
UTME / JAMB 2009
Geometry and Trigonometry
Solution
The probability of a student passing any examination is 2/3. If the students takes three examination, what is the probability that he will not pass any of them?
A
2/3
B
4/9
C
8/27
D
1/27
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
P(pass) = 2/3
P(not pass) = 1 - 2/3 = 1/3
P(not passing any of the 3 Exams) = 1/3 x 1/3 x 1/3
= 1/27
Source: Myschool Past Questions Archive (Record #9366)
Mathematics
UTME / JAMB 2009
Geometry and Trigonometry
Solution
Marks
2
3
4
5
6
7
8
9
No. of students
3
4
1
0
4
5
2
1
The table above shows the distribution of marks of students in a test. Find the probability of passing the test if the pass mark is 5?
A
\(\frac{3}{5}\)
B
\(\frac{2}{5}\)
C
\(\frac{7}{20}\)
D
\(\frac{1}{5}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Total number of students = 20
Marks
No. of students
5
0
6
4
7
5
8
2
9
1
12
Pr(passing) = \(\frac{12}{20}\)
= \(\frac{3}{5}\)
Source: Myschool Past Questions Archive (Record #9491)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Solution
The interior angles of a pentagon are (2x + 5)o, (x + 20)o, xo, (3x - 20)o and (x + 15)o. Find the value of x
A
80o
B
70o
C
65o
D
40o
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
(2x + 5)o + (xo + 20o) + x + (3x - 20)o + (x + 15)o = (n - 2) x 180
8n + 20 = (5 - 2) x 180
where n = 5(Pentagon)
8n + 20 = 3 x 180
8n + 20 = 540
8n = 540 - 20
8n = 520
x = 65o
Source: WAEC / JAMB / NECO Classroom Archive (Record #36451)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Solution
Express the square root of 0.000144 in the standard form
A
1.2 x 10-4
B
1.2 x 10-3
C
1.2 x 10-2
D
1.2 x 10-1
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\sqrt{0.000144}\) = 0.012
= 1.2 x 10-2
Source: Myschool Past Questions Archive (Record #36435)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Solution
The diameter of a bicycle wheel is 42cm. If the wheel makes 16 complete revolution, what will be the total distance covered by the wheel? [Take \(\pi \frac{22}{7}\)
A
672cm
B
1056cm
C
2112cm
D
4224cm
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
d = 42cm
Circumference = \(\pi d = \frac{22}{7} \times 42cm\)
= 22 x 6cm = 132cm
Total distance covered = circumference x number of revolutions
= 132cm x 16 = 2112cm
Source: Myschool Past Questions Archive (Record #36448)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Solution
If M and N are two fixed points in a plane. Find the locus L = [P : PM = PN]
A
a line equal to MN
B
line parallel to MN
C
Perpendicular bisector of MN
D
A circle centre P, radius MN
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Locus L = (P : PM = Pn}
For M and N being two fixed points
Since PM = PN, P is equidistant from Mand N, So L must be the perpendicular bisector of the line MN
Source: Myschool Past Questions Archive (Record #36453)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Solution
Given that sin 60o = \(\frac{\sqrt{3}}{2}\) and cos 60o = \(\frac{1}{2}\), evaluate \(\frac{1 - sin 60^o}{1 + cos 60^o}\)
A
\(\frac{2 + \sqrt{3}}{3}\)
B
\(\frac{1 - \sqrt{3}}{3}\)
C
\(\frac{1 + \sqrt{3}}{3}\)
D
\(\frac{2 - \sqrt{3}}{3}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Sin 60 = \(\frac{\sqrt{3}}{2}\); cos 60o = \(\frac{1}{2}\)
= \(\frac{1 - \sin 60}{1 + \cos 60} = \frac{1 - \frac{\sqrt{3}}{2}}{1 + \frac{1}{2}}\)
= \(\frac{2 - \sqrt{3}}{3}{2}\div \frac{2 + 1}{2}\)
= \(\frac{2 - \sqrt{3}}{2} \times \frac{2}{3}\)
= \(\frac{2 - \sqrt{3}}{3}\)
Source: Myschool Past Questions Archive (Record #36455)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Solution
An arc of a circle subtends an angle of 60o at the centre. If the radius of the circle is 3cm, find , in terms of \(\pi\), the length of the arc
A
\(\pi\)cm
B
2\(\pi\)cm
C
3\(\pi\)cm
D
6\(\pi\)cm
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Length of arc = \(\frac{\theta}{360} \times 2\pi r\)
= \(\frac{60}{360} \times 2\pi \times 3cm\)
= \(\pi\)cm
Source: Myschool Past Questions Archive (Record #36465)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Solution
Each of the interior angles of a regular polygon is 140o. Calculate the sum of all the interior angles of the polygon
A
1080o
B
1260o
C
1800o
D
2160o
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Each interior angle = 140
\(\frac{(n - 2) \times 180}{n} = 140\)
(n - 2) x 180 = 140n
150 - 360 = 140n
180m - 140n = 360
40n - 360
n = \(\frac{360}{40}\)
n = 9
Sum of all interior angles = (n - 2) x 180
= (9 - 2) x 180
= 7 x 180
= 1260
Source: Myschool Past Questions Archive (Record #36469)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Solution
The angles of triangle are (x + 10)o, (2x - 40)o and (3x - 90)o. Which of the following accurately describes the triangle?
A
it is a scalene triangle
B
it is right angled isosceles triangle
C
t is an equilateral triangle
D
It is an isosceles triangle but not right angled
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
(x + 10)o + (2x - 40)o + (3x - 90)o = 180
6x - 120 = 180
6x = 180 + 120
6x = 300
x = \(\frac{300}{6}\)
x = 50
x + 10o = 50o + 10o = 60o
2x - 40 = 2(50o) - 40 = 60o
3x - 90 = 3(50o) - 90o = 60o
Hence, it is an equilateral triangle
Source: Myschool Past Questions Archive (Record #36474)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Diagram
Solution
PQRS is a trapezuim. QR//PS, /PQ/ = 5cm, /OR/ = 6cm, /PS/ = 10cm and angle QPS = 42o. Calculate the perpendicular distance between the parallel sides
Attached Figure / Diagram
A
3.35cm
B
3.73cm
C
4.50cm
D
4.62cm
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
h = ?
\(\frac{h}{5cm}\) = sin 45o
h = 5cm x 0.6691 = 3.35cm
Source: Myschool Past Questions Archive (Record #36483)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Diagram
Solution
PQRS is a trapezium. QR//PS, /PQ/ = 5cm, /OR/ = 6cm, /PS/ = 10cm and angle QPS = 42o. Calculate, correct to the nearest cm2, the area of the trapezium (h = 3.35cm2 )
Attached Figure / Diagram
A
27cm2
B
30cm2
C
36cm2
D
37cm2
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Area = \(\frac{1}{2}(a + b)\)h
where a = QR = 6cm
b = PA = 10cm
Area = \(\frac{1}{2}(6 + 10)\) x 3.35cm2
= \(\frac{1}{2}(16)\) x 3.35cm2
= 26.8cm2
= 27cm2 (approx.)
Source: Myschool Past Questions Archive (Record #36485)
Mathematics
WAEC / WASSCE 2009
Geometry and Trigonometry
Diagram
Solution
In the diagram, /TP/ = 12cm and it is 6cm from O, the centre of the circle, Calculate < TOP
Attached Figure / Diagram
A
120o
B
90o
C
60o
D
45o
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Tan \(\theta = \frac{6}{6} = 1\)
\(\theta\) = tab - 1(1) = 45o
o = 90o
Source: Myschool Past Questions Archive (Record #36486)