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Mathematics
WAEC / WASSCE 1999
Trigonometry, Elevation & Bearings
Solution
(a) Copy and complete the following table of values for \(y = 9 \cos x + 5 \sin x\) to one decimal place.
x
0^\circ
30^\circ
60^\circ
90^\circ
120^\circ
150^\circ
180^\circ
210^\circ
y
10.3
-0.2
-5.3
-10.3
(b) Using a scale of 2cm to 30^\circ on the x- axis and 2 cm to 1 unit on the y- axis, draw the graph of \(y = 9 \cos x + 5 \sin x\) for \(0^\circ \leq x \leq 210^\circ\).
(c) Use your graph to solve the equation: (i) \(9\cos x + 5\sin x = 0\); (ii) \(9\cos x+ 5\sin x = 3.5\), correct to the nearest degree.
(d) Find the maximum value of y correct to one decimal place.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
x
0^\circ
30^\circ
60^\circ
90^\circ
120^\circ
150^\circ
180^\circ
210^\circ
y
9
10.3
8.8
5
-0.2
-5.3
-9
-10.3
(b)
(c)(i) From the graph, \(9\cos x + 5\sin x = 0 \implies x = 119^\circ\)
(d) Maximum value of y = 10.3.
Source: Myschool Past Questions Archive (Record #54428)
Mathematics
WAEC / WASSCE 1999
Plane Geometry, Polygons & Circle Theorems
Solution
In the diagram above, RST is a tangent to circle VSU center O ∠SVU = 50^\circ and UV is a diameter. Calculate ∠RSV.
A
90°
B
50°
C
45°
D
40°
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
< VSU = 90^\circ (angle in a semi-circle)
\(\therefore\) < SUV = 180^\circ - 90^\circ - 50^\circ
= 40^\circ
< RSV = 40^\circ (angle which chord makes tangent = angle in alternate segment)
Source: Myschool Past Questions Archive (Record #54932)
Mathematics
UTME / JAMB 1999
Plane Geometry, Polygons & Circle Theorems
Solution
If \(\frac{^6C_r}{^6P_r}\) = \(\frac{1}{6}\), find the value of r.
A
1
B
3
C
5
D
6
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Given: \(\frac{^6C_r}{^6P_r} = \frac{1}{6}\)
Recall the formulas:
\(^nC_r = \frac{n!}{r!(n-r)!}, \quad ^nP_r = \frac{n!}{r!}\)
Substitute \( n = 6 \):
\(\frac{^6C_r}{^6P_r} = \frac{\dfrac{6!}{r!(6-r)!}}{\dfrac{6!}{r!}} = \frac{6!}{r!(6-r)!} \times \frac{r!}{6!} = \frac{1}{(6-r)!}\)
So the equation simplifies to:
\(\frac{1}{(6-r)!} = \frac{1}{6}\)
Therefore: (6-r)! = 6
We know that 3! = 6, so:
6 - r = 3 \(\implies\) r = 3
Source: Myschool Past Questions Archive (Record #77196)
Mathematics
UTME / JAMB 1999
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
The mean of four numbers is 5, and the mean deviation is 3. Find the fourth number if the mean deviation of the first three numbers is 2
A
6
B
10
C
11
D
12
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Let the four numbers be \(a, b, c, d\).
- Mean of the four numbers = 5
→ a + b + c + d = 20
- Mean deviation of the four numbers = 3
Mean deviation = \(\frac{|a-5| + |b-5| + |c-5| + |d-5|}{4} = 3\)
→ Sum of absolute deviations = 12
- Mean deviation of the first three numbers a, b, c = 2
→ \(\frac{|a-5| + |b-5| + |c-5|}{3} = 2\)
→ Sum of absolute deviations for first three = 6
- Therefore, absolute deviation for the fourth number:
|d - 5| = 12 - 6 = 6
The fourth number = 6 + 5 = 11.
Source: Myschool Past Questions Archive (Record #77197)
Mathematics
UTME / JAMB 1999
Permutations, Combinations & Probability
Solution
A number is selected at random from 0 to 20. What is the probability that the number is odd prime?
A
\(\frac{8}{21}\)
B
\(\frac{1}{3}\)
C
\(\frac{2}{7}\)
D
\(\frac{5}{21}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
The numbers from 0 to 20 inclusive are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.
Total numbers = 21
Odd prime numbers in this range:
Primes must be greater than 1 and have no divisors other than 1 and themselves.
Odd primes (exclude 2, which is even):
3, 5, 7, 11, 13, 17, 19
→ There are 7 odd prime numbers.
3. Probability = \(\frac{\text{Number of favourable outcomes}}{\text{Total outcomes}}\)
= \(\dfrac{7}{21} = \dfrac{1}{3}\)
Source: Myschool Past Questions Archive (Record #77198)
Mathematics
WAEC / WASSCE 1998
Linear, Simultaneous & Quadratic Equations
Solution
Find the root of the equation 2x\(^2\) - 3x - 2 = 0
A
x = -2 or 1/2
B
x = -2 or1
C
x = -2 or 2
D
x = -1 or 2
E
x = -1/2 or 2
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
2x\(^2\) - 3x - 2 = 0
2x\(^2\) - 4x + x - 2 = 0
2x(x - 2) + 1(x - 2) = 0
(2x + 1)(x - 2) = 0
x = \(\frac{-1}{2}\) or x = 2
Source: Myschool Past Questions Archive (Record #22093)
Mathematics
WAEC / WASSCE 1998
Linear, Simultaneous & Quadratic Equations
Solution
What value of k makes the given expression a perfect square ? m\(^2\) - 8m + k = 0
A
2
B
4
C
8
D
16
E
64
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Take half the coefficient of m and square it
=> (-8/2)2 = k = 16
Source: Myschool Past Questions Archive (Record #22094)
Mathematics
WAEC / WASSCE 1998
Indices, Logarithms & Surds
Solution
If log\(_{10}\) q = 2.7078, what is q?
A
5102
B
849.9
C
510.2
D
84.99
E
51.02
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
q = 10\(^{2.7078}\)
Antilog of 2.7078 = 510.2
Source: Myschool Past Questions Archive (Record #22095)
Mathematics
WAEC / WASSCE 1998
Trigonometry, Elevation & Bearings
Solution
Cos x is negative and sin x is negative.Which of the following is true of x?
A
0o o
B
90o o
C
180o o
D
270o o
E
-90o o
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
In the 3rd quadrant, only the tan of angles are positive.
The 3rd quadrant = 180^\circ < x < 270^\circ
Source: Myschool Past Questions Archive (Record #22096)
Mathematics
WAEC / WASSCE 1998
Fractions, Decimals, Percentages & Approximations
Solution
Simplify 0.63954 \div 0.003 giving your answer correct to two significant figures
A
213.18
B
213.00
C
213
D
210
E
21
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
= \(\frac{0.63954}{0.003}\)
moving the decimal places, we have
= \(\frac{639.54}{3}\)
= 213.18
\(\approxeq\) 210 (to 2 s.f.)
Source: Myschool Past Questions Archive (Record #22097)
Mathematics
WAEC / WASSCE 1998
Indices, Logarithms & Surds
Solution
If log\(_{10}\) a = 4; what is a?
A
0.4
B
40
C
400
D
1000
E
10000
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
log\(_{10}\) a = 4
a = 10\(^4\)
= 10000
Source: Myschool Past Questions Archive (Record #22098)
Mathematics
WAEC / WASSCE 1998
Fractions, Decimals, Percentages & Approximations
Solution
A student measured the length of a room and obtained the measurement of 3.99m. If the percentage error of is measurement was 5% and his own measurement was smaller than the length , what is the length of the room?
A
3.78m
B
3.80m
C
4.18m
D
4.20m
E
4.788m
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Let the actual length of the room = y m
\(\therefore \frac{y - 3.99}{y} \times 100 = 5%\)
\(100(y - 3.99) = 5y \implies 100y - 399 = 5y\)
\(100y - 5y = 399 \implies y = \frac{399}{95}\)
y = 4.2 m
Source: Myschool Past Questions Archive (Record #22099)
Mathematics
WAEC / WASSCE 1998
Trigonometry, Elevation & Bearings
Solution
When an aeroplane is 800m above the ground, its angle of elevation from a point P on the ground is 30o. How far is the plane from P by line of sight?
A
400m
B
800m
C
1500m
D
1600m
E
1700m
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
From the diagram, \(\sin 30 = \frac{800}{x}\)
\(x = \frac{800}{\sin 30} \)
= \(\frac{800}{0.5} \)
= 1600 m
Source: Myschool Past Questions Archive (Record #22100)
Mathematics
UTME / JAMB 1998
Plane Geometry, Polygons & Circle Theorems
Diagram
Solution
In diagram, PQ || ST and < PQR = 120º, < RST = 130º, find the angle marked x
Attached Figure / Diagram
A
50
B
65
C
70
D
80
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Produce line PQ to meet line RS at U
< RQU = 180º - 120º = 60º (
In angle QRU: < QPT + RQU = < RUV ( 2 interior angles = one exterior angle)
x + 60º = 130º and x = 70º
Source: WAEC / JAMB / NECO Classroom Archive (Record #33077)
Mathematics
WAEC / WASSCE 1998
Fractions, Decimals, Percentages & Approximations
Solution
Given that \(\frac{6x-y}{x+2y}=2\), find the value of \(\frac{x}{y}\)
A
\(\frac{3}{8}\)
B
\(\frac{5}{8}\)
C
\(\frac{4}{5}\)
D
\(\frac{5}{4}\)
E
\(\frac{8}{5}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{6x-y}{x+2y}=2\)
→ \((6x-y) = (x+2y)2\)
= 6x - y = 2x + 4y
Collect like terms: 6x - 2x = 4y + y
→ 4x = 5y
\(\frac{x}{y} = \frac{5}{4}\)
Source: WAEC / JAMB / NECO Classroom Archive (Record #26338)