Mathematics 2008 Jamb Examination
Mathematics 2008 Jamb Examination
Question 21
A student sitting on a tower 68 metres high observes his principal's car at the angle of depression of 20°. How far is the car from the bottom of the tower to the nearest metre?
Using the right-angled triangle formed, the horizontal distance is the adjacent side of the angle of depression.\[ \tan 20^{\circ} = \dfrac{68}{d} \]\[ d = \dfrac{68}{\tan 20^{\circ}} \approx 186.9 \; \text{m} \]The nearest whole number is 187 m.
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Question 22
Evaluate: \(\displaystyle \int_{1}^{2} (6x^{2} - 2x) \, dx \)
Integrate term by term:\[ \int (6x^{2})\,dx = 2x^{3}, \quad \int(-2x)\,dx = -x^{2} \]So the antiderivative is:\[ F(x) = 2x^{3} - x^{2} \]Evaluate between 1 and 2:\[ F(2) = 2(2^3) - 2^2 = 16 - 4 = 12 \]\[ F(1) = 2(1^3) - 1^2 = 1 \]\[ F(2) - F(1) = 12 - 1 = 11 \]
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Question 23
Evaluate: \(\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx \)
The integral of cosine is sine:\[ \int \cos x\,dx = \sin x \]Evaluate between limits:\[ \sin\left(\dfrac{\pi}{2}\right) - \sin\left(-\dfrac{\pi}{2}\right) = 1 - (-1) = 2 \]
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Question 24
On a pie chart, there are six sectors of which four angles are 30°, 45°, 60°, 90° and the remaining two angles are in the ratio 2 : 1. Find the smaller angle of the remaining two angles.
Sum of given angles:\[ 30 + 45 + 60 + 90 = 225^{\circ} \]Total angle of a circle:\[ 360^{\circ} - 225^{\circ} = 135^{\circ} \]The remaining two angles are in ratio 2 : 1, so:\[ 2k + k = 3k = 135^{\circ} \Rightarrow k = 45^{\circ} \]The smaller angle is 45°.
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Question 25
What is the mean of the data \( t, 2t - 1, t - 2, 2t - 1, 4t, 2t + 2 \)?
Sum the terms:\[ t + (2t - 1) + (t - 2) + (2t - 1) + 4t + (2t + 2) \]Combine like terms:\[ t + 2t + t + 2t + 4t + 2t = 12t \]Constants:\[ -1 - 2 - 1 + 2 = -2 \]Total sum:\[ 12t - 2 \]Mean:\[ \dfrac{12t - 2}{6} = 2t - \dfrac{1}{3} \]
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Question 26
If \( x > 0 \), find the range of the numbers:
x − 3, 3x + 2, x − 1, 4x, 2x − 1, x − 2, 2x − 2, 3x, 3x + 1
x − 3, 3x + 2, x − 1, 4x, 2x − 1, x − 2, 2x − 2, 3x, 3x + 1
Find the minimum value among all expressions:
For \( x > 0 \), the smallest is \( x - 3 \).
Largest value is \( 4x \).
Range:\[ 4x - (x - 3) = 3x + 3 \]
For \( x > 0 \), the smallest is \( x - 3 \).
Largest value is \( 4x \).
Range:\[ 4x - (x - 3) = 3x + 3 \]
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Question 27
In how many ways can the letters of the word ACCEPTANCE be arranged?
The word ACCEPTANCE has 10 letters.
Letter repetitions:
A occurs 3 times,
C occurs 2 times,
E occurs 2 times.
The number of distinguishable arrangements is:\[ \dfrac{10!}{3!\,2!\,2!} \]
Letter repetitions:
A occurs 3 times,
C occurs 2 times,
E occurs 2 times.
The number of distinguishable arrangements is:\[ \dfrac{10!}{3!\,2!\,2!} \]
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Question 28
The probability of picking the letter T from the word OBSTRUCTION.
OBSTRUCTION contains 11 letters.
Count of T's: 2.
Probability is defined as:\( \dfrac{Number of outcomes}{Number of possible outcomes} \)
Probability:\( = \dfrac{2}{11} \)
Count of T's: 2.
Probability is defined as:\( \dfrac{Number of outcomes}{Number of possible outcomes} \)
Probability:\( = \dfrac{2}{11} \)
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Question 29
The result of rolling a fair die 150 times is summarised as:
Number: 1, 2, 3, 4, 5, 6
Frequency: 12, 18, x, 30, 2x, 45
What is the probability of obtaining a 5?
Number: 1, 2, 3, 4, 5, 6
Frequency: 12, 18, x, 30, 2x, 45
What is the probability of obtaining a 5?
Total frequency:\[ 12 + 18 + x + 30 + 2x + 45 = 150 \]\[ 105 + 3x = 150 \Rightarrow x = 15 \]Frequency of 5 is \( 2x = 30 \).
Probability:\[ \dfrac{30}{150} = \dfrac{1}{5} \]
Probability:\[ \dfrac{30}{150} = \dfrac{1}{5} \]
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Question 30
In the diagram, ∠OPQ is:
The triangle inscribed is an isosceles triangle since two sides are equal.
Let one of the unknow sides be 'a'
such that the sum of angles in the given triangle will be:
2a + 74 = 180
2a = 180-74
2a = 106
a = 53
∴∠OPQ = 53o
Let one of the unknow sides be 'a'
such that the sum of angles in the given triangle will be:
2a + 74 = 180
2a = 180-74
2a = 106
a = 53
∴∠OPQ = 53o
