Mathematics 2015 Jamb Examination

To calculate the percentage birth rate, we use the standard formula for percentage: finding the portion relative to the total population and multiplying by 100.
Total inhabitants = 6250
Number of births = 62\[ \text{Percentage Birth Rate} = \left( \dfrac{\text{Births}}{\text{Total Population}} \right) \times 100 \]\[ \text{Percentage Birth Rate} = \left( \dfrac{62}{6250} \right) \times 100 \]\[ \text{Percentage Birth Rate} = 0.00992 \times 100 \]\[ \text{Percentage Birth Rate} = 0.992\% \]Rounding the result to one decimal place as seen in the options gives 1.0%.

To simplify this expression, we apply the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction).
Step 1: Handle the 'of' operation inside the bracket:\[ \dfrac{1}{4} \text{ of } 28 = \dfrac{1}{4} \times 28 = 7 \]Step 2: Solve the division inside the bracket:\[ (2 \div 7) = \dfrac{2}{7} \]Step 3: Convert the mixed fraction 1¼ to an improper fraction:\[ 1\dfrac{1}{4} = \dfrac{(4 \times 1) + 1}{4} = \dfrac{5}{4} \]Step 4: Perform the final division:\[ \dfrac{5}{4} \div \dfrac{2}{7} = \dfrac{5}{4} \times \dfrac{7}{2} \]\[ = \dfrac{35}{8} \]Step 5: Convert the improper fraction back to a mixed fraction:\[ \dfrac{35}{8} = 4\dfrac{3}{8} \]

To factorise the quadratic expression \(x^2 + 9x + 20\), we look for two numbers that multiply to give the constant term (20) and add up to give the coefficient of the middle term (9).
The factors of 20 are (1, 20), (2, 10), and (4, 5).
Checking the sums:
1 + 20 = 21
2 + 10 = 12
4 + 5 = 9 (This is the required middle coefficient)
We can rewrite the expression and factorise by grouping: \[ x^2 + 4x + 5x + 20 \]\[ x(x + 4) + 5(x + 4) \]\[ (x + 5)(x + 4) \]

The ratio of their ages is 18 : 20 : 22. We can simplify this ratio by dividing through by 2 to get 9 : 10 : 11.
Sum of the ratio parts = 9 + 10 + 11 = 30.
The total amount to be shared is ₦120,000.
The second staff's share corresponds to the middle part of the ratio (10):\[ \text{Second Staff Share} = \left( \dfrac{10}{30} \right) \times 120,000 \]\[ \text{Second Staff Share} = \dfrac{1}{3} \times 120,000 \]\[ \text{Second Staff Share} = 40,000 \]The second staff receives ₦40,000.

We use the inclusion-exclusion principle formula for the cardinality of the union of two sets:\[ n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \]Given:
\(n(X) = 15\)
\(n(Y) = 12\)
\(n(X \cap Y) = 7\)
Substituting the values into the formula:\[ n(X \cup Y) = 15 + 12 - 7 \]\[ n(X \cup Y) = 27 - 7 \]\[ n(X \cup Y) = 20 \]

To find the intercepts of the boundary line \(3x - z = 9\):
Step 1: To find the x-intercept, set \(z = 0\):\[ 3x - 0 = 9 \]\[ 3x = 9 \]\[ x = 3 \]So the x-intercept is (3, 0).
Step 2: To find the z-intercept, set \(x = 0\):\[ 3(0) - z = 9 \]\[ -z = 9 \]\[ z = -9 \]So the z-intercept is (0, -9). Looking at the coordinates given in the options, the intercept values are 3 and -9 respectively.

We use the laws of logarithms: the power law \(n\log A = \log A^n\), the product law \(\log A + \log B = \log(AB)\), and the quotient law \(\log A - \log B = \log(A/B)\).
Given: \(\log_{10}1.5 + 3\log_{10}2 - \log_{10}0.3\)
First, apply the power law:\[ 3\log_{10}2 = \log_{10}2^3 = \log_{10}8 \]Now combine the terms:\[ \log_{10}1.5 + \log_{10}8 - \log_{10}0.3 \]\[ = \log_{10} (1.5 \times 8) - \log_{10}0.3 \]\[ = \log_{10} 12 - \log_{10}0.3 \]\[ = \log_{10} \left( \dfrac{12}{0.3} \right) \]\[ = \log_{10} 40 \]

The total surface area (TSA) of a cylinder is given by the formula:\[ \text{TSA} = 2\pi r(r + h) \]Where:
\(r = 5 \; \text{cm}\)
\(h = 7 \; \text{cm}\)
\(\pi = 3.14\)
Substituting the values:\[ \text{TSA} = 2 \times 3.14 \times 5 \times (5 + 7) \]\[ \text{TSA} = 31.4 \times 12 \]\[ \text{TSA} = 376.8 \; \text{cm}^2 \]Note: Option E in the original list was renamed and swapped to D as it was the correct value.

First, expand the given expression:\[ (\sin x + \sin y)^2 - 2\sin x \sin y \]\[ = \sin^2 x + 2\sin x \sin y + \sin^2 y - 2\sin x \sin y \]\[ = \sin^2 x + \sin^2 y \]Since \(x + y = 90^\circ\), the angles are complementary. This means \(\sin y = \cos x\).
Substituting \(\cos x\) for \(\sin y\):\[ \sin^2 x + \cos^2 x \]Using the fundamental trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\):\[ \sin^2 x + \cos^2 x = 1 \]

Taxable income is the portion of the salary subject to tax after subtracting allowances.\[ \text{Taxable Income} = \text{Total Salary} - \text{Allowances} \]\[ \text{Taxable Income} = 2000 - 600 = 1400 \]The income tax is 5% of the taxable income:\[ \text{Income Tax} = \dfrac{5}{100} \times 1400 \]\[ \text{Income Tax} = 0.05 \times 1400 \]\[ \text{Income Tax} = 70 \]The man pays ₦70 in income tax per year.