Mathematics 2015 Jamb Examination

The volume of a cylinder is calculated using the formula: \[ V = \pi r^2 h \] Given that the radius \(r = 5\) cm and the height \(h = 4\) cm, and using \(\pi = 3.14\), we substitute these values into the formula: \[ V = 3.14 \times 5^2 \times 4 \] \[ V = 3.14 \times 25 \times 4 \] \[ V = 3.14 \times 100 \] \[ V = 314 \; \text{cm}^3 \] Thus, the volume of the cylinder is 314 cubic centimetres.

To find the probability of event A or event B occurring, we use the addition law of probability: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given from the question: \(P(A) = \dfrac{1}{5}\) \(P(B) = \dfrac{1}{3}\) \(P(A \cap B) = \dfrac{1}{15}\) Substituting these values into the formula: \[ P(A \cup B) = \dfrac{1}{5} + \dfrac{1}{3} - \dfrac{1}{15} \] To add the fractions, find a common denominator, which is 15: \[ P(A \cup B) = \dfrac{3}{15} + \dfrac{5}{15} - \dfrac{1}{15} \] \[ P(A \cup B) = \dfrac{3 + 5 - 1}{15} = \dfrac{7}{15} \] The probability of either event A or B or both is 7/15.

To convert the decimal number 159.75 to base 6, we separate the integer part and the fractional part.
For the integer part (159): \[ 159 \div 6 = 26 \text{ remainder } 3 \] \[ 26 \div 6 = 4 \text{ remainder } 2 \] \[ 4 \div 6 = 0 \text{ remainder } 4 \] Reading the remainders from the last to the first, the integer part is 423.
For the fractional part (0.75): \[ 0.75 \times 6 = 4.5 \rightarrow 4 \] \[ 0.5 \times 6 = 3.0 \rightarrow 3 \] Reading the resulting integers from top to bottom, the fractional part is .43.
Combining both parts, we get 423.43 in base 6.

To find the loss percentage, we first determine the amount of loss incurred by the man: \[ \text{Loss} = \text{Cost Price} - \text{Selling Price} \] \[ \text{Loss} = \text{₦}800 - \text{₦}520 = \text{₦}280 \] Now, we calculate the loss percentage relative to the cost price: \[ \text{Loss percentage} = \left( \dfrac{\text{Loss}}{\text{Cost Price}} \right) \times 100 \] \[ \text{Loss percentage} = \left( \dfrac{280}{800} \right) \times 100 \] \[ \text{Loss percentage} = \dfrac{280}{8} = 35\% \] The man suffered a 35% loss on the sale of the car.

The expression involves mixed fractions. Let's separate the whole numbers and fractions to simplify: \[ (6 - 2 + 1) + \left( \dfrac{1}{12} - \dfrac{3}{4} + \dfrac{1}{2} \right) \] Calculate the whole numbers: \(6 - 2 + 1 = 5\). Now calculate the fractional parts using a common denominator of 12: \[ \dfrac{1}{12} - \dfrac{9}{12} + \dfrac{6}{12} = \dfrac{1 - 9 + 6}{12} = -\dfrac{2}{12} = -\dfrac{1}{6} \] Combine the results: \[ 5 - \dfrac{1}{6} = 4 \dfrac{5}{6} \] 

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates \((-2, -3)\) and \((-2, 4)\): \[ d = \sqrt{(-2 - (-2))^2 + (4 - (-3))^2} \] \[ d = \sqrt{(0)^2 + (7)^2} \] \[ d = \sqrt{49} = 7.0 \] Since the x-coordinates are the same, the distance is simply the absolute difference in the y-coordinates: \(|4 - (-3)| = 7\).

In a Geometric Progression (GP), terms are found using the common ratio \(r\). Given: First term \(a = 2\) Second term \(ar = 54\) First, find the common ratio \(r\): \[ r = \dfrac{54}{2} = 27 \] Now, calculate the third term \(T_3\): \[ T_3 = ar^2 \] \[ T_3 = 2 \times 27^2 = 2 \times 729 = 1458 \] Alternatively, multiply the second term by the common ratio: \(54 \times 27 = 1458\).

The union of two sets \(S \cup T\) consists of all elements found in either set S or set T. Set S covers values from 0 up to and including 5: \([0, 5]\). Set T covers values from greater than -2 up to 3: \((-2, 3)\). Combining these intervals, we start from just above -2 (the lowest point in T) and go all the way up to 5 (the highest point in S). Therefore, the union is: \[ -2 < x \le 5 \]

The return is proportional to the investment. We can find the rate of return first: \[ \text{Rate} = \dfrac{25,000}{450,000} = \dfrac{1}{18} \] Now, calculate the return for an investment of ₦700,000: \[ \text{Return} = 700,000 \times \dfrac{1}{18} \] \[ \text{Return} = \dfrac{700,000}{18} \approx 38,888.88... \] Rounding to the nearest two decimal places, we get ₦38,888.90.

To evaluate a logarithm with base 7, we apply the change of base formula: \[ \log_7 17 = \dfrac{\log_{10} 17}{\log_{10} 7} \] Using approximate values for base-10 logarithms: \(\log_{10} 17 \approx 1.2304\) \(\log_{10} 7 \approx 0.8451\) \[ \log_7 17 = \dfrac{1.2304}{0.8451} \approx 1.4559 \] Rounding to three decimal places, we get 1.456. The closest option is 1.455.