Mathematics 2013 Jamb Examination

To differentiate the function \(y = (2x + 2)^3\), we use the Chain Rule, which states that for a composite function \(y = f(u)\) where \(u = g(x)\), the derivative is \(\displaystyle \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}\).
Let \(u = 2x + 2\). Then the original function becomes \(y = u^3\).
Step 1: Differentiate \(y\) with respect to \(u\): \[\frac{dy}{du} = 3u^2\]Step 2: Differentiate \(u\) with respect to \(x\): \[\frac{du}{dx} = 2\]Step 3: Multiply the two results together: \[\frac{dy}{dx} = 3u^2 \times 2 = 6u^2\]Substituting the original expression for \(u\) back in: \[\frac{dy}{dx} = 6(2x + 2)^2\]This result shows the rate of change of the cubic function relative to \(x\).

This is a related rates problem. We know the area of a circle is given by \(A = \pi r^2\). To find how the area changes over time, we differentiate both sides with respect to time (\(t\)) using the chain rule: \[\frac{dA}{dt} = 2\pi r \frac{dr}{dt}\]We are given the following information:
Radius (\(r\)) = 7 cm
Rate of change of radius (\(\frac{dr}{dt}\)) = 0.02 cm/s
Substituting these values into our derivative formula: \[\frac{dA}{dt} = 2 \times \frac{22}{7} \times 7 \times 0.02\]The 7s cancel out: \[\frac{dA}{dt} = 2 \times 22 \times 0.02\]\[\frac{dA}{dt} = 44 \times 0.02 = 0.88\]Thus, the area is increasing at a rate of 0.88 cm²/s when the radius reaches 7 cm.

The arithmetic mean of a set of numbers is calculated by summing all the values and dividing by the count of the values. We have four algebraic expressions: \[(t + 2), (2t - 4), (3t + 2), 2t\]Step 1: Calculate the total sum of the expressions: \[\text{Sum} = (t + 2t + 3t + 2t) + (2 - 4 + 2)\]\[\text{Sum} = 8t + 0 = 8t\]Step 2: Divide the total sum by the number of terms (4): \[\text{Mean} = \frac{8t}{4} = 2t\]The average of these four algebraic terms simplifies to 2t.

To find the mode, we first need to find the missing seventh number in the data set. We are given the mean of seven numbers is 10.
Step 1: Calculate the total sum of the seven numbers: \[\text{Total Sum} = 7 \times 10 = 70\]Step 2: Find the sum of the six known numbers: \[2 + 4 + 8 + 14 + 16 + 18 = 62\]Step 3: Subtract the sum of the six numbers from the total sum to find the seventh number (x): \[x = 70 - 62 = 8\]Now we list all seven numbers: 2, 4, 8, 14, 16, 18, and 8. The mode is the number that appears most frequently in a data set. In this list, the number 8 appears twice, while all other numbers appear once. Therefore, the mode is 8.

The median is the middle value of a dataset. For frequency distributions, we first calculate the total frequency (N) and find the position of the median.
Total frequency (N) = \(3 + 5 + 1 + 1 + 2 + 3 = 15\).
Since the total number of people is odd, the median is the value at the \(\frac{N+1}{2}\) position: \[\text{Position} = \frac{15 + 1}{2} = 8 \text{th position}\]Now we find which age contains the 8th person using cumulative frequency:
Age 20: Cumulative Frequency = 3 (covers positions 1-3)
Age 25: Cumulative Frequency = 3 + 5 = 8 (covers positions 4-8)
The 8th person falls exactly within the 'Age 25' category. Therefore, the median age is 25.

Standard deviation is defined as the positive square root of the variance. This question provides extra information regarding the mean and the specific values of the data set, but to find the standard deviation, we only need the value of the variance.
Given:\[\text{Variance } (\sigma^2) = 4\]The formula for standard deviation (\(\sigma\)) is:\[\sigma = \sqrt{\text{Variance}}\]\[\sigma = \sqrt{4} = 2\]The relationship between these two measures of dispersion is constant regardless of the value of \(x\) or the mean of the set. Thus, the standard deviation is 2.

The range of a frequency distribution is the difference between the highest score and the lowest score that actually occurred (i.e., scores with a frequency greater than zero).
From the table:
Highest score with frequency > 0 is 10.
Lowest score with frequency > 0 is 3.
The calculation for the range is:\[\text{Range} = \text{Maximum Score} - \text{Minimum Score}\]\[\text{Range} = 10 - 3 = 7\]This represents the spread of the marks obtained by the 20 students.

This problem involves choosing a subset of items where the order of selection does not matter, which is a combination. The number of ways to select \(r\) items from a set of \(n\) items is given by the formula:\[^nC_r = \frac{n!}{r!(n - r)!}\]In this case, the total number of subjects \(n = 5\) and the number to be selected \(r = 2\). Substituting these into the formula:\[^5C_2 = \frac{5!}{2!(5 - 2)!}\]\[^5C_2 = \frac{5!}{2!3!}\]This represents the mathematical expression for the number of distinct pairs of subjects possible.

This is a permutation problem because the order in which people occupy the seats matters (different people in different seats constitutes a different arrangement). The number of ways to arrange \(r\) objects from \(n\) available objects is:\[^nP_r = \frac{n!}{(n - r)!}\]Here, \(n = 5\) (people) and \(r = 3\) (seats):\[^5P_3 = \frac{5!}{(5 - 3)!} = \frac{5!}{2!}\]\[^5P_3 = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1}\]\[^5P_3 = 5 \times 4 \times 3 = 60\]Alternatively, for the first seat there are 5 choices, for the second seat 4 choices, and for the third seat 3 choices: \(5 \times 4 \times 3 = 60\).

For a number to be divisible by both 2 and 3, it must be a multiple of their least common multiple (LCM). The LCM of 2 and 3 is 6. Therefore, we are looking for multiples of 6 within the range 1 to 25.
The multiples of 6 in this range are:
6, 12, 18, 24.
The number of successful outcomes (n) is 4.
The total number of possible outcomes (N) is 25 (integers from 1 to 25).
The probability \(P\) is calculated as:\[P = \frac{\text{Number of successful outcomes}}{\text{Total number of outcomes}}\]\[P = \frac{4}{25}\]There is a 16% chance of selecting such an integer.