This problem involves calculating the time taken to complete a task when two individuals work alternately. We need to determine their individual work rates, combine their work over a two-day cycle, and then calculate how many such cycles are needed to complete the work, accounting for any remaining work.
Calculate individual work rates:
Brazen can paint the wall in 12 days. So, Brazen's one day's work is:
\[ \text{Brazen's work per day} = \frac{1}{12} \text{ of the wall} \]Ballary can paint the wall in 21 days. So, Ballary's one day's work is:
\[ \text{Ballary's work per day} = \frac{1}{21} \text{ of the wall} \]Calculate work done in one cycle (2 days):
They work alternately, starting with Brazen. So, in a 2-day cycle:
Work done in 2 days (1 cycle) = Brazen's work + Ballary's work
\[ \text{Work in 2 days} = \frac{1}{12} + \frac{1}{21} \]To add these fractions, find the Least Common Multiple (LCM) of 12 and 21. The LCM of 12 (3 x 4) and 21 (3 x 7) is \(3 \times 4 \times 7 = 84\).
\[ \text{Work in 2 days} = \frac{7}{84} + \frac{4}{84} = \frac{11}{84} \text{ of the wall} \]Determine the number of full cycles:
Let \(N\) be the number of 2-day cycles. We want to find how many cycles are needed to complete as much work as possible without exceeding the total work (1 whole wall).
Work done in \(N\) cycles = \(N \times \frac{11}{84}\)
We need to find the largest \(N\) such that \(N \times \frac{11}{84} < 1\).
If \(N = 7\), work done = \[7 \times \frac{11}{84} = \frac{77}{84}\].
If \(N = 8\), work done = \[8 \times \frac{11}{84} = \frac{88}{84}\], which is greater than 1, so 8 cycles are too many.
Thus, 7 full cycles (14 days) will complete \(\frac{77}{84}\) of the work.
Calculate remaining work and time for the last day:
After 7 cycles (14 days), the remaining work is:
\[ \text{Remaining work} = 1 - \frac{77}{84} = \frac{84 - 77}{84} = \frac{7}{84} \text{ of the wall} \]On the 15th day, it's Brazen's turn (since he starts the cycle). Brazen's work rate is \(\frac{1}{12}\) or \(\frac{7}{84}\) of the wall per day.
Since the remaining work is exactly \(\frac{7}{84}\), Brazen can complete this work in 1 day.
Total time:
Total days = Days in 7 cycles + Days for remaining work
Total days = \(14 \text{ days} + 1 \text{ day} = 15 \text{ days}\)
Correct Option: B) 15 days
The given series is an alternating series, where two different patterns are interleaved. We need to identify these two patterns to find the missing term.
D) 17 is the correct answer. When 17 is placed in the blank, the series splits into two alternating sub-series with clear patterns:
To determine if a number is divisible by 9, we use the divisibility rule for 9. This rule states that a number is divisible by 9 if the sum of its digits is divisible by 9. We will calculate the sum of the digits for each given option and check if the sum is a multiple of 9.
D) 31239 — The sum of its digits is \(3+1+2+3+9 = 18\), which is divisible by 9. Therefore, 31239 is divisible by 9.
This problem involves calculating compound interest. Since the interest is compounded every four months, we need to adjust the annual interest rate and the number of compounding periods per year accordingly. Then, we use the compound interest formula to find the final amount.
Since interest is compounded every four months, there are \(12 \text{ months} / 4 \text{ months} = 3\) compounding periods in a year.
The annual interest rate is 30%. For each compounding period, the rate will be:
\[ r = \frac{R}{n} = \frac{0.30}{3} = 0.10 \]For 1 year, with 3 compounding periods per year:
\[ N = T \times n = 1 \times 3 = 3 \]The formula for the amount (\(A\)) after compound interest is:
\[ A = P \left(1 + r\right)^N \]Correct Option: A) 3993
To determine the day of the week for a given date, we use the concept of "odd days". Odd days are the number of days remaining after dividing the total number of days by 7. We calculate the total odd days from a reference date (usually 1600 or 2000) up to the day before the target date, and then add the odd days for the target year's months and days. Each day of the week corresponds to a specific number of odd days (e.g., 0 for Sunday, 1 for Monday, etc.).
A) Sunday. Based on the calculation of total odd days being 0, the day was Sunday.
The question asks about the age limit for banning child labor as per the Child Labour Amendment Act, 2016. This requires knowledge of the specific provisions of this Act.
Correct Option: C) 14 — The Child Labour Amendment Act, 2016, completely bans the employment of children below 14 years of age in all occupations and processes.
This question tests knowledge of trigonometric identities, specifically complementary angle identities. We need to recall the relationship between sine and cosine for complementary angles.
Correct Option: C) cos \(\theta\)
How many factors of 28 × 32 × 53 × 75 are even numbers?
To find the number of even factors of a given number, we first express the number in its prime factorization form. A factor is even if and only if it contains at least one factor of 2. We can find the total number of factors and subtract the number of odd factors, or directly calculate the number of even factors by ensuring the power of 2 in the factor is at least 1.
The given number is \(N = 2^8 \times 3^2 \times 5^3 \times 7^5\).
A factor of \(N\) will be of the form \(2^a \times 3^b \times 5^c \times 7^d\), where \(0 \le a \le 8\), \(0 \le b \le 2\), \(0 \le c \le 3\), and \(0 \le d \le 5\).
For a factor to be an even number, the power of 2 in its prime factorization must be at least 1. That is, \(a\) must be greater than or equal to 1.
So, the possible values for \(a\) are \(1, 2, 3, 4, 5, 6, 7, 8\). This gives 8 choices for \(a\).
The possible values for \(b\) are \(0, 1, 2\). This gives \((2+1) = 3\) choices for \(b\).
The possible values for \(c\) are \(0, 1, 2, 3\). This gives \((3+1) = 4\) choices for \(c\).
The possible values for \(d\) are \(0, 1, 2, 3, 4, 5\). This gives \((5+1) = 6\) choices for \(d\).
The total number of even factors is the product of the number of choices for each exponent:
\[ \text{Number of even factors} = (\text{choices for } a) \times (\text{choices for } b) \times (\text{choices for } c) \times (\text{choices for } d) \]\[ \text{Number of even factors} = 8 \times 3 \times 4 \times 6 \]\[ \text{Number of even factors} = 24 \times 24 \]\[ \text{Number of even factors} = 576 \]A) 576 is the correct number of even factors, as calculated by ensuring the power of 2 in each factor is at least 1.
This problem requires us to understand the relationship between momentum and kinetic energy. We will use their respective formulas and calculate the percentage change in kinetic energy when momentum increases by a given percentage.
Let the initial momentum of the object be \(P_1\) and its initial kinetic energy be \(KE_1\). The mass of the object is \(m\).
The formula for momentum is:
\[ P = mv \]where \(v\) is the velocity.
The formula for kinetic energy is:
\[ KE = \frac{1}{2}mv^2 \]From the momentum formula, we can write \(v = \frac{P}{m}\). Substitute this into the kinetic energy formula:
\[ KE = \frac{1}{2}m \left(\frac{P}{m}\right)^2 \]\[ KE = \frac{1}{2}m \frac{P^2}{m^2} \]\[ KE = \frac{P^2}{2m} \]The momentum increases by 20%. So, the new momentum \(P_2\) will be:
\[ P_2 = P_1 + 0.20 P_1 \]\[ P_2 = 1.20 P_1 \]Using the relationship \(KE = \frac{P^2}{2m}\), the new kinetic energy \(KE_2\) will be:
\[ KE_2 = \frac{P_2^2}{2m} \]\[ KE_2 = \frac{(1.20 P_1)^2}{2m} \]\[ KE_2 = \frac{1.44 P_1^2}{2m} \]\[ KE_2 = 1.44 \left(\frac{P_1^2}{2m}\right) \]Since \(KE_1 = \frac{P_1^2}{2m}\), we have:
\[ KE_2 = 1.44 KE_1 \]The percentage increase is given by:
\[ \text{Percentage increase} = \frac{KE_2 - KE_1}{KE_1} \times 100\% \]\[ \text{Percentage increase} = \frac{1.44 KE_1 - KE_1}{KE_1} \times 100\% \]\[ \text{Percentage increase} = \frac{(1.44 - 1) KE_1}{KE_1} \times 100\% \]\[ \text{Percentage increase} = (0.44) \times 100\% \]\[ \text{Percentage increase} = 44\% \]C) 44% is the correct answer. When the momentum increases by 20%, the kinetic energy increases by 44%.
This is a direction-based reasoning question. We need to determine the new naming convention for directions based on the given condition and then apply it to find the new name for "West". The core idea is to identify the shift or rotation in the compass directions.
Correct Option: B) North-West — As established in the step-by-step analysis, if North-West is shifted to North (a 45-degree clockwise rotation), then West will shift to North-West.