This problem involves calculating the work done by pipes and a leak. We will first determine the combined filling rate of pipes A and B. Then, we will calculate the time it would take for them to fill the cistern without a leak. Given that the cistern took 30 minutes more to fill due to a leak, we can find the actual time taken. Using this information, we can determine the effective filling rate with the leak and subsequently find the rate of the leak. Finally, we will calculate the time the leak alone would take to empty the full cistern.
Calculate the filling rate of Pipe A:
Pipe A fills the cistern in 6 hours. So, its filling rate is:
\[ \text{Rate of A} = \frac{1}{6} \text{ cistern/hour} \]Calculate the filling rate of Pipe B:
Pipe B fills the cistern in 30 hours. So, its filling rate is:
\[ \text{Rate of B} = \frac{1}{30} \text{ cistern/hour} \]Calculate the combined filling rate of A and B (without leak):
When both pipes are open, their combined filling rate is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{6} + \frac{1}{30} \] \[ \text{Combined Rate} = \frac{5}{30} + \frac{1}{30} = \frac{6}{30} = \frac{1}{5} \text{ cistern/hour} \]Calculate the time taken to fill the cistern by A and B (without leak):
If the combined rate is \( \frac{1}{5} \) cistern/hour, then the time taken to fill the cistern is the reciprocal of the rate:
\[ \text{Time (without leak)} = 5 \text{ hours} \]Calculate the actual time taken to fill the cistern (with leak):
The cistern took 30 minutes more to fill due to the leak. 30 minutes is 0.5 hours.
\[ \text{Actual Time} = 5 \text{ hours} + 0.5 \text{ hours} = 5.5 \text{ hours} \]Calculate the effective filling rate with the leak:
If the actual time taken is 5.5 hours, then the effective filling rate (A + B - Leak) is:
\[ \text{Effective Rate} = \frac{1}{5.5} = \frac{1}{11/2} = \frac{2}{11} \text{ cistern/hour} \]Calculate the rate of the leak:
The effective rate is the combined rate of A and B minus the leak rate:
\[ \text{Effective Rate} = (\text{Rate of A} + \text{Rate of B}) - \text{Rate of Leak} \] \[ \frac{2}{11} = \frac{1}{5} - \text{Rate of Leak} \] \[ \text{Rate of Leak} = \frac{1}{5} - \frac{2}{11} \] \[ \text{Rate of Leak} = \frac{11}{55} - \frac{10}{55} = \frac{1}{55} \text{ cistern/hour} \]Calculate the time taken for the leak to empty the full cistern:
If the leak rate is \( \frac{1}{55} \) cistern/hour, then the time taken to empty the full cistern is the reciprocal of the leak rate:
\[ \text{Time to empty} = 55 \text{ hours} \]D) 55 hours
The question asks to identify the predominant climate type of the Indian subcontinent. This requires knowledge of the major climate classifications and their geographical distribution, specifically focusing on the characteristics of the Indian climate.
Correct Option: D) Tropical Monsoon — The Indian subcontinent is predominantly characterized by a Tropical Monsoon climate. This is due to its geographical location, the influence of the Himalayas, and the seasonal reversal of winds (monsoons) that bring heavy rainfall during the summer months and lead to a dry winter.
To find the median of a set of observations, we first need to arrange the observations in ascending order. Then, we identify the middle value(s). If the number of observations is odd, the median is the single middle value. If the number of observations is even, the median is the average of the two middle values.
D) 8 — The median of the given observations is 8, as calculated by arranging the numbers in ascending order and taking the average of the two middle values.
To find the missing number in the series, we need to identify the pattern or relationship between consecutive terms. We will calculate the differences between adjacent numbers and look for a sequence in these differences.
Write down the given series: 21, 22, 26, 35, 51, 76, ?
Calculate the difference between consecutive terms:
Difference between 22 and 21: \(22 - 21 = 1\)
Difference between 26 and 22: \(26 - 22 = 4\)
Difference between 35 and 26: \(35 - 26 = 9\)
Difference between 51 and 35: \(51 - 35 = 16\)
Difference between 76 and 51: \(76 - 51 = 25\)
Observe the sequence of differences: 1, 4, 9, 16, 25. This sequence consists of perfect squares:
\(1 = 1^2\)
\(4 = 2^2\)
\(9 = 3^2\)
\(16 = 4^2\)
\(25 = 5^2\)
Following this pattern, the next difference should be the next perfect square, which is \(6^2\).
\(6^2 = 36\)
Add this next difference to the last term in the given series to find the missing number:
Missing number \( = 76 + 36 = 112\)
Correct Option: C) 112
To determine the direction of electrons, we need to use Fleming's Left-Hand Rule. This rule relates the direction of magnetic field, current, and force. Since the question asks for the direction of electrons, we must remember that the conventional current direction is opposite to the direction of electron flow.
Correct Option: A) west to east
A sum of Rs 4,800 is divided between A, B and C such that the ratio of the share of A to the combined share of B and C is 3 : 5 and C receives \( \frac{5}{7} \) of what A and B together receive. The difference (in Rs) of A’s share and B’s share is:
The problem involves distributing a total sum among three individuals (A, B, C) based on given ratios. We will use the given ratios to establish relationships between the shares of A, B, and C, and then solve for their individual shares. Finally, we will calculate the difference between A's and B's shares.
Define variables and total sum:
Let the shares of A, B, and C be \(A\), \(B\), and \(C\) respectively.
Total sum \(S = 4,800\).
Use the first ratio:
The ratio of A's share to the combined share of B and C is \(3:5\).
\[ \frac{A}{B+C} = \frac{3}{5} \]
This implies \(5A = 3(B+C)\).
We know that \(A + B + C = S = 4,800\). So, \(B+C = 4,800 - A\).
Substitute this into the ratio equation:
\[ 5A = 3(4,800 - A) \]
\[ 5A = 14,400 - 3A \]
\[ 8A = 14,400 \]
\[ A = \frac{14,400}{8} \]
\[ A = 1,800 \]
So, A's share is Rs 1,800.
Use the second ratio:
C receives \( \frac{5}{7} \) of what A and B together receive.
\[ C = \frac{5}{7}(A+B) \]
This implies \(7C = 5(A+B)\).
We know that \(A+B+C = 4,800\). So, \(A+B = 4,800 - C\).
Substitute this into the equation:
\[ 7C = 5(4,800 - C) \]
\[ 7C = 24,000 - 5C \]
\[ 12C = 24,000 \]
\[ C = \frac{24,000}{12} \]
\[ C = 2,000 \]
So, C's share is Rs 2,000.
Calculate B's share:
We know \(A + B + C = 4,800\).
Substitute the values of A and C:
\[ 1,800 + B + 2,000 = 4,800 \]
\[ 3,800 + B = 4,800 \]
\[ B = 4,800 - 3,800 \]
\[ B = 1,000 \]
So, B's share is Rs 1,000.
Calculate the difference between A's and B's shares:
Difference \( = A - B \)
\[ = 1,800 - 1,000 \]
\[ = 800 \]
The difference between A's share and B's share is Rs 800.
B) 800
This question tests the understanding of Snell's Law and the behavior of light when it passes from one medium to another, specifically from air to glass and then from glass back to air. We need to analyze the relationship between the angle of incidence and the angle of refraction at each surface based on the refractive indices of the media.
Here, light travels from air (rarer medium) to glass (denser medium).
According to Snell's Law:
\[n_1 \sin i_1 = n_2 \sin r_1\]\[1 \cdot \sin i_1 = n_2 \sin r_1\]Since \(n_2 > 1\), for the equality to hold, \(\sin r_1\) must be less than \(\sin i_1\). This implies that \(r_1 < i_1\). Therefore, the angle of refraction at surface AB is less than the angle of incidence.
Here, light travels from glass (denser medium) to air (rarer medium).
According to Snell's Law:
\[n_1 \sin i_2 = n_2 \sin r_2\]\[n_1 \sin i_2 = 1 \cdot \sin r_2\]Since \(n_1 > 1\), for the equality to hold, \(\sin r_2\) must be greater than \(\sin i_2\). This implies that \(r_2 > i_2\). Therefore, the angle of refraction at surface AC is more than the angle of incidence.
C) Neither (a) nor (b)
This problem involves calculating averages and ratios. We will use the formula for average marks and set up equations based on the given information to find the average marks of the girls.
Let \(A_C\) be the average marks of the class, \(A_B\) be the average marks of boys, and \(A_G\) be the average marks of girls.
Given: \(A_C = 70\).
Let \(N_B\) be the number of boys and \(N_G\) be the number of girls.
Given the ratio of boys to girls is \(3:2\), we can write \(N_B = 3x\) and \(N_G = 2x\) for some common factor \(x\).
The average marks obtained by boys is 20% more than the average marks obtained by girls.
So, \(A_B = A_G + 0.20 A_G = 1.2 A_G\).
The total marks of the class can be expressed in two ways:
Total marks \( = A_C \times (N_B + N_G) \)
Total marks \( = (A_B \times N_B) + (A_G \times N_G) \)
Equating these two expressions:
\[ A_C \times (N_B + N_G) = (A_B \times N_B) + (A_G \times N_G) \]Substitute the known values and expressions:
\[ 70 \times (3x + 2x) = (1.2 A_G \times 3x) + (A_G \times 2x) \] \[ 70 \times (5x) = 3.6 A_G x + 2 A_G x \] \[ 350x = 5.6 A_G x \]Since \(x\) cannot be zero (as there are students), we can divide both sides by \(x\):
\[ 350 = 5.6 A_G \]Now, solve for \(A_G\):
\[ A_G = \frac{350}{5.6} \] \[ A_G = \frac{3500}{56} \]Divide by 7:
\[ A_G = \frac{500}{8} \]Divide by 4:
\[ A_G = \frac{125}{2} \] \[ A_G = 62.5 \]Correct Option: A) 62.5
If the cost price of an item is Rs 4,500 and its selling price is Rs 3,500, then the loss percentage is:
44\( \frac{2}{9} \)%
55\( \frac{2}{9} \)%
33\( \frac{2}{9} \)%
22\( \frac{2}{9} \)%
To calculate the loss percentage, we first need to find the actual loss by subtracting the selling price from the cost price. Then, we divide the loss by the cost price and multiply by 100 to express it as a percentage. The result needs to be converted into a mixed fraction.
D) The calculated loss percentage is \( 22\frac{2}{9} \% \), which matches option D.
E, F, G, H, I and J live on six different floors of the same building. The lowermost floor in the building is numbered 1, the floor above it is numbered 2, and so on till the topmost floor is numbered 6.
G lives on the lowermost floor. Only J lives between the floors occupied by H and I. E lives on floor number 2. F lives on an even-numbered floor, with I as an immediate neighbour.
Who lives on floor number 4?
This is a logical reasoning problem involving arrangement on floors. We need to deduce the position of each person based on the given clues and then identify who lives on floor number 4.
Correct Option: D) J