To find the power used by the fan, we need to use the formula that relates power, voltage, and current. This is a fundamental concept in electrical physics.
Electrical power (\(P\)) is given by the product of voltage (\(V\)) and current (\(I\)).
\[ P = V \times I \]The unit of power is Watt (W).
Correct Option: A) 110 Watt
The power used by the fan is 110 Watts, calculated by multiplying the voltage (220 V) by the current (0.50 A).
Find the arithmetic mean of the following data (correct to two places of decimals).
| Class-interval | 5-15 | 15-25 | 25-35 | 35-45 | 45-55 | 55-65 |
| Frequency | 12 | 18 | 8 | 14 | 16 | 8 |
To find the arithmetic mean of grouped data, we use the formula for the mean, which involves calculating the mid-point of each class interval, multiplying it by its corresponding frequency, summing these products, and then dividing by the total frequency.
A) 33.68 is the correct arithmetic mean when calculated and rounded to two decimal places.
The problem presents an analogy between letter-clusters. We need to find the pattern and relationship between the first pair of letter-clusters and apply the same logic to the second pair to find the missing clusters. This involves analyzing the alphabetical positions of the letters and the changes between them.
The pattern observed is that each letter in the first cluster is shifted forward by 3 positions in the alphabet to get the corresponding letter in the second cluster. Applying this to OUQ, we get R (+3), X (+3), T (+3), resulting in RXT. Applying the same pattern to FLH, we get I (+3), O (+3), K (+3), resulting in IOK. Thus, # = OUQ and % = IOK.
The question asks to identify an ephemeral river from the given options. An ephemeral river is one that flows only during certain times of the year, typically after rainfall or snowmelt, and remains dry for the rest of the time. We need to evaluate each option based on this definition.
Correct Option: D) The Luni River is an ephemeral river, flowing only during the monsoon season and often drying up in its lower reaches due to the arid conditions of the Thar Desert.
A natural monopoly arises when a single firm can supply a good or service to an entire market at a lower cost than two or more firms could. This often occurs in industries with high fixed costs and significant economies of scale, meaning the average cost of production decreases as output increases. We need to identify which of the given options best fits this definition.
Correct Option: B) Railways. The railway industry is a classic example of a natural monopoly due to its extremely high fixed costs for infrastructure (tracks, stations, etc.) and significant economies of scale, making it most efficient for a single provider to operate the service.
To find the original number, we need to understand what "increased by 50%" means. If a number is increased by 50%, it becomes 150% of its original value. We can set up an equation where 150% of the unknown number is equal to 2760 and then solve for the number.
Let the unknown number be \(x\).
When the number is increased by 50%, it means we add 50% of the number to the original number. Mathematically, this can be written as:
\[ x + 50\% \text{ of } x \]Or, in terms of decimals or fractions:
\[ x + \frac{50}{100}x = x + 0.5x = 1.5x \]Alternatively, an increase of 50% means the new number is \(100\% + 50\% = 150\%\) of the original number.
\[ 150\% \text{ of } x = \frac{150}{100}x = 1.5x \]According to the problem, this increased number is 2760. So, we can set up the equation:
\[ 1.5x = 2760 \]To find \(x\), divide both sides of the equation by 1.5:
\[ x = \frac{2760}{1.5} \]Perform the division:
\[ x = \frac{2760}{\frac{3}{2}} = 2760 \times \frac{2}{3} \] \[ x = \frac{5520}{3} \] \[ x = 1840 \]A) 1840 is the correct number. When 1840 is increased by 50%, it becomes \(1840 + (0.50 \times 1840) = 1840 + 920 = 2760\).
B) 3680 — If the number were 3680, an increase of 50% would be \(3680 + (0.50 \times 3680) = 3680 + 1840 = 5520\), which is not 2760.
C) 5520 — If the number were 5520, an increase of 50% would be \(5520 + (0.50 \times 5520) = 5520 + 2760 = 8280\), which is not 2760. This option might arise if one incorrectly multiplies 2760 by 2.
D) 920 — If the number were 920, an increase of 50% would be \(920 + (0.50 \times 920) = 920 + 460 = 1380\), which is not 2760. This option might arise if one incorrectly calculates 50% of 1840 (which is 920) as the original number.
To expand the given expression \((5a + 6b + 8c)^2\), we will use the algebraic identity for the square of a trinomial: \((x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx\).
Identify \(x\), \(y\), and \(z\):
In our expression \((5a + 6b + 8c)^2\), we can identify:
Apply the identity:
Substitute these values into the identity \((x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx\):
\[ (5a + 6b + 8c)^2 = (5a)^2 + (6b)^2 + (8c)^2 + 2(5a)(6b) + 2(6b)(8c) + 2(8c)(5a) \]Calculate the squared terms:
Calculate the cross-product terms:
Combine all terms:
Add all the calculated terms together:
\[ (5a + 6b + 8c)^2 = 25a^2 + 36b^2 + 64c^2 + 60ab + 96bc + 80ac \]Correct Option: C) \(25a^2 + 36b^2 + 64c^2 + 60ab + 96bc + 80ac\)
This problem involves calculating the combined work rate of multiple taps (some filling, some emptying) to determine the total time required to fill a cistern. We will find the individual work rates of each tap and then combine them to find the net filling rate of the cistern.
Calculate the filling rate of the first tap:
The first tap fills the cistern in 2 hours. So, in 1 hour, it fills \(\frac{1}{2}\) of the cistern.
Calculate the filling rate of the second tap:
The second tap fills the cistern in 45 hours. So, in 1 hour, it fills \(\frac{1}{45}\) of the cistern.
Calculate the emptying rate of the third tap:
The third tap empties the cistern in 45 hours. So, in 1 hour, it empties \(\frac{1}{45}\) of the cistern. Since it empties, its contribution to filling is negative.
Calculate the net filling rate when all three taps are opened together:
The net part of the cistern filled in 1 hour is the sum of the filling rates minus the emptying rate.
Notice that the filling rate of the second tap and the emptying rate of the third tap cancel each other out.
\[ \text{Net filling rate} = \frac{1}{2} \]\]So, in 1 hour, \(\frac{1}{2}\) of the cistern is filled.
Calculate the total time to fill the cistern:
If \(\frac{1}{2}\) of the cistern is filled in 1 hour, then the total time to fill the entire cistern is the reciprocal of the net filling rate.
Correct Option: A) 2
The reactivity of metals is determined by their ability to displace other metals from their salt solutions. A more reactive metal can displace a less reactive metal from its solution. We will use the given displacement reactions to establish the order of reactivity among metals A, B, C, and D.
Now, let's combine these inequalities to establish the overall order of reactivity:
From this order, it is clear that C is the most reactive metal.
A) C is the most reactive metal because it can displace A, B, and D from their solutions, as established by the combined reactivity series \( \text{C} > \text{A} > \text{B} > \text{D} \).
The question asks to identify the specific part of the pistil where pollen grains are deposited during pollination. This requires knowledge of the structure of a flower's reproductive organs and the process of pollination.
B) Stigma — The stigma is the receptive surface of the pistil, specifically adapted to receive and hold pollen grains during pollination.