This problem involves calculating the combined work rate of two taps, one filling and one emptying a tank. We will determine the individual rates of filling and emptying, and then find the net rate when both are open. The time taken to fill the tank will be the reciprocal of the net filling rate.
The first tap fills the tank in 8 hours. So, in one hour, it fills \(\frac{1}{8}\) of the tank.
\[ \text{Filling rate} = \frac{1}{8} \text{ tank/hour} \]The second tap empties the tank in 10 hours. So, in one hour, it empties \(\frac{1}{10}\) of the tank.
\[ \text{Emptying rate} = \frac{1}{10} \text{ tank/hour} \]Since one tap fills and the other empties, their effects are opposite. The net rate will be the filling rate minus the emptying rate.
\[ \text{Net rate} = \text{Filling rate} - \text{Emptying rate} \]\[ \text{Net rate} = \frac{1}{8} - \frac{1}{10} \]To subtract the fractions, find a common denominator, which is 40.
\[ \text{Net rate} = \frac{5}{40} - \frac{4}{40} \]\[ \text{Net rate} = \frac{5 - 4}{40} \]\[ \text{Net rate} = \frac{1}{40} \text{ tank/hour} \]If the net rate is \(\frac{1}{40}\) of the tank per hour, it means the tank will be filled in 40 hours.
\[ \text{Time} = \frac{1}{\text{Net rate}} \]\[ \text{Time} = \frac{1}{\frac{1}{40}} \]\[ \text{Time} = 40 \text{ hours} \]A) 40 is the correct answer. When both taps are opened simultaneously, the tank will be filled in 40 hours.
A bullet of 40 gm is fired horizontally with a velocity of 160 ms–1 from a pistol weighing 2 kg. What is the rebound velocity of the pistol?
This problem can be solved using the principle of conservation of momentum. In an isolated system (like a pistol firing a bullet), the total momentum before the event is equal to the total momentum after the event. Since the pistol and bullet are initially at rest, the initial momentum is zero. After firing, the bullet moves forward and the pistol recoils backward, conserving the total momentum.
The total momentum before firing is equal to the total momentum after firing.
\[ (m_p + m_b) \times u = m_p v_p + m_b v_b \]Where \(u\) is the initial velocity (which is \(0\)), \(v_p\) is the recoil velocity of the pistol, and \(v_b\) is the velocity of the bullet.
The negative sign indicates that the pistol recoils in the opposite direction to the bullet's motion.
Correct Option: B) –3.2 ms –1
The chemical formula of unslaked lime (quicklime) is _________.
The question asks for the chemical formula of unslaked lime, also known as quicklime. This requires knowledge of common chemical names and their corresponding formulas.
Correct Option: D) \(\text{CaO}\) is the chemical formula for unslaked lime (quicklime).
Square of 1\( \frac{3}{11} \) is:
\( \frac{16}{111} \)
\( \frac{225}{112} \)
\( \frac{196}{121} \)
\( \frac{9}{121} \)
To find the square of a mixed fraction, first convert the mixed fraction into an improper fraction. Then, square the resulting improper fraction by squaring both its numerator and its denominator.
The given mixed fraction is \(1 \frac{3}{11}\). To convert this to an improper fraction, multiply the whole number part by the denominator and add the numerator. Keep the same denominator.
\[ 1 \frac{3}{11} = \frac{(1 \times 11) + 3}{11} = \frac{11 + 3}{11} = \frac{14}{11} \]Now, we need to find the square of \( \frac{14}{11} \). To square a fraction, we square both the numerator and the denominator.
\[ \left( \frac{14}{11} \right)^2 = \frac{14^2}{11^2} \]Combine the squared numerator and denominator.
\[ \frac{196}{121} \]Correct Option: C) \( \frac{196}{121} \)
The question asks to identify the recipient of the 2018 Padma Bhushan award in the field of art-music from the given options. This requires knowledge of prominent awardees and their respective fields and years of recognition.
Correct Option: A) Sharda Sinha was awarded the Padma Bhushan in 2018 for her significant contributions to the field of art-music, particularly folk music.
Select the odd one
The task is to identify the odd one out from a set of diagrams. Each option represents a different arrangement of lines within a square. The key is to identify the pattern or characteristic that sets one option apart from the others.
Correct Option: A) Option A is the odd one out because it does not follow the typical parallel or intersecting line pattern seen in the other options.
To find the ratio A:C from the given ratios A:B and B:C, we need to make the common term 'B' equal in both ratios. Once 'B' is equalized, we can directly combine the ratios to find A:B:C, and then extract A:C.
Write down the given ratios:
We are given:
\[ A : B = 5 : 8 \]and
\[ B : C = 18 : 25 \]Equalize the common term 'B':
The value of B in the first ratio is 8, and in the second ratio, it is 18. To make them equal, we find the Least Common Multiple (LCM) of 8 and 18.
The LCM of 8 and 18 is 72.
Now, we adjust both ratios so that the 'B' term becomes 72.
Combine the ratios and find A:C:
Now that B is common (72) in both ratios, we can write the combined ratio A:B:C:
\[ A : B : C = 45 : 72 : 100 \]From this combined ratio, we can find A:C:
\[ A : C = 45 : 100 \]Simplify the ratio A:C:
Both 45 and 100 are divisible by 5. Divide both parts by 5:
\[ A : C = \frac{45}{5} : \frac{100}{5} = 9 : 20 \]Correct Option: B) 9:20
The question asks for the position of Indian Railways among the world's largest railways as of 2018. This requires knowledge of the ranking of railway networks globally based on factors like route length or operational length.
Correct Option: A) 4
The question asks about the winner of the majority of seats in the Gujarat Assembly Election 2017. To answer this, one needs to recall the results of that specific election.
Correct Option: D) Bharatiya Janata Party
To find the altitude to the longest side of a triangle, we first need to determine if the triangle is a right-angled triangle. If it is, we can use the area formula in two ways. If not, we can use Heron's formula to find the area and then use the standard area formula to find the altitude.
The altitude to the longest side is 9.6 m.
B) 9.6 m is the correct altitude calculated using the area of the right-angled triangle.