We are given two conditions about two numbers: their sum and their difference. We can set up a system of two linear equations with two variables and solve for the numbers. Alternatively, we can test the given options against the conditions.
Correct Option: B) 53, 37
To find the distance covered by the train, we need to use the basic formula relating speed, distance, and time. The key is to ensure that all units are consistent. Since the speed is given in km/hr, the time should also be in hours before calculating the distance.
Since there are 60 minutes in 1 hour, we convert 24 minutes to hours:
\[ \text{Time in hours} = \frac{\text{Time in minutes}}{60} \]So,
\[ \text{Time} = \frac{24}{60} \text{ hours} = \frac{2}{5} \text{ hours} = 0.4 \text{ hours} \]The formula for distance is:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]Substitute the given speed and the time in hours:
\[ \text{Distance} = 82 \text{ km/hr} \times 0.4 \text{ hours} \]\[ \text{Distance} = 32.8 \text{ km} \]
Correct Option: C) 32.8 km
This problem involves calculating the amount of water in a sugar solution given the total volume and the percentage of sugar. We need to find the volume of sugar first and then subtract it from the total volume to get the volume of water.
The volume of sugar is \(6\%\) of the total solution volume.
\[ \text{Volume of sugar} = 6\% \times 350 \text{ ml} \]\[ \text{Volume of sugar} = \frac{6}{100} \times 350 \text{ ml} \]\[ \text{Volume of sugar} = \frac{2100}{100} \text{ ml} \]\[ \text{Volume of sugar} = 21 \text{ ml} \]The total volume of the solution is the sum of the volume of sugar and the volume of water.
\[ \text{Total volume} = \text{Volume of sugar} + \text{Volume of water} \]\[ 350 \text{ ml} = 21 \text{ ml} + \text{Volume of water} \]\[ \text{Volume of water} = 350 \text{ ml} - 21 \text{ ml} \]\[ \text{Volume of water} = 329 \text{ ml} \]D) The volume of water in the solution is \(329 \text{ ml}\).
To find the total resistance of resistors connected in parallel, we use the formula for parallel resistances. For two resistors \(R_1\) and \(R_2\) in parallel, the total resistance \(R_{total}\) is given by the reciprocal of the sum of the reciprocals of individual resistances, or more simply, by the product-over-sum formula.
Correct Option: A) 1.75 Ξ©
Which of the following statements is correct for a body in an elevator moving downwards with uniform velocity?
The Weight of a body in an elevator moving downwards decreases while its Mass remains the same.
The Weight as well as the Mass of a body in an elevator moving downwards decreases.
The Weight of the body in an elevator moving downwards increases while its mass remains the same.
The Weight and the Mass of the body in an elevator moving downwards do not change.
This question tests the understanding of fundamental physics concepts: mass, weight, and the effect of motion on these quantities. We need to analyze how mass and weight behave when an object is in an elevator moving with uniform velocity.
D) The Weight and the Mass of the body in an elevator moving downwards do not change. This is correct because mass is an intrinsic property and remains constant. Since the elevator is moving with uniform velocity, its acceleration is zero. Therefore, the apparent weight (normal force) on the body remains equal to its actual weight (\(mg\)), meaning the weight also does not change.
To find the next number in the series, we need to identify the pattern or rule governing the sequence of numbers. We will examine the operations between consecutive terms.
So, the next number in the series is 60.
Correct Option: B) 60
This question asks us to identify which assumption is implicit in the given statement. An implicit assumption is something that is not stated directly but must be true for the statement to make sense or be valid. We need to analyze each assumption in relation to the statement: "It is important to involve children in various co-curricular activities."
Correct Option: A) Only assumption I is implicit
Which of the following is the greatest fraction?
\( \frac{5}{14} \)
\( \frac{29}{77} \)
\( \frac{25}{66} \)
\( \frac{8}{21} \)
To find the greatest fraction among the given options, we can convert each fraction into a decimal or find a common denominator. Converting to decimals is often simpler for comparison.
Comparing the decimal values:
The largest decimal value is \( 0.3809 \).
The decimal \( 0.3809 \) corresponds to the fraction \( \frac{8}{21} \).
We can compare fractions pairwise. Let's compare A and B first:
Therefore, \( \frac{8}{21} \) is the greatest fraction.
Correct Option: D) \( \frac{8}{21} \)
The focal length of a mirror is determined by its curvature and is assigned a sign based on the Cartesian sign convention. For spherical mirrors, the principal focus of a convex mirror is always located behind the mirror.
Correct Option: C) positive
The focal length of a convex mirror is always positive because its principal focus is a virtual focus located behind the mirror, on the side opposite to the incident light, which is considered the positive direction according to the Cartesian sign convention.
To find the combined time taken by two pipes to fill a tank, we first determine the individual filling rates of each pipe. Then, we add their rates to find the combined rate. Finally, we take the reciprocal of the combined rate to find the total time taken to fill the tank.
Pipe A fills the tank in 10 hours. So, in 1 hour, Pipe A fills \(\frac{1}{10}\) of the tank.
\[ \text{Rate of A} = \frac{1}{10} \text{ tank/hour} \]
Pipe B fills the tank in 15 hours. So, in 1 hour, Pipe B fills \(\frac{1}{15}\) of the tank.
\[ \text{Rate of B} = \frac{1}{15} \text{ tank/hour} \]
When both pipes work together, their rates add up.
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} \]
\[ \text{Combined Rate} = \frac{1}{10} + \frac{1}{15} \]
To add these fractions, find a common denominator, which is 30.
\[ \text{Combined Rate} = \frac{3}{30} + \frac{2}{30} \]
\[ \text{Combined Rate} = \frac{3+2}{30} = \frac{5}{30} \]
\[ \text{Combined Rate} = \frac{1}{6} \text{ tank/hour} \]
The time taken is the reciprocal of the combined rate.
\[ \text{Time} = \frac{1}{\text{Combined Rate}} \]
\[ \text{Time} = \frac{1}{\frac{1}{6}} = 6 \text{ hours} \]
Correct Option: B) 6