The problem asks for the total length of a journey given the total time taken and the speeds for two equal halves of the distance. We can use the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\) to set up an equation based on the total time.
Let the total length of the journey be \(D\) km.
Aman covers half the distance at 39 km/h. So, the first half of the distance is \(\frac{D}{2}\) km.
The time taken for the first half of the journey is \[T_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{D/2}{39}\] hours.
The rest of the distance is also \(\frac{D}{2}\) km, covered at 45 km/h.
The time taken for the second half of the journey is \[T_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{D/2}{45}\] hours.
The total time taken for the journey is given as 14 hours. So, \(T_1 + T_2 = 14\).
Substitute the expressions for \(T_1\) and \(T_2\) into the total time equation:
\[ \frac{D/2}{39} + \frac{D/2}{45} = 14 \]Factor out \(\frac{D}{2}\):
\[ \frac{D}{2} \left( \frac{1}{39} + \frac{1}{45} \right) = 14 \]Find a common denominator for the fractions inside the parenthesis. The least common multiple of 39 and 45 is \(3 \times 13 \times 3 \times 5 = 585\).
\[ \frac{D}{2} \left( \frac{45}{39 \times 45} + \frac{39}{45 \times 39} \right) = 14 \]\[ \frac{D}{2} \left( \frac{45}{1755} + \frac{39}{1755} \right) = 14 \]\[ \frac{D}{2} \left( \frac{45 + 39}{1755} \right) = 14 \]\[ \frac{D}{2} \left( \frac{84}{1755} \right) = 14 \]Simplify the fraction \(\frac{84}{1755}\). Both are divisible by 3:
\[ \frac{84 \div 3}{1755 \div 3} = \frac{28}{585} \]Substitute the simplified fraction back into the equation:
\[ \frac{D}{2} \left( \frac{28}{585} \right) = 14 \]Now, solve for \(D\):
\[ D \times \frac{28}{2 \times 585} = 14 \]\[ D \times \frac{28}{1170} = 14 \]\[ D = 14 \times \frac{1170}{28} \]\[ D = \frac{14}{28} \times 1170 \]\[ D = \frac{1}{2} \times 1170 \]\[ D = 585 \]The total length of the journey is 585 km.
Correct Option: D) 585
The calculation shows that the total length of the journey is 585 km, which matches option D.
The problem asks us to identify the letter-cluster that does not follow the same pattern as the others, based on the English alphabetical order. We need to find a common rule or relationship between the letters within each cluster and then identify the cluster that deviates from this rule.
Let's convert each letter to its alphabetical position and find the differences:
Comparing the patterns:
Options A, C, and D follow the pattern of differences (7, 9, 5). Option B follows the pattern (7, 9, 6), which is different in the last difference.
B) JQZF — This cluster has differences of (7, 9, 6) between consecutive letters (considering cyclic order for the last letter), while the other clusters have differences of (7, 9, 5).
To find the original number, we can set up an algebraic equation. If a number is increased by 50%, it means the new number is the original number plus 50% of the original number. This can be expressed as 100% of the number + 50% of the number, which equals 150% of the original number.
Correct Option: C) 2020
This question tests knowledge of the thermal decomposition reaction of ammonium dichromate. It's a classic example of a redox reaction that produces specific gaseous and solid products.
Correct Option: A) Nitrogen, water vapour and chromium(III) oxide
The resistance of a wire depends on its length, cross-sectional area, and resistivity. We will use the formula for resistance and substitute the given changes in length and area to find the new resistance.
Recall the formula for resistance:
The resistance \(R\) of a wire is given by the formula:
where \( \rho \) is the resistivity of the material, \(L\) is the length of the wire, and \(A\) is its cross-sectional area.
Identify the initial conditions:
Let the initial resistance be \(R\), initial length be \(L\), and initial cross-sectional area be \(A\). So, the initial resistance is:
Identify the new conditions:
The length is increased by a factor of 8, so the new length \(L'\) is \(8L\).
The cross-sectional area is decreased by a factor of 4, so the new area \(A'\) is \(A/4\).
Calculate the new resistance (R'):
Substitute the new length \(L'\) and new area \(A'\) into the resistance formula:
Simplify the expression for R':
To simplify, we can rewrite the division by a fraction as multiplication by its reciprocal:
Relate R' to R:
We know that \( R = \rho \frac{L}{A} \). Substitute this into the expression for \(R'\):
Thus, the new resistance \(R'\) is 32 times the original resistance \(R\).
C) R' = 32 R — As calculated, the new resistance is 32 times the original resistance.
The problem asks us to determine who sits on the extreme left end of a straight line of 5 people (A, B, C, D, E) facing north. We need to evaluate if the given statements (I and II) are sufficient, either independently or together, to answer this question.
This statement implies two possible arrangements for A and E:
In none of these cases can we definitively determine who sits on the extreme left end (P1). It could be A or E. The other three people (B, C, D) could fill the remaining spots in various ways. Therefore, Statement I alone is not sufficient.
Similar to Statement I, this statement implies two possible arrangements for B and C:
Again, we cannot definitively determine who sits on the extreme left end (P1). It could be B or C. The other three people (A, D, E) could fill the remaining spots. Therefore, Statement II alone is not sufficient.
Let's combine the information. We know there are 5 positions. Let's denote them as 1, 2, 3, 4, 5 from left to right.
From Statement I: Exactly 2 people between A and E. Possible pairs of positions for (A, E) or (E, A) are (1, 4) or (2, 5).
From Statement II: Exactly 2 people between B and C. Possible pairs of positions for (B, C) or (C, B) are (1, 4) or (2, 5).
Consider the possible scenarios when both statements are true:
From these scenarios, the person on the extreme left could be A, E, B, or C. Since we cannot determine a unique person for the extreme left position, even both statements together are not sufficient.
B) Both Statements I and II put together are not sufficient to answer the question.
To find the new average marks, we first calculate the total marks initially. Then, we determine the change in marks due to re-evaluation in three subjects. We add this change to the initial total marks to get the new total marks. Finally, we divide the new total marks by the number of subjects to find the new average.
Given that the average marks of a student in six subjects is 94.
\[ \text{Initial Total Marks} = \text{Average Marks} \times \text{Number of Subjects} \]\[ \text{Initial Total Marks} = 94 \times 6 = 564 \]The number of subjects remains 6.
\[ \text{New Average Marks} = \frac{\text{New Total Marks}}{\text{Number of Subjects}} \]\[ \text{New Average Marks} = \frac{594}{6} = 99 \]Correct Option: A) 99
This is a logical reasoning problem involving arrangement. We need to deduce the floor on which each person lives based on the given clues and then determine the number of people between A and D. We will systematically use each clue to narrow down the possibilities for the floor assignments.
Identify the floors: There are 6 floors, numbered 1 to 6.
Clue 1: E lives on an even numbered floor but not 6.
The even numbered floors are 2, 4, 6. Since E does not live on floor 6, E must live on either floor 2 or floor 4.
Clue 2: The sum of floors on which A and E live is 7.
Let \(F_A\) be the floor of A and \(F_E\) be the floor of E. So, \(F_A + F_E = 7\).
Clue 3: The sum of floors on which E and D live is 3.
Let \(F_D\) be the floor of D. So, \(F_E + F_D = 3\).
From this, we can definitively conclude that E lives on floor 2 and D lives on floor 1.
Update A's floor: Since E lives on floor 2, from Clue 2 (\(F_A + F_E = 7\)), we get \(F_A + 2 = 7 \implies F_A = 5\). So, A lives on floor 5.
Current arrangement:
Floor 6: ?
Floor 5: A
Floor 4: ?
Floor 3: ?
Floor 2: E
Floor 1: D
Clue 4: F lives immediately above B.
This means F and B must occupy two consecutive floors. The available consecutive floors are 3 and 4. So, F lives on floor 4 and B lives on floor 3.
Final arrangement:
Floor 6: C (The only remaining person and floor)
Floor 5: A
Floor 4: F
Floor 3: B
Floor 2: E
Floor 1: D
Answer the question: How many people live between A and D?
A lives on floor 5 and D lives on floor 1. The people living between floor 5 and floor 1 are F (floor 4), B (floor 3), and E (floor 2). There are 3 people between A and D.
C) 3
The question asks about the environmental and societal impact of using thermocol disposables. We need to evaluate each option based on the properties of thermocol and its disposal.
Correct Option: B) we are increasing non-biodegradable waste
Thermocol (expanded polystyrene) is a plastic material. Plastics are known for their non-biodegradable nature, meaning they do not decompose naturally over a short period. When thermocol disposables are used and thrown away, they accumulate in landfills and the environment, contributing significantly to non-biodegradable waste, which poses long-term environmental challenges.
If the mean of the following data is 37, then the value of X is:
| xi | 20 | 30 | 40 | 85 |
| fi | 19 | X | 13 | 24 |
To find the value of X, we will use the formula for the mean of grouped data. The mean is calculated as the sum of the products of each observation and its frequency, divided by the sum of the frequencies.
First, let's list the given data:
| \(x_i\) | \(f_i\) | \(f_i x_i\) |
|---|---|---|
| 20 | 19 | \(20 \times 19 = 380\) |
| 30 | X | \(30 \times X = 30X\) |
| 40 | 13 | \(40 \times 13 = 520\) |
| 85 | 24 | \(85 \times 24 = 2040\) |
Calculate the sum of the frequencies (\(\sum f_i\)):
\[ \sum f_i = 19 + X + 13 + 24 = 56 + X \]Calculate the sum of the products of observations and frequencies (\(\sum f_i x_i\)):
\[ \sum f_i x_i = 380 + 30X + 520 + 2040 = 2940 + 30X \]The formula for the mean (\(\bar{x}\)) is:
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]We are given that the mean is 37. Substitute the values into the formula:
\[ 37 = \frac{2940 + 30X}{56 + X} \]Now, solve for X:
\[ 37(56 + X) = 2940 + 30X \]\[ 37 \times 56 + 37X = 2940 + 30X \]\[ 2072 + 37X = 2940 + 30X \]Group the terms with X on one side and constant terms on the other side:
\[ 37X - 30X = 2940 - 2072 \]\[ 7X = 868 \]Divide by 7 to find X:
\[ X = \frac{868}{7} \]\[ X = 124 \]Correct Option: A) 124