When two individuals work together and earn a certain amount, the earnings should be distributed in proportion to the amount of work they do, or inversely proportional to the time they take to complete the work individually. This means we need to find the ratio of their efficiencies or the ratio of the work done by them in a given unit of time.
Determine the work rate of A:
A can complete the work in 2 weeks. So, A's work rate per week is:
Determine the work rate of B:
B can complete the work in 3 weeks. So, B's work rate per week is:
Find the ratio of their work rates:
The amount earned should be distributed in the ratio of their work rates.
To simplify this ratio, multiply both sides by the least common multiple of 2 and 3, which is 6:
\[ \left(\frac{1}{2} \times 6\right) : \left(\frac{1}{3} \times 6\right) = 3 : 2 \]So, the ratio in which the amount should be distributed is \(3:2\).
Calculate A's share:
The total amount to be distributed is Rs. 300. The sum of the ratio parts is \(3 + 2 = 5\).
Calculate B's share:
\[ \text{B's share} = \frac{2}{5} \times 300 \text{ Rs.} = 2 \times 60 \text{ Rs.} = 120 \text{ Rs.} \]Alternatively, B's share can be calculated as Total amount - A's share:
\[ \text{B's share} = 300 \text{ Rs.} - 180 \text{ Rs.} = 120 \text{ Rs.} \]Correct Option: B) A = Rs. 180, B = Rs. 120
This problem involves understanding relative directions in a standard carrom game setup. In carrom, two players sit opposite each other. If one player is facing a certain direction, the player sitting directly opposite them will be facing the exact opposite direction.
Correct Option: D) Southeast
The question asks about the factors on which the heat generated in a circuit depends, according to Joule's law of heating. We need to recall the formula for Joule's law to identify all the contributing factors.
B) Current, the resistance of conductor and time period for current to flow through it
This option correctly lists all three factors (current, resistance, and time) that determine the heat generated according to Joule's law (\(H = I^2 R t\)).
The problem requires us to simplify a mathematical expression involving multiplication, subtraction, and division. We can use algebraic identities, specifically the difference of squares formula, to simplify the numerator and denominator before performing the division.
The numerator is \( (9.5 \times 9.5 - 2.5 \times 2.5) \). This can be written as \( (9.5^2 - 2.5^2) \).
We use the difference of squares identity: \( a^2 - b^2 = (a - b)(a + b) \).
Here, \( a = 9.5 \) and \( b = 2.5 \).
\[ 9.5^2 - 2.5^2 = (9.5 - 2.5)(9.5 + 2.5) \]\[ = (7.0)(12.0) \]\[ = 7 \times 12 \]\[ = 84 \]The denominator is \( (1.5^2 - 0.25) \).
We know that \( 0.25 = 0.5^2 \).
So, the denominator becomes \( (1.5^2 - 0.5^2) \).
Again, using the difference of squares identity: \( a^2 - b^2 = (a - b)(a + b) \).
Here, \( a = 1.5 \) and \( b = 0.5 \).
\[ 1.5^2 - 0.5^2 = (1.5 - 0.5)(1.5 + 0.5) \]\[ = (1.0)(2.0) \]\[ = 1 \times 2 \]\[ = 2 \]Now, we divide the simplified numerator by the simplified denominator.
\[ \frac{\text{Numerator}}{\text{Denominator}} = \frac{84}{2} \]\[ = 42 \]B) 42 is the correct answer, as calculated by simplifying the expression using the difference of squares identity.
The question asks to complete an analogy. Analogies test the ability to identify the relationship between a given pair of words and then apply the same relationship to a new word to find its corresponding pair. Here, we need to find the relationship between "Hot" and "Burning" and apply it to "Cold".
Correct Option: C) Freezing — "Burning" is an extreme or intense state/action associated with "Hot". Similarly, "Freezing" is an extreme or intense state/action associated with "Cold".
The question asks for evidence of evolutionary relationships. We need to identify which option directly provides historical biological data that can be used to trace the lineage and changes in life forms over time.
Correct Option: B) Fossil
This question asks us to evaluate the strength of two arguments regarding the establishment of new research centers in India. We need to determine which argument(s) provide a compelling reason for or against the proposal.
Correct Option: A) Only argument I is strong. Argument I provides a valid and generally accepted reason for establishing research centers, linking them directly to innovation. Argument II is weak because it makes a broad, unsubstantiated claim about resource scarcity without considering the potential for resource mobilization or the long-term benefits that might justify such an investment.
The question asks to identify which bank was NOT one of the three Presidency Banks established in India during British rule. This requires knowledge of the history of banking in British India, specifically the names of the Presidency Banks.
Correct Option: C) Bank of Travancore
The Bank of Travancore was not one of the three Presidency Banks. It was a state-owned bank established in 1945 in the princely state of Travancore, much later than the Presidency Banks and with a different origin and purpose. It later became a subsidiary of the State Bank of India (State Bank of Travancore) and was eventually merged into the State Bank of India in 2017.
To find the length of the longer piece, we first need to determine the total number of parts in the given ratio. Then, we can calculate the length corresponding to one part and finally multiply it by the larger number in the ratio.
Understand the given information:
Calculate the sum of the ratio parts:
The ratio \(13 : 23\) means the wire is divided into \(13 + 23\) equal parts.
\[ \text{Sum of ratio parts} = 13 + 23 = 36 \]Determine the length of one part:
Divide the total length of the wire by the sum of the ratio parts to find the length represented by one part.
\[ \text{Length of one part} = \frac{\text{Total length}}{\text{Sum of ratio parts}} = \frac{28.8 \text{ m}}{36} \]\[ \text{Length of one part} = 0.8 \text{ m} \]Calculate the lengths of the two pieces:
The lengths of the two pieces will be \(13 \times 0.8 \text{ m}\) and \(23 \times 0.8 \text{ m}\).
Identify the longer piece:
Comparing the two lengths, \(18.4 \text{ m}\) is longer than \(10.4 \text{ m}\).
Correct Option: D) The length of the longer piece is \(18.4 \text{ m}\).
If x = 9, y = \( \sqrt{17} \), then what is the value of (x2-y2)\( \frac{-1}{3} \) -1
\( \frac{3}{5} \)
\( \frac{-3}{4} \)
\( \frac{5}{3} \)
\( \frac{-7}{4} \)
The problem asks us to evaluate an expression involving variables \(x\) and \(y\), given their values. We will substitute the given values of \(x\) and \(y\) into the expression and simplify it step-by-step using algebraic identities and rules of exponents.
First, identify the given values:
\[ x = 9 \]\[ y = \sqrt{17} \]Identify the expression to be evaluated:
\[ (x^2 - y^2)^{\frac{-1}{3}} - 1 \]Calculate \(x^2\):
\[ x^2 = 9^2 = 81 \]Calculate \(y^2\):
\[ y^2 = (\sqrt{17})^2 = 17 \]Substitute \(x^2\) and \(y^2\) into the expression \(x^2 - y^2\):
\[ x^2 - y^2 = 81 - 17 = 64 \]Now, substitute this value back into the main expression:
\[ (64)^{\frac{-1}{3}} - 1 \]Recall the rule of exponents: \(a^{-n} = \frac{1}{a^n}\). So,
\[ (64)^{\frac{-1}{3}} = \frac{1}{(64)^{\frac{1}{3}}} \]Recall that \(a^{\frac{1}{n}} = \sqrt[n]{a}\). So, \((64)^{\frac{1}{3}}\) is the cube root of 64.
\[ (64)^{\frac{1}{3}} = \sqrt[3]{64} \]Find the cube root of 64. We know that \(4 \times 4 \times 4 = 64\), so \(\sqrt[3]{64} = 4\).
Substitute this value back into the expression:
\[ \frac{1}{4} - 1 \]Perform the subtraction:
\[ \frac{1}{4} - 1 = \frac{1}{4} - \frac{4}{4} = \frac{1 - 4}{4} = \frac{-3}{4} \]Thus, the value of the expression is \( \frac{-3}{4} \).
B) \( \frac{-3}{4} \) is the correct answer, as calculated above.