The problem asks for the greatest number that divides 137 and 198, leaving specific remainders. This is a classic application of finding the Highest Common Factor (HCF) or Greatest Common Divisor (GCD). If a number \(N\) divides \(A\) and leaves a remainder \(R_A\), then \(N\) must divide \((A - R_A)\) exactly. We will apply this principle to both given numbers and then find the HCF of the resulting exact divisors.
The greatest number that satisfies the given conditions is 65.
Correct Option: C) 65
To find the profit percentage, we first need to calculate the selling price after the discount. Then, we can use the cost price and selling price to determine the profit and subsequently the profit percentage.
The discount is 50% of the Marked Price.
\[ \text{Discount Amount} = \text{Discount Percentage} \times \text{Marked Price} \\ \text{Discount Amount} = 50\% \times \text{₹}1,032 \\ \text{Discount Amount} = \frac{50}{100} \times 1032 \\ \text{Discount Amount} = \frac{1}{2} \times 1032 \\ \text{Discount Amount} = \text{₹}516 \]The Selling Price is the Marked Price minus the Discount Amount.
\[ \text{Selling Price (SP)} = \text{Marked Price} - \text{Discount Amount} \\ \text{SP} = \text{₹}1,032 - \text{₹}516 \\ \text{SP} = \text{₹}516 \]Profit is the difference between the Selling Price and the Cost Price.
\[ \text{Profit} = \text{Selling Price (SP)} - \text{Cost Price (CP)} \\ \text{Profit} = \text{₹}516 - \text{₹}300 \\ \text{Profit} = \text{₹}216 \]Profit percentage is calculated with respect to the Cost Price.
\[ \text{Profit Percentage} = \frac{\text{Profit}}{\text{Cost Price (CP)}} \times 100\% \\ \text{Profit Percentage} = \frac{216}{300} \times 100\% \\ \text{Profit Percentage} = \frac{216}{3} \% \\ \text{Profit Percentage} = 72\% \]D) 72% is the correct profit percentage.
The problem asks us to identify the highest and lowest numbers from a given set of five three-digit numbers. Then, we need to extract the second digit of the highest number and the second digit of the lowest number. Finally, we subtract the second digit of the highest number from the second digit of the lowest number.
The resultant value is \(-1\).
C) -1 — As calculated in the step-by-step solution, the second digit of the lowest number (6) minus the second digit of the highest number (7) is \(6 - 7 = -1\).
When four numbers are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. We will set up this proportion as an equation and solve for the unknown variable \(y\).
Understand the concept of proportion: If four numbers \(a, b, c,\) and \(d\) are in proportion, it means that \(a:b = c:d\), which can also be written as a fraction: \[\frac{a}{b} = \frac{c}{d}\].
Set up the proportion: Given the numbers 14, 7, 12, and \(y\) are in proportion, we can write this as:
\[ \frac{14}{7} = \frac{12}{y} \]
Simplify the left side of the equation:
\[ \frac{14}{7} = 2 \]
So the equation becomes:
\[ 2 = \frac{12}{y} \]
Solve for \(y\): To isolate \(y\), multiply both sides by \(y\) and then divide by 2:
\[ 2y = 12 \]
\[ y = \frac{12}{2} \]
\[ y = 6 \]
Correct Option: B) 6
The question asks to identify the term for the concluding part of 'Pallavi' in Carnatic music. This requires knowledge of the structure and terminology used in Carnatic music compositions, specifically within the Ragam Tanam Pallavi (RTP) format.
A) Ragamalika — In Carnatic music, particularly in the context of a Ragam Tanam Pallavi (RTP), after the main Pallavi line is presented and elaborated upon, the performer often improvises by singing or playing the Pallavi line in a garland of different ragas. This section, where the Pallavi is rendered in multiple ragas, is known as Ragamalika (literally "garland of ragas"). It serves as a concluding and often climactic part of the Pallavi presentation.
The question asks to identify the leader known as 'Young Raja' in the context of the Khond Uprising in Odisha. This requires knowledge of prominent figures involved in tribal revolts against British rule in India, specifically the Khond Uprising.
Correct Option: B) Chakra Bisoi was the leader known as 'Young Raja' during the Khond Uprising in Odisha. He led the Khonds against the British between 1846 and 1856.
The question asks about the social group that gained increased importance in post-Gupta society due to land grants. This requires knowledge of the social and economic changes that occurred in ancient India, particularly the impact of land grants on different varnas.
Correct Option: D) Brahmins gained increased importance. Land grants, particularly brahmadeya, were primarily given to Brahmins, which enhanced their economic status, social prestige, and often their administrative control over the granted territories. This led to their increased prominence in the post-Gupta period.
The length of a rectangular field is \(\left(\frac{2a+2\sqrt{5}}{4}\right)\) m and breadth is \(\left(\frac{a-\sqrt{5}}{2}\right)\) m. Find the area of the rectangular field.
\(\left(\frac{a^2-5}{2}\right)\) m²
\(\left(\frac{a^2-5}{4}\right)\) m²
\(\left(\frac{a^2+5}{2}\right)\) m²
\(\left(\frac{a^2+5}{4}\right)\) m²
To find the area of a rectangular field, we multiply its length by its breadth. The given expressions for length and breadth involve algebraic terms and square roots. We will use the distributive property of multiplication and the algebraic identity \((x+y)(x-y) = x^2 - y^2\).
Identify the given dimensions:
Length \(L = \left(\frac{2a+2\sqrt{5}}{4}\right)\) m
Breadth \(B = \left(\frac{a-\sqrt{5}}{2}\right)\) m
Simplify the expression for Length:
We can factor out 2 from the numerator of the length expression:
\[ L = \frac{2(a+\sqrt{5})}{4} = \frac{a+\sqrt{5}}{2} \]Calculate the Area:
The area of a rectangle is given by the formula \( \text{Area} = L \times B \).
Substitute the simplified expressions for length and breadth:
\[ \text{Area} = \left(\frac{a+\sqrt{5}}{2}\right) \times \left(\frac{a-\sqrt{5}}{2}\right) \]Perform the multiplication:
Multiply the numerators and the denominators separately:
\[ \text{Area} = \frac{(a+\sqrt{5})(a-\sqrt{5})}{2 \times 2} \]\[ \text{Area} = \frac{(a+\sqrt{5})(a-\sqrt{5})}{4} \]Apply the algebraic identity:
Use the identity \((x+y)(x-y) = x^2 - y^2\), where \(x=a\) and \(y=\sqrt{5}\):
The numerator becomes \(a^2 - (\sqrt{5})^2\).
Since \((\sqrt{5})^2 = 5\), the numerator is \(a^2 - 5\).
Write the final expression for the Area:
\[ \text{Area} = \frac{a^2 - 5}{4} \text{ m}^2 \]B) \(\left(\frac{a^2-5}{4}\right)\) m² is the correct answer, matching our calculated area.
The question asks to identify the Governor-General under whose administration the practice of Sati was officially abolished in British India. This requires knowledge of key social reforms and the administrators associated with them during the British colonial period.
Correct Option: D) Lord William Bentinck
Lord William Bentinck officially abolished the practice of Sati in British India in 1829 through the Bengal Sati Regulation. He is widely recognized for his social reforms, including the suppression of Thuggee and the introduction of English education.
The question asks for the correct sequence of actions to move a block of text from one location to another within an MS Word document. This is a fundamental operation in word processing, and understanding the difference between 'copy' and 'cut' is key.
Correct Option: C) Cut the text → Paste the text at the desired location
This sequence correctly moves the text. The 'Cut' command removes the text from its original position and stores it temporarily in the clipboard. The 'Paste' command then inserts this text from the clipboard to the new desired location, thus completing the move operation.