The given expression \(18 : A :: A : 128\) represents a proportion. In a proportion, the product of the means is equal to the product of the extremes. We will use this property to set up an equation and solve for the positive value of A.
Understand the proportion notation: The notation \(a : b :: c : d\) means that \(a\) is to \(b\) as \(c\) is to \(d\). This can be written as a fraction:
\[ \frac{a}{b} = \frac{c}{d} \]Apply to the given problem: For \(18 : A :: A : 128\), we can write:
\[ \frac{18}{A} = \frac{A}{128} \]Cross-multiply: Multiply both sides by \(A\) and by \(128\) to eliminate the denominators:
\[ 18 \times 128 = A \times A \]\[ 18 \times 128 = A^2 \]Calculate the product:
\[ 18 \times 128 = 2304 \]Form the equation for A:
\[ A^2 = 2304 \]Solve for A: To find \(A\), take the square root of both sides:
\[ A = \sqrt{2304} \]Find the square root: We need to find the number that, when multiplied by itself, gives 2304. We can estimate or use prime factorization. Let's try to estimate. We know \(40^2 = 1600\) and \(50^2 = 2500\). So, \(A\) is between 40 and 50. Since the last digit of 2304 is 4, the last digit of \(A\) must be 2 or 8. Let's try \(48\):
\[ 48 \times 48 = 2304 \]Therefore,\[ A = 48 \]Consider the positive value: The question asks for the positive value of A. Since \(\sqrt{2304}\) can be \(48\) or \(-48\), the positive value is \(48\).
Correct Option: B) 48
The positive value of A that satisfies the proportion \(18 : A :: A : 128\) is 48.
To find the final selling price after successive discounts and an additional cash payment discount, we need to calculate the price after each discount sequentially. We start with the marked price and apply the first discount, then the second discount on the new price, and finally the cash discount on the price after the first two discounts.
D) 243. As calculated, the final selling price after all three discounts is ₹243.
This problem involves decoding a blood relation expression using given symbols. We need to break down the expression 'M + N − O × P ÷ Q' step-by-step, determine the relationship between each pair, and then combine them to find the final relationship between M and Q.
From the above, O is the wife of P, and P is the father of Q. Therefore, O is the mother of Q. Since M is the sister of O (M, N, O are siblings, and O is female), M is the sister of Q's mother. Thus, M is Q's Mother's Sister.
A) Mother's Sister — As derived, M is the sister of O, and O is the mother of Q. Therefore, M is the mother's sister of Q.
To find the price of the costliest item, we first need to determine the total price of the three items using their average price. Then, we can use the given ratio of their prices to find the individual prices and identify the costliest one.
Given that the average price of three items is ₹15,165.
\[ \text{Total Price} = \text{Average Price} \times \text{Number of Items} \]\[ \text{Total Price} = 15,165 \times 3 \]\[ \text{Total Price} = \text{₹}45,495 \]The prices are in the ratio 3 : 5 : 7. Let the prices of the three items be \(3x\), \(5x\), and \(7x\) respectively.
The sum of the individual prices must equal the total price.
\[ 3x + 5x + 7x = 45,495 \]\[ 15x = 45,495 \]\[ x = \frac{45,495}{15} \]\[ x = 3,033 \]The costliest item corresponds to the largest part of the ratio, which is \(7x\).
\[ \text{Price of Costliest Item} = 7x \]\[ \text{Price of Costliest Item} = 7 \times 3,033 \]\[ \text{Price of Costliest Item} = \text{₹}21,231 \]C) ₹21,231 is the price of the costliest item, as calculated.
This is a logical reasoning puzzle involving scheduling. We need to deduce the exam schedule for seven people (P, Q, R, S, T, U, V) over seven days (Monday to Sunday) based on the given clues. We will create a table for the days of the week and fill in the names as we deduce them.
List the days of the week:
| Day | Person |
|---|---|
| Monday | |
| Tuesday | |
| Wednesday | |
| Thursday | |
| Friday | |
| Saturday | |
| Sunday |
Clue 1: R has an exam on Wednesday.
| Day | Person |
|---|---|
| Monday | |
| Tuesday | |
| Wednesday | R |
| Thursday | |
| Friday | |
| Saturday | |
| Sunday |
Clue 2: Exactly three people have an exam between R and Q.
Since R is on Wednesday, we count three days after Wednesday (Thursday, Friday, Saturday) to place Q. This means Q is on Sunday.
| Day | Person |
|---|---|
| Monday | |
| Tuesday | |
| Wednesday | R |
| Thursday | |
| Friday | |
| Saturday | |
| Sunday | Q |
Clue 3: V has an exam immediately after S and T has an exam immediately before S.
This implies a sequence T-S-V. These three people must occupy three consecutive days.
Clue 4: Only three people have an exam between P and S.
Let's look at the remaining empty slots: Monday, Tuesday, Thursday, Friday, Saturday. The sequence T-S-V needs three consecutive slots. The only available consecutive slots are Thursday, Friday, Saturday.
| Day | Person |
|---|---|
| Monday | |
| Tuesday | |
| Wednesday | R |
| Thursday | T |
| Friday | S |
| Saturday | V |
| Sunday | Q |
Now, let's use Clue 4: Only three people have an exam between P and S. S is on Friday. Counting three days before S (Thursday, Wednesday, Tuesday), P would be on Monday.
| Day | Person |
|---|---|
| Monday | P |
| Tuesday | |
| Wednesday | R |
| Thursday | T |
| Friday | S |
| Saturday | V |
| Sunday | Q |
The only remaining person is U, and the only remaining day is Tuesday. So, U has an exam on Tuesday.
| Day | Person |
|---|---|
| Monday | P |
| Tuesday | U |
| Wednesday | R |
| Thursday | T |
| Friday | S |
| Saturday | V |
| Sunday | Q |
The question asks who has an exam on Sunday. From our final schedule, Q has an exam on Sunday.
Correct Option: B) Q
Based on our step-by-step deduction, Q has an exam on Sunday.
To solve this problem, we need to identify the highest and lowest numbers from the given set, then extract specific digits from them as per the question's instructions, and finally perform the requested arithmetic operation.
Correct Option: D) 4
The question asks to identify the singing style to which 'Vilambit' and 'Drut' varieties belong. This requires knowledge of Indian classical music terminology and the characteristics of different vocal forms.
A) Khayal — Khayal is a prominent form of Hindustani classical music where compositions are typically rendered in two main tempo variations: 'Vilambit' (slow tempo) and 'Drut' (fast tempo). The 'Vilambit Khayal' allows for extensive melodic elaboration and improvisation, while the 'Drut Khayal' showcases rhythmic complexity and virtuosity.
We need to find the principal amount given the difference between compound interest (CI) and simple interest (SI) for three years at a specific annual interest rate. We will use the standard formulas for CI and SI and then find their difference to set up an equation to solve for the principal.
C) ₹7,000 is the principal amount.
The question asks to identify the drama that received the 2025 Pulitzer Prize for Drama. This requires knowledge of recent Pulitzer Prize winners in the drama category.
Correct Option: A) Purpose
"Purpose" by Branden Jacobs-Jenkins was awarded the 2025 Pulitzer Prize for Drama. The play explores themes of family, legacy, and political ambition through the lens of a prominent Black political family.
The question asks for the establishment year of the Industrial Finance Corporation of India (IFCI). This requires knowledge of key dates in India's economic history and the establishment of financial institutions.
Correct Option: A) 1948