The given series is an alternating series where the pattern involves both subtraction and addition, and the magnitude of the difference increases with each step. We need to identify the underlying pattern of differences between consecutive terms to predict the next term.
The differences are \(-1, +4, -9, +16\). These numbers are the squares of consecutive integers, with alternating signs:
The pattern is \( -(1^2), +(2^2), -(3^2), +(4^2) \).
Following this pattern, the next difference should be \( -(5^2) \).
Add the predicted difference to the last known term:
\[ 1524 + (-25) = 1524 - 25 = 1499 \]B) 1499 is the correct answer, as it follows the established pattern of alternating subtraction and addition of perfect squares.
This problem involves analyzing logical statements and conclusions, a common type of question in competitive exams. We will use Venn diagrams to visually represent the relationships described in the statements and then check if the conclusions logically follow.
Since both conclusions (I) and (II) logically follow from the given statements, the correct option is B.
B) Both conclusions (I) and (II) follow. As explained in the step-by-step analysis, both conclusions are direct logical consequences of the given statements.
The problem requires us to modify a given number by applying specific rules to its even and odd digits. Then, we need to find the sum of the first and last digits of the new number formed.
D) 16 — The sum of the first (7) and last (9) digits of the new number (735579) is \(7 + 9 = 16\).
This problem involves profit, selling price, and cost price. We will use the basic formulas for profit and profit percentage. We'll set up equations based on the given conditions (initial and changed scenarios) and solve for the initial profit percentage.
Let the initial Cost Price be \(CP\), initial Selling Price be \(SP_1\), and initial Profit be \(P_1\).
The formula for profit is: \[\text{Profit} = \text{Selling Price} - \text{Cost Price}\]
So, for the initial scenario:
\[P_1 = SP_1 - CP \quad \ldots(1)\]According to the problem, the new selling price (\(SP_2\)) is 2 times the initial selling price:
\[SP_2 = 2 \times SP_1\]The new profit (\(P_2\)) is 8 times the initial profit:
\[P_2 = 8 \times P_1\]The cost price (\(CP\)) remains the same. So, for the new scenario:
\[P_2 = SP_2 - CP \quad \ldots(2)\]Substitute the expressions for \(SP_2\) and \(P_2\) from steps 2 and 3 into equation (2):
\[8 \times P_1 = (2 \times SP_1) - CP \quad \ldots(3)\]Now we have a system of two equations (1) and (3):
(1) \(P_1 = SP_1 - CP\)
(3) \(8 P_1 = 2 SP_1 - CP\)
From equation (1), we can express \(CP\) as: \[\text{CP} = SP_1 - P_1\]
Substitute this expression for \(CP\) into equation (3):
\[8 P_1 = 2 SP_1 - (SP_1 - P_1)\]\[8 P_1 = 2 SP_1 - SP_1 + P_1\]\[8 P_1 = SP_1 + P_1\]Rearrange the equation to find the relationship between \(SP_1\) and \(P_1\):
\[8 P_1 - P_1 = SP_1\]\[7 P_1 = SP_1\]Now we need to find the initial profit percentage. The formula for profit percentage is:
\[\text{Profit Percentage} = \frac{\text{Profit}}{\text{Cost Price}} \times 100\%\]We need to find \(\frac{P_1}{CP} \times 100\).
From \(7 P_1 = SP_1\), we can substitute \(SP_1\) back into the expression for \(CP\) from step 6:
\[CP = SP_1 - P_1\]\[CP = 7 P_1 - P_1\]\[CP = 6 P_1\]Now, calculate the initial profit percentage:
\[\text{Initial Profit Percentage} = \frac{P_1}{CP} \times 100\%\]\[\text{Initial Profit Percentage} = \frac{P_1}{6 P_1} \times 100\%\]\[\text{Initial Profit Percentage} = \frac{1}{6} \times 100\%\]\[\text{Initial Profit Percentage} = 16.666...\%\]Rounding to two decimal places, this is \(16.67\%\).
D) 16.67
Simplify: [(26 ÷ \( \frac{4}{2} \)) × {8 ÷ 2 × (\( \frac{12-2}{5} \))} ÷ {4 × 4 ÷ 3}]
To simplify the given expression, we must follow the order of operations, commonly known as BODMAS/PEMDAS. This acronym stands for Brackets (Parentheses), Orders (Exponents), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).
First, simplify the innermost brackets/parentheses:
The expression is: \[[(26 \div \frac{4}{2}) \times \{8 \div 2 \times (\frac{12-2}{5})\} \div \{4 \times 4 \div 3\}\]
Calculate the term inside the first parenthesis: \(\frac{4}{2} = 2\]
Calculate the term inside the second parenthesis: \[\frac{12-2}{5} = \frac{10}{5} = 2\]
The expression becomes: \([(26 \div 2) \times \{8 \div 2 \times 2\} \div \{4 \times 4 \div 3\}\]
Next, simplify the terms inside the square brackets \([\ ]\) and curly braces \(\{\ \}\).
For the first part: \((26 \div 2) = 13\)
For the second part: \(\{8 \div 2 \times 2\}\)
Perform division first: \(8 \div 2 = 4\)
Then multiplication: \(4 \times 2 = 8\)
So, \(\{8 \div 2 \times 2\} = 8\)
For the third part: \(\{4 \times 4 \div 3\}\)
Perform multiplication first: \(4 \times 4 = 16\)
Then division: \(16 \div 3 = \frac{16}{3}\)
So, \(\{4 \times 4 \div 3\} = \frac{16}{3}\)
Substitute these simplified values back into the main expression:
The expression becomes: \([13 \times 8 \div \frac{16}{3}]\)
Finally, perform the remaining operations from left to right.
First, multiplication: \(13 \times 8 = 104\)
Then, division: \(104 \div \frac{16}{3}\)
Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[104 \times \frac{3}{16}\]
Simplify the expression:
\[\frac{104 \times 3}{16}\]
We can divide 104 by 16. \(104 \div 16 = 6.5\)
\[6.5 \times 3 = 19.5\]
Correct Option: B) 19.5
To solve this problem, we need to track Mr. Becks' movements in a coordinate system or by calculating the net displacement in the East-West and North-South directions. We will assume East is positive X and North is positive Y.
A) 5 km to the South. This matches our calculation that Mr. Becks needs to drive 5 km South from Point P to reach Point A.
The question asks to identify the role of 'Rajukas' during the Mauryan period. This requires knowledge of the administrative structure of the Mauryan Empire, particularly the officials mentioned in Ashokan edicts.
A) Provincial officials. Rajukas were important administrative and judicial officers appointed by Emperor Ashoka to govern large sections of the population and territories, making them provincial or district-level officials.
To solve this letter-number series, we need to identify the pattern in the letters and the numbers separately. We will analyze the progression of each letter position and the numerical sequence.
Correct Option: B) LOT58
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 answer the specific question about the number of people living above E.
Identify the floors: There are 6 floors, numbered 1 to 6 from bottom to top.
Clue 1: C lives on the top floor.
So, C is on Floor 6.
| Floor | Person |
|---|---|
| 6 | C |
| 5 | |
| 4 | |
| 3 | |
| 2 | |
| 1 |
Clue 2: B lives on an odd-numbered floor but not 3.
Odd-numbered floors are 1, 3, 5. Since B does not live on 3, B can live on Floor 1 or Floor 5.
Clue 3: The sum of floors on which B and D live is 7.
If B lives on Floor 1, then D lives on Floor \(7 - 1 = 6\). But Floor 6 is occupied by C. So, B cannot be on Floor 1.
Therefore, B must live on Floor 5. Then D lives on Floor \(7 - 5 = 2\).
| Floor | Person |
|---|---|
| 6 | C |
| 5 | B |
| 4 | |
| 3 | |
| 2 | D |
| 1 |
Clue 4: E lives immediately below F.
This means F and E must occupy two consecutive floors. The only consecutive floors remaining are 3 and 4.
| Floor | Person |
|---|---|
| 6 | C |
| 5 | B |
| 4 | F |
| 3 | E |
| 2 | D |
| 1 |
Determine the remaining person: The only person left is A, and the only floor left is 1.
So, A lives on Floor 1.
| Floor | Person |
|---|---|
| 6 | C |
| 5 | B |
| 4 | F |
| 3 | E |
| 2 | D |
| 1 | A |
Answer the question: How many people live above E?
E lives on Floor 3. The people living above Floor 3 are F (on Floor 4), B (on Floor 5), and C (on Floor 6). There are 3 people.
C) 3
The question asks to identify the correct statements regarding "Operation Brahma" launched in March 2025. This requires knowledge of a specific humanitarian aid operation by India. We need to evaluate each statement based on factual accuracy.
Given that "Operation Brahma" in March 2025 for an earthquake in Myanmar and Thailand is not a known real event, the question seems to be testing the candidate's general knowledge of India's HADR capabilities and typical deployments, rather than a specific event. In such questions, often some statements are factually plausible in a general sense, while others contain specific inaccuracies. If we assume the question is based on a hypothetical scenario, we need to identify the most plausible statements.
Let's re-examine the options. The correct answer is B) Only 2 and 3. This implies that statements 2 and 3 are considered correct within the context of the hypothetical operation. This suggests that the deployment of Indian Navy ships like Satpura and Savitri to Yangon with relief material (Statement 2) and the deployment of an Indian Army medical task force from the Shatrujeet Brigade (Statement 3) are plausible actions for India's HADR efforts, even if the operation name and specific disaster are fictional.
Statement 1 is incorrect because there was no such operation named "Brahma" in March 2025 for an earthquake in Myanmar and Thailand. Statement 4 is also incorrect, as setting up a 60-bed medical centre in Thailand as part of this fictional operation is not a known event, and the specific details make it less plausible than the general deployment of ships and medical teams.
B) Only 2 and 3