The problem asks us to identify numbers in a given series that are immediately preceded by a symbol and immediately followed by a symbol. We need to scan the series from left to right and apply this condition to each number.
D) 1 — There is only one number (6) in the series that is immediately preceded by a symbol ('%') and immediately followed by a symbol ('&').
This is a logical reasoning problem involving arrangement. We need to determine the floor on which each person lives based on the given clues and then find the number of people between A and C.
List the floors: There are 6 floors, numbered 1 to 6.
| Floor No. | Person |
|---|---|
| 6 | |
| 5 | |
| 4 | |
| 3 | |
| 2 | |
| 1 |
Clue 1: B lives on a floor that is a prime number.
The prime numbers between 1 and 6 are 2, 3, and 5. So, B can live on floor 2, 3, or 5.
Clue 2: The product of floors on which B and D live is 3.
Since the floor numbers are integers, the only way their product can be 3 is if one lives on floor 1 and the other on floor 3. This means B and D must live on floors 1 and 3 (in some order).
Combining this with Clue 1 (B lives on a prime number floor), B must live on floor 3 (as 1 is not a prime number). Consequently, D must live on floor 1.
| Floor No. | Person |
|---|---|
| 6 | |
| 5 | |
| 4 | |
| 3 | B |
| 2 | |
| 1 | D |
Clue 3: Only 2 people live above C.
If 2 people live above C, it means C lives on the 3rd floor from the top. So, C lives on floor 4 (Floors 5 and 6 are above C).
| Floor No. | Person |
|---|---|
| 6 | |
| 5 | |
| 4 | C |
| 3 | B |
| 2 | |
| 1 | D |
Clue 4: F lives immediately below A.
This means A and F occupy two consecutive floors, with A above F. The only remaining consecutive floors are 5 and 6. So, A lives on floor 6 and F lives on floor 5.
| Floor No. | Person |
|---|---|
| 6 | A |
| 5 | F |
| 4 | C |
| 3 | B |
| 2 | E |
| 1 | D |
The only person left is E, and the only floor left is 2. So E lives on floor 2.
Final Arrangement:
Answer the question: How many people live between A and C?
A is on floor 6 and C is on floor 4. The person living between them is F (on floor 5).
Therefore, 1 person lives between A and C.
B) 1 — As determined in the step-by-step analysis, F lives on floor 5, which is between A (floor 6) and C (floor 4). Thus, there is 1 person between A and C.
The question asks about the final court of appeal for India during British rule. This requires knowledge of the judicial system established by the British in India.
B) Privy Council in London — The Judicial Committee of the Privy Council in London was indeed the highest court of appeal for cases originating from British India. Appeals from the High Courts and later the Federal Court of India could be taken to the Privy Council. This continued until 1949, when the Federal Court of India became the final court of appeal, shortly before the establishment of the Supreme Court of India in 1950.
To find the cost price, we need to work backward from the marked price. First, calculate the selling price after the discount. Then, use the profit percentage to find the cost price.
Identify the given values:
Calculate the Selling Price (SP) after the discount:
The discount is applied to the Marked Price.
\[ \text{Discount Amount} = \text{Discount Percentage} \times \text{MP} \] \[ \text{Discount Amount} = 31\% \times 5180 = \frac{31}{100} \times 5180 = 31 \times 51.80 = 1605.80 \]Selling Price (SP) = Marked Price - Discount Amount
\[ \text{SP} = 5180 - 1605.80 = 3574.20 \]Alternatively, SP can be calculated directly as:
\[ \text{SP} = \text{MP} \times \left(1 - \frac{\text{Discount Percentage}}{100}\right) \] \[ \text{SP} = 5180 \times \left(1 - \frac{31}{100}\right) = 5180 \times \frac{69}{100} = 51.80 \times 69 = 3574.20 \]So, the Selling Price is ₹3,574.20.
Calculate the Cost Price (CP) using the gain percentage:
The shopkeeper gains 5% on the Cost Price. This means the Selling Price is 105% of the Cost Price.
\[ \text{SP} = \text{CP} \times \left(1 + \frac{\text{Gain Percentage}}{100}\right) \] \[ 3574.20 = \text{CP} \times \left(1 + \frac{5}{100}\right) \] \[ 3574.20 = \text{CP} \times \left(\frac{105}{100}\right) \] \[ \text{CP} = \frac{3574.20 \times 100}{105} \] \[ \text{CP} = \frac{357420}{105} \] \[ \text{CP} = 3404 \]So, the Cost Price is ₹3,404.
C) ₹3,404 is the correct cost price after accounting for the discount and the gain.
This problem involves calculating the net filling rate of a cistern when there's an inlet tap filling it and an outlet hole emptying it simultaneously. We will determine the individual rates of filling and emptying, then find the effective filling rate, and finally calculate the total time required to fill the cistern.
C) 10 hours
This problem involves the concepts of speed in still water and the speed of the current. When a swimmer moves downstream, the speed of the current adds to their speed in still water. When moving upstream, the speed of the current subtracts from their speed in still water. We can set up two equations based on these relationships and solve for the speed of the swimmer in still water.
Let the speed of the swimmer in still water be \(S_{\text{swimmer}}\) (in km/hr).
Let the speed of the current be \(S_{\text{current}}\) (in km/hr).
When swimming downstream, the effective speed is the sum of the swimmer's speed in still water and the speed of the current. We are given this speed as 37 km/hr.
\[ S_{\text{swimmer}} + S_{\text{current}} = 37 \quad \text{(Equation 1)} \]When swimming upstream, the effective speed is the difference between the swimmer's speed in still water and the speed of the current. We are given this speed as 27 km/hr.
\[ S_{\text{swimmer}} - S_{\text{current}} = 27 \quad \text{(Equation 2)} \]To find the speed of the swimmer in still water, we can add Equation 1 and Equation 2:
\[ (S_{\text{swimmer}} + S_{\text{current}}) + (S_{\text{swimmer}} - S_{\text{current}}) = 37 + 27 \]\[ 2 \times S_{\text{swimmer}} = 64 \]Now, solve for \(S_{\text{swimmer}}\):
\[ S_{\text{swimmer}} = \frac{64}{2} \]\[ S_{\text{swimmer}} = 32 \text{ km/hr} \]C) 32 km/hr is the correct answer. This is the speed of the swimmer in still water, calculated by averaging the downstream and upstream speeds.
The problem presents a series of letter-number clusters. To find the missing cluster, we need to identify the pattern in the letters and the numbers separately.
Correct Option: A) OU49 — This option correctly follows all three identified patterns: the first letter sequence (G, I, K, M, O), the second letter sequence (M, O, Q, S, U), and the number sequence (17, 25, 33, 41, 49).
To simplify the given algebraic expression, we will expand the squared terms using the algebraic identity \((a-b)^2 = a^2 - 2ab + b^2\) and \((a+b)^2 = a^2 + 2ab + b^2\). Then, we will combine like terms to arrive at the simplified form.
Expand the first term \((3z - 9y)^{2}\) using the identity \((a-b)^2 = a^2 - 2ab + b^2\):
\[ (3z - 9y)^{2} = (3z)^2 - 2(3z)(9y) + (9y)^2 \]\[ = 9z^2 - 54zy + 81y^2 \]Expand the second term \((9z + 3y)^{2}\) using the identity \((a+b)^2 = a^2 + 2ab + b^2\):
\[ (9z + 3y)^{2} = (9z)^2 + 2(9z)(3y) + (3y)^2 \]\[ = 81z^2 + 54zy + 9y^2 \]Substitute these expanded forms back into the original expression:
\[ (9z^2 - 54zy + 81y^2) + (81z^2 + 54zy + 9y^2) - 81z^2 \]Combine the like terms:
The simplified expression is:
\[ 9z^2 + 90y^2 \]B) \( 9z^{2} + 90y^{2} \) is the correct simplified expression after expanding and combining like terms.
This is a syllogism problem. We need to evaluate the given conclusions based on the provided statements. We can use Venn diagrams or logical rules to determine if the conclusions logically follow from the statements.
Let's analyze each statement and conclusion:
Now let's evaluate the conclusions:
C) Both conclusions (I) and (II) follow.
As established above, both conclusion (I) and conclusion (II) can be logically derived from the given statements.
The problem asks us to set up an algebraic equation based on the given information: the product of two consecutive positive integers is 247, and the smaller integer is represented by \(x\). We need to find the correct quadratic equation that describes this relationship.
Identify the smaller integer: The problem states that the smaller of the two consecutive positive integers is \(x\).
Identify the next consecutive integer: Since the integers are consecutive, the integer immediately following \(x\) will be \(x+1\).
Formulate the product: The problem states that the product of these two consecutive integers is 247. So, we can write the equation:
\[ x(x+1) = 247 \]Expand the equation: Distribute \(x\) on the left side of the equation:
\[ x^2 + x = 247 \]Rearrange into standard quadratic form: To match the options, we need to move the constant term to the left side, setting the equation to zero:
\[ x^2 + x - 247 = 0 \]This is the required quadratic equation.
C) \( x^{2} + x - 247 = 0 \)
This option correctly represents the derived quadratic equation from the problem statement.