1. \(P^2 + Q^2\) varies as R.
2. PQ varies as R.
3. \(P^2 - Q^2\) varies as R.
Select the correct answer using the code given below:
Variation Laws Explained: P, Q, and R
This problem delves into the concept of variation, specifically direct variation. When a quantity \(A\) varies as another quantity \(B\), it signifies a direct proportionality, mathematically represented as \(A = kB\), where \(k\) is a non-zero constant of proportionality.
The problem provides the following information:
- \(P^2\) varies as \(R\), meaning \(P^2 = k_1 R\), where \(k_1\) is a constant. Since \(P^2\) is non-negative, \(k_1 ≥ 0\). For variation, we assume \(k_1 ≠ 0\).
- \(Q^2\) varies as \(R\), meaning \(Q^2 = k_2 R\), where \(k_2\) is a constant. Similarly, \(k_2 ≥ 0\), and we assume \(k_2 ≠ 0\).
- Additionally, \(P ≠ Q\).
We need to evaluate the variation of three expressions involving \(P\) and \(Q\) with respect to \(R\).
Analyzing Statements on P, Q, R Variation
Statement 1 Analysis: \(P^2 + Q^2\) and \(R\)
We substitute the given variation relationships:
- \(P^2 = k_1 R\)
- \(Q^2 = k_2 R\)
Adding these equations gives:
\(P^2 + Q^2 = k_1 R + k_2 R\)
Factoring out \(R\) yields:
\(P^2 + Q^2 = (k_1 + k_2) R\)
Let the combined constant be \(k_3 = k_1 + k_2\). Since \(k_1 > 0\) and \(k_2 > 0\), their sum \(k_3\) is also positive (\(k_3 > 0\)). This confirms that \(P^2 + Q^2\) varies directly as \(R\).
Result: Statement 1 is correct.
Statement 2 Analysis: \(PQ\) and \(R\) Variation
From the initial variations, we can write:
- \(P = ±√(k_1 R)\)
- \(Q = ±√(k_2 R)\)
Calculating the product \(PQ\):
\(PQ = (±√(k_1 R)) × (±√(k_2 R))\)
\(PQ = (±√(k_1 k_2)) R\)
Let \(k_4 = ±√(k_1 k_2)\). This \(k_4\) represents a constant value (positive or negative, depending on the signs of \(P\) and \(Q\)). Since \(k_1 > 0\) and \(k_2 > 0\), \(√(k_1 k_2)\) is a real positive number, making \(k_4\) a valid constant.
Therefore, \(PQ = k_4 R\), confirming that \(PQ\) varies directly as \(R\).
Result: Statement 2 is correct.
Statement 3 Analysis: \(P^2 - Q^2\) and \(R\) Variation
Using the given relationships:
- \(P^2 = k_1 R\)
- \(Q^2 = k_2 R\)
Subtracting the second from the first gives:
\(P^2 - Q^2 = k_1 R - k_2 R\)
Factoring out \(R\):
\(P^2 - Q^2 = (k_1 - k_2) R\)
This expression varies as \(R\) only if the constant of proportionality, \((k_1 - k_2)\), is non-zero. However, the problem conditions do not exclude the case where \(k_1 = k_2\). If \(k_1 = k_2\), then \(P^2 - Q^2 = (0) R = 0\). A value that is constantly zero does not exhibit variation in the context of proportionality.
Because this statement is not universally true under the given conditions, it is considered incorrect.
Result: Statement 3 is incorrect.
Overall Conclusion on Variation Statements
Our analysis shows that statements 1 and 2 hold true, while statement 3 does not hold true in all cases allowed by the problem statement.
Thus, the correct choice involves only statements 1 and 2.
Similar Questions
- If \(\frac{p+q}{q+r}=\frac{r+s}{s+p}\); \((q+r) \neq 0\), \((s+p) \neq 0\), then which one of the following is correct?
If \(a:b:c:d = \sqrt{4}: \sqrt{3}: \sqrt{2}: \sqrt{1}\), then what is the value of \(\frac{(-a^2 + b^2 + c^2 + d^2)}{(a^2 - b^2 + c^2 - d^2)}\) ?
- If \(m: n = 1:2\) and \(p:q=3:4\), then what is \((2m + 4p): (n + 3q)\) equal to ?
- If \((a+b): (b+c): (c+a) = 5:7:6\), then what is the value of \((a-b+c):(a+b-c)\)?
- ₹ 9400 is distributed among P, Q, R in such a way that if ₹ 93, ₹ 24, ₹ 55 are deducted from their respective shares, then they have money in the ratio \(3:4:5\). What is the share of P?
\(p\) varies directly as \((x^2 + y^2 + z^2)\). When \(x = 1, y = 2, z = 3\), then \(p = 70\). What is the value of \(p\) when \(x = -1, y = 1\), \(z = 5\)?
- A locomotive engine can go 40 km/hr. Its speed gets reduced by a quantity that varies directly as the square root of the number of wagons attached. It is known that its speed becomes 16 km/hr if 36 wagons are attached. What is the smallest number of wagons with which the engine is unable to move ?
- Two students X and Y appeared in a test. The score of X is 20 more than that of Y. If the score of X is 75% of the sum of the scores of X and Y, then what is the ratio of score of X to score of Y?
- If \(x\) varies directly as \(y\), then which of the following is/are correct ?
I. \((x^2+y^2)\) varies directly as \(y^2\).
II. \(\left(\frac{x^3}{y^2}\right)\) varies directly as \(y\).
Select the answer using the code given below : - Two numbers P and Q are such that P : Q = x : y. If 1 is added to both the numbers, the ratio becomes 7 : 10. If 3 is added to both the numbers, the ratio becomes 3 : 4. What is (x + y) : (x - y) equal to ?
Important Questions from Ratio and proportion
A sum of Rs. 2,500 is distributed among X, Y and Z in the ratio 1/2 : 3/4 : 5/6 . What is the difference between the maximum share and the minimum share?
In a class, there are three groups A, B and C. If one student from group A and two students from group B are shifted to group C, then what happens to the average weight of the students of the class?
How many different sums can be formed with the denominations Rs. 50, Rs. 100, Rs. 200, Rs. 500 and Rs. 2,000 taking at least three denominations at a time?
In a 500 metres race, B starts 45 metres ahead of A, but A wins the race while B is still 35 metres behind. What is the ratio of the speeds of A to B assuming that both start at the same time?
The monthly incomes of Peter and Paul are in the ratio of 4 : 3. Their expenses are in the ratio of 3 : 2. If each saves Rs. 6,000 at the end of the month, their monthly incomes respectively are (in Rs.)
