The passage discusses how inflation is measured using aggregate indices like WPI and CPI. It highlights that a single inflation number is an average across various commodities and services, and that price rises differ for different items. Crucially, it points out that consumption baskets vary significantly for different economic classes (poor, middle, rich), making a single CPI an average of these diverse baskets. The question asks for the most logical, rational, and crucial message conveyed by the passage.
Correct Option: B) The present calculation of inflation rate does not correctly measure price rise of individual item/commodity.
The question asks to identify the statement that best reflects the crux of the provided passage. To do this, we need to read the passage carefully, identify its main theme and key arguments, and then evaluate each option against this understanding.
D) Trust is important though it entails risk.
This statement accurately captures the dual nature of trust as presented in the passage: its essential role in various domains (entrepreneurship, global markets) and the inherent vulnerability or risk of betrayal that comes with it.
To determine if Team P wins the match, we need to know the final scores of both Team P and Team Q. The question provides information about Team P being behind by 3 goals with 10 minutes remaining. We need to evaluate if the given statements provide enough information to calculate the final scores and compare them.
Analyze the initial situation:
With 10 minutes remaining, Team P was behind by 3 goals. Let \(P_{\text{initial}}\) be the score of Team P and \(Q_{\text{initial}}\) be the score of Team Q at that moment.
We know that \(Q_{\text{initial}} = P_{\text{initial}} + 3\).
Evaluate Statement I: Team P scored 4 goals in the last 10 minutes.
If Team P scored 4 goals, its final score would be \(P_{\text{final}} = P_{\text{initial}} + 4\).
However, we do not know how many goals Team Q scored in the last 10 minutes. Let \(Q_{\text{last 10 min}}\) be the goals scored by Team Q in the last 10 minutes.
Then, \(Q_{\text{final}} = Q_{\text{initial}} + Q_{\text{last 10 min}}\).
Since we don't know \(P_{\text{initial}}\) or \(Q_{\text{last 10 min}}\), we cannot determine the final scores of both teams, nor can we compare them to see if Team P won. Therefore, Statement I alone is not sufficient.
Evaluate Statement II: Team Q scored a total of 4 goals in the match.
This statement gives us Team Q's final score: \(Q_{\text{final}} = 4\).
However, we do not know how many goals Team P scored in total. We only know that at some point, Team P was 3 goals behind. This statement does not tell us Team P's score at that time or how many goals Team P scored in total. Therefore, Statement II alone is not sufficient.
Evaluate both Statements I and II together:
From Statement II, we know \(Q_{\text{final}} = 4\).
From Statement I, we know Team P scored 4 goals in the last 10 minutes.
We still don't know the scores of Team P and Team Q before the last 10 minutes. Let's assume \(P_{\text{initial}}\) and \(Q_{\text{initial}}\) are the scores before the last 10 minutes.
We know \(Q_{\text{initial}} = P_{\text{initial}} + 3\).
Team P's final score: \(P_{\text{final}} = P_{\text{initial}} + 4\).
Team Q's final score: \(Q_{\text{final}} = Q_{\text{initial}} + (\text{goals scored by Q in last 10 min})\).
We are given \(Q_{\text{final}} = 4\). So, \(4 = Q_{\text{initial}} + (\text{goals scored by Q in last 10 min})\).
We have too many unknowns. We don't know \(P_{\text{initial}}\) (and thus \(Q_{\text{initial}}\)), nor do we know how many goals Team Q scored in the last 10 minutes. For example:
Since we can get different outcomes (draw or win) depending on the unknown initial scores and goals scored by Q in the last 10 minutes, even both statements together are not sufficient to answer the question.
D) The Question cannot be answered even using any of the Statements.
The question asks whether the expression \((p+q)^2 - 4pq\) is positive, given that \(p\) and \(q\) are natural numbers. We need to simplify the expression first and then evaluate if the given statements are sufficient to determine its sign.
Simplify the expression:
The given expression is \((p+q)^2 - 4pq\).
Expand \((p+q)^2\):
\[ (p+q)^2 = p^2 + 2pq + q^2 \]Substitute this back into the expression:
\[ (p^2 + 2pq + q^2) - 4pq = p^2 - 2pq + q^2 \]Recognize this as the square of a difference:
\[ p^2 - 2pq + q^2 = (p-q)^2 \]So, the question is equivalent to asking: Is \((p-q)^2\) positive?
Analyze the simplified expression:
We know that \(p\) and \(q\) are natural numbers. Natural numbers are positive integers \(\{1, 2, 3, \dots\}\).
The square of any real number is always non-negative. That is, \((p-q)^2 \ge 0\).
For \((p-q)^2\) to be positive, we need \((p-q)^2 > 0\). This occurs if and only if \(p-q \ne 0\), which means \(p \ne q\).
Evaluate Statement I: \(p \ne q\)
Statement I says \(p \ne q\). If \(p \ne q\), then \(p-q \ne 0\). Consequently, \((p-q)^2\) must be strictly positive (since it cannot be negative).
Therefore, Statement I alone is sufficient to answer the question (Yes, it is positive).
Evaluate Statement II: \(q \ne p\)
Statement II says \(q \ne p\). This is algebraically identical to \(p \ne q\).
If \(q \ne p\), then \(p-q \ne 0\). Consequently, \((p-q)^2\) must be strictly positive.
Therefore, Statement II alone is also sufficient to answer the question (Yes, it is positive).
B) The Question can be answered by using either Statement alone.
Both Statement I and Statement II individually provide the condition \(p \ne q\), which is necessary and sufficient for \((p-q)^2\) to be positive. Thus, either statement alone is sufficient to answer the question.
To find the maximum value of the product \(p \times q\) given \(p + q = 10\), we can express \(q\) in terms of \(p\) (or vice versa) and then analyze the resulting quadratic function. The maximum value of a quadratic function \(f(x) = ax^2 + bx + c\) occurs at \(x = -b/(2a)\) if \(a < 0\). We will apply this principle for both Value-I and Value-II, considering the given constraints on \(p\) and \(q\).
Given \(p + q = 10\), we can write \(q = 10 - p\). The product \(P = p \times q\) becomes:
\[ P = p(10 - p) = 10p - p^2 \]This is a quadratic function \(P(p) = -p^2 + 10p\), which is a downward-opening parabola (since the coefficient of \(p^2\) is negative). Its maximum value occurs at the vertex.
For a quadratic function \(ax^2 + bx + c\), the x-coordinate of the vertex is \(-b/(2a)\). Here, \(a = -1\) and \(b = 10\).
\[ p_{\text{vertex}} = -\frac{10}{2(-1)} = -\frac{10}{-2} = 5 \]When \(p = 5\), then \(q = 10 - 5 = 5\).
The vertex occurs at \(p = 5\). Since \(p=5\) is a positive integer, and \(q=5\) is also a positive integer, this point is within the allowed domain. The maximum product is:
\[ \text{Value-I} = 5 \times 5 = 25 \]We still have \(P(p) = -p^2 + 10p\), and the vertex is at \(p = 5\). We need to check if this \(p\) value satisfies the given constraints. If \(p=5\), then \(q = 10 - 5 = 5\).
Since the vertex \(p=5\) (which gives the global maximum of the quadratic) falls within the allowed range for \(p\) and \(q\), the maximum value of \(p \times q\) under these constraints is also the value at the vertex.
\[ \text{Value-II} = 5 \times 5 = 25 \]We found that Value-I = 25 and Value-II = 25.
Therefore, Value-I = Value-II.
Correct Option: C) Value-I = Value-II
We need to calculate two values: Value-I, the minimum average of 11 consecutive integers greater than or equal to -5, and Value-II, the minimum product of 11 consecutive non-negative integers. Then we will compare these two values.
We have a set of 11 consecutive integers, and they must be \(\ge -5\). To minimize their average, we need to choose the smallest possible integers. The smallest possible starting integer is -5.
So, the set of 11 consecutive integers will be:
\[ \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\} \]The average of a set of consecutive integers is equal to the median (the middle term) if the number of terms is odd. Here, there are 11 terms, so the 6th term is the median.
The 6th term in the set \(\{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}\) is 0.
Alternatively, we can sum the terms and divide by 11:
\[ \text{Sum} = (-5) + (-4) + \dots + 4 + 5 \]The negative terms cancel out the positive terms: \((-5+5) + (-4+4) + \dots + (-1+1) + 0 = 0\).
So, the average is:
\[ \text{Average} = \frac{0}{11} = 0 \]Therefore, Value-I = 0.
We need the minimum value of the product of 11 consecutive non-negative integers. Non-negative integers include 0, 1, 2, 3, ...
To minimize the product, we must include 0 in the set of consecutive integers. If 0 is one of the numbers in the set, the product of all numbers will be 0.
The smallest possible set of 11 consecutive non-negative integers that includes 0 would be:
\[ \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]The product of these numbers is:
\[ \text{Product} = 0 \times 1 \times 2 \times \dots \times 10 = 0 \]Any other set of 11 consecutive non-negative integers (e.g., starting from 1: \(\{1, 2, \dots, 11\}\)) would result in a positive product, which is greater than 0.
Therefore, Value-II = 0.
We found Value-I = 0 and Value-II = 0.
Thus, Value-I = Value-II.
C) Value-I = Value-II. Both calculated values are 0.
This problem involves understanding the definition of an average and using algebraic manipulation to find the value of an unknown variable. We will set up equations based on the given information and solve for 'r'.
Define the average: The average of three numbers p, q, and r is k. This can be written as:
\[ \frac{p + q + r}{3} = k \]Express the sum in terms of k: Multiply both sides by 3:
\[ p + q + r = 3k \]Use the given relationship between p, q, and the average: We are told that p is as much more than the average (k) as q is less than the average (k). This can be expressed as:
\[ p - k = k - q \]Rearrange the relationship: Add q to both sides and add k to both sides to group p and q on one side and k on the other:
\[ p + q = k + k \]\[ p + q = 2k \]Substitute into the average equation: Now substitute the value of \(p + q\) from step 4 into the equation from step 2:
\[ (p + q) + r = 3k \]\[ 2k + r = 3k \]Solve for r: Subtract \(2k\) from both sides:
\[ r = 3k - 2k \]\[ r = k \]Thus, the value of r is k.
Correct Option: A) k
The problem asks us to determine the correctness of two statements involving a real number \(x\) such that \(0 < x < 1\). We will analyze each statement individually by using properties of numbers between 0 and 1 or by testing a specific value of \(x\).
Let's analyze each statement:
Statement I: \(x^2 > x^3\)
Statement II: \(x > \sqrt{x}\)
Based on the analysis, only Statement I is correct.
A) I only. Statement I is correct, and Statement II is incorrect.
The problem asks us to determine the number of natural numbers divisible by 5 that lie between two natural numbers whose difference is 10. We need to consider different pairs of natural numbers with a difference of 10 to see if the count of numbers divisible by 5 remains constant or varies.
Correct Option: C) There can be more than one such number.