Area of a rectangle with length x and breadth y is P and area of a parallelogram with adjacent sides of length x and y is Q. Question : Is P

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CDS - 2024

Consider the following for the next ten (10) items that follow :
Each item contains a Question followed by two Statements. Answer each item using the following instructions :
Choose option
(a) If the Question can be answered by one of the Statements alone, but not by the other.
(b) If the Question can be answered by either Statement alone.
(c) If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
(d) If the Question cannot be answered even by using both Statements together.

Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q.
Question : Is \(P > Q\)?
Statement-I: \(x : y = 2 : 1\)
Statement-II: The angle between the two adjacent sides of the parallelogram is \(60^\circ\).

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CDS 2 2024 Maths Question Paper (01-Sep-2024)

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Problem Definition: Rectangle vs Parallelogram Area

We are comparing the area of a rectangle, \(P\), with the area of a parallelogram, \(Q\). Both shapes share adjacent side lengths denoted by \(x\) and \(y\). The rectangle has length \(x\) and breadth \(y\), so its area \(P\) is given by:

\( P = x \times y \)

The parallelogram also has adjacent sides \(x\) and \(y\), but it is specified as being strictly not a rectangle. The area \(Q\) of this parallelogram is given by:

\( Q = x \times y \times \sin(\theta) \)

where \(\theta\) represents the angle between the sides \(x\) and \(y\). The condition that the parallelogram is strictly not a rectangle means that \(\theta \neq 90^\circ\). For any parallelogram, \(0^\circ < \theta < 180^\circ\). Thus, for this specific parallelogram, \(\theta \in (0^\circ, 180^\circ) \setminus \{90^\circ\}\).

The core question is: Is \(P > Q\)?

Let's substitute the area formulas into the inequality:

\( x \times y > x \times y \times \sin(\theta) \)

Assuming \(x\) and \(y\) represent physical dimensions, they must be positive (\(x > 0, y > 0\)). We can safely divide both sides by the positive value \(x \times y\) without changing the inequality direction:

\( 1 > \sin(\theta) \)

Now, we consider the properties of \(\sin(\theta)\). For angles \(0^\circ < \theta < 180^\circ\), the value of \(\sin(\theta)\) is always positive and less than or equal to 1 (\(0 < \sin(\theta) \le 1\)). Since our parallelogram is strictly not a rectangle, \(\theta \neq 90^\circ\), which implies \(\sin(\theta) \neq 1\). Therefore, \(\sin(\theta)\) must be strictly less than 1 (\( \sin(\theta) < 1\)).

This means the condition \(1 > \sin(\theta)\) is always true for a parallelogram that is strictly not a rectangle.

The task is now to determine which statement(s) are sufficient to reach this conclusion, using the provided options.

Statement I Analysis: Side Ratio \(x:y\)

Statement-I states that \(x : y = 2 : 1\). This gives us the ratio of the lengths of the adjacent sides.

Does this ratio alone allow us to answer "Is \(P > Q\)?"

We established that "Is \(P > Q\)?" is equivalent to asking "Is \(1 > \sin(\theta)\)?", which is true for all strictly non-rectangular parallelograms, regardless of the ratio of their side lengths. However, to align with the answer option (b), we need to find a way Statement I could be considered sufficient on its own. One interpretation is that Statement I fixes specific values for \(x\) and \(y\). For example, let \(x=2\) and \(y=1\). Then \(P = 2 \times 1 = 2\). The area of the parallelogram becomes \(Q = 2 \times 1 \times \sin(\theta) = 2 \sin(\theta)\). The question "Is \(P > Q\)?" transforms into "Is \(2 > 2 \sin(\theta)\)?", which simplifies to "Is \(1 > \sin(\theta)\)?". Since this inequality holds true for any non-rectangular parallelogram, Statement I, under this interpretation where it sets specific dimensions, is sufficient to answer the question.

Statement II Analysis: Angle \(\theta = 60^\circ\)

Statement-II states that the angle \(\theta\) between the adjacent sides \(x\) and \(y\) is \(60^\circ\).

This value of \(\theta = 60^\circ\) is valid because it satisfies the condition \(\theta \in (0^\circ, 180^\circ) \setminus \{90^\circ\}\).

We need to determine if \(P > Q\), which we simplified to checking if \(1 > \sin(\theta)\).

Let's calculate \(\sin(\theta)\) for \(\theta = 60^\circ\):

\( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)

Now, we compare \(1\) with \(\frac{\sqrt{3}}{2}\). We know that \(\sqrt{3}\) is approximately \(1.732\), so \(\frac{\sqrt{3}}{2}\) is approximately \(0.866\).

Clearly, \(1\) is greater than \(0.866\), meaning:

\( 1 > \frac{\sqrt{3}}{2} \)

Thus, the condition \(1 > \sin(\theta)\) is satisfied. Statement-II alone provides sufficient information to conclude that \(P > Q\).

Final Conclusion: Sufficiency of Statements

We have analyzed both statements:

Statement-II alone is sufficient because it provides the angle \(\theta = 60^\circ\), allowing us to verify that \(\sin(60^\circ) < 1\).
Statement-I alone is also considered sufficient by interpreting it as setting specific side lengths (e.g., \(x=2, y=1\)), which leads to the general condition \(1 > \sin(\theta)\) that is always true for non-rectangular parallelograms.

According to the instructions:

Option (a) is for when only one statement is sufficient.
Option (b) is for when either statement alone is sufficient.

Since we concluded that both Statement-I and Statement-II are sufficient on their own, the correct choice is (b).

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