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).