Question

Match the following:

• A. P–I, Q–III, R–II
• B. P–II, Q–I, R–III
• C. P–III, Q–II, R–I
• D. P–I, Q–II, R–III

Hint:

In matching questions, you often do not need to solve all parts.
Start with the easiest component. Here, matching P to I immediately eliminates Option (B) and Option (C).
Then, evaluating either Q or R will instantly lead to the unique correct option, saving precious exam time.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:

This is a matching-type question where we need to evaluate three different mathematical expressions in List-I and match them with their corresponding results in List-II.

The expressions involve matrix determinants, indefinite integration, and differentiation.

Step 2: Key Formula or Approach:

We will evaluate each part independently using standard mathematical formulas:

1. For \(P\), the determinant of a \(2 \times 2\) matrix \[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] is: \[ ad - bc \]

2. For \(Q\), the power rule of integration is: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

3. For \(R\), the power rule of differentiation is: \[ \frac{d}{dx}(x^n) = n x^{n-1} \]

Step 3: Detailed Explanation:

Let us solve each item in List-I step-by-step.

Evaluating \(P\):

We need to find the determinant of the given identity matrix of order 2:

\[ A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]

Using the determinant formula:

\[ \det(A) = (1 \cdot 1) - (0 \cdot 0) = 1 - 0 = 1 \]

So, \(P\) matches with \(I\).

Evaluating \(Q\):

We need to evaluate the indefinite integral:

\[ \int 2x \, dx \]

Using the constant multiple rule and power rule of integration:

\[ \int 2x \, dx = 2 \int x^1 \, dx = 2 \left( \frac{x^{1+1}}{1+1} \right) + C = 2 \left( \frac{x^2}{2} \right) + C = x^2 + C \]

So, \(Q\) matches with \(III\).

Evaluating \(R\):

We need to find the derivative of \(x^2\) with respect to \(x\):

\[ \frac{d}{dx}(x^2) \]

Using the power rule of differentiation:

\[ \frac{d}{dx}(x^2) = 2x^{2-1} = 2x \]

So, \(R\) matches with \(II\).

Combining the Matches:

\(P \rightarrow I\)
\(Q \rightarrow III\) 
\(R \rightarrow II\)

Therefore, the correct matching combination is:

\(P\text{–}I,\; Q\text{–}III,\; R\text{–}II\)

Step 4: Final Answer:

The correct answer is Option (A).