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.
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 our matches:
P matches with I.
Q matches with III.
R matches with II.
This gives the combination: P–I, Q–III, R–II. Step 4: Final Answer:
The correct matching combination is Option (A).
