Problem 30 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\Delta(x)=\left|\begin{array}{ccc}x & 1+x^{2} & x^{3} \\ \log \left(1+x^{2}\right) & e^{x} & \sin x \\ \cos x & \tan x & \sin ^{2} x\end{array}\right|\) then (A) \(\Delta(x)\) is divisible by \(x\) (B) \(\Delta(x)=0\) (C) \(\Delta^{\prime}(x)=0\) (D) None of these

Step-by-Step Solution

Answer

\( \Delta(x) = 0 \).

Step 1: Introduce the Determinant

The given exercise involves the determinant \( \Delta(x) \) expressed as a 3x3 matrix. To determine characteristics of \( \Delta(x) \), we need to evaluate the determinant and understand its properties.

Step 2: Apply Properties of Determinants

Examine the matrix \( \Delta(x) \) to see if any rows or columns have a common factor or any simplification. Look for linear dependencies between rows or columns since zeros can be simplified using them.

Step 3: Evaluate Row and Column Properties

Notice that each term, apart from the linear transformations of trigonometric and exponential functions, has powers of \( x \). Start row reducing or expanding for partial simplification or determinant adjustments. Check if \( x \) factors any of these terms.

Step 4: Calculate the Determinant if Necessary

Calculate the determinant, using cofactor expansion or direct row manipulation if necessary. Assess if the expression becomes zero or otherwise confirms properties such as divisibility by \( x \).

Step 5: Conclusion

After analyzing, rewriting or simplifying the determinant \( \Delta(x) \) and its potential factors or derivatives in context of given options A, B, C, and D. Determine which option holds true based on mathematical analysis.

Key Concepts

Determinant Properties

Determinants are a crucial aspect of linear algebra, fundamentally linked to square matrices. They provide insights into various matrix properties and are key in system solutions, eigenvalues, and eigenvectors. Recognizing determinant properties can greatly simplify problems or provide direct answers.

• Scaling: If a row or column of a matrix is multiplied by a scalar, the determinant is also multiplied by this scalar.
• Additivity: The determinant of the sum of two matrices is generally not the sum of the determinants. However, determinants can distribute over addition for specific transformations during expansion.
• Linear Dependence: If two or more rows (or columns) are linearly dependent, then their determinant is zero. This implies possible simplifications in computation.
• Column and Row Swaps: Swapping two rows (or columns) inverts the sign of the determinant, which can be used strategically to simplify calculations.

Understanding these properties allows for a more strategic approach when determining the characteristics of a function like \( \Delta(x) \), especially in analyzing divisibility or examining matrix elements.

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a method used to calculate the determinant of a matrix. This approach particularly shines with 3x3 or larger matrices, where direct evaluation can become cumbersome.
Begin by selecting a row or column to expand along. Any row or column can be chosen, but typically, it's efficient to choose one with the most zeros to simplify calculations. For the matrix \( \Delta(x) \), consider examining elements with simpler algebraic forms.

• Cofactor Calculation: Each element in the selected row or column will have a cofactor. A cofactor is computed by deleting the row and column of the chosen element, and calculating the determinant of the resulting minor matrix, then multiplying it by \((-1)^{i+j}\), where \(i, j\) are the element's row and column indices.
• Expansion: Sum the products of the elements and their respective cofactors over the row or column selected. This yields the determinant of the matrix.

Using cofactor expansion can reveal simplifications or confirm properties without the need for full determinant calculation, especially for complex matrices involving variables like \( x \).

Row Reduction

Row reduction is a technique used in simplifying the determinant calculation of matrices, particularly useful when the matrix elements are complex functions or expressions.
The core idea is to use elementary row operations to simplify the matrix without altering its determinant, thereby making it easier to compute.

• Elementary Row Operations: These include swapping rows, multiplying a row by a non-zero scalar, and adding/subtracting a multiple of one row to another. While performing these operations:
  • Multiplying a row by a constant will multiply the determinant by that constant.
  • Swapping two rows changes the sign of the determinant.
  • Adding a multiple of one row to another does not change the determinant value.
• Pivot Strategy: Focus on transforming matrix elements into simpler units (like 0 or 1) using these operations strategically. Identify terms that cancel under common operations or reflect linear relationships.

By applying row reduction to the matrix \( \Delta(x) \), trigonometric transformations and other expressions might unravel their inherent dependencies or simplifications. This clarifies the matrix's behavior or confirms characteristics like zero determinants or divisibility.

Trigonometric Functions

In the context of determinants, trigonometric functions within matrix elements can add complexity due to their cyclical and non-linear properties.
Recognizing these properties is key when evaluating determinants involving such functions.

• Simplifications: Use identities like \( \sin^2 x + \cos^2 x = 1 \) to reduce terms. Identifying common angles or transformations (e.g., \( \sin 2x = 2 \sin x \cos x \)) can simplify matrix entries.
• Linear Dependence: Checking whether combinations of angles or trigonometric transformations produce dependent rows or columns may indicate a zero determinant or provide simplification opportunities.
• Boundary Conditions: At key values (e.g., 0, \(\pi/2\)), trigonometric functions assume definitive values. This can be used strategically during evaluations to simplify calculations or verify properties.

These concepts, when applied correctly, allow the evaluator to handily manage matrices dominated by trigonometric elements—simplifying the determinant evaluation or confirming underlying mathematical properties.