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

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

\(\Delta(x)\) is divisible by \(x\); option (A) is correct.

Step 1: Write the Determinant

We begin by writing the determinant of the matrix \( \Delta(x) \) which is \( \Delta(x) = \left|\begin{array}{ccc}x & 1+x^{2} & x^{3} \ \log \left(1+x^{2}\right) & e^{x} & \sin x \ \cos x & \tan x & \sin^{2} x\end{array}\right| \).

Step 2: Analyze Rows and Columns

Observe the elements in the first row of the determinant: they contain \(x\), \(x^2\), and \(x^3\). Factor \(x\) out from the first row to see if \(\Delta(x)\) is divisible by \(x\).

Step 3: Factor the Determinant by x

Factor out \(x\) from the first row:\[ \Delta(x) = x \left| \begin{array}{ccc} 1 & \frac{1+x^2}{x} & x^2 \ \log (1+x^2) & e^x & \sin x \ \cos x & \tan x & \sin^2 x \end{array} \right| \]This shows that the entire determinant is divisible by \(x\).

Step 4: Conclusion

Since \(\Delta(x)\) can be factored such that \(x\) is common in the first row, \(\Delta(x)\) is divisible by \(x\). This means the correct answer is option (A).

Key Concepts

Matrix Properties

Mathematical matrices play a crucial role in various applications from computer science to physics. A matrix is essentially a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Here, we'll explore the key properties of matrices relevant to the given problem.

One important property is the **determinant**, which is a scalar value that can be computed from a square matrix. It offers insights into matrix characteristics, such as whether it's invertible or not. A determinant of zero implies the matrix is not invertible, which can have significant implications in solving systems of linear equations.

**Determinants** have specific calculation methods depending on the matrix size. For a 3x3 matrix, the determinant is calculated through the cross multiplication of elements. This often involves the use of the formula:

\[\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]

where each letter represents elements of the matrix. Understanding these basic matrix properties is essential for examining conditions like invertibility or evaluating expressions such as the given problem's determinant.

Divisibility in Determinants

Divisibility within determinants involves checking if there's a common factor in one or more rows or columns, which simplifies the determinant calculation.

In the exercise, the problem involved factoring the first row of the determinant to check for divisibility by a particular variable, here, **x**. This method requires:

• Identifying elements that share a common factor.
• Factoring this out, which simplifies and might completely cancel terms in some rows or columns, as seen by the ability to factor out **x**.

By recognizing that the first row of the determinant \( \Delta(x) \) includes terms like \( x \), \( x^2 \), and \( x^3 \), it becomes apparent that these terms can be factored by **x**. This reveals that \( \Delta(x) \) can be expressed as \( x \times \text{det}(A') \) where \( A' \) is a simplified matrix. Divisibility by **x** then results in the zero determinant if factoring cancels all elements, allowing the solution to assert that \( \Delta(x) \) is divisible by **x**.

Factorization Techniques

Factorization is a mathematical process of breaking down expressions into a product of simpler expressions which can provide beneficial insights into problems involving determinants.

In matrix problems, especially when looking at divisibility, factorization can simplify the overall problem-solving process. This is primarily due to factorization's ability to reduce complexity and reveal underlying properties that are not immediately obvious.

**Steps in Factorization**:

• Identify terms with a common variable or factor. In \( \Delta(x) \), the presence of terms like \( x \), \( x^2 \), and \( x^3 \) in the first row was a strong indicator.
• Extract the common factor. In this exercise, factoring out \( x \) aided in simplifying the determination of divisibility.
• Simplify the remaining matrix. Often, extracting the factor simplifies operations and can show divisibility or other useful properties.

Such factorization techniques are not only useful in determinants but also across various branches of algebra, making it a versatile tool in mathematical analysis.