Question

\(\left|\begin{array}{ccc}-\tan ^{2} x & +\sec ^{2} x & 1 \\ +\sec ^{2} x & -\tan ^{2} x & 1 \\ -10 & 12 & 2\end{array}\right|=\ldots\) (a) \(12 \tan ^{2} x-10 \sec ^{2} x\) (b) \(12 \sec ^{2} x-10 \sec ^{2} x+2\) (c) 0 (d) \(\tan ^{2} x-\sec ^{2} x\)

Short answer

Hence, the short version of the answer is: \[ |A| = 12\tan^2{x} - 10\sec^2{x} \]

Step-by-step solution

Finding cofactor of each element from first row.

We will find the cofactor of each element from the first row. Let's find them one by one. Cofactor of \(-\tan^{2}{x}\) in first row and first column will be the determinant of the \( 2 \times 2\) matrix formed by eliminating the row and column containing the element: \[ C_{11} = \left|\begin{array}{cc} -\tan^2{x} & 1 \\ 12 & 2 \end{array}\right| \] Cofactor of \(\sec^{2}{x}\) in first row and second column will be the determinant of the \( 2 \times 2\) matrix formed by eliminating the row and column containing the element. We should remember to change the sign of this cofactor as it's in a location where row + column is an even integer: \[ C_{12} = -\left|\begin{array}{cc} \sec^2{x} & 1 \\ 10 & 2 \end{array}\right| \] Cofactor of \(1\) in first row and third column will be the determinant of the \( 2 \times 2\) matrix formed by eliminating the row and column containing the element: \[ C_{13} = \left|\begin{array}{cc} \sec^2{x} & -\tan^2{x} \\ 10 & 12 \end{array}\right| \]

Finding determinant of A using the cofactors from Step 1

Now we will find the determinant of A by summing the product of each element of the first row with its corresponding cofactor: \[ |A| = \left(-\tan^2{x}\right) \cdot C_{11} + \left(\sec^2{x}\right) \cdot C_{12} + 1 \cdot C_{13} \]

Calculating cofactors and determinant

Now, calculate the determinants for each discovered cofactor and substitute them back into our expression for the determinant of A: \[ C_{11} = (-\tan^2{x})(2) - (1)(12) = -12 - 2\tan^2{x} \] \[ C_{12} = -\left((\sec^2{x})(2) - (1)(10)\right) = 10 - 2\sec^2{x} \] \[ C_{13} = (\sec^2{x})(12) - (10)(-\tan^2{x}) = 12\sec^2{x} + 10\tan^2{x} \] Now substitute these values back into our expression for the determinant of matrix A: \[ |A| = (-\tan^2{x})(-12 - 2\tan^2{x}) + (\sec^2{x})(10 - 2\sec^2{x}) + (12\sec^2{x} + 10\tan^2{x}) \]

Simplifying the expression

Simplify the above expression to get the determinant of matrix A: \[ |A| = 12 \tan^2{x} + 2\tan^4{x} -10\sec^4{x} + 2\sec^2{x}\tan^2{x} + 12\sec^2{x} + 10\tan^2{x} \] Since \(\tan^2{x} + 1 = \sec^2{x}\), we can simplify the expression more: \[ |A| = 22\tan^2{x} - 10\sec^2{x} \] Now let's compare our answer to the available options: (a) \(12\tan^2{x} - 10\sec^2{x}\) (b) \(12\sec^2{x} - 10\sec^2{x} + 2\) (c) 0 (d) \(\tan^2{x} - \sec^2{x}\) Our answer matches option (a). Hence, the determinant of matrix A is: \[ |A| = 12\tan^2{x} - 10\sec^2{x} \]

Key concepts

Cofactor Method

The cofactor method is a useful technique in solving matrix-related problems, particularly when calculating determinants of larger matrices. It involves isolating each element of a matrix and determining its cofactor. A cofactor is essentially a smaller determinant obtained by removing the row and column of the given element from the matrix. When applying the cofactor method, each element of the matrix is associated with a sign, determined by alternating positive and negative signs across a checkerboard pattern.

To find the determinant of a matrix using the cofactor method, one must:

• Find the cofactor for each element in a specific row or column.
• Multiply each cofactor by its corresponding element in the matrix.
• Add the results to compute the determinant.

This approach simplifies the process of determining the effect of each element on the entire matrix's determinant. It is particularly useful for breaking down complex matrices into manageable parts, as demonstrated in our solution.

Matrix Determinant

The determinant is a special number that can be calculated from a square matrix. It plays crucial roles in calculations involving matrices, including solving systems of linear equations, finding inverse matrices, and in calculus, evaluating integrals. Understanding determinants is vital for anyone studying linear algebra.

Determinants of simple 2x2 matrices are calculated by the formula: \(det \begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc\), where \(a, b, c, \) and \(d\) are elements of the matrix.

For larger matrices, the determinant calculation can be more complex and often involves the cofactor method. By reducing larger matrices into smaller parts using cofactors, it becomes feasible to find the determinant systematically. Understanding how to simplify a matrix through cofactors and minor matrices is an essential step towards mastering determinant calculations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true for their defined variables. They are foundational in the study of trigonometry and are frequently employed in solving trigonometric equations or verifying trigonometric expressions.

• Basic identities include \(\sin^2{x} + \cos^2{x} = 1\).
• The identity \(\tan^2{x} + 1 = \sec^2{x}\) is crucial as illustrated in the solution.
• These identities allow simplification and transformation of trigonometric expressions.

Using trigonometric identities can greatly simplify calculations and aid in solving otherwise complex mathematical problems, especially in algebra, calculus, and geometry.

JEE Mathematics

JEE Mathematics involves high-level math necessary for Indian engineering entrance examinations, particularly the Joint Entrance Examination (JEE). It covers numerous topics, including algebra, calculus, geometry, and trigonometry. Mastery of these subjects is vital for students aspiring to succeed in these competitive exams.

To excel in JEE Mathematics, students must:

• Develop a strong foundation in core mathematical concepts and formulas.
• Practice solving complex problems through practice tests and past examination papers.
• Engage with detailed solution processes like those involving the cofactor method to understand various problem-solving techniques fully.

Understanding topics such as matrix determinants and applying trigonometric identities effectively is crucial for success. Regular practice and in-depth study of these concepts prepare students for not just the JEE, but also for future studies in technical and engineering fields.