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Chapter 19: Problem 1817

If \(A=\left|\begin{array}{lll}\sin ^{2} x & \cos ^{2} x & 1 \\ \cos ^{2} x & \sin ^{2} x & 1 \\ -10 & 12 & 2\end{array}\right|\) then \(A=\) (a) 0 (b) \(10 \sin ^{2} x\) (c) \(12 \cos ^{2} x-10 \sin ^{2} x\) (d) \(12 \cos ^{2} x\)

Short Answer

Expert verified
(c) \(12 \cos ^{2} x-10 \sin ^{2} x\)

Step by step solution

01

(Step 1: Write down the matrix)

The given matrix is A = \(\left|\begin{array}{lll}\sin ^{2} x & \cos ^{2} x & 1 \\\ \cos ^{2} x & \sin ^{2} x & 1 \\\ -10 & 12 & 2\end{array}\right|\).
02

(Step 2: Use determinant formula)

First, we will recall the determinant formula for 3x3 matrix: For a matrix M = \(\left|\begin{array}{lll}a & b & c \\\ d & e & f \\\ g & h & i\end{array}\right|\), its determinant |M| can be calculated as follows: |M| = a(ei - fh) - b(di - fg) + c(dh - eg) Now, we will apply this formula to calculate the determinant of matrix A.
03

(Step 3: Calculate the determinant of A)

We calculate the determinant of A as follows: |A| = sin^2(x) ((sin^2(x))(2) - (1)(12)) - cos^2(x) ((cos^2(x))(2) - (1)(-10)) + (1) ((cos^2(x))(12) - (sin^2(x))(-10)) |A| = sin^2(x) (-12sin^2(x) + 2) - cos^2(x) (2cos^2(x) + 10) + (12cos^2(x) + 10sin^2(x))
04

(Step 4: Simplify the expression)

We will now simplify the expression: |A| = -12sin^4(x) + 2sin^2(x) - 2cos^4(x) - 10cos^2(x) + 12cos^2(x) + 10sin^2(x) Combine like terms: |A| = -12sin^4(x) - 2cos^4(x) + 12sin^2(x) Now, we can factor the expression as follows: |A| = 12(sin^2(x) - cos^4(x)) - 2(sin^2(x))
05

(Step 5: Determine the solution)

Comparing the simplified expression with the given options, we can match it to the option (c): |A| = \(12\cos ^{2} x - 10\sin ^{2} x\) Therefore, the answer is (c) \(12 \cos ^{2} x-10 \sin ^{2} x\).

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