Question

If `Delta_(1) = |{:(x, "sin"theta, "cos"theta), (-"sin"theta, -x, 1), ("cos"theta, 1, x):}|`
`"and "Delta_(2) = |{:(x, "sin"2theta, "cos"2theta), (-"sin"2theta, -x, 1), ("cos"2theta, 1, x):}|, x ne,` then for all `theta in (0, (pi)/(2))`
A. `Delta_(1) + Delta_(2) = -2(x^(3) + x -1)`
B. `Delta_(1) - Delta_(2) = -2x^(3)`
C. `Delta_(1) + Delta_(2) = -2x^(3)`
D. `Delta_(1) - Delta_(2) = x("cos"2 theta - "cos" 4 theta)`

Answer 1

Correct Answer - C
Given determinants are
`Delta_(1) =|{:(x, "sin"theta, "cos"theta), (-"sin"theta, -x, 1), ("cos"theta, 1, x):}|`
`=-x^(3) + "sin"theta"cos"theta - "sin"theta"cos"theta + x"cos"^(2)theta - x + x"sin"^(2)theta`
`=-x^(3)`
`"and "Delta_(2) = |{:(x, "sin"2theta, "cos"2theta), (-"sin"2theta, -x, 1), ("cos"2theta, 1, x):}|,x ne 0`
` =-x^(3)` (similarly as `Delta_(1)`)
So, according to options, we get `Delta_(1) + Delta_(2) = -2x^(3)`