The parameter on which the value of the determinant `|1a a^2"cos"(p-d)xcosp x"cos"(p+d)x"sin"(p-d)xsinp x"sin"(p+d)x|` does not depend is `a` b. `p` c. `d` d. `x`

Options

A. a
B. p
C. d
D. x

Explanation

Let ` Delta = |{:(" "1 , " "a, " "a^(2)),("cos"(p-d)x, "cos "px, "cos"(p+d)x),("sin"(p-d)x, "sin "px, "sin"(p+d)x):}|`
Applying `C_(1) to C_(1) + C_(3)`
`rArr Delta = |{:(" "1+a^(2) , " "a, " "a^(2)),("cos"(p-d)x+"cos"(p+d)x, "cos "px, "cos"(p+d)x),("sin"(p-d)x + "sin"(p+d)x, "sin "px, "sin"(p+d)x):}|`
`rArr Delta = |{:(" "1+a^(2) , " "a, " "a^(2)),(2"cos"px"cos"dx, "cos"px, "cos"(p+d)x),(2"sin"px "cos"dx, "sin"px, "sin"(p+d)x):}|`
Applying `C_(1) to C_(1)-2 "cos"dxC_(2)`
`rArr Delta = |{:(1+a^(2)-2a"cos"dx , " "a, " "a^(2)),(0, "cos"px, "cos"(p+d)x),(0, "sin"px, "sin"(p+d)x):}|`
`rArr Detlta = (1+a^(2)-2a"cos"dx)["sin"(p+d)x"cos"px-"sin"px "cos"(p+d)x]`
`rArr Delta = (1+a^(2)-2a " cos "dx)"sin "dx`
which is independent of p.