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Determinants — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDeterminants1 Mark
MCQ
The value of the determinant $\begin{vmatrix}\text{a}^2&\text{a}&1\\\cos\text{nx}&\cos(\text{n}+1)\text{x}&\cos(\text{n}+2)\text{x}\\\sin\text{nx}&\sin(\text{n}+1)\text{x}&\sin(\text{n}+2)\text{x}\end{vmatrix}$ is independent of:
  • ✓
    $n$
  • B
    $a$
  • C
    $x$
  • D
    None of these.
✓

Answer

Correct option: A.
$n$
Let $A = nx, B = (n - 1)x, C = (n + 2)x$
$\Rightarrow C - B = x, B - A = x, C - A = 2x$
Thus, the given determinant is
$\begin{vmatrix}\text{a}^2&\text{a}&1\\\cos\text{A}&\cos\text{B}&\cos\text{C}\\\sin\text{A}&\sin\text{B}&\sin\text{C}\end{vmatrix}$
$=\text{a}^2(\cos\text{B}\sin\text{C}-\cos\text{C}\sin\text{B})-\text{a}\times(\cos\text{A}\sin\text{C}-\cos\text{C}\sin\text{A})+1\times(\cos\text{A}\sin\text{B}-\sin\text{A}\cos\text{B})$
$=\text{a}^2\sin(\text{C}-\text{B})-\text{a}\sin(\text{C}-\text{A})+\sin(\text{B}-\text{A})$
$=\text{a}^2\sin{\text{x}}-\text{a}\sin2\text{x}+\sin\text{x} [$Independent of $n]$

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