3 and 4 .Determinants and Matrices
medium

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\{\cos (nx)}&{\cos (n + 1)x}&{\cos (n + 2)x}\\{\sin (nx)}&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}\,} \right|$ निर्भर नहीं करता है

A

$x$ पर

B

$ n$  पर

C

$x$ तथा $n$ दोनों पर

D

इनमें से कोई नहीं

Solution

(b) $\Delta = \left| {\,\begin{array}{*{20}{c}}1&1&1\\{\cos nx}&{\cos (n + 1)x}&{\cos (n + 2)x}\\{\sin nx}&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}\,} \right|$

${C_1} \to {C_1} + {C_3} – (2\cos x){C_2}$ के प्रयोग से,

$\Delta = \left| {\,\begin{array}{*{20}{c}}{2(1 – \cos x)}&1&1\\0&{\cos (n + 1)x}&{\cos (n + 2)x}\\0&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}\,} \right|$

$\Delta = 2(1 – \cos x)[\cos (n + 1)x\sin (n + 2)x$ $ – \cos (n + 2)x\sin (n + 1)x]$

$\Delta = 2(1 – \cos x)\,[\sin (n + 2 – n – 1)x]$ $ = 2\sin x(1 – \cos x)$

अर्थात  $\Delta $, $n$. से स्वतंत्र है ।

Standard 12
Mathematics

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