If Δ(x)=exsin2xtanx2ln (1+x)cosxsinxcosx2ex−1sinx2=A+Bx+Cx2+...., then B is equal to

Δ'(x)=|ex2cos2x2xsec2x2ln(1+x)cosxsinxcosx2ex−1sinx2|+|exsin2x2xsec2x21(1+x)−sinxcosxcosx2ex−1sinx2|+|exsin2xtanx2ln(1+x)cosxsinx−2xsinx2ex2xcosxe|=B+2Cx+......Put x=0,B=|120010100|+|10010−1100|+|100010010|=0

If Δ(x)=exsin2xtanx2ln (1+x)cosxsinxcosx2ex−1sinx2=A+Bx+Cx2+...., then B is equal to

Δ'(x)=|ex2cos2x2xsec2x2ln(1+x)cosxsinxcosx2ex−1sinx2|+|exsin2x2xsec2x21(1+x)−sinxcosxcosx2ex−1sinx2|+|exsin2xtanx2ln(1+x)cosxsinx−2xsinx2ex2xcosxe|=B+2Cx+......Put x=0,B=|120010100|+|10010−1100|+|100010010|=0

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If $Δ (x) = |\begin{matrix} e^{x} & \sin 2 x & \tan x^{2} \\ \ln (1 + x) & \cos x & \sin x \\ \cos x^{2} & e^{x} - 1 & \sin x^{2} \end{matrix}| = A + B x + C x^{2} + ....,$ then B is equal to

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Detailed Solution

$\begin{array}{l} Δ' (x) = | \begin{matrix} e^{x} & 2 \cos 2 x & 2 x \sec^{2} x^{2} \\ \ln (1 + x) & \cos x & \sin x \\ \cos x^{2} & e^{x} - 1 & \sin x^{2} \end{matrix} | + | \begin{matrix} e^{x} & \sin 2 x & 2 x \sec^{2} x^{2} \\ \frac{1}{(1 + x)} & - \sin x & \cos x \\ \cos x^{2} & e^{x} - 1 & \sin x^{2} \end{matrix} | + | \begin{matrix} e^{x} & \sin 2 x & \tan x^{2} \\ \ln (1 + x) & \cos x & \sin x \\ - 2 x \sin x^{2} & e^{x} & 2 x \cos x^{e} \end{matrix} | \\ = B + 2 C x + ...... \end{array}$

Put $x = 0, B = | \begin{matrix} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} | + | \begin{matrix} 1 & 0 & 0 \\ 1 & 0 & - 1 \\ 1 & 0 & 0 \end{matrix} | + | \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{matrix} | = 0$