If Wigtriangleup - Vegin vmatrix e - x -sin x - 1cos x -ln1-x-2 - 1 x - x-2- 1lend vmatrix -a-b x-c x -2 then the value of b isA- 0B- -1C- -2D- None of these

The correct option is B 1 △ '= e^x amp; sinx amp; 1 sinx amp; ln(1+x^2) amp; 1 1 amp; x^2 amp; 1 + e^x amp; cosx amp; 1 cosx ...

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Byju's Answer

Standard XII

Mathematics

Algebra of Derivatives

If Wigtriangl...

Question

If $△= ∣ ∣ ∣ ∣ ∣ \begin{matrix} e^{x} & s i n x & 1 c o s x & l n (1 + x^{2}) & 1 x & x^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = a + b x + c x^{2}$ then the value of b is

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-1

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-2

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None of these

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Solution

The correct option is B
-1

$△^{'} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} e^{x} & s i n x & 1 - s i n x & l n (1 + x^{2}) & 1 1 & x^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} e^{x} & c o s x & 1 c o s x & \frac{2 x}{1 + x^{2}} & 1 x & 2 x & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ + 0 = b + 2 c x$
Now use x=0
$b = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 1 0 & 0 & 1 1 & 0 & 1 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 1 & 0 & 1 0 & 0 & 1 \end{matrix} ∣ ∣ ∣ ∣ = - 1$

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∣ ∣ ∣ ∣ \begin{matrix} x & 2 & x x^{2} & x & 6 x & x & 6 \end{matrix} ∣ ∣ ∣ ∣ = A x^{4} + B x^{3} + C x^{2} + D x + E

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If $ω \neq 1$ is a cube root of unity and $∣ ∣ ∣ ∣ ∣ \begin{matrix} x + ω^{2} & ω & 1 ω & ω^{2} & 1 + x 1 & x + ω & ω^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$ , then value of x is

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If △=∣∣ ∣ ∣∣exsinx1cosxln(1+x2)1xx21∣∣ ∣ ∣∣=a+bx+cx2 then the value of b is

The correct option is B -1 △′=∣∣ ∣ ∣∣exsinx1−sinxln(1+x2)11x21∣∣ ∣ ∣∣+∣∣ ∣ ∣∣excosx1cosx2x1+x21x2x1∣∣ ∣ ∣∣+0=b+2cx Now use x=0 b=∣∣ ∣∣101001101∣∣ ∣∣+∣∣ ∣∣111101001∣∣ ∣∣=−1

If $△= ∣ ∣ ∣ ∣ ∣ \begin{matrix} e^{x} & s i n x & 1 c o s x & l n (1 + x^{2}) & 1 x & x^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = a + b x + c x^{2}$ then the value of b is