Evaluate $∣ ∣ ∣ ∣ ∣ \begin{matrix} c o s (x + x^{2}) & s i n (x + x^{2}) & - c o s (x + x^{2}) s i n (x - x^{2}) & c o s (x - x^{2}) & s i n (x - x^{2}) s i n 2 x & 0 & s i n 2 x^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$

s i n (2 x + 2 x^{2})

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- s i n (2 x + 2 x^{2})

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c o s (2 x + 2 x^{2})

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- c o s (2 x + 2 x^{2})

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Solution

The correct option is A $s i n (2 x + 2 x^{2})$
$∣ ∣ ∣ ∣ ∣ \begin{matrix} c o s (x + x^{2}) & s i n (x + x^{2}) & - c o s (x + x^{2}) s i n (x - x^{2}) & c o s (x - x^{2}) & s i n (x - x^{2}) s i n 2 x & 0 & s i n 2 x^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
expanding along $R_{3}$ gives
$= sin 2 x (sin (x + x^{2}) sin (x - x^{2}) + cos (x + x^{2}) cos (x - x^{2})) + sin 2 x^{2} (cos (x + x^{2}) cos (x - x^{2}) - sin (x + x^{2}) sin (x - x^{2}))$
$= sin 2 x cos 2 x^{2} + sin 2 x^{2} cos 2 x$
$= sin (2 x + 2 x^{2})$
Hence, option A.

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Similar questions

f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} cos (x + x^{2}) & sin (x + x^{2}) & - cos (x + x^{2}) sin (x - x^{2}) & cos (x - x^{2}) & sin (x - x^{2}) sin (2 x) & 0 & sin (2 x^{2}) \end{matrix} ∣ ∣ ∣ ∣ ∣

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f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} cos (x + x^{2}) & sin (x + x^{2}) & - cos (x + x^{2}) sin (x - x^{2}) & cos (x - x^{2}) & sin (x - x^{2}) sin 2 x & 0 & sin ({2 x}^{2}) \end{matrix} ∣ ∣ ∣ ∣ ∣

f^{'} (0)

\int (x^{3} - 2 x^{2} + 3 x - 1) c o s 2 x d x = \frac{s i n 2 x}{4} u (x) + \frac{c o s 2 x}{8} v (x) + c

lim x \to π / 2 \frac{1 + cos 2 x}{(π - 2 x)^{2}}

1) $c o s 2 x = c o s^{2} x - s i n^{2} x$

2) $1 + s i n 2 x = (c o s x + s i n x)^{2}$

3) $1 - s i n 2 x = (c o s x - s i n x)^{2}$

How many of the above statements are correct?

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