Choose the correct answer from the given four options.For any two…

$\left| {\begin{array}{*{20}{c}}
{{{\log }_e}\,a_1^ra_2^k}&{{{\log }_e}\,a_2^ra_3^k}&{{{\log }_e}\,a_3^ra_4^k} \\
{{{\log }_e}\,a_4^ra_5^k}&{{{\log }_e}\,a_5^ra_6^k}&{{{\log }_e}\,a_6^ra_7^k} \\
{{{\log }_e}\,a_7^ra_8^k}&{{{\log }_e}\,a_8^ra_9^k}&{{{\log }_e}\,a_9^ra_{10}^k}
\end{array}} \right| = 0$

Then the number of elements in $S$, is

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Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function given by

$f(x)= \begin{cases}\frac{1-\cos 2 x}{x^2} & , x<0 \\ \alpha & , x=0, \text { where } \alpha, \beta \in R \text {. If } \\ \frac{\beta \sqrt{1-\cos x}}{x} & , x>0\end{cases}$

$f$ is continuous at $\mathrm{x}=0$, then $\alpha^2+\beta^2$ is equal to :

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Which of the following holds true for a vector quantity:

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The scalar product of $5i + j - 3k$ and $3i - 4j + 7k$ is:

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Let $f(\theta)=3\left(\sin ^4\left(\frac{3 \pi}{2}-\theta\right)+\sin ^4(3 \pi+\theta)\right)-2\left(1-\sin ^2 2 \theta\right)$ and $S=\left\{\theta \in[0, \pi]: f^{\prime}(\theta)=-\frac{\sqrt{3}}{2}\right\}$. If $4 \beta=\sum_{\theta \in S} \theta$ then $f(\beta)$ is equal to

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The solution set of $f ' (x) > g ' (x),$ where $ f(x)$$ = \frac{1}{2} (5^{2x + 1}) $ $\,\, \& \,\, g(x) = 5^x + 4x (ln \,\, 5)$ is :

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The value of $x$ for which $\sin (\cot^{-1} (1 + x)) = \cos(\tan^{-1} \,x)$ is

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