Let the function
\[{\text{f}}\left( \theta \right) = \left| {\begin{array}{*{20}{c}}
{\sin \theta }&{\cos \theta }&{\tan \theta } \\
{\sin \left( {\frac{\pi }{6}} \right)}&{\cos \left( {\frac{\pi }{6}} \right)}&{\tan \left( {\frac{\pi }{6}} \right)} \\
{\sin \left( {\frac{\pi }{3}} \right)}&{\cos \left( {\frac{\pi }{3}} \right)}&{\tan \left( {\frac{\pi }{3}} \right)}
\end{array}} \right|\]
where \[\theta \in \left[ {\frac{\pi }{6},\,\frac{\pi }{3}} \right]\] and \[{\text{f'}}\left( \theta \right)\] denote the derivative of f with respect to \[\theta \]. Which of the following statements is/are TRUE?
I. There exists \[\theta \in \left( {\frac{\pi }{6},\,\frac{\pi }{3}} \right)\] such that \[{\text{f'}}\left( \theta \right) = 0.\]
II. There exists \[\theta \in \left( {\frac{\pi }{6},\,\frac{\pi }{3}} \right)\] such that
\[{\text{f'}}\left( \theta \right) \ne 0\]
A. l only
B. ll only
C. Both l and ll
D. Neither l nor ll
Answer: Option C
This Question Belongs to Engineering Maths >> Calculus
The Taylor series expansion of 3 sinx + 2 cosx is . . . . . . . .
A. 2 + 3x - x2 - \[\frac{{{{\text{x}}^3}}}{2}\] + ...
B. 2 - 3x + x2 - \[\frac{{{{\text{x}}^3}}}{2}\] + ...
C. 2 + 3x + x2 + \[\frac{{{{\text{x}}^3}}}{2}\] + ...
D. 2 - 3x - x2 + \[\frac{{{{\text{x}}^3}}}{2}\] + ...
B. \[\infty \]
C. \[\frac{1}{2}\]
D. \[ - \infty \]
A. \[1 + \frac{{{{\left( {{\text{x}} - \pi } \right)}^2}}}{{3!}} + ...\]
B. \[ - 1 - \frac{{{{\left( {{\text{x}} - \pi } \right)}^2}}}{{3!}} + ...\]
C. \[1 - \frac{{{{\left( {{\text{x}} - \pi } \right)}^2}}}{{3!}} + ...\]
D. \[ - 1 + \frac{{{{\left( {{\text{x}} - \pi } \right)}^2}}}{{3!}} + ...\]
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