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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Let $S = \{\lambda ,\mu \} \in R \times R:f\left( t \right) = \left( {\left| \lambda  \right|{e^{\left| t \right|}} - \mu } \right)$. $\sin \left( {2\left| t \right|} \right),t \in R$ , is a differentiable function$\}$ . Then $S$ is a subest of?
  • ✓
    $R \times \left[ {0,\infty } \right)$
  • B
    $\left( { - \infty ,0} \right) \times R$
  • C
    $\left[ {0,\infty } \right) \times R$
  • D
    $R \times \left( { - \infty ,0} \right)$
✓

Answer

Correct option: A.
$R \times \left[ {0,\infty } \right)$
a
$S = \left\{ {\lambda ,\mu } \right\} \in R \times R:f\left( t \right) = \left( {\left| \lambda  \right|{e^{\left| t \right|}} - \mu } \right)\sin \left( {2\left| t \right|} \right),$

$t \in R$

$f\left( t \right) = \left( {\left| \lambda  \right|{e^{\left| t \right|}} - \mu } \right)\sin \left( {2\left| t \right|} \right)$

$\left\{ \begin{array}{l} \left( {\left| \lambda  \right|{e^t} - \mu } \right)\sin 2t\,\,\,\,\,\,\,\,t > 0\\ \left( {\left| \lambda  \right|{e^{ - t}} - \mu } \right)\left( { - \sin 2t\,} \right)\,\,\,\,\,\,\,t < 0 \end{array} \right.$

$f'(t)$

$ = \left\{ \begin{array}{l} \left( {\left| \lambda  \right|{e^t}} \right)\sin 2t + \left( {\left| \lambda  \right|{e^t} - \mu } \right)\left( {2\cos 2t} \right)\,\,\,t > 0\\ \left| \lambda  \right|{e^{ - t}}\sin 2t + \left( {\left| \lambda  \right|{e^{ - t}} - \mu } \right)\left( { - 2\cos 2t} \right)\,\,\,\,t < 0 \end{array} \right.$

As, $f(t)$ is differentiable

$\therefore LHD = RHD$ at $t=0$

$\left| \lambda  \right|.\sin 2\left( 0 \right) + \left( {\left| \lambda  \right|{e^0} - \mu } \right)2\cos \left( 0 \right)$

$ = \left| \lambda  \right|{e^0}.\sin 2\left( 0 \right) - 2\cos \left( 0 \right)\left( {\left| \lambda  \right|{e^0} - \mu } \right)$

$ \Rightarrow 0 + \left( {\left| \lambda  \right| - \mu } \right)2 = 0 - 2\left( {\left| \lambda  \right| - \mu } \right)$

                  $4\left( {\left| \lambda  \right| - \mu } \right) = 0$

                  $\left| \lambda  \right| = \mu $

So, $S \equiv \left( {\lambda ,\mu } \right) = \left\{ {\lambda  \in R\;and\;\mu  \in \left[ {0,\infty } \right)} \right\}$

Therefore set $S$ is subset of $R \times \left[ {0,\infty } \right)$

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