MCQ
If $\sin y = x\sin (a + y),$ then ${{dy} \over {dx}} = $
- A${{{\sin }^2}(a + y)}$
- B${{{{\sin }^2}(a + y)} \over {\sin (a + 2y)}}$
- ✓${{{{\sin }^2}(a + y)} \over {\sin a}}$
- D${{{{\sin }^2}(a + y)} \over {\cos a}}$
==> $1 = \frac{{\cos y.\frac{{dy}}{{dx}}.\sin (a + y) - \sin y\cos (a + y)\frac{{dy}}{{dx}}}}{{{{\sin }^2}(a + y)}}$
$ = \frac{{\frac{{dy}}{{dx}}.\sin (a + y - y)}}{{{{\sin }^2}(a + y)}} $
$\Rightarrow \frac{{dy}}{{dx}} = \frac{{{{\sin }^2}(a + y)}}{{\sin a}}$.
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If the derivative $f^{\prime}$ of $f$ satisfies the equation $f ^{\prime}( x )=\frac{ f ( x )}{ b ^2+ x ^2}$ for all $x \in R$, then which of the following statements is/are TRUE?
$(A)$ If $b>0$, then $f$ is an increasing function
$(B)$ If $b<0$, then $f$ is a decreasing function
$(C)$ $(x)(-x)=1$ for all $x \in R$
$(D)$ $(x)-f(-x)=0$ for all $x \in R$