MCQ
Solution of differential equation $\frac{{dy}}{{dx}} + ay = {e^{mx}}$ is
- A$(a + m)\,y = {e^{mx}} + c$
- B$y{e^{ax}} = m{e^{mx}} + c$
- C$y = {e^{mx}} + c{e^{ - ax}}$
- ✓$(a + m)y = {e^{mx}} + c{e^{ - ax}}(a + m)$
$\therefore $Required solution is given by
$y.\,{e^{ax}} = \int_{}^{} {{e^{mx}}.{e^{ax}}} dx = \frac{{{e^{(a + m)x}}}}{{a + m}} + C$
==> $y = \frac{{{e^{mx}}}}{{a + m}} + C{e^{ - ax}}$
==> $y(a + m) = {e^{mx}} + C(a + m){\rm{ }}{e^{ - ax}}$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2} .$
Match the statements in Column $I$ with the statements in Column $II$ and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the $ORS$.
| Column $I$ | Column $II$ |
| $(A)$ If $a=1$ and $b=0$, then ( $x, y$ ) | $(p)$ lies on the circle $x^2+y^2=1$ |
| $(B)$ If $a=1$ and $b=1$, then $(x, y)$ | $(q)$ lies on $\left(x^2-1\right)\left(y^2-1\right)=0$ |
| $(C)$ If $a=1$ and $b=2$, then ( $x, y)$ | $(r)$ lies on $y=x$ |
| $(D)$ If $a=2$ and $b=2$, then $(x, y)$ | $(s)$ lies on $\left(4 x^2-1\right)\left(y^2-1\right)=0$ |
$2\left(a_1+a_2+\ldots .+a_n\right)=b_1+b_2+\ldots . .+b_n$
holds for some positive integer $n$, is. . . . . . .