Question

Match the statements/expressions in List 1 with the values given in List 2.

Answer 1

A) $x.e^{\sin x}-\cos x=0$

$\Rightarrow e^{\sin x}=\frac{\cos x}{x}$
Let $f\left(x\right)=\frac{\cos x}{x},g\left(x\right)=e^{\sin x}$
Then from figure, number of solutions is $1$.
B) $\left|\begin{array}{ccc}k& 4& 1\\ 4& k& 2\\ 2& 2& 1\end{array}\right|=0\Rightarrow k=2,4$
C) $|x-1|+|x-2|+|x+1|+|x+2|=4k$
From figure 2 we have $k=2,3,4,5$, so number of solution is $4$
D) $\frac{dy}{dx}=y+1\Rightarrow \int \frac{dy}{y+1}=\int dx\Rightarrow \ln (y+1)=x+c$
$\Rightarrow y+1=e^{(x+c)}\Rightarrow y=e^{(x+c)}-1$
For $x=0,y=1\Rightarrow 1=e^c-1\Rightarrow e^c=2$
$y=e^{(x+\ln 2)}-1$
And for $x=\ln 2$
$y=e^{2\ln 2}-1=3$