Question

Match of the following :
List - I List - II
1. Max. value of $3\sin x+4\cos x$ $a.24$
2. Min. positive value of $\sin x+cosecx$ $b.5$
3. Min. value of $9{\sin }^{2}x+16{cosec}^{2}x$ $c.2$
4. Max. value of ${\sin }^{4}x+{\cos }^{4}x$ $d.1$

• A. $1-b,2-c,3-a,4-d$
• B. $1-d,2-a,3-b,4-c$
• C. $1-c,2-d,3-a,4-b$
• D. $1-c,2-b,3-d,4-a$

Answer 1

The correct option is A $1-b,2-c,3-a,4-d$

1. $f\left(x\right)=3\sin x+4\cos x$
we know, max value of $a\sin x\pm b\cos x$ is $+\sqrt{a^2+b^2}$
$\therefore$ max. value of $3\sin x+4\cos x$ is 5
2. $\sin x+cosecx=\sin x+\frac{1}{\sin x}$
put $\sin x=4\Rightarrow f\left(y\right)=y+\frac{1}{y}$
$f^{\prime }\left(y\right)=1-\frac{1}{y^2}=0$
$y=1$
at $\sin x=1,f\left(x\right)=1+\frac{1}{1}=2.$
at $\sin x=-1,f\left(x\right)=-2$
at $\sin x=0,f\left(x\right)=\infty$
min. positive value of $\sin x+cosecx=2.$
3. $f\left(x\right)=9{\sin }^{2}x+\frac{16}{{\sin }^{2}x}$
put $y=\sin x,yϵ[-1,1]$
$f\left(y\right)=9y^2+\frac{16}{y^2}$
$f^{\prime }\left(y\right)=18y-\frac{32}{y^2}=0$
$\Rightarrow y^2=\frac{4}{3}$
$\therefore 9\times \frac{4}{3}+16\times \frac{3}{4}$
$f\left(x\right)=24$

$f\left(n\right)ϵ(24,\infty )$
4. $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x=1-\frac{(\sin 2x{)}^2}{2}$
$f\left(x\right)ϵ[\frac{1}{2},1]$