Question

Assertion :Let $y=\sin x$ and $y_r$ represents $r^{th}$ derivative of $y$ with respect to $x$.
STATEMENT-1 :$\left|\begin{array}{ccc}{y}_{102}& y_{103}& y_{104}\\ y_{109}& y_{111}& y_{113}\\ y_{117}& y_{119}& y_{125}\end{array}\right|=0$ Reason: STATEMENT-2 : $y_{4n+k}=y_{4(n+1)+k}$ , where $k=0,1,2,3$ and $n$ in $N$.

• A. Statement -1 is True, Statement -2 is True ; Statement -2 is a correct explanation for Statement -1
• B. Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1
• C. Statement -1 is True, Statement -2 is False
• D. Statement -1 is False, Statement -2 is True

Answer 1

The correct option is A Statement -1 is True, Statement -2 is True ; Statement -2 is a correct explanation for Statement -1

$y=\sin x$
$y_1=\cos x=\sin [x+\frac{\pi }{2}]$
$y_2=\cos [x+\frac{\pi }{2}]=\sin [x+2\frac{\pi }{2}]$
$.$
$.$
$.$
$y_n=\sin [x+\frac{n\pi }{2}]$
$y_{4(n+1)+k}=\sin [x+(4(n+1)+k\left)\frac{\pi }{2}\right]$
$=\sin [x+(4n+k)\frac{\pi }{2}+2\pi ]=\sin [x+(4n+k\left)\frac{\pi }{2}\right]=y_{4n+k}$
$\therefore$ Statement-2 is true
Now,$\left|\begin{array}{ccc}{y}_{102}& y_{103}& y_{104}\\ y_{109}& y_{111}& y_{113}\\ y_{117}& y_{119}& y_{125}\end{array}\right|$
Here, according to statement 2,
$y_{117}=y_{109+8}=y_{109}$
$y_{119}=y_{111+8}=y_{111}$
$y_{125}=y_{113+12}=y_{113}$
$=\left|\begin{array}{ccc}{y}_{102}& y_{103}& y_{104}\\ y_{109}& y_{111}& y_{113}\\ y_{109}& y_{111}& y_{113}\end{array}\right|=0$
Hence,statement-1 is true