MCQ
${\sin ^6}\theta + {\cos ^6}\theta + 3{\sin ^2}\theta {\cos ^2}\theta = $
- A$0$
- B$-1$
- ✓$1$
- DNone of these
$ = {({\sin ^2}\theta + {\cos ^2}\theta )^3} - 3{\sin ^2}\theta {\cos ^2}\theta + 3{\sin ^2}\theta {\cos ^2}\theta = 1.$
Trick : Put $\theta = {0^o},$
we get the value of expression equal to $1. $
Again put $\theta = {45^o},$ the value remains $1,$ it means that the expression is independent of $\theta$ and is equal to $1.$
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$L _1: x \sqrt{2}+ y -1=0$ and $L _2: x \sqrt{2}- y +1=0$
For a fixed constant $\lambda$, let $C$ be the locus of a point $P$ such that the product of the distance of $P$ from $L_1$ and the distance of $P$ from $L_2$ is $\lambda^2$. The line $y=2 x+1$ meets $C$ at two points $R$ and $S$, where the distance between $R$ and $S$ is $\sqrt{270}$.
Let the perpendicular bisector of $RS$ meet $C$ at two distinct points $R ^{\prime}$ and $S ^{\prime}$. Let $D$ be the square of the distance between $R ^{\prime}$ and $S ^{\prime}$.
($1$) The value of $\lambda^2$ is
($2$) The value of $D$ is
$I$. For any $n$, the roots are distinct.
$II$. There are infinitely many values of $n$ for which both roots are real.
$III$. The product of the roots is necessarily an integer.