MCQ
If $^{56}{P_{r + 6}}{:^{54}}{P_{r + 3}} = 30800:1$, then $r = $
- A$31$
- ✓$41$
- C$51$
- DNone of these
$ = \frac{{30800}}{1} \Rightarrow 56 \times 55 \times (51 - r) = 30800$
$ \Rightarrow $ $r = 41$.
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Let $f( x )=\sin (\pi \cos x )$ and $g ( x )=\cos (2 \pi \sin x )$ be two functions defined for $x >0$. Define the following sets whose elements are written in the increasing order :
$X =\{ x : f( x )=0\}, Y =\left\{ x : f^{\prime}( x )=0\right\}$
$Z =\{ x : g ( x )=0\}, W =\left\{ x : g ^{\prime}( x )=0\right\}.$
$List-I$ contains the sets $X , Y , Z$ and $W$. $List -II$ contains some information regarding these sets.
| $List-I$ | $List-II$ |
| $(I)$ $X$ | $(P)$ $\supseteq\left\{\frac{\pi}{2}, \frac{3 \pi}{2}, 4 \pi, 7 \pi\right\}$ |
| $(II)$ $Y$ | $(Q)$ an arithmetic progression |
| $(III)$ $Z$ | $(R)$ $NOT$ an arithmetic progression |
| $(IV)$ $W$ | $(S)$ $\supseteq\left\{\frac{\pi}{6}, \frac{7 \pi}{6}, \frac{13 \pi}{6}\right\}$ |
| $(T)$ $\supseteq\left\{\frac{\pi}{3}, \frac{2 \pi}{3}, \pi\right\}$ | |
| $( U )$ $\supseteq\left\{\frac{\pi}{6}, \frac{3 \pi}{4}\right\}$ |
($1$) Which of the following is the only $CORRECT$ combination?
$(1) (II), (R), (S)$ $(2) (I), (P), (R)$ $(3) (II), (Q), (T)$ $(4) (I), (Q), (U)$
($2$) Which of the following is the only $CORRECT$ combinations?
$(1) (IV), (Q), (T)$ $(2) (IV), (P), (R), (S)$ $(3) (III), (R), (U)$ $(4) (III), (P), (Q), (U)$
Give the answer the quetion ($1$) and ($2$)