MCQ
$16 \sin \left(20^{\circ}\right) \sin \left(40^{\circ}\right) \sin \left(80^{\circ}\right)$ is equal to
- A$\sqrt{3}$
- ✓$2 \sqrt{3}$
- C$3$
- D$4 \sqrt{3}$
$=16 \sin 40^{\circ} \sin 20^{\circ} \sin 80^{\circ}$
$=4(4 \sin (60-20) \sin (20) \sin (60+20))$
$=4 \times \sin \left(3 \times 20^{\circ}\right)$
${[\because \sin 3 \theta=4 \sin (60-\theta) \times \sin \theta \times \sin (60+\theta)]}$
$=4 \times \sin 60^{\circ}$
$=4 \times \frac{\sqrt{3}}{2}=2 \sqrt{3}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$\binom{s}{r}=\left\{\begin{array}{ll}\frac{s!}{r!(s-r)!} & \text { if } r \leq s \\ 0 & \text { if } r>s\end{array}\right.$
For positive integers $m$ and $n$, let
$(m, n) \sum_{ p =0}^{ m + n } \frac{ f ( m , n , p )}{\binom{ n + p }{ p }}$
where for any nonnegative integer $p$,
$f(m, n, p)=\sum_{i=0}^{ p }\binom{m}{i}\binom{n+i}{p}\binom{p+n}{p-i}$
Then which of the following statements is/are $TRUE$?
$(A)$ $(m, n)=g(n, m)$ for all positive integers $m, n$
$(B)$ $(m, n+1)=g(m+1, n)$ for all positive integers $m, n$
$(C)$ $(2 m, 2 n)=2 g(m, n)$ for all positive integers $m, n$
$(D)$ $(2 m, 2 n)=(g(m, n))^2$ for all positive integers $m, n$