MCQ
The total number of three-digit numbers, divisible by $3$ , which can be formed using the digits $1,3,5,8$ , if repetition of digits is allowed, is:
- ✓$22$
- B$18$
- C$21$
- D$20$
$(5,5,8) \quad(8,8,5) \quad(1,3,5) \quad(1,3,8)$
$\text { Total number }=1+1+1+1+\frac{3 !}{2 !}+\frac{3 !}{2 !}+3 !+3 !=22$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$f(n)=\frac{\sum_{k=0}^n \sin \left(\frac{k+1}{n+2} \pi\right) \sin \left(\frac{k+2}{n+2} \pi\right)}{\sum_{k=0}^n \sin ^2\left(\frac{k+1}{n+2} \pi\right)}$
Assuming $\cos ^{-1} x$ takes values in $[0, \pi]$, which of the following options is/are correct ?
$(1)$ $\sin \left(7 \cos ^{-1} f(5)\right)=0$
$(2)$ $f(4)=\frac{\sqrt{3}}{2}$
$(3)$ $\lim _{n \rightarrow \infty} f(n)=\frac{1}{2}$
$(4)$ If $\alpha=\tan \left(\cos ^{-1} f(6)\right)$, then $\alpha^2+2 \alpha-1=0$