JEE PYQ: Sequence And Series Question 9
Question 9 - 2021 (24 Feb Shift 1)
Let $A = {n \in N : n \text{ is a 3-digit number}}$, $B = {9k + 2 : k \in N}$ and $C = {9k + \ell : k \in N}$ for some $\ell$ $(0 < \ell < 9)$.
If the sum of all the elements of the set $A \cap (B \cup C)$ is $274 \times 400$, then $\ell$ is equal to _______.
Show Answer
Answer: (5)
Solution
3 digit number of the form $9K + 2$ are ${101, 109, \ldots, 992}$
$\Rightarrow$ Sum equal to $\frac{100}{2}(1093) = s_1 = 54650$
$274 \times 400 = s_1 + s_2$
$274 \times 400 = \frac{100}{2}[101 + 992] + s_2$
$274 \times 400 = 50 \times 1093 + s_2$
$s_2 = 109600 - 54650$
$s_2 = 54950$
$s_2 = 54950 = \frac{100}{2}[(99 + \ell) + (990 + \ell)]$
$1099 = 2\ell + 1089$
$\ell = 5$
BETA
