TANCET 2025 Solved Papers

Concept:For an arithmetic sequence, the general term is $a_n = a + (n-1)d$.The 9th term is $a_9 = a + 8d$, so the values of both $a$ and $d$ are required.Explanation:Let $a$ be the first term and $d$ be the common difference.Statement (1): The sum from the 5th to the 12th terms consists of 8 terms.$S = \frac{8}{2}(a_5 + a_{12}) = 4[(a+4d)+(a+11d)] = 4(2a+15d) = 77$.This gives only one equation in $a$ and $d$, so it cannot determine the 9th term alone.Statement (2): The sum from the 6th to the 10th terms consists of 5 terms.$S = \frac{5}{2}(a_6 + a_{10}) = \frac{5}{2}[(a+5d)+(a+9d)] = 5(a+7d) = 108$.This also gives only one equation in $a$ and $d$, so it is not sufficient alone.Combining both statements gives two independent linear equations in $a$ and $d$.These can be solved to obtain unique values of $a$ and $d$, and therefore a unique value of $a_9 = a+8d$.Hence, both statements together are sufficient, but neither statement alone is sufficient.Answer:Option C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.