KVPY Sample Paper KVPY Stream-SX Model Paper-3

Let \[{{a}_{1}},\]\[{{a}_{2,}}\]...\[{{a}_{30}}\] be an A.P., \[S=\sum\limits_{i=1}^{30}{ai}\] and \[\sum\limits_{i=1}^{15}{{{a}_{{{(2i-1)}^{.}}}}}\] If \[{{a}_{5}}=27\] and \[\text{S}-2T=75,\] then \[{{a}_{10}}\] is equal to:

A) 57

B) 47

C) 42

D) 52

Correct Answer: D

Solution :

\[{{a}_{1}},\]\[{{a}_{2}},\] ?. \[{{a}_{30}},\] A.P., let d be common difference ?d?

\[\text{S}=\sum\limits_{i=l}^{30}{{{a}_{i}}},\] \[\text{T}=\sum\limits_{i=l}^{18}{{{a}_{2i-\,1}}}\]

\[{{a}_{5}}=27\]

\[\Rightarrow \] \[{{a}_{1}}+4d=27\]

\[\Rightarrow \] \[\text{S}\,-\,2\text{T}=75\]

\[({{a}_{2}}-{{a}_{1}})+({{a}_{4}}-{{a}_{3}})\,+...+({{a}_{30}}-{{a}_{29}})=75\]

\[15d=75\]

\[d=5\]

by (1), \[{{a}_{1}}=5\]

\[\therefore \] \[{{a}_{10}}={{a}_{1}}+9d=7+45=52.\]