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Question

# If (10)9+2(11)1(10)8+3(11)2(10)7+...+10(11)9=k(10)9, then k is equals to: (IIT JEE Main 2014)

A

12110

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B

441100

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C

100

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D

110

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Solution

## The correct option is D 110 We have, (10)9+2(11)1(10)8+3(11)2(10)7+...+10(11)9−k(10)9 Dividing by (10)9 on both sides, we get 1+2(1110)+3(1110)2+...+10(1110)9=k ...(i) Multiplying (1110)on both side of above equation we get, (1110)+2(1110)2+3(1110)3+...+10(1110)10−(1110)k ...(ii) Subtracting (ii) from (i), we get k−(1110)k−1+(1110)+(1110)2+...+(1110)9−10(1110)10 ⇒k(1−(1110))=1((1110)10−1)(1110)−1−10(1110)10 ⇒k(1−(1110))=−10 ⇒k=100

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