11. The common differenced of the A.P. in which T, = 9 and T,T,T, is least...

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11. The common differenced of the A.P. in which T, = 9 and T,T,T, is least is %3D 33 (A) 2 5 (B) 33 (C) 20 (D) None of these 11.ANS:C 12. If 1, 2, 3.... are first terms; 1, 3, 5.... are common differences and S,, S2, S3... are sums of n terms of given p AP's; then S, + S, + S3 + .... + S, is equal to np(np + (A)- 1) (B)- 2 n(np+ 1) np(p +1) (C) np(np - 1) (D)- 12.ANS:A 13. If positive numbers a, b, c are in A.P. and a, b², c are in H.P., then (A) a = b = c (B) 2b = a + c ac (C) b² = 8. (D) None of these 13.ANS:AB 14. Show that in any arithmetic progression a,, a,, a3.. 2 2 2 2 a, - a, + az [K/(2K - 1)] (a, 2 + a 2 K - 1 - a + 2K a 2k). %3D V Edit with WPS Office Scanned with CamScanne 11/ 12 15. If n is a root of the equation x° (1 – ac) – x (a² + c?) - (1 + ac) = 0 & if n HM's are inserted between a & c, show that the difference between the first & the last mean is equal to ac (a – c). 2 11. The common differenced of the A.P. in which T, = 9 and T,T,T, is least is %3D 33 (A) 2 5 (B) 33 (C) 20 (D) None of these 11.ANS:C 12. If 1, 2, 3.... are first terms; 1, 3, 5.... are common differences and S,, S2, S3... are sums of n terms of given p AP's; then S, + S, + S3 + .... + S, is equal to np(np + (A)- 1) (B)- 2 n(np+ 1) np(p +1) (C) np(np - 1) (D)- 12.ANS:A 13. If positive numbers a, b, c are in A.P. and a, b², c are in H.P., then (A) a = b = c (B) 2b = a + c ac (C) b² = 8. (D) None of these 13.ANS:AB 14. Show that in any arithmetic progression a,, a,, a3.. 2 2 2 2 a, - a, + az [K/(2K - 1)] (a, 2 + a 2 K - 1 - a + 2K a 2k). %3D V Edit with WPS Office Scanned with CamScanne 11/ 12 15. If n is a root of the equation x° (1 – ac) – x (a² + c?) - (1 + ac) = 0 & if n HM's are inserted between a & c, show that the difference between the first & the last mean is equal to ac (a – c). 2