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PASSAGE  1
Let V_{r} denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1).
Let T_{r }= V_{r + 1} – V_{r – 2} and Q_{r }= T_{r + 1} – T_{r} for r = 1, 2, ...
1. The sum V_{1} + V_{2} + ... + V_{n} is (2007 4 marks)
(a)
(b)
(c)
(d)
Ans. Sol.
(b) V_{1 }+ V_{2} + ....
2. Tr is always (2007 4 marks)
(a) an odd number
(b) an even number
(c) a prime number
(d) a composite number
Ans. Sol.
(d) T_{r }= V_{r +1} V_{r 2}
= 3r ^{2} + 2r+1
T_{r} = (r + 1) (3r – 1)
For each r, T_{r} has two different factors other than 1 and itself.
∴ T_{r} is always a composite number.
3. Which one of the following is a correct statement ? (2007 4 marks)
(a) Q_{1}, Q_{2}, Q_{3}, ... are in A.P. with common difference 5
(b) Q_{1}, Q_{2}, Q_{3}, ... are in A.P. with common difference 6
(c) Q_{1}, Q_{2}, Q_{3}, ... are in A.P. with common difference 11
(d) Q_{1}= Q_{2} = Q_{3} = ....
Ans. Sol.
(b) ∴ Q_{r +1 } Q_{r} = T_{r +2 } T_{r +1 } (T_{r+1}T_{r}) = T_{r + 2}  2T_{r+1}+Tr
= (r+ 3)(3r + 5)  2(r + 2)(3r + 2)+ (r + 1)(3r 1)
∵Q_{r + 1} – Q_{r }= 6 (r + 1) + 5 – 6r – 5 = 6 (constant)
∴ ∵1, Q_{2}, Q_{3}, .... are in AP with common difference 6.
PASSAGE 2
Let A_{1}, G_{1}, H_{1} denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, Let A_{n – 1 }and H_{n – 1} have arithmetic, geometric and harmonic means as A_{n}, G_{n}, H_{n }respectively.
4. Which one of the following statements is correct ? (a) G_{1} > G_{2} > G_{3} > ... (2007 4 marks) (b) G_{1} < G_{2} < G_{3} < ... (c) G_{1 }= G_{2 }= G_{3} = ... (d) G_{1} < G_{3 }< G_{5} < ... and G_{2 }> G_{4} > G_{6} > ....
Ans. Sol.
(c) Given
also
Similarly we can prove
AnHn = A_{n–1}H_{n–1 }= A_{n–2} H_{n–2} = .... = A_{1}H_{1}
⇒ A_{n}Hn_{ =} ab
∴ G_{1}^{2 }= G_{2}^{2 } = G_{3}^{2 } ....= ab
⇒ G_{1} = G_{2} = G_{3} ...
5. Which one of the following statements is correct ? (a) A_{1} > A_{2} > A_{3} > ... (2007 4 marks) (b) A_{1} < A_{2} < A_{3 }< ... (c) A_{1} > A_{3} > A_{5 }> ... and A_{2} < A_{4 }< A_{6} < ... (d) A_{1} < A_{3} < A_{5 }< ... and A_{2} > A_{4} > A_{6} > ...
Ans. Sol.
(a) We have
⇒ A_{n} < A_{n–1 }or A_{n–1 }> A_{n} ∴ We can conclude that A_{1} > A_{2} > A_{3} > ....
6. Which one of the following statements is correct ? (a) H_{1} > H_{2 }> H_{3} > ... (2007 4 marks) (b) H_{1} < H_{2} < H_{3} < ... (c) H_{1} > H_{3} > H_{5 }> ... and H_{2} < H_{4} < H_{6} < ... (d) H_{1} < H_{3 }< H_{6} < ... and H_{2} > H_{4} > H_{6} > ...
Ans. Sol.
(b) We have A_{n} H_{n} = ab
∴ H_{1 }< H_{2} < H_{3} < .......
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