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IIT JEE Advanced 2007 Paper 1

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This section contains 2 paragraphs. Based on each paragraph, there are 3 questions. Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Question 58–60:

Let

V_r

done the sum of the 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 , ...

Section: Mathematics

Question : 59 of 66

Marks: +1, -0

T_r

is always

[JEE Adv 2007 P1]

an odd number.
an even number.
a prime number.
a composite number.

Validate

Solution:

We have

v_r

\frac{1}{2} (2r^3 - r^2 + r)

v_{r+1}

\frac{1}{2}[2(r+1)^3 - (r+1)^2 + (r+1)]

T_r

v_{r+1} - v_r

(r+1)^3 - \frac{1}{2} [(r+1)^2 - r^2] + \frac{1}{2} (1)

2r^2 + 2r - 1

= (r + 1) (3r - 1)

which is a composite number.

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