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1

IIT-JEE 2007

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 $${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r = 1,2,...$$

$${T_r}$$ is always

A
an odd number
B
an even number
C
a prime number
D
a composite number
2

IIT-JEE 2007

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 $${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r = 1,2,...$$

The sum $${V_1}$$+$${V_2}$$ +...+$${V_n}$$ is

A
$${1 \over {12}}n(n + 1)\,(3{n^2} - n + 1)$$
B
$${1 \over {12}}n(n + 1)\,(3{n^2} + n + 2)$$
C
$${1 \over 2}n(2{n^2} - n + 1)$$
D
$${1 \over 3}(2{n^3} - 2n + 3)$$
3

IIT-JEE 2005 Screening

In the quadratic equation $$\,\,a{x^2} + bx + c = 0,$$ $$\Delta$$ $$= {b^2} - 4ac$$ and $$\alpha + \beta ,\,{\alpha ^2} + {\beta ^2},\,{\alpha ^3} + {\beta ^3},$$ are in G.P. where $$\alpha ,\beta$$ are the root of $$\,\,a{x^2} + bx + c = 0,$$ then
A
$$\Delta \ne 0$$
B
$$b\Delta = 0$$
C
$$c\Delta = 0$$
D
$$\Delta = 0$$
4

IIT-JEE 2004 Screening

An infinite G.P. has first term '$$x$$' and sum '$$5$$', then $$x$$ belongs to
A
$$x < - 10$$
B
$$- 10 < x < 0$$
C
$$0 < x < 10$$
D
$$x > 10$$

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