1
IIT-JEE 2007
+4
-1
Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $$n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}$$ have arithmetic, geometric and harminic means as $${A_n},{G_n}\,,{H_n}$$ repectively.

Which one of the following statements is correct ?

A
$${A_1} > {A_2}\, > {A_3} > ...$$
B
$${A_1} < {A_2}\, < {A_3} < ...$$
C
$${A_1} > {A_2}\, > {A_3} > ...$$ and $${A_1} < {A_2}\, < {A_3} < ...$$
D
$${A_1} < {A_2}\, < {A_3} < ...$$ and $${A_1} > {A_2}\, > {A_3} > ...$$
2
IIT-JEE 2007
+4
-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 $${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r = 1,2,...$$

Which one of the following is a correct statement?

A
$${Q_1},\,\,{Q_2},\,\,{Q_3},...$$ are A.P. with common difference 5
B
$${Q_1},\,\,{Q_2},\,\,{Q_3},...$$ are A.P. with common difference 6
C
$${Q_1},\,\,{Q_2},\,\,{Q_3},...$$ are A.P. with common difference 11
D
$${Q_1} = \,\,{Q_2} = \,\,{Q_3} = ...$$
3
IIT-JEE 2007
+4
-1
Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $$n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}$$ have arithmetic, geometric and harminic means as $${A_n},{G_n}\,,{H_n}$$ repectively.

Which one of the following statements is correct ?

A
$${G_1} > {G_2}\, > {G_3} > ...$$
B
$${G_1} < {G_2}\, < {G_3} < ...$$
C
$${G_1} = {G_2}\, = {G_3} = ...$$
D
$${G_1} < {G_2}\, < {G_3} < ...$$ and $${G_1} > {G_2}\, > {G_3} > ...$$
4
IIT-JEE 2005 Screening
+2
-0.5
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
$$\Delta = 0$$
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