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1

IIT-JEE 1997

Fill in the Blanks
Let $$p$$ and $$q$$ be roots of the equation $${x^2} - 2x + A = 0$$ and let $$r$$ and $$s$$ be the roots of the equation $${x^2} - 18x + B = 0.$$ If $$p < q < r < s$$ are in arithmetic progression, then $$A = \,..........$$ and $$B = \,..........$$

Answer

$$-3,77$$
2

IIT-JEE 1996

Fill in the Blanks
For any odd integer $$n$$ $$ \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3} = ........$$

Answer

$${1 \over 4}{\left( {n + 1} \right)^2}\left( {2n - 1} \right)$$
3

IIT-JEE 1992

Fill in the Blanks
Let the harmonic mean and geometric mean of two positive numbers be the ratio 4 : 5. Then the two number are in the ratio .........

Answer

4 : 1 or 1 : 4
4

IIT-JEE 1988

Fill in the Blanks
The sum of the first n terms of the series $${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + .........$$ is
$$n\,\,{\left( {n + 1} \right)^2}/2,$$ when $$n$$ is even. When $$n$$ is odd, the sum is .............

Answer

$${{{n^2}\left( {n + 1} \right)} \over 2}$$

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