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1

IIT-JEE 1997

Fill in the Blanks
The sum of all the real roots of the equation $${\left| {x - 2} \right|^2} + \left| {x - 2} \right| - 2 = 0$$ is ............................

Answer

4
2

IIT-JEE 1996

Fill in the Blanks
Let n and k be positive such that $$n \ge {{k(k + 1)} \over 2}$$ . The number of solutions $$\,({x_1},\,{x_2},\,.....{x_k}),\,{x_1}\,\, \ge \,1,\,{x_2}\, \ge \,2,.......,{x_k} \ge k$$, all integers, satisfying $${x_1} + {x_2} + \,..... + {x_k} = n,\,$$ is......................................

Answer

$${{\left[ {k + \left( {n - {{k(k + 1)} \over 2}} \right) - 1} \right]!} \over {\left[ {n - {{k(k + 1)} \over 2}} \right]!\,(k - 1)\,!}}$$
3

IIT-JEE 1990

Fill in the Blanks
If $$\,x < 0,\,\,y < 0,\,\,x + y + {x \over y} = {1 \over 2}$$ and $$(x + y)\,{x \over y} = - {1 \over 2}$$, then x =..........and y =.........

Answer

$$ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}$$ ,$$ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}$$
4

IIT-JEE 1986

Fill in the Blanks
The solution of equation $${\log _7}\,{\log _5}\,\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$ is ........................

Answer

4

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