If ^@ \alpha ^@ and ^@ \beta ^@ are the zeros of polynomial ^@ x^2-2x-3,^@ find a polynomial whose zeros are ^@ \dfrac{ \alpha^2 }{ \beta^2 } ^@ and ^@ \dfrac{ \beta^2 }{ \alpha^2 }. ^@

If ^@9^@ is added to the numerator and denominator of a fraction, the fraction becomes ^@\dfrac { 21 } { 25 } ^@ and if ^@6^@ is subtracted from its numerator and denominator it becomes ^@\dfrac { 6 } { 10 } ^@. Find the fraction.

^@ D^@, ^@E^@ and ^@F^@ are respectively the midpoints of sides ^@AB^@, ^@BC^@ and ^@CA^@ of ^@\Delta ABC^@. Find the ratio of the areas of ^@\Delta DEF^@ and ^@\Delta ABC^@.

The angle of elevation of the top of a vertical tower from a point on the ground is ^@ 60^\circ ^@. From another point, ^@ 3 \space m ^@ vertically above the first, its angle of elevation is ^@ 30^\circ ^@. What is the height of the tower?

In the given figure, ^@ PQ ^@ is a chord of a circle with centre ^@ O ^@ and ^@ PT ^@ is a tangent at point ^@ P ^@. If ^@\angle QPT ^@ = ^@ 40^\circ ^@, find ^@ \angle PRQ ^@.

There are a cylinder and a cone with height, ^@h^@ and radius ^@r^@. There is also a sphere of the same radius. The height, ^@h^@ of the cylinder and the cone is ^@ 6 ^@ times the radius, ^@r^@. If ^@V_1^@ is the volume of the cylinder, ^@V_2^@ is the volume of the cone and ^@V_3^@ is the volume of the sphere, then which of the following is true?

A piggy bank contains ^@15^@ €1 coins, ^@21^@ €2 coins, ^@31^@ €5 coins, and ^@38^@ €10 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, find the probability the coin falling out will be a ^@€1^@ or ^@€2^@ or ^@€5^@ coin.

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