Let $$\alpha {\mkern 1mu} {\text{,}}{\mkern 1mu} \beta $$  be the roots of $${x^2} - x + p = 0\,$$   and $$\gamma ,\delta $$  be the roots of $${x^2} - 4x + q = 0.\,$$   If $$\alpha {\mkern 1mu} ,{\mkern 1mu} \beta {\mkern 1mu} ,{\mkern 1mu} \gamma ,{\mkern 1mu} \delta $$   are in G.P., then the integral values of $$p$$ and $$q$$ respectively, are

Solution :
$$\eqalign{ & \alpha \,,\beta \,{\text{ are the roots of }}{x^2} - x + p = 0 \cr & \therefore \,\,\,\,\alpha + \beta = 1\,\,\,\,\,\,.....\left( 1 \right) \cr & \,\,\,\,\,\,\,\alpha \beta = p\,\,\,\,\,\,\,\,.....\left( 2 \right) \cr & \gamma \,,\delta \,\,{\text{are the roots of }}{x^2} - 4x + q = 0 \cr & \therefore \,\,\,\,\,\gamma + \delta = 4\,\,\,\,\,.....\left( 3 \right) \cr & \,\,\,\,\,\,\,\,\,\gamma \delta = q\,\,\,\,\,\,\,\,.....\left( 4 \right) \cr & \alpha ,\,\,\beta ,\,\,\gamma ,\,\,\delta \,\,{\text{are in G}}{\text{.P}}{\text{.}} \cr & \therefore \,\,\,{\text{Let }}\alpha = a;\beta = ar,\gamma = a{r^2},\delta = a{r^3}. \cr & {\text{Substituting these values in equations }}\left( {\text{1}} \right){\text{,}}\left( {\text{2}} \right){\text{, }}\left( 3 \right){\text{ and}}\,\left( 4 \right),\,{\text{we get }} \cr & \,\,\,\,a + ar = 1\,\,\,\,\,\,\,\,.....\left( 5 \right) \cr & \,\,\,\,{a^2}r = p\,\,\,\,\,\,\,\,\,\,\,\,.....\left( 6 \right) \cr & \,\,\,\,a{r^2} + a{r^3} = 4\,\,\,.....\left( 7 \right) \cr & \,\,\,\,{a^2}{r^5} = q\,\,\,\,\,\,\,\,\,\,\,.....\left( 8 \right) \cr & {\text{Dividing (7) by (5) we get}} \cr & \frac{{a{r^2}\left( {1 + r} \right)}}{{a\left( {1 + r} \right)}} = \frac{4}{1} \cr & \Rightarrow {r^2} = 4 \cr & \Rightarrow r = 2, - 2 \cr & \left( 5 \right)\,\,\,\,\,\,\,\,\, \Rightarrow a = \frac{1}{{1 + r}} = \frac{1}{{1 + 2}}{\text{ or }}\frac{1}{{1 - 2}} = \frac{1}{3}{\text{ or }} - 1 \cr & {\text{As }}p{\text{ is an integer (given), }}r{\text{ is also an integer}}\left( {{\text{2 or}} - 2} \right) \cr & \therefore \,\left( 6 \right)\,\,\,\,\, \Rightarrow a \ne \frac{1}{3}\,{\text{ Hence }}a = - 1{\text{ and }}r = - 2. \cr & \therefore \,\,\,\,\,p = {\left( { - 1} \right)^2} \times \left( { - 2} \right) = - 2 \cr & q = {\left( { - 1} \right)^2} \times {\left( { - 2} \right)^5} = - 32 \cr} $$