$\alpha+\beta=\frac{31}{14} \ldots .(1) \text { and } \alpha \beta=\frac{3 \lambda}{14} \ldots .(2)$
$35 x^2-53 x+4 \lambda=0$
$\alpha+\gamma=\frac{53}{35} \ldots(3) \text { and } \alpha \gamma=\frac{4 \lambda}{35} \ldots(4)$
$\frac{(2)}{(4)} \Rightarrow \frac{\beta}{\gamma}=\frac{3 \times 35}{4 \times 14}=\frac{15}{8} \Rightarrow \beta=\frac{15}{8} \gamma$
$\frac{15}{8} \gamma-\gamma=\frac{7}{10} \Rightarrow \gamma=\frac{4}{5}$
$\Rightarrow \beta=\frac{15}{8} \times \frac{4}{5}=\frac{3}{2}$
$\Rightarrow \alpha=\frac{31}{14}-\beta=\frac{31}{14}-\frac{3}{2}=\frac{5}{7}$
$\Rightarrow \lambda=\frac{14}{3} \alpha \beta=\frac{14}{3} \times \frac{5}{7} \times \frac{3}{2}=5$
$\text { so, sum of roots } \frac{3 \alpha}{\beta}+\frac{4 \alpha}{\gamma}=\left(\frac{3 \alpha \gamma+4 \alpha \beta}{\beta \gamma}\right)$
$=\frac{\left(3 \times \frac{4 \lambda}{35}+4 \times \frac{3 \lambda}{14}\right)}{\beta \gamma}=\frac{12 \lambda(14+35)}{14 \times 35 \beta \gamma}$
$=\frac{49 \times 12 \times 5}{490 \times \frac{3}{2} \times \frac{4}{5}}=5$
Product of roots
$=\frac{3 \alpha}{\beta} \times \frac{4 \alpha}{\gamma}=\frac{12 \alpha^2}{\beta \gamma}=\frac{12 \times \frac{25}{49}}{\frac{3}{2} \times \frac{4}{5}}=\frac{250}{49}$
So, required equation is $x^2-5 x+\frac{250}{49}=0$
$\Rightarrow 49 x^2-245 x+250=0$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$(S1) :$ જો $P(A) =0$ હોય, તો $A =\emptyset$
$(S2) :$ જો $P ( A )=1$ હોય, તો $A =\Omega$
તો