JEE MAIN - Mathematics (2022 - 28th June Evening Shift - No. 25)
If the system of linear equations
$$2x - 3y = \gamma + 5$$,
$$\alpha x + 5y = \beta + 1$$, where $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R has infinitely many solutions then the value
of | 9$$\alpha$$ + 3$$\beta$$ + 5$$\gamma$$ | is equal to ____________.
$$2x - 3y = \gamma + 5$$,
$$\alpha x + 5y = \beta + 1$$, where $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R has infinitely many solutions then the value
of | 9$$\alpha$$ + 3$$\beta$$ + 5$$\gamma$$ | is equal to ____________.
Answer
58
Explanation
If 2x $$-$$ 3y = $$\gamma$$ + 5 and $$\alpha$$x + 5y = $$\beta$$ + 1 have infinitely many solutions then
$${2 \over \alpha } = {{ - 3} \over 5} = {{\gamma + 5} \over {\beta + 1}}$$
$$ \Rightarrow \alpha = - {{10} \over 3}$$ and $$3\beta + 5\gamma = - 28$$
So $$|9\alpha + 3\beta + 5\gamma | = | - 30 - 28| = 58$$
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