Let the points lie on or inside the triangle with sides x + y = 11, x + 2y = 16 and 2x + 3y = 29. Then the product of the smallest and the largest values of is equal to : [2025]

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Solution

Q.
Let the points $(\frac{11}{2}, α)$ lie on or inside the triangle with sides x + y = 11, x + 2y = 16 and 2x + 3y = 29. Then the product of the smallest and the largest values of $α$ is equal to : [2025]

1 22

2 55

3 33

4 44

Ans.
(3)

Given that a triangle bounded by lines $L_{1}$ : x + y = 11, $L_{2}$ : x + 2y =16 and $L_{3}$ : 2x + 3y = 29.

The region is given by

Using $L_{1}$ : When $x = \frac{11}{2}$ , y = 11 – x

$\Rightarrow α = 11 - \frac{11}{2} = \frac{11}{2}$ (minimum)

Using $L_{3}$ : when x = 11/2, 3y = 29 – 2x

$\Rightarrow 3 α = 29 - 2 \times \frac{11}{2} \Rightarrow 3 α = 18$

$\Rightarrow α = 6$ (maximum)

$∴$ Product of the smallest and the largest value of $α = \frac{11}{2} \times 6 = 33$ .

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