The area of the region ( x , y ) : 0 ≤ x ≤ 9 4 , 0 ≤ y ≤ 1 , x ≥ 3 y , x + y ≥ 2 is

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MathematicsArea Under CurvesArea bounded by Miscellaneous CurvesEasy2 minPYQ_2021

MathematicsEasysingle choice

The area of the region
$\{(x, y) : 0 \leq x \leq \frac{9}{4}, 0 \leq y \leq 1, x \geq 3 y, x + y \geq 2\}$ is

Question diagram: The area of the region ( x , y ) : 0 ≤ x ≤ 9 4 , 0 ≤ y ≤ 1 ,

Options:

Answer:

Solution:

The region is a quadrilateral $A B C D$

Now, area of the quadrilateral $A B C D =$ area of triangle $A B D +$ area of the triangle $B C D$

So, area $= \frac{1}{2} |\begin{matrix} \frac{3}{2} & \frac{1}{2} & 1 \\ 2 & 0 & 1 \\ \frac{9}{4} & \frac{3}{4} & 1 \end{matrix}| + \frac{1}{2} \times (\frac{9}{4} - 2) \times \frac{3}{4}$

$= \frac{1}{2} \{- 2 (\frac{1}{2} - \frac{3}{4}) - (\frac{9}{8} - \frac{9}{8})\} + \frac{1}{2} \times \frac{1}{4} \times \frac{3}{4}$

$= \frac{1}{4} + \frac{3}{32} = \frac{11}{32}$

Stream:JEE_ADVSubject:MathematicsTopic:Area Under CurvesSubtopic:Area bounded by Miscellaneous Curves

⏱ 2mℹ️ Source: PYQ_2021

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