JEE MAIN - Mathematics (2021 - 16th March Morning Shift - No. 15)
Let the curve y = y(x) be the solution of the differential equation, $${{dy} \over {dx}}$$ = 2(x + 1). If the numerical value of area bounded by the curve y = y(x) and x-axis is $${{4\sqrt 8 } \over 3}$$, then the value of y(1) is equal to _________.
Answer
2
Explanation
Given, $${{dy} \over {dx}}$$ = 2(x + 1)
Integrating both sides, we get
$$y = {x^2} + 2x + c$$
Let the two roots of the quadratic equation $$\alpha $$ and $$\beta $$
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As parabola intercept the x axis so D > 0
From figure, AB = |$$\alpha $$ - $$\beta $$| = $${{\sqrt D } \over {\left| a \right|}}$$ = $$\sqrt D $$
and BC = $$ - {D \over {4a}}$$ = $$ - {D \over 4}$$
$$ \therefore $$ Area of rectangle (ABCD) = AB $$ \times $$ BC = $$\sqrt D \times {D \over 4}$$
From property we know,
Area of parabola with the x axis = $${2 \over 3}$$(Area of rectangle)
$$ \Rightarrow $$ $${{4\sqrt 8 } \over 3}$$ = $${2 \over 3} \times \sqrt D \times {D \over 4}$$
$$ \Rightarrow $$ $$D\sqrt D $$ = $$8\sqrt 8 $$
$$ \Rightarrow $$ D = 8
$$ \therefore $$ b2 - 4ac = 8
$$ \Rightarrow $$ 4 - 4c = 8
$$ \Rightarrow $$ 1 $$-$$ c = 2 $$ \Rightarrow $$ c = $$-$$ 1
Equation of f(x) = x2 + 2x $$-$$ 1
$$ \therefore $$ f(1) = 1 + 2 $$-$$ 1 = 2
Integrating both sides, we get
$$y = {x^2} + 2x + c$$
Let the two roots of the quadratic equation $$\alpha $$ and $$\beta $$
As parabola intercept the x axis so D > 0
From figure, AB = |$$\alpha $$ - $$\beta $$| = $${{\sqrt D } \over {\left| a \right|}}$$ = $$\sqrt D $$
and BC = $$ - {D \over {4a}}$$ = $$ - {D \over 4}$$
$$ \therefore $$ Area of rectangle (ABCD) = AB $$ \times $$ BC = $$\sqrt D \times {D \over 4}$$
From property we know,
Area of parabola with the x axis = $${2 \over 3}$$(Area of rectangle)
$$ \Rightarrow $$ $${{4\sqrt 8 } \over 3}$$ = $${2 \over 3} \times \sqrt D \times {D \over 4}$$
$$ \Rightarrow $$ $$D\sqrt D $$ = $$8\sqrt 8 $$
$$ \Rightarrow $$ D = 8
$$ \therefore $$ b2 - 4ac = 8
$$ \Rightarrow $$ 4 - 4c = 8
$$ \Rightarrow $$ 1 $$-$$ c = 2 $$ \Rightarrow $$ c = $$-$$ 1
Equation of f(x) = x2 + 2x $$-$$ 1
$$ \therefore $$ f(1) = 1 + 2 $$-$$ 1 = 2
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