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Question

# Let the curve y=f(x) passing through (4,−2) satisfies the differential equation y(x+y3)dx=x(y3−x)dy and let y=g(x)=sin2x∫1/8sin−1(√t)dt+cos2x∫1/8cos−1(√t)dt, 0≤x≤π2 be the second curve, then

A
If the equation of curve y=f(x) satisfies ay3+bx=0, then a+b=3;where a and b are positive integers and co-prime.
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B
If the equation of curve y=f(x) satisfies ay3+bx=0, then a=b;where a and b are positive integers.
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C
Area of the region bounded by y=f(x),y=g(x) and x=0 is 18(3π8)4 sq. units.
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D
Area of the region bounded by y=f(x),y=g(x) and x=0 is 18(3π16)4 sq. units.
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Solution

## The correct options are A If the equation of curve y=f(x) satisfies ay3+bx=0, then a+b=3;where a and b are positive integers and co-prime. D Area of the region bounded by y=f(x),y=g(x) and x=0 is 18(3π16)4 sq. units. y(x+y3)dx=x(y3−x)dy ⇒(y4dx−xy3dy)+(xydx+x2dy)=0 ⇒x2y3(ydx−xdyx2)+x(xdy+ydx)=0 ⇒yx.d(yx)−d(xy)x2y2=0 ⇒12(yx)2+1xy=C y=f(x) passes through (4,−2). ⇒C=0 So, y3=−2x ⇒y=f(x)=−3√2x g(x)=sin2x∫1/8sin−1(√t)dt+cos2x∫1/8cos−1(√t)dt g′(x)=sin−1(sin x)×sin2x+cos−1(cos x)×(−sin2x)=0 ⇒g′(x)=0 ⇒g(x)=C At x=π4,g(x)=3π16 So, g(x)=3π16 Required shaded area =3π/16∫0y32dy=18(3π16)4

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