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1)

Let the straight line x = b divide the area enclosed by   y=(1x)2,y=0 and x=0 into two parts R1(0xb)and R2(bx1)   such that R1R2=14


A) 34

B) 12

C) 13

D) 14

Answer:

Option B

Explanation:

Here area between 0 to b is R1 and b to 1 is R2

  \int_{0}^{b} (1-x)^{2}dx-\int_{b}^{1} (1-x)^{2}dx=\frac{1}{4}

\Rightarrow  \left(\frac{(1-x)^{3}}{-3}\right)_{0}^{b}-\left(\frac{(1-x)^{3}}{-3}\right)^{1}_{b}=\frac{1}{4}

\Rightarrow  -\frac{1}{3}{(1-b)^{3}-1})+\frac{1}{3}(0-(1-b)^{3})=\frac{1}{4}

\Rightarrow   -\frac{2}{3}(1-b)^{3}=-\frac{1}{3}+\frac{1}{4}=-\frac{1}{12}

\Rightarrow   (1-b)^{3}=\frac{1}{8}

\Rightarrow   (1-)=\frac{1}{2} \Rightarrow  b= \frac{1}{2}