Q137 / 225 · JEE Main 2002 · Mathematics

If $y=f(x)$ makes $+v e$ intercept of 2 and 0 unit on $x$ and $y$ axes and encloses an area of $3 / 4$ square unit with the axes then $\int_0^2 x f^{\prime}(x) d x ~~is$

  • A
    3/2
  • B
    1
  • C
    5/4
  • Correct answer
    D
    -3/4

Solution

We have $\int_0^2 f(x) d x=\frac{3}{4} ;$ Now, $\int_0^2 x f^{\prime}(x) d x=x \int_0^2 f^{\prime}(x) d x-\int_0^2 f(x) d x$ $=[x f(x)]_0^2-\frac{3}{4}=2 f(2)-\frac{3}{4}=0-\frac{3}{4}(\because f(2)=0)=-\frac{3}{4}$

Hence, the correct answer is option: 'D'.

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