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JEE Main 28-Jan-2025 Paper - Shift 2 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 28-Jan-2025 Paper - Shift 24 Marks
MCQ
Let $f$ be a real valued continuous function defined on the positive real axis such that $g ( x )=\int_0^{ x } t f( t ) dt$.
If $g\left(x^3\right)=x^6+x^7$, then value of $\sum_{r=1}^{15} f\left(r^3\right)$ is:
  • A
    320
  • B
    340
  • C
    270
  • D
    310
✓

Answer

D.
$g(x)=x 2+x^{\frac{7}{3}}$
$g^{\prime}(x)=2 x+\frac{7}{3} x^{\frac{4}{3}}$
$f(x)=\frac{g^{\prime}(x)}{x}$
$f(x)=2+\frac{7}{3} x^{\frac{1}{3}}$
$f\left(r^3\right)=2+\frac{7 r}{3}$
$\sum_{r=1}^{15}\left(1+\frac{7}{3} r\right)=310$

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