Let f be a real valued continuous function defined on the positiv… - Vidyadip

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

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "SECTION - A [MATHS - MCQ]" from JEE Main 28-Jan-2025 Paper - Shift 2→JEE Main 28-Jan-2025 Paper - Shift 2 — all question types→JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

An angle between the lines whose direction cosines are given by the equations, $l+ 3m + 5n\, = 0$ and $5lm -2mn + 6nl = 0$ , is

Let $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4$ be the solution of the equation $4 x^4+8 x^3-17 x^2-12 x+9=0$ and $\left(4+x_1^2\right)\left(4+x_2^2\right)\left(4+x_3^2\right)\left(4+x_4^2\right)=\frac{125}{16} m$. Then the value of $\mathrm{m}$ is..........

When ${2^{301}}$ is divided by $5$, the least positive remainder is

If $f(x) = \left\{ \begin{array}{l}\frac{1}{x}\sin {x^2},\,x \ne 0\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,x = 0\end{array} \right.$, then

The sum of the roots of the equation $x+1-2 \log _{2}\left(3+2^{x}\right)+2 \log _{4}\left(10-2^{-x}\right)=0$, is :

The number of positive integers $n$ in the set $\{1,2,3$, $\ldots \ldots . ., 100\}$ for which the number $\frac{1^2+2^2+3^2+\ldots .+n^2}{1+2+3+\ldots+n}$ is an integer is

If $A = \left[ {\begin{array}{*{20}{c}}i&0\\0&i\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}0&{ - i}\\{ - i}&0\end{array}} \right]$, then $(A + B)(A - B)$ is equal to

Both the roots of the given equation$(x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a) = 0$ are always

Let $f: \mathbf{R}-\{0\} \rightarrow(-\infty, 1)$ be a polynomial of degree 2, satisfying $f(\mathrm{x}) f\left(\frac{1}{\mathrm{x}}\right)=f(\mathrm{x})+f\left(\frac{1}{\mathrm{x}}\right)$. If $f(\mathrm{~K})=-2 \mathrm{~K}$, then the sum of squares of all possible values of K is :

If $\left\{a_{i}\right\}_{i=1}^{n}$ where $n$ is an even integer, is an arithmetic progression with common difference $1$ , and $\sum \limits_{ i =1}^{ n } a _{ i }=192, \sum \limits_{ i =1}^{ n / 2} a _{2 i }=120$, then $n$ is equal to