Let p ( x ) be a function defined on R such that p' ( x ) = p' ( … - Vidyadip

Question

Let $p\left( x \right)$ be a function defined on $R$ such that $p'\left( x \right) = p'\left( {1 - x} \right),$ for all $x \in \left[ {0,1} \right]$ $p\left( 0 \right) = 1,p\left( 1 \right) = 41.$ then $\mathop \smallint \limits_0^1 p\left( x \right)dx = $

✓

Answer

a
$p^{\prime}(x)=p^{\prime}(1-x)$

$\Rightarrow p(x)=-p(1-x)+c$

at $x=0$ $p(0)=-p(1)+c \quad $

$ \Rightarrow 42=c$

now $p(x)=-p(1-x)+42$

$\Rightarrow p(x)+p(1-x)=42$

$I = \int\limits_0^1 p (x)dx = \int\limits_0^1 p (1 - x)dx$

$I = \int\limits_0^1 {\left( {42} \right)dx} \,\,\,\,\,\,\,\, \Rightarrow I = 21.$

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If the function $f(x)=\left\{\begin{array}{cl}\frac{1}{|x|} & ,|x| \geq 2 \\ a x^2+2 b, & |x|<2\end{array}\right.$ is differentiable on $R$, then $48(a+b)$ is equal to___________.

Let $S$ be the set of all passwords which are six to eight characters long, where each character is either an alphabet from $\{ A , B , C , D , E \}$ or a number from $\{1,2,3,4,5\}$ with the repetition of characters allowed. If the number of passwords in $S$ whose at least one character is a number from $\{1,2,3,4,5\}$ is $\alpha \times 5^{6}$, then $\alpha$ is equal to $.......$

If $ a$ and $b $ be parallel vectors, then $[a c b] =$

What is the sum of the squares of roots of ${x^2} - 3x + 1 = 0$

In a series of $2n$ observations, half of them equal to $a$ and remaining half equal to $-a$. If the standard deviation of the observations is $2$, then $|a|$ equals

Let a line perpendicular to the line 2x – y = 10 touch the parabola $y^2=4(x-9)$ at the point $P.$ The distance of the point $P$ from the centre of the circle$x^2+y^2-14 x-8 y+56=0$ is $.....$

The obtuse angle between the lines $y = - \;2$ and $y = x + 2$ is ....$^o$

The equations of the sides $AB , BC$ and $CA$ of a triangle $ABC$ are $2 x + y =0, x + py =15 a$ and $x-y=3$ respectively. If its orthocentre is $(2, a)$, $-\frac{1}{2}< a <2$, then $p$ is equal to$...$

If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{\ell}}\right)^9$ is $-84$ and the Coefficient of $x^{-3 \ell}$ is $2^\alpha \beta$, where $\beta < 0$ is an odd number, Then $|\alpha \ell-\beta|$ is equal to

Let $a$, $b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x ^{2}-8 ax +2 a =0$ and $q$ and $s$ are the roots of the equation $x^{2}+12 b x+6 b$ $=0$, such that $\frac{1}{ p }, \frac{1}{ q }, \frac{1}{ r }, \frac{1}{ s }$ are in A.P., then $a ^{-1}- b ^{-1}$ is equal to $......$