QJEEM10Mab93c: JEE Main 2010 Mathematics - Class 12 JEEM10Mab93c Solution: Let be a function defined on such that for all and Then equals :

Question

Verified Solution • 2025–26 Session moderate

Let

p(x)

be a function defined on

R

such that

p'(x)=p'(1-x),

for all

x \in \left[ {0,1} \right],p\left( 0 \right) = 1

and

p(1)=41.

Then

\int\limits_0^1 {p\left( x \right)dx}

equals :

Accepted Answer

p'\left( x \right) = p'\left( {1 - x} \right)

\Rightarrow p\left( x \right) = - p\left( {1 - x} \right) + c

at

x=0

p\left( 0 \right) = - p\left( 1 \right) + c \Rightarrow 42 = c

Now,

p\left( x \right) = - p\left( {1 - x} \right) + 42

\Rightarrow p\left( x \right) + p\left( {1 - x} \right) = 42

\Rightarrow I = \int\limits_0^1 {p\left( x \right)dx\,\,\,\,\,\,\,\,\,\,...\left( i \right)}

\Rightarrow I = \int\limits_0^1 {p\left( {1 - x} \right)} dx\,\,\,\,\,\,...\left( {ii} \right)

on adding

(i)

and

(ii),

\,\,\,\,\,\,\,\,\,\,\,

2I = \int\limits_0^1 {\left( {42} \right)dx \Rightarrow I = 21}

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