QJEEM20Ma9732: JEE Main 2020 Mathematics - Class 12 JEEM20Ma9732 Solution: If for all real triplets (a, b, c), ƒ(x) = a + bx + cx 2 ; then …

Question

Verified Solution • 2025–26 Session moderate

If for all real triplets (a, b, c), ƒ(x) = a + bx + cx²; then

\int\limits_0^1 {f(x)dx}

is equal to :

Accepted Answer

ƒ(x) = a + bx + cx²

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

=

\left[ {ax + {{b{x^2}} \over 2} + {{c{x^3}} \over 3}} \right]_0^1

=

{a + {b \over 2} + {c \over 3}}

=

{1 \over 6}\left[ {6a + 3b + c} \right]

f(1) = a + b + c

f(0) = a

f\left( {{1 \over 2}} \right) = a + {b \over 2} + {c \over 4}

By checking each option you have to find the solution.

{1 \over 6}\left\{ {f(0) + f(1) + 4f\left( {{1 \over 2}} \right)} \right\}

=

{1 \over 6}\left[ {a + a + b + c + 4\left( {a + {b \over 2} + {c \over 4}} \right)} \right]

=

{1 \over 6}\left[ {6a + 3b + c} \right]

\therefore

Option (A) is correct option.

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