For m , n 0 , let ( m , n ) = 0 2 t m ( 1 + 3 t ) n d t . If 11 ( 10 , 6 ) + 18 ( 11 , 5 ) = p ( 14 ) 6 , then p is equal to ______ . [2023]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
For $m, n > 0$ , let $α (m, n) = \int_{0}^{2} t^{m} {(1 + 3 t)}^{n} d t$ . If $11 α (10, 6) + 18 α (11, 5) = p {(14)}^{6}$ , then $p$ is equal to ______ . [2023]

Ans.
(32)

We have, $α (m, n) = \int_{0}^{2} t^{m} {(1 + 3 t)}^{n} d t$

Now, $11 α (10, 6) + 18 α (11, 5)$

$= 11 \int_{0}^{2} t^{10} {(1 + 3 t)}^{6} d t + 18 \int_{0}^{2} t^{11} {(1 + 3 t)}^{5} d t$

$= 11 {[{(1 + 3 t)}^{6} \cdot \frac{t^{11}}{11}]}_{0}^{2} - 11 \int_{0}^{2} 6 {(1 + 3 t)}^{5} \cdot$ $3 \frac{t^{11}}{11} d t + 18 \int_{0}^{2} t^{11} {(1 + 3 t)}^{5} d t$

$= {[t^{11} {(1 + 3 t)}^{6}]}_{0}^{2}$ $= 2^{11} {(7)}^{6} = 2^{5} {(14)}^{6}$

$= 32 {(14)}^{6} = p {(14)}^{6}$ $[∵ Given]$

$\Rightarrow p = 32$

Want Us to Email you About Special Offers & Updates?