Exams Joint Entrance Examination Mathematics Binomial Theorem Question 296 Question 296 of 347 Medium MCQ
If n is the degree of the polynomial,
\[{\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8} + \] \[{\left[ {{2 \over {\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8}\]
and m is the coefficient of xn in it, then the ordered pair (n, m) is equal to :
Solutions · 1 Post your solution Official solution
Given,
\[{\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8}\] + \[{\left[ {{2 \over {\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8}\]
= \[{\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }} \times {{\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} } \over {\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8}\]
+ \[{\left[ {{2 \over {\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }} \times {{\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} } \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8}\]
= \[{\left[ {{{2(\sqrt {5x^3 + 1} + \sqrt {5x^3 - 1} } \over 2}} \right]^8}\] + \[{\left[ {{{2(\sqrt {5x^3 - 1} - \sqrt {5x^3 - 1} } \over 2}} \right]^8}\]
= (\[{\sqrt {5{x^3} + 1} }\] + \[{\sqrt {5{x^3} - 1} }\])8 + (\[{\sqrt {5{x^3} - 1} }\] - \[{\sqrt {5{x^3} - 1} }\])8
= 2 \[\left[ {{}^8{C_0}} \right.{(\sqrt {5{x^3} + 1} )^8}\] +
+ \[{}^8{C_2}{(\sqrt {5{x^3} + 1} )^6}\sqrt {5{x^3} - 1} \]
+ \[{}^8{C_4}{(\sqrt {5{x^3} + 1} )^4}{(5{x^3} - 1)^2}\]
+ \[{}^8{C_6}{(\sqrt {5{x^3} + 1} )^2}{(5{x^3} - 1)^3}\]
+ \[\left. {{}^8{C_8}{{(5{x^3} - 1)}^4}} \right]\]
= 2\[[{}^8{C_0}{(5{x^3} + 1)^4}\]
+ 8 C 2 ( 5 x 3 + 1 ) 3 ( 5 x 3 − 1 ) {}^8{C_2}{(5{x^3} + 1)^3}(5{x^3} - 1) 8 C 2 ( 5 x 3 + 1 ) 3 ( 5 x 3 − 1 )
+ 8 C 4 ( 5 x 3 + 1 ) 2 ( 5 x 3 − 1 ) 2 {}^8{C_4}(5{x^3} + 1)^2{(5{x^3} - 1)^2} 8 C 4 ( 5 x 3 + 1 ) 2 ( 5 x 3 − 1 ) 2
+ 8 C 6 ( 5 x 3 + 1 ) ( 5 x 3 − 1 ) 3 {}^8{C_6}(5{x^3} + 1){(5{x^3} - 1)^3} 8 C 6 ( 5 x 3 + 1 ) ( 5 x 3 − 1 ) 3
+ \[{}^8{C_8}{(5{x^3} - 1)^4}]\]
Here maximum power of x is 12
\[ \therefore \] Degree of polynomial = 12
Coefficient of x12
= 2 [8C0 54 + 8C2
\[\times\] 53 \[\times\] 5 + 8C4
\[\times\] 52 \[\times\] 52 + 8C6 \[\times\] 5 \[\times\] 53 + 8C8 \[\times\] 54]
= 160000 = (20)4 Answer the question to unlock