If the number of terms in the expansion of $${\left( {1 - {2 \over x} + {4 \over {{x^2}}}} \right)^n},\,x \ne 0,$$ is 28, then the sum of the coefficients of all the terms in this expansion, is :
A
243
B
729
C
64
D
2187
Explanation
Total no of terms in $${\left( {1 - {2 \over x} + {4 \over {{x^2}}}} \right)^n}$$ = $${}^{n + 2}{C_2}$$ = 28
(n+2)(n+1) = 56
$$ \Rightarrow n = 6$$
Sum of coefficient = (1 - 2 + 4)6 = 36 = 729
2
JEE Main 2015 (Offline)
MCQ (Single Correct Answer)
The sum of coefficients of integral power of $$x$$ in the binomial expansion $${\left( {1 - 2\sqrt x } \right)^{50}}$$ is :
A
$${1 \over 2}\left( {{3^{50}} - 1} \right)$$
B
$${1 \over 2}\left( {{2^{50}} + 1} \right)$$
C
$${1 \over 2}\left( {{3^{50}} + 1} \right)$$
D
$${1 \over 2}\left( {{3^{50}}} \right)$$
Explanation
$${\left( {1 - 2\sqrt x } \right)^{50}}$$
= $${}^{50}{C_0} + {}^{50}{C_1}.\left( { - 2\sqrt x } \right) + {}^{50}{C_2}.{\left( { - 2\sqrt x } \right)^2} + ....$$
Now we need to find out those coefficient where degree of x is integer and you can see at odd terms power of x is integer.
Let $${\left( {1 - 2\sqrt x } \right)^{50}}$$ = Odd(A) - Even(B)
So $${\left( {1 + 2\sqrt x } \right)^{50}}$$ = A + B
$$\therefore$$ 2A = $${\left( {1 + 2\sqrt x } \right)^{50}}$$ + $${\left( {1 - 2\sqrt x } \right)^{50}}$$
$$ \Rightarrow A = {1 \over 2}\left[ {{{\left( {1 + 2\sqrt x } \right)}^{50}} + {{\left( {1 - 2\sqrt x } \right)}^{50}}} \right]$$
Now to find sum of coefficient of A, put x = 1.
$$\therefore$$ Sum of coefficient of A = $${1 \over 2}\left[ {{{\left( {1 + 2} \right)}^{50}} + {{\left( {1 - 2} \right)}^{50}}} \right]$$
If the coefficints of $${x^3}$$ and $${x^4}$$ in the expansion of $$\left( {1 + ax + b{x^2}} \right){\left( {1 - 2x} \right)^{18}}$$ in powers of $$x$$ are both zero, then $$\left( {a,\,b} \right)$$ is equal to:
Solving (1) and (2), we get $$a$$ = 16, b = $${{172} \over 3}$$
4
JEE Main 2013 (Offline)
MCQ (Single Correct Answer)
The term independent of $$x$$ in expansion of
$${\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}$$ is