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Question

In the expansion of (35x/4+3x/4)n the sum of binomial coefficient is 64. If the term with greatest binomial coefficient exceeds the third term by (n1), then the number of value(s) of x is 
  1. 0
  2. 3
  3. 1
  4. 2


Solution

The correct option is D 2
Given that the sum of coefficients
2n=64n=6
The term with greatest binomial coefficient is T4,
Given condition is
T4T3=(n1)=61=5 6C3(35x/4)3(3x/4)3 6C2(35x/4)4(3x/4)2=520(33x)15(39x/2)=533x[43(33x/2)]=1
Assuming 33x/2=t
t2[43t]=13t34t2+1=0
By observation, t=1 is a root of the equation, 
(t1)(3t2t1)=0t=1,3t2t1=0
Now, checking the discriminant, we get
D=1+12=13>0
Now, 
t=1,1±136
As 33x/2>0,  xR
So, 
t=1,1+136

Hence, there are 2 possible values of x.

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