Question

# In the expansion of (35x/4+3−x/4)n the sum of binomial coefficient is 64. If the term with greatest binomial coefficient exceeds the third term by (n−1), then the number of value(s) of x is 0312

Solution

## The correct option is D 2Given that the sum of coefficients 2n=64⇒n=6 The term with greatest binomial coefficient is T4, Given condition is T4−T3=(n−1)=6−1=5⇒ 6C3(35x/4)3(3−x/4)3− 6C2(35x/4)4(3−x/4)2=5⇒20(33x)−15(39x/2)=5⇒33x[4−3(33x/2)]=1 Assuming 33x/2=t ⇒t2[4−3t]=1⇒3t3−4t2+1=0 By observation, t=1 is a root of the equation,  ⇒(t−1)(3t2−t−1)=0⇒t=1,3t2−t−1=0 Now, checking the discriminant, we get D=1+12=13>0 Now,  t=1,1±√136 As 33x/2>0,  ∀x∈R So,  t=1,1+√136 Hence, there are 2 possible values of x.

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