If in the expansion of ((1/x)+x tan x)5 the ratio of the 4 th term to the 2 nd is (2/27) π4, then the value of x is

Question

If in the expansion of ((1/x)+x tan x)5 the ratio of the 4 th term to the 2 nd is (2/27) π4, then the value of x is (A) (π/3) (B) (π/6) (C) (π/2) (D) (π/4)

Tardigrade Admin · Accepted Answer

T 2= 5 C 1((1/ x ))4 ⋅ x tan x =(5 tan x / x 3), T 4= 5 C 3((1/ x ))2 ⋅ x 3 tan 3 x =10 x tan 3 x ( T 4/ T 2)=2 x 4 tan 2 x =(2 π4/27) or x 4 tan 2 x =((π/3))4(√3)2 ∴ x =(π/3)

Q. If in the expansion of $(\frac{1}{x} + x tan x)^{5}$ the ratio of the 4 th term to the 2 nd is $\frac{2}{27} π^{4}$ , then the value of $x$ is

$\frac{π}{3}$

$\frac{π}{6}$

$\frac{π}{2}$

$\frac{π}{4}$

$T_{2} =^{5} C_{1} (\frac{1}{x})^{4} \cdot x tan x = \frac{5 t a n x}{x ^{3}}, T_{4} =^{5} C_{3} (\frac{1}{x})^{2} \cdot x^{3} tan^{3} x = 10 x tan^{3} x$
$\frac{T _{4}}{T _{2}} = 2 x^{4} tan^{2} x = \frac{2 π ^{4}}{27}$ or $x^{4} tan^{2} x = (\frac{π}{3})^{4} (3)^{2}$
$∴ x = \frac{π}{3}$

Q. If in the expansion of (x1​+xtanx)5 the ratio of the 4 th term to the 2 nd is 272​π4, then the value of x is

3π​

6π​

2π​

4π​