Question

13. If x+1/x= 2 cos 20°, then the value
х^3+1/x^3
1
of x' +
3.
х
1) 1/4
2) 1/2
3) 1
4) 1/8​

Answer 1

Solution:-

x+1/x = 2cos20°

LHS=

=> x³+1/x³= (x+1/x)(x²-x.1/x+1/x²)

= (2cos20°) [( x+1/x)²-3x.1/x]

= (2cos20°) [( 2cos20°)²-3]

= 2×0.94) ( 4×0.88-3)

= 1.88×( 3.52-3)

= 1.88×0.52

= 0.98

= 1 (approximately)

=> option (3) 1 is the correct answer

Hope this helps you

Answer 2

The value of $x^{3}$+1/$x^{3}$ is 1. (Option 3)

Given:

x+1/x=2 cos 20°

To find:

The value of $x^{3}$+1/$x^{3}$

Solution:

We will calculate the value by cubing the given sum.

The given sum=x+1/x=2 cos 20°

On cubing the equation, we get

$(x+1/x)^{3}$=$(2 cos 20)^{3}$

Using the identity $(a+b)^{3}$=$a^{3} +b^{3} +3ab(a+b)$,

$x^{3}$ + 1/$x^{3}$ +3×(x)×(1/x)×(x+1/x)=8×$Cos^{3}20$

$x^{3}$ + 1/$x^{3}$ +3(2 Cos 20)=8×$Cos^{3}20$

$x^{3}$ + 1/$x^{3}$ =8 $Cos^{3}20$ - 6 Cos 20

$x^{3}$ + 1/$x^{3}$ = 2(4 $Cos^{3}20$ - 3 Cos 20)

Now, we know that 4 $Cos^{3}$θ -3Cosθ =Cos 3θ.

$x^{3}$ + 1/$x^{3}$ = 2 Cos 3×20°

$x^{3}$ + 1/$x^{3}$ =2 Cos 60°

$x^{3}$ + 1/$x^{3}$ =2×1/2

$x^{3}$ + 1/$x^{3}$ =1

Therefore, the value of $x^{3}$+1/$x^{3}$ is 1.

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