AP EAMCET Engineering 2013 Solved Paper

Given system of equation is
x+y=

2π

3

.......(i)
and cos‌x+cos‌y=

3

2

, where x,y are real.
⇒2‌cos(

x+y

2

).cos(

x−y

2

)=

3

2

⇒cos(

1

2

.

2π

3

).cos(

x−y

2

)=

3

4

[from Eq. (i)]
⇒ cos(

π

3

).cos(

x−y

2

)=

3

4

⇒

1

2

‌cos(

x−y

2

)=

3

4

⇒ cos(

x−y

2

)=

3

2

.......(ii)
Now, we have
cos(x−y)=2cos2(

x−y

2

)−1
=2×

9

4

−1=

9

2

−1=

7

2

[from Eq. (ii)]
and cos(x−y)=1−2sin2(

x−y

2

)
⇒ sin(

x−y

2

)<0
So, system of equation have empty set of solution.