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Question

Solve the following equations:
x3+y3+z3=a3,x2+y2+z2=a2,x+y+z=a.

A
x=(a,0,0);y=(0,a,0);z=(0,0,a)
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B
x=(2a,a,0);y=(0,a,0);z=(0,a,a)
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C
x=(a,0,2a);y=(a,a,0);z=(0,0,a)
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D
None of the above
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Solution

The correct option is A x=(a,0,0);y=(0,a,0);z=(0,0,a)
x3+y3+z3=a3 ......(i)
x2+y2+z2=a2 .....(ii)
x+y+z=a .........(iii)
(x+y+z)2=x2+y2+z2+2xy+2yz+2zxa2=a2+2xy+2yz+2zxxy+yz+zx=0 ........(iv)
x3+y3+z33xyz=(x+y+z)(x2+y2+z2xyyzzx)a33xyz=a(a20)xyz=0x=0,y=0,z=0
When x=0
z2+y2=a2z+y=a
Solving both the sides, we get
z=a,0;y=0,a
When y=0
x2+z2=a2y+z=ax=a,0;z=0,a
When z=0
x2+y2=a2x+y=ax=0,a;y=a,0
So, complete solution is x=(a,0,0);y=(0,a,0);z=(0,0,a).

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