SSC CGL 202031)If \(x^2-2\sqrt5x +1=0\) , then what is the value of \((x^5+{1\over x^5})\) ?
\(610\sqrt5\)
\(x^2-2\sqrt5x +1=0\) ; ⇒ \(x^2+1=2\sqrt5x\) ;
⇒ \({x^2+1\over x}={2\sqrt5x\over x}\) ; ⇒ \(x+{1\over x}=2\sqrt5\) ___(i) ; \(x^2+{1\over x^2}= 18\) ____(ii) ;
Again, \((x+{1\over x})^3= (2\sqrt5)^3\); ⇒ \(x^3+{1\over x^3}=34\sqrt5\) ___(iii) ;
Multiplying equation (ii) by (iii), we have \((x^2+{1\over x^2})(x^3+{1\over x^3}) = 18\times 34\sqrt5\) ;
⇒ \(x^5+{1\over x^5}+x+{1\over x}=612\sqrt5\) ; \(x^5+{1\over x^5} = 610\sqrt5\)
SSC CGL 202032)If \(16a^4 + 36a^2b^2 \)\(+ 81b^4 = 91\) and \(4a^2 + 9b^2\)\( - 6ab = 13\), then what is the value of 3ab ?
\(-{3\over2}\)
\(4a^2 + 9b^2 - 6ab = 13\) ;
⇒ \((4a^2 + 9b^2 - 6ab)^2 = (13)^2\);
\((\because(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ac))\)
⇒\( (4a^2)^2 + (9b^2)^2 + (6ab)^2 +2(4a^2.9b^2 - 9b^2.6ab - 6ab.4a^2) = 169\) ;
⇒ \(16a^4 + 36a^2b^2 + 81b^4 + 2(36a^2b^2 - 54ab^3 - 24a^3b) = 169\) ;
⇒ \(36a^2b^2 - 54ab^3 - 24a^3b = \frac{169 - 91}{2}\) ;
⇒ \(6ab(6ab - 9b^2 - 4a^2) = 39\) ;
⇒ 6ab = -3 ; ⇒ 3ab = -3/2
SSC CGL 202033)If \(P={x^3+y^3\over (x-y)^2+3xy}\) , \(Q={(x+y)^2-3xy\over x^3-y^3}\) and \(R={(x+y)^2+(x-y)^2\over x^2-y^2}\), then what is the value of \((P\div Q)\times R\) ?
\(2(x^2+y^2)\)
\((P\div Q)\times R\) \(={x^3+y^3\over (x-y)^2+3xy}\div {(x+y)^2-3xy\over x^3-y^3}\times {(x+y)^2+(x-y)^2\over x^2-y^2}\)
\(={(x+y)(x^2+y^2-xy)\over x^2+y^2-2xy+3xy}\times {(x-y)(x^2+y^2+xy)\over x^2+y^2+2xy-3xy}\times {x^2+y^2+2xy+x^2+y^2-2xy\over (x+y)(x-y)}\)
= \(2(x^2+y^2)\)