| Statement I: The quadratic equation \[a{{x}^{2}}+bx+c=0\] has two distinct real roots, if\[{{b}^{2}}-4ac>0\]. |
| Statement II: The quadratic equation \[2({{a}^{2}}+{{b}^{2}}){{x}^{2}}+2(a+b)x+1=0\]has no real roots, when \[a\ne b\]. |
A) Both Statement - I and Statement - II are true.
B) Statement - I is true but Statement - II is false.
C) Statement - I is false but Statement - II is true.
D) Both Statement - I and Statement - II are false.
Correct Answer: C
Solution :
Statement - I is false, since the quadratic equation \[a{{x}^{2}}+bx+c=0\]has two distinct real roots, if\[{{b}^{2}}-4ac>0\]. Also, given equation is \[2({{a}^{2}}+{{b}^{2}}){{x}^{2}}+2(a+b)x+1=0\] \[D={{b}^{2}}-4ac={{(2(a+b))}^{2}}-4(2{{a}^{2}}+2{{b}^{2}})(1)\] \[=4{{a}^{2}}+4{{b}^{2}}+8ab-8{{a}^{2}}-8{{b}^{2}}\] \[=-4{{a}^{2}}-4{{b}^{2}}+8ab=-4{{(a-b)}^{2}}<0\] \[\therefore \]Given equation has no real roots. Hence, statement - II is true.
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