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Question

Equation $$2x^2 - 2(2a + 1)x + a(a + 1) = 0$$ has one root less than $$a$$ and other root greater than $$a$$, then which of the following is true


A
0<a<1
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B
1<a<0
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C
a>0
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D
a<1
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Solution

The correct options are
A $$0 < a < 1$$
B $$a > 0$$
C $$a < -1$$
Quadratic equation : $$2{ x }^{ 2 }-2\left( 2a+1 \right) x+a\left( a+1 \right) =0$$
Both roots are real, discriminant $$= D>0$$
Comparing with the standard quadratic equation $$A{ x }^{ 2 }+Bx+C=0$$, we get $$A=2>0$$.
Since, $$x=a$$ lies between the two roots and A is positive, we have $$f(a)<0$$
$$\Rightarrow $$ $$2{ \left( a \right)  }^{ 2 }-2\left( 2a+1 \right) \left( a \right) +a\left( a+1 \right) <0$$
$$\Rightarrow $$ $$-{ a }^{ 2 }-a<0$$
$$\Rightarrow $$ $${ a }^{ 2 }+a>0$$
$$\Rightarrow $$ $${ a }\in \left( -\infty ,-1 \right) \cup \left( 0,\infty  \right) $$
Hence, options (A), (C) and (D) are correct.

Mathematics

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