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

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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