The number of real roots of the equation - x2-1 2 x3 x- 2 x3 x x2-1- x x2-1 2 x3 -0 is

The correct option is B 1a amp;b amp;c b amp;c amp;a c amp;a amp;b= (a^3+b^3+c^3 3abc) If a^3+b^3+c^3 3abc=0 ⇒ a+b+c=0 or a=b=c ⇒ x^2+1=2x^3=x ( N ...

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Byju's Answer

Standard XII

Mathematics

Relation between Roots and Coefficients for Quadratic

The number of...

Question

The number of real roots of the equation
$∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + 1 & 2 x^{3} & x 2 x^{3} & x & x^{2} + 1 x & x^{2} + 1 & 2 x^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$ is

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Solution

The correct option is B $1$
$∣ ∣ ∣ ∣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ∣ ∣ ∣ ∣ = - (a^{3} + b^{3} + c^{3} - 3 a b c) If a^{3} + b^{3} + c^{3} - 3 a b c = 0 \Rightarrow a + b + c = 0 or a = b = c \Rightarrow x^{2} + 1 = 2 x^{3} = x (Not possible) or, x^{2} + 1 + 2 x^{3} + x = 0 ⇓ f^{'} (x) = 6 x^{2} + 2 x + 1 (D < 0)$
$∴ f (x) = 0$ has exactly one real root.

Suggest Corrections

The number of real roots of the equation ∣∣ ∣ ∣∣x2+12x3x2x3xx2+1xx2+12x3∣∣ ∣ ∣∣=0 is

The correct option is B 1∣∣ ∣∣abcbcacab∣∣ ∣∣=−(a3+b3+c3−3abc)If a3+b3+c3−3abc=0⇒a+b+c=0 or a=b=c⇒x2+1=2x3=x ( Not possible ) or, x2+1+2x3+x=0 ⇓f′(x)=6x2+2x+1 (D<0) ∴f(x)=0 has exactly one real root.

The number of real roots of the equation
$∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + 1 & 2 x^{3} & x 2 x^{3} & x & x^{2} + 1 x & x^{2} + 1 & 2 x^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$ is