If A- a b c d - where a- b- c- d are the roots of the equation x4-13x3-px2-qx-64-0- p-q- whose three roots are positive and one is negative- then the minimum positive value of detA is

A=[ nbsp; nbsp;a amp; b nbsp; nbsp; nbsp;c amp; d nbsp; ] ∴| A |=ad bc x^4 13x^3+px^2+qx 64=0 Roots of the above equation are a,b,c,d A ...

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Byju's Answer

Standard XII

Mathematics

Multiplication of Matrices

If A=[ a b...

Question

If $A = [\begin{matrix} a & b c & d \end{matrix}],$ where $a, b, c, d$ are the roots of the equation $x^{4} - 13 x^{3} + p x^{2} + q x - 64 = 0,$ $p, q \in R$ whose three roots are positive and one is negative, then the minimum positive value of $det (A)$ is

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Solution

The correct option is B $16$
$A = [\begin{matrix} a & b c & d \end{matrix}]$
$∴ | A | = a d - b c$

$x^{4} - 13 x^{3} + p x^{2} + q x - 64 = 0$
Roots of the above equation are $a, b, c, d$
Applying A.M. $\geq$ G.M. on $a d$ and $- b c$
$\frac{a d + (- b c)}{2} \geq \sqrt{a d (- b c)} \Rightarrow \frac{a d - b c}{2} \geq \sqrt{64} \Rightarrow a d - b c \geq 16$
$\Rightarrow | A | \geq 16$

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If A=[abcd], where a,b,c,d are the roots of the equation x4−13x3+px2+qx−64=0, p,q∈R whose three roots are positive and one is negative, then the minimum positive value of det(A) is

The correct option is B 16A=[abcd] ∴|A|=ad−bc x4−13x3+px2+qx−64=0 Roots of the above equation are a,b,c,d Applying A.M. ≥ G.M. on ad and −bc ad+(−bc)2≥√ad(−bc)⇒ad−bc2≥√64⇒ad−bc≥16 ⇒|A|≥16

If $A = [\begin{matrix} a & b c & d \end{matrix}],$ where $a, b, c, d$ are the roots of the equation $x^{4} - 13 x^{3} + p x^{2} + q x - 64 = 0,$ $p, q \in R$ whose three roots are positive and one is negative, then the minimum positive value of $det (A)$ is