The total number of distinct x ∈ R for which x x 2 1 + x 3 2 x 4 x 2 1 + 8 x 3 3 x 9 x 2 1 + 27 x 3 = 10 is

x 3 1 1 1 + x 3 2 4 1 + 8 x 3 3 9 1 + 27 x 3 = 10 ⇒ x 3 1 1 1 1 2 4 1 3 9 + x 3 . x 3 1 1 1 2 4 8 3 9 27 = 10 ⇒ x 3 25 - 23 + 6 x 6 . 2 = 10 ⇒ 6 x 6 + x 3 - 5 = 0 ⇒ x 3 = 5 6 , - 1 Two real solutions for x

Mathematics - Determinant Question with Solution | TestHub

MathematicsDeterminantProperties of det.Medium2 minPYQ_2016

MathematicsMediumnumerical

The total number of distinct $x \in R$ for which $|\begin{matrix} x & x^{2} & 1 + x^{3} \\ 2 x & 4 x^{2} & 1 + 8 x^{3} \\ 3 x & 9 x^{2} & 1 + 27 x^{3} \end{matrix}| = 10$ is

Answer:

2.00

Solution:

$x^{3} |\begin{matrix} 1 & 1 & 1 + x^{3} \\ 2 & 4 & 1 + 8 x^{3} \\ 3 & 9 & 1 + 27 x^{3} \end{matrix}| = 10$
$\Rightarrow$ $x^{3} |\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{matrix}| + x^{3} . x^{3} |\begin{matrix} 1 & 1 & 1 \\ 2 & 4 & 8 \\ 3 & 9 & 27 \end{matrix}| = 10$
$\Rightarrow$ $x^{3} (25 - 23) + 6 x^{6} . 2 = 10$
$\Rightarrow$ $6 x^{6} + x^{3} - 5 = 0$
$\Rightarrow$ $x^{3} = \frac{5}{6}, - 1$
Two real solutions for x

Stream:JEE_ADVSubject:MathematicsTopic:DeterminantSubtopic:Properties of det.

⏱ 2mℹ️ Source: PYQ_2016

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