For how many values of x in the closed interval - -4-1 - the matrix - 3 -x-1 2- 3 -1 x-2- x-3 -1 2 - is singular-

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Standard XII

Mathematics

Singular and Non Singualar Matrices

For how many ...

Question

For how many values of $x$ in the closed interval $[- 4, - 1]$ the matrix $⎡ ⎢ ⎣ \begin{matrix} 3 & - x - 1 & 2 3 & - 1 & x + 2 x + 3 & - 1 & 2 \end{matrix} ⎤ ⎥ ⎦$ is singular.

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Solution

The correct option is A $1$
A matrix is singular only if its determinant is zero
$⟹ ∣ ∣ ∣ ∣ \begin{matrix} 3 & - x - 1 & 2 3 & - 1 & x + 2 x + 3 & - 1 & 2 \end{matrix} ∣ ∣ ∣ ∣ = 0 ⟹ 3 (- 2 + x + 2) + (x + 1) (6 - (x + 2) (x + 3)) + 2 (- 3 + x + 3) = 0 ⟹ 3 x + (x + 1) (- x^{2} - 5 x) + 2 x = 0 ⟹ x^{3} + 6 x^{2} = 0 ⟹ x = 0, - 6$
$0 \in [- 4, - 1]$
Hence, option A is correct.

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Similar questions

Q. For how many values of x in the closed interval

[- 4, - 1]

is the matrix

⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 + x & 2 3 & - 1 & x + 2 x & - 1 & 2 \end{matrix} ⎤ ⎥ ⎦

singular ?

For how many values of ‘x’ in the closed interval [-4, -1] is the matrix $⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 + x & 2 3 & - 1 & x + 2 x + 3 & - 1 & 2 \end{matrix} ⎤ ⎥ ⎦$ singular?

Q. The number of value of

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[- 4, - 1]

, the matrix

⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 + x & 2 3 & - 1 & x + 2 x + 3 & - 1 & 2 \end{matrix} ⎤ ⎥ ⎦

is singular is

Q. Number of value(s) of

x

in the interval

[- 4, - 1]

for which the matrix

A

is singular.

A = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 + x & 2 3 & - 1 & x + 2 x + 3 & - 1 & 2 \end{matrix} ⎤ ⎥ ⎦

Q. The values of

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[- 4, - 1]

for which matrix

∣ ∣ ∣ ∣ \begin{matrix} 3 & x - 1 & 2 3 & - 1 & x + 2 x + 3 & - 1 & 2 \end{matrix} ∣ ∣ ∣ ∣

is singular is

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For how many values of x in the closed interval [−4,−1] the matrix ⎡⎢⎣3−x−123−1x+2x+3−12⎤⎥⎦ is singular.

The correct option is A 1A matrix is singular only if its determinant is zero⟹∣∣ ∣∣3−x−123−1x+2x+3−12∣∣ ∣∣=0⟹3(−2+x+2)+(x+1)(6−(x+2)(x+3))+2(−3+x+3)=0⟹3x+(x+1)(−x2−5x)+2x=0⟹x3+6x2=0⟹x=0,−60∈[−4,−1]Hence, option A is correct.

For how many values of $x$ in the closed interval $[- 4, - 1]$ the matrix $⎡ ⎢ ⎣ \begin{matrix} 3 & - x - 1 & 2 3 & - 1 & x + 2 x + 3 & - 1 & 2 \end{matrix} ⎤ ⎥ ⎦$ is singular.