Let Delta-lbeginvmatrix Ax - x - 2 - 111By - y - 2 - 1Cz - z - 2 - 1A-B-CWx - y - z z y - z x - x ylend vmatrix thenA- -1-B- -1-C- -1-D- -1-2 -

The correct option is B Δ_1= ΔΔ_1= A amp;B amp;C x amp;y amp;z zy amp;zx amp;xy ⇒Δ_1=1/xyz Ax amp;By amp;Cz x^2 amp;y^2 amp;z^2 xyz a ...

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Byju's Answer

Standard XII

Mathematics

Summation of Determinant

Let Delta=lbe...

Question

Let $Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} A x & x^{2} & 1 B y & y^{2} & 1 C z & z^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ and $Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} A & B & C x & y & z z y & z x & x y \end{matrix} ∣ ∣ ∣ ∣$ then

Δ_{1} = - Δ

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Δ_{1} = Δ

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Δ_{1} \neq Δ

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Δ_{1} = 2 Δ

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Solution

The correct option is B $Δ_{1} = Δ$
$Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} A & B & C x & y & z z y & z x & x y \end{matrix} ∣ ∣ ∣ ∣$
$\Rightarrow Δ_{1} = \frac{1}{x y z} ∣ ∣ ∣ ∣ ∣ \begin{matrix} A x & B y & C z x^{2} & y^{2} & z^{2} x y z & x y z & x y z \end{matrix} ∣ ∣ ∣ ∣ ∣ (C_{1} \to C_{1} x, C_{2} \to C_{2} y, C_{3} \to C_{3} z)$

$\Rightarrow Δ_{1} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} A x & B y & C z x^{2} & y^{2} & z^{2} 1 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} A x & x^{2} & 1 B y & y^{2} & 1 C z & z^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = Δ$

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Similar questions

Q. If

Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} x & sin θ & cos θ - sin θ & - x & 1 cos θ & 1 & x \end{matrix} ∣ ∣ ∣ ∣

and

Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} x & sin 2 θ & cos 2 θ - sin 2 θ & - x & 1 cos 2 θ & 1 & x \end{matrix} ∣ ∣ ∣ ∣

x \neq 0

; then for all

θ \in (0, \frac{π}{2})

$Δ_{1}$ , $∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{2} & b^{2} & c^{2} \end{matrix} ∣ ∣ ∣ ∣$ , $Δ_{2}$ = $∣ ∣ ∣ ∣ \begin{matrix} 1 & b c & a 1 & c a & b 1 & a b & c \end{matrix} ∣ ∣ ∣ ∣$ then

Q. If

∆_{1} = |\begin{matrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{matrix}|, ∆_{2} = |\begin{matrix} 1 & b c & a \\ 1 & c a & b \\ 1 & a b & c \end{matrix}|, then

(a)

∆_{1} + ∆_{2} = 0

(b)

∆_{1} + 2 ∆_{2} = 0

(c)

∆_{1} = ∆_{2}

(d) none of these

Let $Δ_{1}$ = $∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣$ and
$Δ_{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} α_{1} & β_{1} & γ_{1} α_{2} & β_{2} & γ_{3} α_{3} & β_{3} & γ_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣$ , and $Δ_{1} \times Δ_{2}$ can be expressed as the sum of $n$ determinants, then $n =$

Q. Let

Δ_{1} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & cos α & cos β cos α & 1 & cos γ cos β & cos γ & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

and

Δ_{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & cos α & cos β cos α & 0 & cos γ cos β & cos γ & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣

Δ_{1} = Δ_{2}

, find

{sin}^{2} α + {sin}^{2} β + {sin}^{2} γ

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Let Δ=∣∣ ∣ ∣∣Axx21Byy21Czz21∣∣ ∣ ∣∣ and Δ1=∣∣ ∣∣ABCxyzzyzxxy∣∣ ∣∣ then

The correct option is B Δ1=ΔΔ1=∣∣ ∣∣ABCxyzzyzxxy∣∣ ∣∣ ⇒Δ1=1xyz∣∣ ∣ ∣∣AxByCzx2y2z2xyzxyzxyz∣∣ ∣ ∣∣ (C1→C1x, C2→C2y, C3→C3z) ⇒Δ1=∣∣ ∣ ∣∣AxByCzx2y2z2111∣∣ ∣ ∣∣=∣∣ ∣ ∣∣Axx21Byy21Czz21∣∣ ∣ ∣∣=Δ

Let $Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} A x & x^{2} & 1 B y & y^{2} & 1 C z & z^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ and $Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} A & B & C x & y & z z y & z x & x y \end{matrix} ∣ ∣ ∣ ∣$ then