Let Delta1- 18283 cr8 - 9 - 10 Cr15 - 16817 cr31 - 34 - 37 cr889810 cr15 - 16 - 17 crlend vmatrix -If -1- k -2- then the value of k is

Δ_2= 31 amp;34 amp;37 8 amp;9 amp;10 15 amp;16 amp;17 = 1+2(15) amp;2+2(16) amp;3+2(17) 8 amp;9 amp;10 15 amp;16 amp;17 = 1 amp;2 am ...

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Byju's Answer

Standard XII

Mathematics

Evaluation of a Determinant

Let Delta1= 1...

Question

Let $Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 3 8 & 9 & 10 15 & 16 & 17 \end{matrix} ∣ ∣ ∣ ∣$ and $Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} 31 & 34 & 37 8 & 9 & 10 15 & 16 & 17 \end{matrix} ∣ ∣ ∣ ∣$ .
If $Δ_{1} = k Δ_{2}$ , then the value of $k$ is _____

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Solution

$Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} 31 & 34 & 37 8 & 9 & 10 15 & 16 & 17 \end{matrix} ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ \begin{matrix} 1 + 2 (15) & 2 + 2 (16) & 3 + 2 (17) 8 & 9 & 10 15 & 16 & 17 \end{matrix} ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 3 8 & 9 & 10 15 & 16 & 17 \end{matrix} ∣ ∣ ∣ ∣ + 2 ∣ ∣ ∣ ∣ \begin{matrix} 15 & 16 & 17 8 & 9 & 10 15 & 16 & 17 \end{matrix} ∣ ∣ ∣ ∣$
$= Δ_{1} + 0 = Δ_{1}$

$\Rightarrow Δ_{1} = Δ_{2}$
$∴ k = 1$

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Similar questions

I f x^{a} y^{b} = e^{m}, x^{c} y^{d} = e^{n}, Δ_{1} = ∣ ∣ ∣ \begin{matrix} m & b n & d \end{matrix} ∣ ∣ ∣ Δ_{2} = ∣ ∣ ∣ \begin{matrix} a & m c & n \end{matrix} ∣ ∣ ∣

and

Δ_{3} = ∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣,

then the values of x and y are respectively

Q. If

a e^{x} + b e^{y} = c, p e^{x} + q e^{y} = d

and

Δ_{1} = ∣ ∣ ∣ \begin{matrix} a & b p & q \end{matrix} ∣ ∣ ∣, Δ_{2} = ∣ ∣ ∣ \begin{matrix} c & b d & q \end{matrix} ∣ ∣ ∣, Δ_{3} = ∣ ∣ ∣ \begin{matrix} a & c p & d \end{matrix} ∣ ∣ ∣

,where

Δ_{1}, Δ_{2}, Δ_{3}

are all positive, then the value of

(x, y)

is:

Q. Let

D_{k} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a & 2^{k} & 2^{16} - 1 b & 3 (4^{k}) & 2 (4^{16} - 1) c & 7 (8^{k}) & 4 (8^{16} - 1) \end{matrix} ∣ ∣ ∣ ∣ ∣

, then the value of

16 \sum k = 1 D_{k}

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. The value of following determinant

∣ ∣ ∣ ∣ \begin{matrix} 1 & 4 & 9 4 & 9 & 16 9 & 16 & 25 \end{matrix} ∣ ∣ ∣ ∣

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Let Δ1=∣∣ ∣∣1238910151617∣∣ ∣∣ and Δ2=∣∣ ∣∣3134378910151617∣∣ ∣∣. If Δ1=kΔ2, then the value of k is _____

Δ2=∣∣ ∣∣3134378910151617∣∣ ∣∣ =∣∣ ∣∣1+2(15)2+2(16)3+2(17)8910151617∣∣ ∣∣ =∣∣ ∣∣1238910151617∣∣ ∣∣+2∣∣ ∣∣1516178910151617∣∣ ∣∣ =Δ1+0=Δ1 ⇒Δ1=Δ2 ∴k=1

Let $Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 3 8 & 9 & 10 15 & 16 & 17 \end{matrix} ∣ ∣ ∣ ∣$ and $Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} 31 & 34 & 37 8 & 9 & 10 15 & 16 & 17 \end{matrix} ∣ ∣ ∣ ∣$ .
If $Δ_{1} = k Δ_{2}$ , then the value of $k$ is _____