If - 1-111 abca2b 2 c 2- - 2-1 bca1cablabc- thena - 1- 2-0b - 1-2 - 2-0c - 1- 2d none of these

(a) #8710;1+ #8710;2=0 #916;2 #160;=1 #160; #160; #160;bc #160; #160; #160;a1 #160; #160; #160;ca #160; #160; #160; #160;b1 ...

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Byju's Answer

Standard XII

Mathematics

In Radius

If Δ 1=111 ab...

Question

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

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Solution

(a) $∆_{1} + ∆_{2} = 0$

$Δ_{2 =} |1 b c a 1 c a b 1 a b c| = \frac{1}{a b c} |a a b c a^{2} b b c a b^{2} c c a b c^{2}| [R_{1}, R_{2}, R_{3} are multiplied by a, b and c respectively, therefore we divide by abc] = \frac{a b c}{a b c} |a 1 a^{2} b 1 b^{2} c 1 c^{2}| [Taking a b c common from C_{2}] = - |1 a a^{2} 1 b b^{2} 1 c c^{2}| C_{1} \leftrightarrow C_{2} We know that the value of a determinant remains unchanged if its rows and columns are interchanged . So, ∆_{2} = - |1 1 1 a b c a^{2} b^{2} c^{2}| = - Δ_{1} \Rightarrow Δ_{1} + Δ_{2} = 0$

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

Q. If

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})

Q. If

Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} a & b & c x & y & z p & q & r \end{matrix} ∣ ∣ ∣ ∣

and

Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} y & b & q x & a & p z & c & r \end{matrix} ∣ ∣ ∣ ∣

then

Δ_{1}

is equal to

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

Q. If

Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} 7 & x & 2 - 5 & x + 1 & 3 4 & x & 7 \end{matrix} ∣ ∣ ∣ ∣, Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} x & 2 & 7 x + 1 & 3 & - 5 x & 7 & 4 \end{matrix} ∣ ∣ ∣ ∣

then

Δ_{1} - Δ_{2} = 0

for

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If ∆1=111abca2b2c2, ∆2=1bca1cab1abc, then (a) ∆1+∆2=0 (b) ∆1+2 ∆2=0 (c) ∆1=∆2 (d) none of these

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