NDA 2021
Exam NDA
Year 2021
Subject Mathematics
Chapter Determinants NDA 2021 · Mathematics · Determinants
If Δ=| array lll a & b & c d & e & f g & h & i array | then what is...
If
\Delta=\left| a b c d e f g h i \begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array} a d g b e h c f i \right| then what is \left| 3 d + 5 g 4 a + 7 g 6 g 3 e + 5 h 4 b + 7 h 6 h 3 f + 5 i 4 c + 7 i 6 i \begin{array}{ccc}3 d+5 g & 4 a+7 g & 6 g \\ 3 e+5 h & 4 b+7 h & 6 h \\ 3 f+5 i & 4 c+7 i & 6 i\end{array} 3 d + 5 g 3 e + 5 h 3 f + 5 i 4 a + 7 g 4 b + 7 h 4 c + 7 i 6 g 6 h 6 i \right| equal to?
Options A Δ \Delta Δ B 7 Δ 7 \Delta 7Δ C 72 Δ 72 \Delta 72Δ D − 72 Δ -72 \Delta − 72Δ Correct answer D. − 72 Δ -72 \Delta − 72Δ
Step-by-step solution
∣ 3 d + 5 g 4 a + 7 g 6 g 3 e + 5 h 4 b + 7 h 6 h 3 f + 5 i 4 c + 7 i 6 i ∣ = 6 ∣ 3 d + 5 g 4 a + 7 g g 3 e + 5 h 4 b + 7 h h 3 f + 5 i 4 c + 7 i i ∣ Apply c 1 → c 1 − 5 c 3 and c 2 → c 2 − 7 c 3 = 6 ∣ 3 d 4 a g 3 e 4 b h 3 f 4 c i ∣ = 72 ∣ d a g e b h f c i ∣ = − 72 ∣ a d g b e h c f i ∣ = − 7 q Δ [ ∴ ∣ A ∣ = ∣ A ′ ∣ ] . \begin{aligned}
& \left|\begin{array}{ccc}
3 d+5 g & 4 a+7 g & 6 g \\
3 e+5 h & 4 b+7 h & 6 h \\
3 f+5 i & 4 c+7 i & 6 i
\end{array}\right| \\
& =6\left|\begin{array}{lll}
3 d+5 g & 4 a+7 g & g \\
3 e+5 h & 4 b+7 h & h \\
3 f+5 i & 4 c+7 i & i
\end{array}\right| \\
& \text{ Apply } c_1 \rightarrow c_1-5 c_3 \text{ and } c_2 \rightarrow c_2-7 c_3 \\
& =6\left|\begin{array}{lll}
3 d & 4 a & g \\
3 e & 4 b & h \\
3 f & 4 c & i
\end{array}\right|=72\left|\begin{array}{lll}
d & a & g \\
e & b & h \\
f & c & i
\end{array}\right| \\
& =-72\left|\begin{array}{lll}
a & d & g \\
b & e & h \\
c & f & i
\end{array}\right| \\
& =-7 q \Delta\left[\therefore|\mathrm{ A}|=\left|\mathrm{A}^{\prime}\right|\right].
\end{aligned} 3 d + 5 g 3 e + 5 h 3 f + 5 i 4 a + 7 g 4 b + 7 h 4 c + 7 i 6 g 6 h 6 i = 6 3 d + 5 g 3 e + 5 h 3 f + 5 i 4 a + 7 g 4 b + 7 h 4 c + 7 i g h i Apply c 1 → c 1 − 5 c 3 and c 2 → c 2 − 7 c 3 = 6 3 d 3 e 3 f 4 a 4 b 4 c g h i = 72 d e f a b c g h i = − 72 a b c d e f g h i = − 7 q Δ [ ∴ ∣ A ∣ = ∣ A ′ ∣ ] .
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