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JEE Advanced 2011Paper 1
Exam
JEE Advanced
Year
2011
Paper
Paper 1
Subject
Mathematics
Chapter
Matrices

JEE Advanced 2011 · Paper 1 · Mathematics · Matrices

Paragraph: Let a , b and c be three real numbers satisfying [ array...

Paragraph: Let aa, b and cc be three real numbers satisfying \left[abc\begin{array}{lll}a & b & c\end{array}\right]\left[197827737\begin{array}{lll}1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0\end{array}$$\right]Question: If the point P(a,b,c)P(a, b, c), with reference to Eq. (E), lies on the plane 2x+y+z=12 x+y+z=1, then the value of 7 a+b+c is

Options

  1. A0
  2. B12
  3. C7
  4. D6

Correct answer

D. 6

Step-by-step solution

 Given, [abc]1×3[197827737]3×3=[000]⇒[a+8b+7c9a+2b+3c7a+7b+7c]=[000]⇒a+8b+7c=0⇒9a+2b+3c=0⇒a+b+c=0\begin{aligned}& \text{ Given, }[a b c]_{1 \times3}\left[\begin{array}{lll}1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7 \end{array}\right]_{3 \times3}=\left[\begin{array}{lll}0 & 0 & 0 \end{array}\right] \\ & \Rightarrow \quad\left[\begin{array}{c}a+8 b+7 c \\ 9 a+2 b+3 c \\ 7 a+7 b+7 c \end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0 \end{array}\right] \\ & \Rightarrow \quad a+8 b+7 c=0 \\ & \Rightarrow 9 a+2 b+3 c=0 \\ & \Rightarrow \quad a+b+c=0 \\ & \end{aligned}On multiplying Eq. (iii) by 2 , then subtract from Eq. (ii), we get 7 a+c=0 Again multiplying Eq. (iii) by 3 , then subtract from Eq. (ii), we get 6a−b=0∴b=6a and c=−7a\begin{array}{ll}& 6 a-b=0 \\ \therefore \quad & b=6 a \text{ and }c=-7 a \end{array}As (a,b,c)(a, b, c) lies on 2x+y+z=12 x+y+z=1 ⇒2a+b+c=1⇒2a+6a−7a=1⇒a=1,b=6 and c=−7∴7a+b+c=7+6−7=6\begin{array}{lc} \Rightarrow& 2 a+b+c=1 \\ \Rightarrow & 2 a+6 a-7 a=1 \\ \Rightarrow & a=1, b=6 \text{ and }c=-7 \\ \therefore & 7 a+b+c=7+6-7=6 \end{array}
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