Question

If a point $P (\alpha, \beta, \gamma)$ satisfying $(\alpha\,\, \beta\,\, \gamma) \begin{pmatrix} 2 & 10 & 8 \\9 & 3 & 8 \\8 & 4 & 8\end{pmatrix}=(0\,\,0\,\,0) $ lies on the plane $2 x+4 y+3 z=5$, then $6 \alpha+9 \beta+7 \gamma$ is equal to :

• A. $\frac{5}{4}$
• B. $-1$
• C. 11
• D. $\frac{11}{5}$

Hint:

When solving systems of linear equations, substitution is an effective method. Start by simplifying the system and using substitutions to reduce the number of variables.

Answer 1

The Correct Option is C

$2\alpha +4\beta +3\gamma =5$.........(1)
$2\alpha +9\beta +8\gamma =0$.........(2)
$10\alpha +3\beta +4\gamma =0$.........(3)
$8\alpha +8\beta +8\gamma =0$.........(4)
Subtract (4) from (2)
$-6\alpha +\beta =0$
$\beta =6\alpha$.........(5)
From equation (4)
$8\alpha +48\alpha +8\gamma =0$
$\gamma =-7\alpha$.........(6)
From equation (1)
$2\alpha +24\alpha -21\alpha =5$
$5\alpha =5$
$\alpha =1$
$\beta =+6,\gamma =-7$
$\therefore 6\alpha +9\beta +7\gamma$
$=6+54-49$
$=11$
So, the correct option is (C) : 11

Answer 2

Step 1: Write down the matrix equation: \[ 2\alpha + 4\beta + 3\gamma = 5 \quad \cdots (1) \] \[ 2\alpha + 9\beta + 8\gamma = 0 \quad \cdots (2) \] \[ 10\alpha + 3\beta + 4\gamma = 0 \quad \cdots (3) \] \[ 8\alpha + 8\beta + 8\gamma = 0 \quad \cdots (4) \] Step 2: Subtract equation (4) from equation (2): \[ 2\alpha + 9\beta + 8\gamma - (8\alpha + 8\beta + 8\gamma) = 0 \] Simplifying: \[ -6\alpha + \beta = 0 \quad \Rightarrow \quad \beta = 6\alpha \quad \cdots (5) \] Step 3: Substitute equation (5) into equation (4): \[ 8\alpha + 8(6\alpha) + 8\gamma = 0 \] Simplifying: \[ 8\alpha + 48\alpha + 8\gamma = 0 \quad \Rightarrow \quad \gamma = -7\alpha \quad \cdots (6) \] Step 4: Substitute equations (5) and (6) into equation (1): \[ 2\alpha + 4(6\alpha) + 3(-7\alpha) = 5 \] Simplifying: \[ 2\alpha + 24\alpha - 21\alpha = 5 \quad \Rightarrow \quad 5\alpha = 5 \quad \Rightarrow \quad \alpha = 1 \] Step 5: Now substitute \( \alpha = 1 \) into equations (5) and (6): \[ \beta = 6(1) = 6 \] \[ \gamma = -7(1) = -7 \] Step 6: Now calculate \( 6\alpha + 9\beta + 7\gamma \): \[ 6(1) + 9(6) + 7(-7) = 6 + 54 - 49 = 11 \] Thus, the value of \( 6\alpha + 9\beta + 7\gamma \) is \( 11 \).