If a point P(,,) satisfying the equation pmatrix 2 & 10 & 8 9 & 3 & 8 8 & 4 & 8 pmatrix pmatrix pmatrix = pmatrix 0 0 0 pmatrix lies on the plane 2x + 4y + 3z = 5, then 6 + 9 + 7 is equal to:

Given: P(,,) satisfies the matrix equation and lies on the plane 2x + 4y + 3z = 5. Find: The value of 6 + 9 + 7. From the solution, the equations used are 2 + 4 + 3 = 5 2 + 9 + 8 = 0 10 + 3 + 4 = 0 8 + 8 + 8 = 0 Subtract equation (4) from equation (2): 2 + 9 + 8 - (8 + 8 + 8) = 0 -6 + = 0 So, = 6 Using this in equation (4): 8 + 8 (6) + 8 = 0 8 + 48 + 8 = 0 = -7 Now substitute into equation (1): 2 + 4 (6) + 3 (-7) = 5 2 + 24 - 21 = 5 5 = 5 = 1 Hence, = 6, = -7 Now compute 6 + 9 + 7 = 6 (1) + 9 (6) + 7 (-7) = 6 + 54 - 49 = 11 Therefore, the value of 6 + 9 + 7 is 11. The correct option is D.

If a point P(,,) satisfying the equation pmatrix 2 & 10 & 8 9 & 3… — Solution | JEE Mathematics

MCQMediumJEE 2023Solving Linear Equations (Matrix Method)

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

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

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: $P(\alpha, \beta, \gamma)$ satisfies the matrix equation and lies on the plane $2x + 4y + 3z = 5$ .

Find: The value of $6\alpha + 9\beta + 7\gamma$ .

From the solution, the equations used are

2\alpha + 4\beta + 3\gamma = 5

2\alpha + 9\beta + 8\gamma = 0

10\alpha + 3\beta + 4\gamma = 0

8\alpha + 8\beta + 8\gamma = 0

Subtract equation $\left(4\right)$ from equation $\left(2\right)$ :

2\alpha + 9\beta + 8\gamma - \left(8\alpha + 8\beta + 8\gamma\right) = 0

-6\alpha + \beta = 0

So,

\beta = 6\alpha

Using this in equation $\left(4\right)$ :

8\alpha + 8\left(6\alpha\right) + 8\gamma = 0

8\alpha + 48\alpha + 8\gamma = 0

\gamma = -7\alpha

Now substitute into equation $\left(1\right)$ :

2\alpha + 4\left(6\alpha\right) + 3\left(-7\alpha\right) = 5

2\alpha + 24\alpha - 21\alpha = 5

5\alpha = 5

\alpha = 1

Hence,

\beta = 6, \quad \gamma = -7

Now compute

6\alpha + 9\beta + 7\gamma = 6\left(1\right) + 9\left(6\right) + 7\left(-7\right)

= 6 + 54 - 49 = 11

Therefore, the value of $6\alpha + 9\beta + 7\gamma$ is $11$ . The correct option is D.

Common mistakes

Using the matrix entries in the wrong order to form linear equations is incorrect because rows must multiply the column vector $\begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix}$ . Always write each equation from a row of the matrix, not from a column.
Subtracting equations carelessly can give the wrong relation between $\alpha$ and $\beta$ . For example, from $\left(2\right) - \left(4\right)$ , the correct result is $-6\alpha + \beta = 0$ , so $\beta = 6\alpha$ .
Forgetting to use the plane condition $2x + 4y + 3z = 5$ at $P(\alpha, \beta, \gamma)$ is wrong because it provides the non-homogeneous equation needed to determine the actual values of $\alpha, \beta, \gamma$ . Without it, only proportional relations are obtained.

Practice more Solving Linear Equations (Matrix Method) questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trial More on Solving Linear Equations (Matrix Method)

Answer

Step-by-step solution

Common mistakes

Practice more Solving Linear Equations (Matrix Method) questions

Related questions