JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If the equation of the plane passing through the point $(1,1,2)$ and perpendicular to the line $x - 3y + 2z - 1 = 0 = 4x - y + z$ is $Ax + By + Cz = 1$ , then $140(C - B + A)$ is equal to

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Explanation

Solution

1. Normal vector of the plane ( $\vec{n}$ ):

\vec{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -3 & 2 \\ 4 & -1 & 1 \end{vmatrix}

\vec{n} = \hat{i}(-3 - (-2)) - \hat{j}(1 - 8) + \hat{k}(-1 - (-12))

\vec{n} = -\hat{i} + 7\hat{j} + 11\hat{k}

2. Equation of the plane:

-1(x - 1) + 7(y - 1) + 11(z - 2) = 0

-x + 1 + 7y - 7 + 11z - 22 = 0

-x + 7y + 11z = 28

3. Converting to $Ax + By + Cz = 1$ form:

\frac{-x}{28} + \frac{7y}{28} + \frac{11z}{28} = 1

A = -\frac{1}{28}, \quad B = \frac{7}{28}, \quad C = \frac{11}{28}

4. Final Calculation:

140(C - B + A) = 140 \left( \frac{11}{28} - \frac{7}{28} + \left( -\frac{1}{28} \right) \right)

= 140 \left( \frac{11 - 7 - 1}{28} \right)

= 140 \left( \frac{3}{28} \right)

= 5 \times 3 = 15

Final Answer: 15

Choose the correct answer:

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