Key Concepts and Formulas:

1. Midpoint Formula in 3D: The coordinates of the midpoint $M$ of a line segment connecting two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are given by: $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$
2. Direction Ratios (DRs) of a Line in 3D: The direction ratios of a line passing through two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are given by $(x_2-x_1, y_2-y_1, z_2-z_1)$. Any set of numbers proportional to these values can also represent the DRs.
3. Condition for a Line Equally Inclined to Coordinate Axes: If a line is equally inclined to the coordinate axes (X-axis, Y-axis, and Z-axis), its direction cosines (DCs) are $\left(\pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}\right)$. This implies that its direction ratios $(a, b, c)$ must have equal absolute values, i.e., $|a|=|b|=|c|$.

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Step-by-Step Solution:

1. Understand the Geometry and Identify the Median

• We are given a triangle ABC with vertices A(2, 3, 5), B(−1, 3, 2) and C($\lambda$, 5, $\mu$).
• The problem refers to the median through A. A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
• Therefore, the median through A is the line segment AD, where D is the midpoint of the side BC.

2. Calculate the Coordinates of Point D (Midpoint of BC)

• Why this step? To find the direction ratios of the median AD, we first need the coordinates of both A and D. Since D is the midpoint of BC, we use the midpoint formula.
• Given B(−1, 3, 2) and C($\lambda$, 5, $\mu$).
• Using the midpoint formula: $D = \left(\frac{-1+\lambda}{2}, \frac{3+5}{2}, \frac{2+\mu}{2}\right)$ $D = \left(\frac{\lambda-1}{2}, \frac{8}{2}, \frac{\mu+2}{2}\right)$ $D = \left(\frac{\lambda-1}{2}, 4, \frac{\mu+2}{2}\right)$

3. Determine the Direction Ratios (DRs) of the Median AD

• Why this step? The problem states that the median AD is equally inclined to the coordinate axes. The condition for this involves the line's direction ratios.
• We have A(2, 3, 5) and $D\left(\frac{\lambda-1}{2}, 4, \frac{\mu+2}{2}\right)$.
• The direction ratios of AD are $(x_D - x_A, y_D - y_A, z_D - z_A)$.
  • First DR: $x_D - x_A = \frac{\lambda-1}{2} - 2 = \frac{\lambda-1-4}{2} = \frac{\lambda-5}{2}$
  • Second DR: $y_D - y_A = 4 - 3 = 1$
  • Third DR: $z_D - z_A = \frac{\mu+2}{2} - 5 = \frac{\mu+2-10}{2} = \frac{\mu-8}{2}$
• So, the direction ratios of the median AD are $\left(\frac{\lambda-5}{2}, 1, \frac{\mu-8}{2}\right)$.

4. Apply the Condition for a Line Equally Inclined to Coordinate Axes

• Why this step? This is the crucial piece of information given in the problem statement that allows us to form equations for $\lambda$ and $\mu$.
• Since the median AD is equally inclined to the coordinate axes, its direction ratios must have equal absolute values. Let the direction ratios be $(a, b, c) = \left(\frac{\lambda-5}{2}, 1, \frac{\mu-8}{2}\right)$.
• For the line to be equally inclined, we must have $|a|=|b|=|c|$.
• Since one of the direction ratios, $b$, is $1$, we must have: $\left|\frac{\lambda-5}{2}\right| = 1 \quad \text{and} \quad \left|\frac{\mu-8}{2}\right| = 1$

5. Solve for $\lambda$ and $\mu$

• Why this step? We have derived two equations from the condition of equal inclination, which can now be solved for the unknowns $\lambda$ and $\mu$.

• From the first equation: $\frac{\lambda-5}{2} = \pm 1$ $\lambda-5 = \pm 2$ This gives two possibilities for $\lambda$: $\lambda = 5+2 = 7$ or $\lambda = 5-2 = 3$.

• From the second equation: $\frac{\mu-8}{2} = \pm 1$ $\mu-8 = \pm 2$ This gives two possibilities for $\mu$: $\mu = 8+2 = 10$ or $\mu = 8-2 = 6$.

• We need to choose the combination of $\lambda$ and $\mu$ that leads to one of the options. Let's test the possibilities:

  • If $\lambda=7$ and $\mu=10$: $\lambda^3 + \mu^3 + 5 = 7^3 + 10^3 + 5 = 343 + 1000 + 5 = 1348$. (This matches option B).
  • If $\lambda=3$ and $\mu=10$: $\lambda^3 + \mu^3 + 5 = 3^3 + 10^3 + 5 = 27 + 1000 + 5 = 1032$. (Not an option).
  • If $\lambda=7$ and $\mu=6$: $\lambda^3 + \mu^3 + 5 = 7^3 + 6^3 + 5 = 343 + 216 + 5 = 564$. (Not an option).
  • If $\lambda=3$ and $\mu=6$: $\lambda^3 + \mu^3 + 5 = 3^3 + 6^3 + 5 = 27 + 216 + 5 = 248$. (Not an option).

  Based on the provided options, the only consistent choice for $\lambda$ and $\mu$ is $\lambda=7$ and $\mu=10$.

6. Calculate the Final Expression ($\lambda^3 + \mu^3 + 5$)

• Why this step? This is the final value requested by the problem.
• Substitute the values $\lambda=7$ and $\mu=10$ into the expression $\lambda^3 + \mu^3 + 5$: $\lambda^3 + \mu^3 + 5 = (7)^3 + (10)^3 + 5$ $= 343 + 1000 + 5$ $= 1343 + 5$ $= 1348$

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Common Mistakes & Tips:

• Careful with Midpoint vs. Section Formula: Ensure you use the correct formula for the midpoint, not a general section formula.
• Direction Ratios vs. Direction Cosines: Remember that direction ratios are proportional to direction cosines. For a line equally inclined to the axes, its direction ratios $(a,b,c)$ must satisfy $|a|=|b|=|c|$. This means they can be $(k,k,k)$ or $(k,-k,k)$, etc., for some non-zero $k$. Since one of our DRs is $1$, this implies $k$ must be $1$ or $-1$.
• Algebraic Precision: Double-check your arithmetic, especially when solving for $\lambda$ and $\mu$ and in the final cubic calculation.

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Summary:

This problem tests fundamental 3D geometry concepts: calculating the midpoint of a line segment, finding the direction ratios of a line segment, and applying the specific condition for a line being equally inclined to the coordinate axes. The solution involves a sequence of logical steps: first, identifying the midpoint of the side opposite the vertex from which the median originates, then computing the direction ratios of this median, and finally using the given condition to solve for the unknown coordinates. The condition "equally inclined to coordinate axes" implies that the absolute values of the direction ratios are equal. By systematically evaluating the possible values for $\lambda$ and $\mu$, we find the combination that matches one of the given options.

The final answer is $\boxed{1348}$, which corresponds to option (B).