Three Dimensional Geometry

To verify the given assertion and reason, we calculate the determinant of matrix :

Using cofactor expansion along the first row:

1. Compute the first minor:

2. Compute the second minor:

3. Compute the third minor:

Substitute back into the determinant: Simplify:

Since , we have: Thus, the determinant varies as:

Verification of Assertion (A): The determinant lies in the interval , so the assertion is true.
Verification of Reason (R): The cosine function satisfies for all , so the reason is also true.
Conclusion: Both Assertion (A) and Reason (R) are true, and the Reason (R) correctly explains the Assertion (A).

The given matrices are:

1. Equating Corresponding Elements: - From the first row, first column: - From the second row, second column:

2. Expression to Evaluate: We need to find:

Simplify the expression using the identity :

3. Substitute Known Values: - , - . Substitute these values:

Simplify:

Hence, the value of is (D) 18.

A line in three-dimensional space cannot be perpendicular to all three axes simultaneously. If a line is perpendicular to all three axes, the direction cosines would all be zero, which would violate the fundamental relation of direction cosines:

The given equation ensures that at least one of the direction cosines is non-zero, indicating that the line cannot be simultaneously perpendicular to , , and axes.

Conclusion: Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

To determine the number of corner points of the feasible region, let us analyze the constraints: 1. :

This represents the region to the right of the -axis, including the -axis itself. 2. :

This represents the region above the -axis, including the -axis itself. 3. :

This represents the region above the line .

Rearranging, , which has intercepts at and .

The feasible region is the intersection of these constraints, which lies in the first quadrant ( , ) and above the line . The feasible region is unbounded but has two corner points:

Intersection of with : , - Intersection of with : . Hence, the number of corner points is , and the correct answer is (C).

The distance of a point from the y-axis is the perpendicular distance from to the y-axis.

The y-axis corresponds to the line where and .

The perpendicular distance is:

Therefore, the correct answer is (C) .

Intersection of Two Lines in 3D

Step 1: Represent the lines in vector form

Line 1 (from symmetric form):

Line 2 (from symmetric form):

Step 2: Assume the lines intersect

If the lines intersect, then:

Equating components:

• (i)
• (ii)
• (iii)

Step 3: Solve the system

From (iii): Substitute into (i):

Now, from (iii):

Check in (ii):

✅ Final Answer:

The lines intersect at the point:

Three points are collinear if the vectors formed by them are parallel.
Let us consider:

Since the vectors and are in the same direction (i.e., collinear), one must be a scalar multiple of the other: Comparing components:

Now solve the equation:

Final Answer:

Given Vector Equations of Lines

Line 1: Line 2:

Step 1: Direction Vectors

Since the direction vectors are scalar multiples ⇒ lines are parallel.

Step 2: Check for Coincidence

Take position vectors of a point on each line: Compute: Check if this is a scalar multiple of : It is not, so lines are parallel but not coincident.

Step 3: Find Shortest Distance Between Lines

Use formula: Let: Compute cross product:

✅ Final Conclusion:

• Lines are parallel
• They are not coincident
• Shortest distance between them is approximately:

Find the Vector Equation of a Line

Given:

We are to find the vector equation of a line passing through the point and perpendicular to both the lines:

Line 1: Line 2:

Step 1: Extract Direction Vectors

Direction vector of Line 1: Direction vector of Line 2:

Step 2: Find Cross Product

The required line is perpendicular to both lines, so we compute the cross product:

Step 3: Use Point (1, 2, -3)

Line passes through point

Step 4: Vector Equation of the Required Line

Step 5: Parametric Form

✅ Final Answer:

The vector equation is: