Three Dimensional Geometry

Concept: 3D Geometry - Cross Product and Unit Vectors.

Step 1: Identify the direction vectors of the two lines. The direction vector of a line in the form is given by . For line , the direction vector is . For line , the direction vector is .

Step 2: Calculate the cross product to find a perpendicular vector. A vector perpendicular to both and is obtained by taking the cross product of their direction vectors ( ). Set up the determinant: .

Step 3: Expand the determinant. Expand along the first row: . . . This vector is .

Step 4: Calculate the magnitude of the resulting vector. The magnitude of the cross product vector is . . Simplify the radical: .

Step 5: Form the final unit vector. The unit vector is found by dividing the vector by its magnitude: . Substituting our results: . $ $

Concept: 3D Geometry - Direction Cosines and Line-Plane Intersection.

Step 1: Determine the direction cosines of the line. The line makes equal angles with the coordinate axes, so its direction angles satisfy . Consequently, the direction cosines are also equal: . Using the fundamental identity , we get . Since the direction cosines are given as positive, . Thus, the direction cosines are .

Step 2: Formulate the equation of the line. The equation of a line passing through a point with direction cosines is . Substitute point and our derived direction cosines: . Multiplying the denominators by yields equivalent direction ratios of . The simplified line equation is .

Step 3: Define the coordinates of the general point Q. Set the line equation equal to a scalar constant : . Express in terms of to find the coordinates of any point on the line, which we will call Q: .

Step 4: Find the intersection point with the plane. Point Q lies on the given plane . Substitute the coordinates of Q into the plane's equation: . Expand and simplify: .
Combine like terms: . Substitute back into our point definition to get the exact coordinates: .

Step 5: Calculate the length of the line segment PQ. Use the 3D distance formula to find the length between and : . . . $ $

The foot of perpendicular from point to the plane is and the foot of perpendicular from point to the plane in
Let the equation of plane passing through the point be

Also it is paring through the point and

and

and


and
From E (i),
or

Given equation of line is or Hence, of a line are Now , of a line are or

Answer (a)

Step 1: Identify direction vectors.
Direction vector of first line: Direction vector of second line: Step 2: Use perpendicularity condition.
For perpendicular lines, Step 3: Compute dot product.
Step 4: Solve for .
Step 5: Conclusion.
The value of is .

Step 1: Find the midpoint of .
Step 2: Find the direction ratios of .
Step 3: Use perpendicular bisector condition.
The required plane is perpendicular to , hence its normal vector is .
Equation of plane: Step 4: Simplify the equation.
Dividing by :

Step 1: Understand the XOZ plane.
The XOZ plane is defined by the equation The normal vector to this plane is along the -axis, i.e.,

Step 2: Direction of the required line.
A line perpendicular to the XOZ plane must be parallel to the normal vector . Hence, the direction ratios of the line are .

Step 3: Use the point–direction form of a line.
The given point is . Therefore, the parametric equations of the line are

Step 4: Final conclusion.
The equation of the required line is

Step 1: Write the plane in Cartesian form.
Given This represents the plane
Step 2: Use the formula for distance from origin to a plane.
The distance of the plane from the origin is
Step 3: Substitute the values.
Here .

Step 1: Formula for the foot of the perpendicular.
The foot of the perpendicular from a point to a plane is given by the formula: Here, and the equation of the plane is . Thus, .
Step 2: Apply the formula.
Substituting the values into the formula, we get:
Step 3: Conclusion.
Thus, the foot of the perpendicular is , corresponding to option (B).

Step 1: Use the condition of the plane.
Since the line lies in the plane, the direction cosine along the -axis is zero. Step 2: Use the angle with the -axis.
If the line makes an angle of with the -axis, then Step 3: Use the relation between direction cosines.
Step 4: Solve for .
Step 5: Conclusion.
Hence, the direction cosines are

Step 1: Use the volume formula of a tetrahedron.
The volume of a tetrahedron with vertices is given by Step 2: Find the vectors.
Step 3: Form the determinant.
Step 4: Evaluate the determinant.
Step 5: Find the volume.
Step 6: Conclusion.
The volume of the tetrahedron is cubic units.

Step 1: Simplifying the given equation.

Step 2: Using sine equality condition.
For , either
Step 3: Using the given condition.
It is given that and are not supplementary, so
Step 4: Conclusion.
Hence, the only possible solution is

Step 1: Write parametric equations of the line.


Step 2: Substitute in the plane equation.


Step 3: Find the point of intersection.


Step 4: Find the distance from .


Step 5: Conclusion.

Step 1: Find the direction ratios of the lines.
For the first line, the direction ratios are and for the second line, the direction ratios are .
Step 2: Use the formula for the angle between two lines.
The formula for the angle between two lines with direction ratios and is: Step 3: Calculation.
Substitute the direction ratios into the formula, we get , so .
Step 4: Conclusion.
The correct answer is (B) .

Step 1: Use the trigonometric identity.
We are given that . Using trigonometric identities, we know that:
Step 2: Apply the result.
Since , we directly apply the known identity to conclude that the value of the expression is 1.
Step 3: Conclusion.
Thus, the value of the expression is 1.

Step 1: Identify direction ratios and normal vector.
The direction ratios of the given line are obtained from , hence The normal vector of the plane is
Step 2: Use the formula for angle between a line and a plane.
If is the angle between the line and the plane, then
Step 3: Compute dot product and magnitudes.

Step 4: Final calculation.

Step 1: Parametrize the equations.
For the first line, the direction ratios are , and for the second line, the direction ratios are .
Step 2: Use the formula for the shortest distance.
The formula for the shortest distance between two skew lines is: By substituting the values of direction ratios, we calculate the shortest distance to be units.
Step 3: Conclusion.
The correct answer is (C) 2 units.

Step 1: Write the family of planes.
A plane passing through the intersection of is given by Step 2: Find the direction ratios of the given line.
From , let the common value be .
Then, the direction ratios of the line are Step 3: Impose the parallel condition.
For the plane to be parallel to the given line, the direction ratios of the line must satisfy the plane equation.
Substituting into the plane family condition gives a suitable value of .
Step 4: Obtain the required plane.
After simplification, the required plane equation is

Step 1: Find the direction ratios of the given line.
The direction ratios of the given line are the coefficients of , , and in the equation . These direction ratios are:
Step 2: Use the point and direction ratios to write the equation of the line.
For the line passing through the point and parallel to the given line, the equation is:
Step 3: Conclusion.
Thus, the equation of the line is , corresponding to option (B).

Step 1: Understanding the integral.
The given integral is of the form , which can be solved using substitution. Let’s use the substitution , so .

Step 2: Substituting in the integral.
We rewrite the integral: The integral of is , so we have:
Step 3: Applying the limits.
Now, we apply the limits of integration to :
Step 4: Conclusion.
Thus, the value of the integral is , which makes option (B) the correct answer.

Step 1: Direction ratios of the lines.
For the first line, the direction ratios are given by: For the second line, the direction ratios are:
Step 2: Condition for perpendicularity.
Two lines are perpendicular if the dot product of their direction ratios is zero. Therefore, we compute the dot product of and : Simplifying:
Step 3: Conclusion.
Thus, , corresponding to option (C).

Step 1: Finding direction vectors of the lines.
From the vector equation, the direction vector of the first line is From the symmetric form, the direction vector of the second line is
Step 2: Using the dot product formula.

Step 3: Interpreting the result.
Since the dot product is zero, the two direction vectors are perpendicular.
Step 4: Conclusion.
Hence, the angle between the two lines is .

Step 1: Write the equation in standard form.
The given equation is . First, rearrange the equation to make it look like the standard form of a parabola: Complete the square for to get: Now, the equation is in the form .
Step 2: Find the length of the latus rectum.
The length of the latus rectum for the parabola is given by . In this case, , so the length of the latus rectum is 2.
Step 3: Conclusion.
Thus, the correct answer is 2, corresponding to option (B).

Step 1: Use vector representation.
Let the position vectors of points be respectively.
Since is a parallelogram, Step 2: Find the position vector of point .
Given is the midpoint of , Step 3: Parametric form of the lines.
Point lies on : Point also lies on : Step 4: Equate and simplify.
Substitute and use .
On simplifying, we obtain Step 5: Determine the ratio.
Thus,

Step 1: Write the direction vectors of the given lines.
From the first line, the direction vector is From the second line, the direction vector is

Step 2: Find a point on each line.
For the first line, taking , a point is For the second line, taking , a point is

Step 3: Use the formula for shortest distance between two skew lines.
After calculation,

Step 4: Conclusion.
Hence, the shortest distance between the given lines is

Step 1: Substituting and simplifying.
We are given the integral . The most straightforward method to solve this is to apply integration by parts. Let:
Step 2: Applying integration by parts.
Using the integration by parts formula , we calculate the result:
Step 3: Conclusion.
Thus, the correct answer is , which makes option (C) the correct answer.

Step 1: Checking for discontinuities in the first piece of the function.
The function is discontinuous at , since the denominator becomes zero at this point.

Step 2: Checking for discontinuities in the second piece of the function.
The function is discontinuous at , but since is not in the domain , we only need to check , where the piecewise function transitions. The function is continuous at as the limit from both sides exists and is finite.

Step 3: Conclusion.
Thus, the points of discontinuity in the domain are and , making option (B) the correct answer.

Step 1: Use the given points and the condition of parallelism to the y-axis.
To find the equation of the plane, first calculate the direction ratios of the two points given: Since the plane is parallel to the y-axis, the direction ratio along is zero.
Step 2: Write the equation of the plane.
The equation of the plane is given by: This corresponds to option (C).

Step 1: Use the identity for .
We can simplify using the half-angle identity:
Step 2: Substitute into the integral.
Substitute this identity into the integral:
Step 3: Integrate.
We now integrate term by term: The first term gives: The second term is:
Step 4: Conclusion.
Thus, the value of the integral is .

Step 1: Writing the general term of the Arithmetic progression.
Let the -th term of the Arithmetic progression be given by: where is the first term and is the common difference.

Step 2: Applying the condition.
We are given that 9 times the ninth term is equal to 13 times the thirteenth term: Substituting the formula for the -th term: Simplifying this equation, we find that .

Step 3: Finding the twenty-second term.
Now that we know , the twenty-second term is: Thus, the value of the twenty-second term is 0, which makes option (A) the correct answer.

Step 1: Use the condition for direction cosines.
The direction cosines of a line are given by . For the line, we are given . According to the condition: Step 2: Solve for .
Simplifying the equation: Step 3: Conclusion.
The correct answer is (C) .

Step 1: Use the centroid formula.
The centroid of a triangle is the average of the coordinates of its vertices. For the centroid to be at the origin, we must have: Step 2: Solve the system of equations.
Solving these equations, we find: Step 3: Conclusion.
The correct answer is (D) .

Step 1: Find the direction cosines of the line OP.
The coordinates of the point are , and the coordinates of the point are . The direction ratios of the line OP are the differences in the coordinates of and , which gives:
Step 2: Use the direction ratios to find the angles with the planes.
The angles made by the line with the coordinate planes are given by the direction cosines. The direction cosines of the line are given by: The acute angles with the planes are the angles between the direction ratios and the normal to the planes. Using the formula for the angle between the line and the plane, we find the angles to be . Thus, the correct answer is option (D).

Step 1: Use the distance formula.
The distance from a point to the plane is given by: We apply this formula for both points and .
Step 2: Set the distances equal.
Since the points are equidistant, we equate the distances from the plane for the two points. After solving for , we find that .
Step 3: Conclusion.
The correct answer is (B) 1.

Step 1: Simplify the given expression for .
The given expression for is:
Step 2: Apply the derivative formula.
The derivative of is . In our case, the expression for involves , so we differentiate it to obtain:
Step 3: Conclusion.
The correct answer is .

Step 1: Write the equation of plane .
Using intercept form,

Step 2: Equation of a parallel plane.
A plane parallel to has the same coefficients:

Step 3: Substitute the given point .


Step 4: Final equation.

Step 1: Understand the total number of cards.
In a standard deck of 52 playing cards, there are 4 kings. The total number of non-king cards is 52 - 4 = 48.
Step 2: Calculate the odds.
The odds in favour of drawing a king are the ratio of the number of kings to the number of non-kings:
Step 3: Conclusion.
Thus, the odds in favour of drawing a king are 1:12, corresponding to option (A).

Step 1: Analyzing the given equation.
We are given the equation: This suggests a relationship between the angles , , and in the triangle.
Step 2: Verifying the progression.
By solving the equation and comparing the relationship between the angles, we find that , , and are in arithmetic progression.
Step 3: Conclusion.
Thus, the angles , , and are in arithmetic progression, which makes option (D) the correct answer.

Step 1: Recall the condition on direction cosines.
For a line with direction cosines ,

Step 2: Use the given condition.
Since the line makes equal angles with the coordinate axes,

Step 3: Substitute in the identity.


Step 4: Sign of direction cosines.
As the line lies in the first octant, all direction cosines are positive.

Step 5: Conclusion.

Step 1: Find the direction ratios of the line.
The direction ratios of the line are , derived from the given equation of the line.
Step 2: Use the formula for the distance from a point to a plane.
The distance from a point to a plane measured along the direction ratios is given by: Substitute the point and the plane equation into the formula.
Step 3: Calculate the distance.
The distance from the point to the plane along the direction ratios is 7 units.
Step 4: Conclusion.
Thus, the distance is 7 units.

Step 1: Solve the differential equation.
We are given that . This is a first-order linear differential equation. Rearranging the terms, we get: Integrating both sides: This gives: Thus:
Step 2: Apply the initial condition.
We are also given that . Substituting into the equation for , we get: Thus, , so .
Step 3: Find .
Now that we know , we substitute into the expression for :
Step 4: Conclusion.
Thus, the value of is .

Step 1: Use the formula for the angle between two lines.
The direction ratios of the lines are given by: - For the first line, the direction ratios are . - For the second line, the direction ratios are . The formula for the cosine of the angle between two lines is: where and are the direction ratios of the lines.
Step 2: Calculate the cosine of the angle.
Substitute the values of and into the formula: This simplifies to:
Step 3: Conclusion.
The angle between the lines is .

Step 1: Write parametric form.
From we get
Step 2: Identify position vector and direction vector.
At : point on the line is Direction ratios are proportional to , i.e. .
Step 3: Write the vector equation.

Step 1: Find the direction ratios.
The direction ratios of the line are given by the difference between the coordinates of points and :
Step 2: Write the parametric equations.
The parametric equations of the line are given by: where is the point , and is the direction ratio vector . Thus, the parametric equations of the line are:
Step 3: Conclusion.
The parametric equations are , , and , which corresponds to option (A).

Step 1: Find the direction ratios of the line and the normal to the plane.
The direction ratios of the line are , and the normal vector to the plane is , as given by the coefficients of in the plane equation.
Step 2: Use the formula for the angle between a line and a plane.
The formula for the angle between a line with direction ratios and a plane with normal vector is: Substituting the values for the direction ratios of the line and the normal vector of the plane, we get: Simplifying, we get: Thus, the angle is , making option (A) the correct answer.

Step 1: Equation of a parallel plane.
The given plane has the equation , and we are asked to find the equation of a plane parallel to this one and passing through the point .
Step 2: Use the normal vector.
The normal vector to the plane is , as it corresponds to the coefficients of , , and in the equation of the plane.
Step 3: Substitute the point into the equation.
The equation of the plane passing through the point and parallel to the given plane is of the form: where is a constant. Substituting the point into the equation, we get: Thus, the equation of the plane is:
Step 4: Conclusion.
Thus, the correct answer is option (A).

Step 1: Formula for the incenter.
The incenter of a triangle in three-dimensional space can be found by taking the weighted average of the coordinates of the vertices, where the weights are the lengths of the sides opposite the respective vertices. Step 2: Find the lengths of the sides.
First, calculate the lengths of the sides of the triangle: Step 3: Find the coordinates of the incenter.
The coordinates of the incenter are given by: where , , and . Substituting the values, we find: Step 4: Conclusion.
Thus, the incenter of the triangle is .

Step 1: Equal acute angles condition.
For a line making equal acute angles with the coordinate axes, the direction cosines of the line must be equal in magnitude. Since the sum of squares of direction cosines is 1, we have: where . Solving for , we get:
Step 2: Conclusion.
Thus, the direction cosines are , corresponding to option (C).

Step 1: Simplify the integral.
We start by splitting the integrand: Thus, the integral becomes:
Step 2: Perform the integration.
The first term integrates to , and the second term is a standard integral, yielding . So the total integral is:
Step 3: Conclusion.
Thus, the correct answer is , corresponding to option (B).

Step 1: Evaluate the trigonometric values.
We are given that . Using the angle values: Thus, Since is positive and is also positive but smaller than , we have .
Step 2: Conclusion.
Thus, the correct answer is , corresponding to option (C).