Direction ratios Practice Questions - NDA Mathematics Question Bank

Question 1 of 10 Authentic UPSC NDA Question

Consider the following statements :

1. The direction ratios of y-axis can be < 0, 4, 0 >

2 . The direction ratios of a line perpendicular to z-axis can be < 5, 6, 0 >.

Which of the statements given above is /are correct ?

Step-by-Step Logic & Solution

Solution:

Statement 1:

Direction cosines of y-axis:

⟨0, 1, 0⟩

Direction ratios can be proportional.

⟨0, 4, 0⟩ is valid.

Statement 1 is correct.

Statement 2:

Direction cosines of z-axis:

⟨0, 0, 1⟩

For perpendicular lines:

a₁a₂ + b₁b₂ + c₁c₂ = 0

Checking ⟨5, 6, 0⟩ with z-axis:

5(0) + 6(0) + 0(1)

= 0

Therefore line is perpendicular to z-axis.

Statement 2 is correct.

Hence option (c).

Question 2 of 10 Authentic UPSC NDA Question

If l, m, n are the direction cosines of the line x − 1 = 2(y + 3) = 1 − z, then what is l⁴ + m⁴ + n⁴ equal to?

Step-by-Step Logic & Solution

Question 3 of 10 Authentic UPSC NDA Question

What is the angle between the two lines having direction ratios ⟨6, 3, 6⟩ and ⟨3, 3, 0⟩?

Step-by-Step Logic & Solution

Question 4 of 10 Authentic UPSC NDA Question

For the next two (02) items that follow:

A line L passing through the point (−1, 2, −3) is perpendicular to the plane P given by

2x + 3y + z + 5 = 0.

What are the direction ratios of a line M parallel to the plane P?

Step-by-Step Logic & Solution

Solution

Option (d) is correct.

Explanation:

The direction ratios of the normal to the plane \( P: 2x + 3y + z + 5 = 0 \) are given by the coefficients of \( x, y, z \). \[ \text{Normal Direction Ratios: } \langle 2, 3, 1 \rangle \] A line \( M \) that is parallel to the plane \( P \) must have a direction vector that is orthogonal to the normal vector. If the direction ratios of the line \( M \) are \( \langle a, b, c \rangle \), then the dot product must equal zero: \[ 2a + 3b + c = 0 \] Testing the options: - For option A: \(\langle -3, 2, 1 \rangle\) \[ 2(-3) + 3(2) + 1 = -6 + 6 + 1 = 1 \quad (\text{not zero}) \] - For option B: \(\langle 3, 2, -6 \rangle\) \[ 2(3) + 3(2) - 6 = 6 + 6 - 6 = 6 \quad (\text{not zero}) \] - For option C: \(\langle 1, 3, 2 \rangle\) \[ 2(1) + 3(3) + 2 = 2 + 9 + 2 = 13 \quad (\text{not zero}) \] - For option D: \(\langle 2, 2, -10 \rangle\) \[ 2(2) + 3(2) - 10 = 4 + 6 - 10 = 0 \quad (\text{is zero}) \] Hence, option (d) is indeed the correct choice. The direction ratios \( \langle 2, 2, -10 \rangle \) align with the required orthogonality.

📌 Hints / Properties Used:

Direction ratios of a plane are derived from its coefficients.
Orthogonality condition: The dot product of vectors must equal zero.

📊 Visual Diagram Suggestion:

A coordinate system where the plane \( P \) is represented as a surface, and the normal vector is depicted as an arrow perpendicular to this surface. The direction vectors \( \langle a, b, c \rangle \) should illustrate how they lie in the plane, confirming they are orthogonal to the normal.

📖 Factual Verification & Reference:

Verified against standard texts on Analytical Geometry and Vector Algebra.

Question 5 of 10 Authentic UPSC NDA Question

For the following two (02) items:

A plane P is parallel to the line having direction ra- tios ⟨1, 3, 2⟩ and contains the line of intersection of the planes 6x + 4y - 5z = 2 and x - 2y + 3z = 0.

Which of the following are the direction ratios of the line of intersection of the given planes?

Step-by-Step Logic & Solution

Question 6 of 10 Authentic UPSC NDA Question

Direction: Consider the following for the next two (02)
items that follow:
Let L: x + y + z + 4 = 0 = 2x – y – z + 8 be a
line and P: x+2y + 3z + 1 = 0 be a plane.

What are the direction ratios of the line?

Step-by-Step Logic & Solution

Question 7 of 10 Authentic UPSC NDA Question

Direction: Consider the following for the next two (02) items that follow:
Let S be the line of intersection of two planes
x + y + z = 1 and 2x + 3y – 4z = 8.

Which of the following are the direction ratios of S?

Step-by-Step Logic & Solution

).

Question 8 of 10 Authentic UPSC NDA Question

For the next two (02) items:

Consider two lines whose direction ratios are (2, −1, 2) and (k, 3, 5).

They are inclined at an angle π/4.

What are the direction ratios of a line which isperpendicular to both the lines -

Step-by-Step Logic & Solution

Question 9 of 10 Authentic UPSC NDA Question

If a line has direction ratios ⟨a + b, b + c, c + a⟩, then what is the sum of the squares of its direction cosines?

Step-by-Step Logic & Solution

Solution

Option (d) is correct.

Explanation:

\[ \text{Let the direction ratios be } \langle a+b, b+c, c+a \rangle \]

\[ l = \frac{a+b}{\sqrt{(a+b)^2 + (b+c)^2 + (c+a)^2}}, \; m = \frac{b+c}{\sqrt{(a+b)^2 + (b+c)^2 + (c+a)^2}}, \; n = \frac{c+a}{\sqrt{(a+b)^2 + (b+c)^2 + (c+a)^2}} \]

\[ l^2 + m^2 + n^2 = \frac{(a+b)^2 + (b+c)^2 + (c+a)^2}{(a+b)^2 + (b+c)^2 + (c+a)^2} = 1 \]

Thus, the sum of the squares of its direction cosines is 1.

📌 Hints / Properties Used:

Direction Cosines: \( l, m, n \) represent the cosines of the angles with the coordinate axes.
The sum of the squares of the direction cosines equals 1 for any line in 3D space.

📊 Visual Diagram Suggestion:

A 3D coordinate system illustrating a line with given direction ratios, indicating the angles between the line and the coordinate axes.

📖 Factual Verification & Reference:

Verified against NCERT Class XII Mathematics, Chapter on 3D Geometry.

Question 10 of 10 Authentic UPSC NDA Question

Consider the following statements:

1. A line in space can have infinitely many direction ratios.

2. It is possible for a certain line that the sum of squares of direction cosines can be equal to sum of its direction cosines.

Which of the above statements are correct?

Step-by-Step Logic & Solution

Direction ratios Questions

Solution

Solution