This set of Mathematics Written Test Questions for IIT JEE Exam focuses on “Three Dimensional Geometry – Angle between a Line and a Plane”.

1. _____ is the complement of the angle between the line L and a normal line to the plane π.

a) Normal between a plane and a line

b) The angle between a line and a plane

c) Tangent between a plane and a line

d) Distance between a plane and a line

View Answer

Explanation: The angle between a line and a plane is the complement of the angle between the line L and a normal line to the plane π. If θ is the angle between line whose ratios are a

_{1}, b

_{1}, c

_{1}and the plane ax + by + cz + d = 0 then sin θ=\(\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}\).

2. If θ is the angle between line whose ratios are a_{1}, b_{1}, c_{1} and the plane ax + by + cz + d = 0 then

cos θ=\(\frac {a1a2.b1b2.c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt {a2^2+b2^2+c2^2 }}\).

a) True

b) False

View Answer

Explanation: A mathematical symbol θ is used to find the angle between line and a normal line to the plane π along with a trigonometric function called sine. Hence, the formula

sin θ=\(\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}\).

3. Find the angle between the planes x + 2y + 3z + 1 = 0 and (4, 1, -7).

a) – 29.34

b) 3.43

c) 11.23

d) – 17.54

View Answer

Explanation: Angle between a plane and a line sin θ=\(\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}\)

sin θ = – 0.49

θ = sin

^{-1}(- 0.49)

θ = – 29.34

4. Which trigonometric function is used to find the angle between a line and a plane?

a) Tangent

b) Cosecant

c) Secant

d) Sine

View Answer

Explanation: The trigonometric function is used to find the angle between a line and a plane is sine. If θ is the angle between line whose ratios are a

_{1}, b

_{1}, c

_{1}and the plane ax + by + cz + d = 0 then sin θ=\(\frac {a1a2.b1b2.c1c2}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}\).

5. What is the plane equation involved in the formula sinθ=\(\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}\)?

a) a_{1}x – b_{1}y + c_{1}z + d_{1} = 0

b) a_{1}x^{2} + b_{1}y^{2} + c_{1}z^{2} + d_{1} = 0

c) ax + by + cz+ d = 0

d) a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and ax + by + cz + d = 0

View Answer

Explanation: The angle between a line and a plane is the complement of the angle between the line L and a normal line to the plane π. If θ is the angle between line whose ratios are a

_{1}, b

_{1}, c

_{1}and the plane ax + by + cz + d = 0 then sin θ=\(\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}\).

6. What is the relation between the plane ax + by + cz + d = 0 and a_{1}, b_{1}, c_{1} the direction ratios of a line, if the plane and line are perpendicular to each other?

a) \(\frac {a1}{b1} = \frac{a2}{c1} = \frac{c2}{b2}\)

b) \(\frac {a1}{a2} = \frac{b1}{c2} = \frac{c1}{b2}\)

c) \(\frac {a}{a1} = \frac{b}{b1} = \frac{c}{c1}\)

d) \(\frac {c1}{a2} = \frac{b1}{b2} = \frac{a1}{c2}\)

View Answer

Explanation: θ = 90 degrees

The relation between the plane ax + by + cz + d = 0 and a

_{1}, b

_{1}, c

_{1}the direction ratios of a line, if the plane and line are perpendicular to each other is \(\frac {a}{a1} = \frac{b}{b1} = \frac{c}{c1}\).

7. What is the relation between the plane ax + by + cz + d = 0 and a_{1}, b_{1}, c_{1} the direction ratios of a line, if the plane and line are parallel to each other?

a) a_{1}a_{2} . b_{1}b_{2} . c_{1}c_{2} = 0

b) a_{1}a + b_{1}b + c_{1}c = 0

c) a_{1}a_{2} + b_{1}b_{2} – c_{1}c_{2} = 0

d) a_{1}a_{2} + b_{1}b_{2} – c_{1}c_{2} = 0

View Answer

Explanation: The relation between the plane ax + by +cz + d = 0 and a

_{1}, b

_{1}, c

_{1}the direction ratios of a line, if the plane and line are parallel to each other is a

_{1}a + b

_{1}b + c

_{1}c = 0.

8. A plane and a line having an angle of 90 degrees between them are called _____

a) Orthogonal

b) Tangential

c) Normal

d) Parallel

View Answer

Explanation: A plane and A line which are perpendicular to each other or a plane and a line having an angle 90 degrees between them are called orthogonal. θ is equal to 90 degrees in sin θ=\(\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}\).

9. The condition a_{1}a + b_{1}b + c_{1}c = 0 is for a plane and a line are _____ to each other.

a) integral

b) parallel

c) perpendicular

d) concentric

View Answer

Explanation: The relation between the plane ax + by +cz + d = 0 and a

_{1}, b

_{1}, c

_{1}the direction ratios of a line, if the plane and line are parallel to each other is a

_{1}a + b

_{1}b + c

_{1}c = 0.

10. The condition \(\frac {a}{a1} = \frac{b}{b1} = \frac{c}{c1}\) is for a plane and a line are _____ to each other.

a) perpendicular

b) parallel

c) differential

d) tangential

View Answer

Explanation: θ = 90 degrees

The relation between the plane ax + by + cz + d = 0 and a

_{1}, b

_{1}, c

_{1}the direction ratios of a line, if the plane and line are perpendicular to each other is \(\frac {a}{a1} = \frac{b}{b1} = \frac{c}{c1}\).

11. Find the angle between 2x + 3y – 2z + 4 = 0 and (2, 1, 1).

a) 38.2

b) 19.64

c) 89.21

d) 29.34

View Answer

Explanation: Angle between a plane and a line sin θ=\(\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}\)

sinθ = 0.49

θ = sin

^{-1}(0.49)

θ = 29.34

12. Find the angle between x + 2y + 7z + 2 = 0 and (2, 4, 6).

a) 69.69

b) 84.32

c) 66.92

d) 83.25

View Answer

Explanation: Angle between a plane and a line sin θ=\(\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}\)

sinθ = 0.92

θ = 66.92

13. The plane 5x + y + kz + 1 = 0 and directional ratios of a line (3, -1, 1) are parallel, find k.

a) 4

b) -14

c) 6

d) -8

View Answer

Explanation: The condition for a plane and a line are parallel to each other is a

_{1}a + b

_{1}b + c

_{1}c = 0.

5(3) + 1(-1) + k(1) = 0

K(1) = -14

K = -14

14. Find the angle between the planes 5x + 2y + 3z + 1 = 0 and (1, 1, -2).

a) 30.82

b) 3.43

c) 11.23

d) 7.54

View Answer

Explanation: Angle between a plane and a line sin θ=\(\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}\)

sinθ = 0.06

θ = 3.43

15. Find k for the given plane x + 2y + kz + 2 = 0 and directional ratios of a line (8, 3, 2), if they are parallel to each other.

a) 21

b) -17

c) 12

d) -7

View Answer

Explanation: The condition for a plane and a line are parallel to each other is a

_{1}a + b

_{1}b + c

_{1}c = 0.

8(1) + 3(2) + 2(k) = 0

2(k) = -14

k = -7

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