Angle between a line and plane MCQ Quiz in தமிழ் - Objective Question with Answer for Angle between a line and plane - இலவச PDF ஐப் பதிவிறக்கவும்

Last updated on Apr 20, 2025

பெறு Angle between a line and plane பதில்கள் மற்றும் விரிவான தீர்வுகளுடன் கூடிய பல தேர்வு கேள்விகள் (MCQ வினாடிவினா). இவற்றை இலவசமாகப் பதிவிறக்கவும் Angle between a line and plane MCQ வினாடி வினா Pdf மற்றும் வங்கி, SSC, ரயில்வே, UPSC, மாநில PSC போன்ற உங்களின் வரவிருக்கும் தேர்வுகளுக்குத் தயாராகுங்கள்.

Latest Angle between a line and plane MCQ Objective Questions

Top Angle between a line and plane MCQ Objective Questions

Angle between a line and plane Question 1:

If the angle θ between the line x+11=y12=z22 and plane 2x - y + √λ z + 4 = 0 is such that sin θ = 13, then value of λ is:

  1. 13
  2. 53
  3. 43
  4. More than one of the above
  5. none of the above

Answer (Detailed Solution Below)

Option 2 : 53

Angle between a line and plane Question 1 Detailed Solution

Concept:

If the equation of a line is xx1a1=yy1b1=zz1c1 and the equation of the plane is  a2x+b2y+c2z+d=0

Then the angle between line and plane can be obtained by:

sinθ=|a1a2+b2b2+c1c2a12+b12+c12a22+b22+c22|,

Where (a1,b1,c1)and(a2,b2,c2) are d.r's of line and normal to the plane.

Calculation:

Given:

x+11=y12=z22

Direction ratio of line = (1, 2, 2)

Equation of plane is  2x - y + √λ z + 4 = 0 

Direction ratio of normal to the plane = (2, -1, √λ)

sin θ=13

Now,

sinθ=|1×2+2×(1)+2×λ12+22+2222+(1)2+(λ)2|13=|2λ3×5+λ|

|5+λ|=|2λ|

Squaring both sides, we get

5+λ=4λ3λ=5λ=53

Angle between a line and plane Question 2:

If the angle θ between the line x+11=y12=z22 and plane 2x - y + √λ z + 4 = 0 is such that sin θ = 13, then value of λ is:

  1. 13
  2. 23
  3. 43
  4. More than one of the above
  5. None of the above

Answer (Detailed Solution Below)

Option 5 : None of the above

Angle between a line and plane Question 2 Detailed Solution

Concept:

If the equation of a line is xx1a1=yy1b1=zz1c1 and the equation of the plane is  a2x+b2y+c2z+d=0

Then the angle between line and plane can be obtained by:

sinθ=|a1a2+b2b2+c1c2a12+b12+c12a22+b22+c22|,

Where (a1,b1,c1)and(a2,b2,c2) are d.r's of line and normal to the plane.

Calculation:

Given:

x+11=y12=z22

Direction ratio of line = (1, 2, 2)

Equation of plane is  2x - y + √λ z + 4 = 0 

Direction ratio of normal to the plane = (2, -1, √λ)

sin θ=13

Now,

sinθ=|1×2+2×(1)+2×λ12+22+2222+(1)2+(λ)2|13=|2λ3×5+λ|

|5+λ|=|2λ|

Squaring both sides, we get

5+λ=4λ3λ=5λ=53

Angle between a line and plane Question 3:

The sine of the angle between the straight line x23=y34=z45 and the plane 2x - 2y + z = 5.

  1. √2/10
  2. 3/√50
  3. 3/50
  4. 4/5√2
  5. None of the above

Answer (Detailed Solution Below)

Option 1 : √2/10

Angle between a line and plane Question 3 Detailed Solution

Given line is

 x23=y34=z45
 
and plane is 2x - 2y + z = 5

So, sinθ=|bn|b||n||

sinθ=|(3ı^+4ȷ^+5k^)(2ı^2ȷ^+k^)9+16+254+4+1|

=68+5509=152=152×22=210

Angle between a line and plane Question 4:

The angle between the line x+23=y32=z+56 and the plane 2x + 10y - 11z = 5 is:

  1. cos1(821)
  2. sin1(821)
  3. cos1(2182)
  4. sin1(2182)

Answer (Detailed Solution Below)

Option 1 : cos1(821)

Angle between a line and plane Question 4 Detailed Solution

Concept:

  • The angle between a line and a plane is the complement of the angle between the line’s direction vector and the normal vector of the plane.
  • If θ is the angle between the line and the plane, then: sinθ = (|n · d|) / (|n| × |d|), where n is the normal vector of the plane, and d is the direction vector of the line.
  • Direction vector of the line x+2/3 = y−3/2 = z+5/6 is given by: d = 〈3, 2, 6〉.
  • Normal vector of the plane 2x + 10y − 11z = 5 is: n = 〈2, 10, −11〉.

 

Calculation:

Given,

Direction vector of line, d = 〈3, 2, 6〉

Normal vector of plane, n = 〈2, 10, −11〉

⇒ n · d = 2×3 + 10×2 + (−11)×6 = 6 + 20 − 66 = −40

⇒ |n · d| = 40

⇒ |n| = √(2² + 10² + (−11)²) = √(4 + 100 + 121) = √225 = 15

⇒ |d| = √(3² + 2² + 6²) = √(9 + 4 + 36) = √49 = 7

⇒ sinθ = 40 / (15 × 7) = 40 / 105 = 8 / 21

∴ The angle between the line and the plane is sin⁻¹(8 / 21)

Angle between a line and plane Question 5:

The sine of the angle between the straight line x23=y34=z45 and the plane 2x - 2y + z = 5.

  1. √2/10
  2. 3/√50
  3. 3/50
  4. 4/5√2

Answer (Detailed Solution Below)

Option 1 : √2/10

Angle between a line and plane Question 5 Detailed Solution

Given line is

 x23=y34=z45
 
and plane is 2x - 2y + z = 5

So, sinθ=|bn|b||n||

sinθ=|(3ı^+4ȷ^+5k^)(2ı^2ȷ^+k^)9+16+254+4+1|

=68+5509=152=152×22=210

Angle between a line and plane Question 6:

The sine of the angle between the straight line x23=y34=4z5 and the plane 2x - 2y + z = 5 is

  1. 152
  2. 252
  3. 350
  4. 350

Answer (Detailed Solution Below)

Option 1 : 152

Angle between a line and plane Question 6 Detailed Solution

Concept Used:

The sine of the angle between a line and a plane can be found using the direction ratios of the line and the normal to the plane.

sinθ=|a1b1+a2b2+a3b3|a12+a22+a32b12+b22+b32

Calculation:

Given:

The equation of the straight line: x23=y34=4z5

The equation of the plane: 2x2y+z=5

Direction ratios of the line: (3,4,5)

Normal to the plane: (2,2,1)

|a1b1+a2b2+a3b3|=|(32)+(42)+(51)|=|685|=|3|=3

a12+a22+a32=32+42+(5)2=9+16+25=50

b12+b22+b32=22+(2)2+12=4+4+1=9=3

sinθ=3350=152 

∴ The sine of the angle between the straight line and the plane is 152 

Hence, option 1 is correct.

Angle between a line and plane Question 7:

The sine of the angle between the straight line x23=y34=z45 and the plane 2x − 2y + z = 5 is

  1. 1065
  2. 452
  3. 235
  4. More than one of the above
  5. None of the above

Answer (Detailed Solution Below)

Option 5 : None of the above

Angle between a line and plane Question 7 Detailed Solution

Concept:

If there exists a line xx1a=yy1b=zz1c, having direction ratios (a, b, c) and a plane Ax + By + Cz + D = 0 having direction ratios (A, B, C) then the acute angle θ between the line and the plane is given by:

sin θ = |aA+bB+cCa2+b2+c2A2+B2+C2|

Calculation:

We have the plane 2x – 2y + z – 5 = 0 and line x23=y34=z45

∴ Direction ratios of the plane = (A, B, C) = (2, -2, 1)

and, direction ratios of the line = (a, b, c) = (3, 4, 5)

∴ sin θ = |aA+bB+cCa2+b2+c2A2+B2+C2|

|3×2+4×(2)+5×132+42+5222+(2)2+12|

|68+59+16+254+4+1|

|3509|

352×3

152

210

⇒ sin θ 210

Therefore, the sine of the angle between the given line and plane is 210.

The correct answer is option 5.

Angle between a line and plane Question 8:

Find the angle between the line x+12=y3=z36 and the plane 10x + 2y - 11z - 3 = 0.

  1. sin1(821)
  2. sin1(221)
  3. sin1(521)
  4. More than one of the above
  5. None of the above

Answer (Detailed Solution Below)

Option 1 : sin1(821)

Angle between a line and plane Question 8 Detailed Solution

Concept:

If θ is the angle between the line xx1a1=yy1b1=zz1c1 and the plane a2 x + b2 y + c2 z + d = 0 then sinθ=|a1a2+b1b2+c1c2|(a12+b12+c12)(a22+b22+c22)

Calculation:

Given: Equation of line is x+12=y3=z36 and equation of plane is 10x + 2y - 11z - 3 = 0

As we know that the angle between the line xx1a1=yy1b1=zz1c1 and the plane a2 x + b2 y + c2 z + d = 0 is given by: sinθ=|a1a2+b1b2+c1c2|(a12+b12+c12)(a22+b22+c22)

Here, a1 = 2, b1 = 3, c1 = 6, a2 = 10, b2 = 2 and c2 = - 11.

⇒ a1 ⋅ a2 + b1 ⋅ b2 + c1 ⋅ c2 = 20 + 6 - 66 = - 40

a12+b12+c12=7anda22+b22+c22=15

sinθ=407×15=821

θ=sin1(821)

Hence, option A is the correct answer.

Angle between a line and plane Question 9:

If the angle θ between the line x+11=y12=z22 and plane 2x - y + √λ z + 4 = 0 is such that sin θ = 13, then value of λ is:

  1. 13
  2. 23
  3. 43
  4. 53
  5. None of the above

Answer (Detailed Solution Below)

Option 4 : 53

Angle between a line and plane Question 9 Detailed Solution

Concept:

If the equation of a line is xx1a1=yy1b1=zz1c1 and the equation of the plane is  a2x+b2y+c2z+d=0

Then the angle between line and plane can be obtained by:

sinθ=|a1a2+b2b2+c1c2a12+b12+c12a22+b22+c22|,

Where (a1,b1,c1)and(a2,b2,c2) are d.r's of line and normal to the plane.

Calculation:

Given:

x+11=y12=z22

Direction ratio of line = (1, 2, 2)

Equation of plane is  2x - y + √λ z + 4 = 0 

Direction ratio of normal to the plane = (2, -1, √λ)

sin θ=13

Now,

sinθ=|1×2+2×(1)+2×λ12+22+2222+(1)2+(λ)2|13=|2λ3×5+λ|

|5+λ|=|2λ|

Squaring both sides, we get

5+λ=4λ3λ=5λ=53

Angle between a line and plane Question 10:

If the plane 2x - y + z = 0 is parallel to the line 2x12=2y2=z+1a , then value of a is

  1. 4
  2. -4
  3. 2
  4. More than one of the above
  5. None of the above

Answer (Detailed Solution Below)

Option 2 : -4

Angle between a line and plane Question 10 Detailed Solution

Concept:

If a plane is parallel to a line then the normal vector to the plane will be perpendicular to the direction of the line.

The direction of the line,

b=(2,2,a)

The vector perpendicular to the given plane  2x - y + z = 0,

η=(2,1,1)

Calculation:

Given:

Since, the plane 2x - y + z = 0 is parallel to the line x1/21=y22=z+1a

⇒ 2 (1) + (-1) (-2) + 1 (a) = 0 

⇒ a + 4 = 0 ⇒ a = - 4.

Get Free Access Now
Hot Links: all teen patti game teen patti app teen patti comfun card online