Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

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Q.1. A vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev lies in the plane of the vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev If Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev bisects the angle between Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then    (2020)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(3)
We have,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
On comparing with Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
We get α = 4 and β = 4
Therefore, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now, again consider Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
On comparing withPrevious year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev we get
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Therefore, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.2. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be three unit vectors such that Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then the ordered pair, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is equal to    (2020)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(4)
Given
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(1)
Taking square of both sides, we get
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev (since Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are unit vectors)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.3. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be two vectors. If Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is a vector such that Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is equal to    (2020)
(1) -3/2
(2) 1/2
(3) -1/2
(4) -1
Ans.
(3)
Given Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(1)
Now, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Substituting these values in Eq. (1), we get
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.4. The projection of the line segment joining the points (1, −1, 3) and (2, −4, 11) on the line joining the points (−1, 2, 3) and (3, −2, 10) is _____.    (2020)
Ans.
(8.00)
We have
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, the required projection is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.5. If the vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are coplanar and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then the value of λ is _____.    (2020)
Ans.
(1.00)
The vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are coplanar vectors, then
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Therefore, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(2)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(3)
Now, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(4)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(5)
Hence, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.6. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be three vectors such that Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and the angle between Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev If Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is perpendicular to the vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is equal to ____.    (2020)
Ans.
(30.00)
Given, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is perpendicular to the vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.7. If the distance between the plane 23x - 10y - 2z + 48 =0 and the plane containing the lines Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is equal to Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then k is equal to _____.    (2020)
Ans.
(3.00)
The given lines must be intersecting

Therefore,
(2s - 1, 3s + 3, 8s - 1) = (2t - 3t, t - 2, λt + 1)
Now, 2s - 1 = 2t - 3 ...(1)
3s + 3 = t - 2 ...(2)
8s - 1 = λt + 1 ...(3)
On solving above equations, we get  
t = -1, s = -2 and λ = 18
Distance of plane contains given lines from given plane is same as distance between point (–3, –2, 1) from given plane. So,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.8. Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then, the image of R in the plane P is    (2020)
(1) (6, 5, 2)
(2) (6, 5, −2)
(3) (4, 3, 2)
(4) (3, 4, −2)
Ans.
(2)
The equation of the plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ (x - 2)(1) - (y - 1)(2 - 3) + z(-2) = 0
⇒ x + y - 2z = 3
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Let I and F are respectively image and foot of perpendicular of point R in the plane. So, the equation of RI is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev (say)
⇒ x = k + 2, y = k + 1 and z = -2k + 6
Hence, the coordinates of point F is Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev 
Point F lies in the plane, then
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ 3k = 12 ⇒ k = 4
Hence, the image I is (6, 5, -2).

Q.9. If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (α, 7, 1) is Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev  then α is equal to _____.
Ans.
(4.00)
We have,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Direction ratio of BP Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Direction ratio of AP =Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Since AP and BP are perpendicular, then
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.10. The shortest distance between the lines Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is    (2020)
(1) 2√30
(2) 7/2 (√30)
(3) 3√30
(4) 3
Ans.
(3)
The equations of the lines are
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev  ...(1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(2)
The distance between the lines is given by
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.11. Let the volume of a parallelepiped whose coterminous edges are given by Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be 1 cu. unit. If θ be the angle between the edges Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then cos θ can be    (2020)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) 5/7
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(2)
The volume of parallelepiped is 1 cu. unit. Therefore,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ λ = 2, 4
Now, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.12. The mirror image of the point (1, 2, 3) in a plane is Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev Which of the following points lies on this plane?    (2020)
(1) (1, 1, 1)
(2) (1, -1, 1)
(3) (-1, -1, 1)
(4) (-1, -1, -1)
Ans.
(2)
Direction ratios of normal to the plane is
<1 + (7/3), 2 + (4/3),3 + (1/3)> = <1, 1, 1>
The coordinates of mid- point of image and point is Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
The equation of plane is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ x + y + z = 1
Hence, the point (1, −1, 1) lies on the plane.

Q.13. If for some α and β in Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevthe intersection of the following three planes
x + 4y - 2z =1
x + 7y - 5z = β
x + 5y - αz = 5
is a line in Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevthen α + β is equal to    (2020)
(1) 0
(2) 10
(3) 2
(4) −10
Ans.
(2)
We have
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev=0
⇒ (7α + 25) - (4α + 10) + (-20 + 14) = 0
⇒ 3α + 9 = 0 ⇒ α = -3
and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, α + β = -3 + 13 = 10

Q.14. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be a vector such that Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is equal to:    (2019)
(1) 19/2
(2) 9
(3) 8
(4) 17/2
Ans.
(1)
 Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.15. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be three vectors such that the projection vector of Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
If Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is perpendicular to Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is equal to:    (2019)
(1) √32
(2) 6
(3) √22
(4) 4
Ans.
(2)
Projection of Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
According to question Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
 b1 + b2 = 2    ...(1)
Since, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is perpendicular toPrevious year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ 8 + 5b1 + b2 + 2 = 0 ...(2)
From (1) and (2),
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.16. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be three vectors such that Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is perpendicular to Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev Then a possible value of (λ1, λ2, λ3) is:    (2019)
(1) (1, 3, 1)
(2) (-1/2, 4, 0)
(3) (1/2, 4, -2)
(4) (1, 5, 1)
Ans.
(2)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
∴ 3 - λ2 = 2λ1    ...(1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(2)
Since, (-1/2, 4, 0) satisfies equation (1) and (2). Hence, one of possible value of
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.17. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be two given vectors where vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevare non-collinear. The value of λ for which vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are collinear, is:    (2019)
(1) -4
(2) -3
(3) 4
(4) 3
Ans.
(1)
Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are collinear for same k
i.e., Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
But Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are non-collinear, then
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
λ = -4

Q.18. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevand Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be coplanar vectors. Then the non-zero vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevis:    (2019)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans. 
(4)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are coplanar
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
For Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev(Rejected)

Q.19. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is 3/√2, then the sum of all possible values of β is:    (2019)
(1) 4 
(2) 3
(3) 2
(4) 1
Ans.
(4)
Since, the angle bisector of acute angle between OA and OB would be y = x
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Since, the distance of C from bisector = 3/√2
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, the sum of all possible value of β = 2 + (-1) = 1

Q.20. The sum of the distinct real values of μ, for which the vectors, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are co-planar, is:    (2019)
(1) -1
(2) 0
(3) 1
(4) 2
Ans.
(1)
∵ Three vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevand Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are copalnar.
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Therefore, sum of all real values = 1 - 2 = -1

Q.21. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be three unit vectors, out of which vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevare non-parallel. If α and β are the angles which vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev makes with vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev respectively and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is equal to:    (2019)
(1) 30°
(2) 90°
(3) 60°
(4) 45°
Ans.
(1)
Since, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are three unit vectors
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Then, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ β = 60° and α = 90°
Hence, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.22. The magnitude of the projection of the vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev on the vector perpendicular to the plane containing the vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is:    (2019)
(1) √3/2
(2) √6
(3) 3√6
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans. 
(4)
Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
∴ Vector perpendicular to Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now, projection of vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.23. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev for some real x. Then Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is possible if:    (2019)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(2)
Given Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.24. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevwhere Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is parallel to Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevis perpendicular to Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is equal to:    (2019)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(3)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(1)
Since, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is perpendicular to Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Cross product with Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev in equation (1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.25. If a unit vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev makes angles π/3 with Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and θ ∈ (0, π) with Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then a value of 0 is:    (2019)
(1) 5π/6
(2) π/4
(3) 5π/12
(4) 2π/3
Ans.
(4)
Let cos α, cos β, cos γ be direction cosines of a.
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.26. The distance of the point having position vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev from the straight line passing through the point (2, 3, -4) and parallel to the vector, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is:    (2019)
(1) 7
(2) 4√3
(3) 6
(4) 2√13
Ans. 
(1)
Equation of the line is Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Let the point M on the line l is (6λ + 2, 3λ + 3, -4λ - 4)
Direction ratio's of PM is (6λ + 3, 3λ + 1, - 4λ- 10)
⇒ (6λ + 3) (6) + (3λ + 1) (3) + (-4λ.- 10) (- 4) = 0
⇒ λ = - 1 ⇒ M = (- 4, 0, 0)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.27. Let a = 3i + 2j + 2k and b = i + 2j - 2k be two vectors. If a vector perpendicular to both the vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev has the magnitude 12 then one such vector is:    (2019)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(2)
Let vector be Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Given that magnitude of the vector is 12.
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.28. Let α ∈ R and the three vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevPrevious year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev Then the set Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev    (2019)
(1) is singleton
(2) is empty
(3) contains exactly two positive numbers
(4) contains exactly two numbers only one of which is positive
Ans.
(2)
Let, three vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are coplanar, then Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev 
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
∵ no real value of ‘α’ exist,
∴ set S is an empty set.

Q.29. The equation of the line passing through (-4, 3, 1), parallel to the plane x + 2y - z - 5 = 0 and intersecting the line Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevis:    (2019)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(3)
Let any point on the intersecting line
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
is (-3λ - 1, 2λ + 3, -λ + 2)
Since, the above point lies on a line which passes through the point (-4, 3, 1)
Then, direction ratio of the required line
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Since, line is parallel to the plane
x + 2y - z - 5 = 0
Then, perpendicular vector to the line is Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now (-3λ + 3)(1) + (2λ)(2) + (-λ + 1)(-1) = 0
⇒  λ = - 1
Now direction ratio of the required line = <6, - 2, 2 > or <3,-1, 1>
Hence required equation of the line is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.30. The plane through the intersection of the planes x + y + z = 1 and 2x + 3y-z + 4 = 0 and parallel to j-axis also passes through the point:    (2019)
(1) (-3, 0, -1)
(2) (-3, 1, 1)
(3) (3, 3, -1)
(4) (3, 2, 1)
Ans.
(4)
Since, equation of plane through intersection of planes
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
But, the above plane is parallel to y-axis then
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ λ = -3
Hence, the equation of required plane is
-x - 4z + 7 = 0
⇒ x + 4z - 7 = 0
Therefore, (3, 2, 1) the passes through the point.

Q.31. The equation of the plane containing the straight line Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and perpendicular to the plane containing the straight lines Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is:    (2019)
(1) x - 2y + z = 0
(2) 3x + 2y - 3z = 0
(3) x + 2y - 2z = 0
(4) 5x + 2y - 4z = 0
Ans.
(1)
Let the direction ratios of the plane containing lines
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
∴ Direction ratio of plane = <-8, 1, 10>.
Let the direction ratio of required plane is <l, m, n>
Then-81 + m + 10n = 0    ...(1)
and 2l + 3m + 4n = 0    ...(2)
From (1) and (2),
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
∴ D.R.s are <1, -2, 1>

∴ Equation of plane: x - 2y + z = 0

Q.32. The plane passing through the point (4, -1, 2) and parallel to the lines Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev also passes through the point:    (2019)
(1) (1, 1, -1)
(2) (1, 1, 1)
(3) (-1, -1,-1)
(4) (-1, -1, 1)
Ans. 
(2)
Equation of required plane is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(x - 4)(-3 -4) - (y + 1)(9 - 2) + (z - 2) (6 + 1) = 0
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
∵ point (1, 1, 1) satisfies this equation
∴ point (1, 1, 1) lies on the plane

Q.33. Let A be a point on the line Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and B(3, 2, 6) be a point in the space. Then the value of μ for which the vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is parallel to the plane x - 4y + 3z = 1 is:    (2019)
(1) 1/4
(2) 1/8
(3) 1/2
(4) -1/4
Ans. 
(1)
∵ A be a point on given line.
∴ Position vector of A
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Equation of plane is: x - 4y + 3z = 1
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is parallel to this plane.
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ μ = 1/4

Q.34. The plane which bisects the line segment joining the points (- 3, - 3, 4) and (3, 7, 6) at right angles, passes through which one of the following points?    (2019)
(1) (-2, 3, 5)
(2) (4, -1, 7)
(3) (2, 1, 3)
(4) (4, 1, -2)
Ans.
(4)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Q(0, 2, 5)
Since, direction ratios of normal to the plane is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Then, equation of the plane is
(x - 0)6 + (y - 2)10 + (z - 5)2 = 0
3x + 5y - 10 + z - 5 = 0
3x + 5y + z = 15    ...(1)
Since, plane (1) satisfies the point (4, 1, -2)
Hence, required point is (4, 1, -2)

Q.35. On which of the following lines lies the point of inter-section of the line, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevand the plane, x + y + z = 2?    (2019)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(3)
Let any point on the line Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev be A(2λ + 4, 2λ + 5, λ, + 3) which lies on the plane x +y + z = 2
 2λ + 4 + 2λ + 5 + λ + 3 = 2
 5λ = -10 ⇒ λ = - 2
Then, the point of intersection is (0, 1, 1)
which lies on the line Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.36. The plane containing the line Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and also containing its projection on the plane 2x + 3y- z = 5, contains which one of the following points?    (2019)
(1) (2, 2, 0) 
(2) (-2, 2, 2)
(3) (0, -2, 2)
(4) (2, 0, -2)
Ans.
(4)
Let normal to the required plane is Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is perpendicular to both vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevPrevious year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ Equation of the required plane is
 (x - 3)(-8) + (y + 2) x 8 + (z - 1) x 8 = 0
⇒ (x - 3) (-1) + (y + 2) x 1 + (z - 1) x 1 = 0
 x - 3 - y - 2 - z + 1 =0
∵ x - y - z = 4 passes through (2, 0, -2)
∴ plane contains (2, 0, -2).

Q.37. The direction ratios of normal to the plane through the points (0, -1, 0) and (0,0, 1) and making an angle π/4 with the plane y - z + 5 = 0 are:    (2019)
(1) 2, -1, 1
(2) 2, √2, -√2
(3) √2, 1, -1
(4) 2√3, 1, -1
Ans.
(2, 3)
Let the d.r’s of the normal be (a, b, c)
Equation of the plane is
a(x - 0) + b(y + 1) + c(z - 0) = 0
∵ It passes through (0, 0, 1)
∴ b + c
Also Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
∴ The d.r’s are √2, 1, -1 or 2, √2, -√2

Q.38. Two lines Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev intersect at the point R. The reflection of R in the xy-plane has coordinates:    (2019)
(1) (2, -4, -7)
(2) (2, 4, 7)
(3) (2, -4, 7)
(4) (-2, 4, 7)
Ans.
(1)
Let the coordinate of P with respect to line
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
and coordinate of P w.r.t.
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
From above equation : λ = -1, μ = 1 .
∴ Coordinate of point of intersection R = (2, -4, 1).
Image of R w.r.t. xy plane = (2, -4, -7).

Q.39. If the point (2, α, β) lies on the plane which passes through the points (3,4, 2) and (7, 0, 6) and is perpendicular to the plane 2x-5y = 15, then 2α - 3β is equal to:    (2019)
(1) 12
(2) 7
(3) 5
(4) 17
Ans.
(2)
Let the normal to the required plane is n, then
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
∴ Equation of the plane
(x - 3) x 20 + (y - 4) x 8 + (z - 2) x (-12) = 0
5x - 15 + 2y - 8 - 3z + 6 = 0
5x + 2y - 3z - 17 = 0    ...(1)
Since, equation of plane (1) passes through (2, α, β), then
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.40. The perpendicular distance from the origin to the plane containing the two lines, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is:    (2019)
(1) 11√6
(2) 11/√6
(3) 11
(4) 6/√11
Ans.
(2)
∵ plane containing both lines.
∴ D.R. of plane Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now, equation of plane is,
7(x - 1) - 14(y - 4) + 7 (z + 4) = 0
 x - 1 - 2y + 8 + z + 4 = 0
⇒ x - 2y + z + 11 = 0
Hence, distance from (0, 0, 0) to the plane,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.41. If an angle between the line, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and the plane, x - 2y - kx = 3 is  Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevthen a value of k is    (2019)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) -3/5
(4) -5/3
Ans.
(1)
Let angle between line and plane is θ, then
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Since, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Then, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.42. Let S be the set of all real values of λ such that a plane passing through the points {-λ2, 1, 1), (1, -λ2, 1) and (1, 1, -λ2) also passes through the point-(-1,-1, 1). Then S is equal to:    (2019)
(1) {√3}
(2) {√3, -√3}
(3) {1, -1}
(4) {3, -3}
Ans.
(2)
Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev D(-1, -1, 1) lie on same plane, then
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ (λ2 + 1)((1 -λ2)-4) = 0
 (3 - λ2)(λ+ 1) = 0 ⇒ λ2 = 3
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, S = {-√3, √3}

Q.43. The equation of a plane containing the line of intersection of the planes 2x - y - 4 = 0 and y + 2z - 4 = 0 and passing through the point (1, 1, 0) is:    (2019)
(1) x- 3y- 2z = -2
(2) 2x - z = 2
(3) x - y - z = 0
(4) x + 3y + z = 4
Ans.
(3)
Let the equation of required plane be;
(2x - y - 4) + λ(y + 2z - 4) = 0
∴  This plane passes through the point (1, 1, 0) then (2 - 1 - 4) + λ(1 + 0 - 4) = 0
⇒ λ = -1
Then, equation of required plane is,
(2x - y - 4) - (y + 2z - 4) = 0
⇒ 2x - 2y — 2z = 0 ⇒ x - y - z = 0

Q.44. The length of the perpendicular from the point (2, -1, 4) on the straight line, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevis:    (2019)
(1) greater than 3 but less than 4
(2) less than 2
(3) greater than 2 but less than 3
(4) greater than 4
Ans.
(1)
Let P be the foot of perpendicular from point T (2, -1, 4) on the given line. So P can be assumed as P(10λ - 3, -7λ + 2, λ)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
DR's of TP is proportional to 10λ - 5, - 7λ + 3, λ - 4
∵ TP and given line are perpendicular, so
10(10λ - 5) -7 (- 7λ + 3)+ 1(λ - 4) = 0
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, the length of perpendicular is greater than 3 but less than 4.

Q.45. The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0 is:    (2019)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(4)
Equation of the plane passing through the line of intersection of x + y + z = 1 and 2x + 3y + 4z = 5 is (2x + 3y + 4z - 5) + λ (x + y + z - 1) = 0
∵ plane (i) is perpendicular to the plane x-y + z = 0
∴ (2 + λ)(1) + (3 + λ)(- 1) + (4 + λ)(1) = 0
2 + λ - 3 - λ + 4 + λ = 0 
⇒ λ = -3
Hence, equation of required plane is -x + z- 2 = 0 or x - z + 2 = 0
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.46. If a point R(4, y, z) lies on the line segment joining the points P(2, -3, 4) and Q(8, 0, 10), then distance of R from the origin is:    (2019)
(1) 2√14
(2) 2√21
(3) 6
(4) √53
Ans.
(1)
Here, P, Q, R are collinear
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now, or = Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.47. If the line, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the origin is:    (2019)
(1) √5 / 2
(2) 2 / √5
(3) 9/2
(4) 7/2
Ans.
(3)
Let point on line be P (2k + 1, 3k - 1, 4k + 2)
Since, point P lies on the plane x + 2y + 3z = 15
∴ 2k + 1 + 6k - 2 + 12k + 6= 15
⇒ k = 1/2
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Then the distance of the point P from the origin is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.48. A plane passing through the points (0, -1, 0) and (0, 0, 1) and making an angle π/4 with the plane y - z + 5 = 0, also passes through the point:    (2019)
(1) (-√2, 1, -4)
(2) (-√2, -1, 4)
(3) (-√2, -1, -4)
(4) (√2, 1, 4)
Ans.
(4)
Let the required plane passing through the points (0,-1, 0) and (0, 0, 1) be Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and the given plane is y - z + 5 = 0
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Then, the equation of plane is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Then the point (√2, 1, 4) satisfies the equation of plane -√2x - y + z = 1

Q.49. The vertices B and C of a ΔABC lie on the line, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev such that BC = 5 units. Then the area (in sq. units) of this triangle, given that the point A (1, -1, 2), is:    (2019)
(1) 5√17
(2) 2√34
(3) 6
(4) √34
Ans.
(4)
Let a point D on BC = (3λ - 2, 1, 4λ)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Area of triangle Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.50. Let P be the plane, which contains the line of intersection of the planes, x + y + z - 6 = 0 and 2x + 3y + z + 5 = 0 and it is perpendicular to the xy-plane. Then the distance of the point (0, 0, 256) from P is equal to:    (2019)
(1) 17/√5
(2) 63√5
(3) 205√5
(4) 11/√5
Ans.
(4)
Let the plane be
P = (2x + 3y + z + 5) + λ(x: +y + z- 6) = 0
∵ above plane is perpendicular to xy plane.
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, the equation of the plane is,
P = x + 2y+ 11 = 0
Distance of the plane P from (0, 0, 256)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.51. If Q (0, -1, -3) is the image of the point P in the plane 3x - y + 4z = 2 and R is the point (3, -1, -2), then the area (in sq. units) of ΔPQR is:    (2019)
(1) 2√13
(2) √91/4
(3) √91/2
(4) √65/2
Ans.
(3)
Image of Q (0, -1, -3) in plane is,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.52. Let A(3, 0, -1), B(2, 10, 6) and C (1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G divides BM in the ratio, 2 : 1, then cos (∠GOA)(O being the origin) is equal to:    (2019)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) 1/√15
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) 1/√30
Ans.
(2)
G is the centroid of ΔABC.
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.53. If the length of the perpendicular from the point (β, 0, β) (β ≠ 0) to the line, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then β is equal to:    (2019)
(1) 1
(2) 2
(3) -1
(4) -2
Ans.
(3)
Given, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev (let) and point P (β, 0, β)
Any point on line A = (p, 1, -p -1)
Now, DR of AP ≡ <p - β, 1 - 0 - p - 1 - β>
Which is perpendicular to line.
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Given that distance AP Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
∴ β = -1

Q.54. A perpendicular is drawn from a point on the line Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev to the plane x + y + z = 3 such that the foot of the perpendicular Q also lies on the plane x - y + z = 3. Then the co-ordinates of Q are:    (2019)
(1) (1, 0, 2)
(2) (2, 0, 1)
(3) (-1, 0, 4)
(4) (4, 0, -1)
Ans. 
(2)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Let co-ordinates of Q be (α, β, γ), then
α + β + γ = 3    ...(i)
α - β + γ = 3    ...(ii)
⇒ α + γ = 3 and β = 0
Equating direction ratio's of PQ, we get
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Substituting the values of a and y in equation (i), we get
⇒ 5λ + 3 = 3 ⇒ λ = 0
Hence, point is Q (2, 0, 1)

Q.55. If the plane 2x - y + 2z + 3 =0 has the distances 1/2 and 2/3 units from the planes 4x - 2y + 4z + λ = 0 and 2x - y + 2z + μ = 0, respectively, then the maximum value of λ + μ is equal to:    (2019)
(1) 9
(2) 15
(3) 5 
(4) 13
Ans.
(4)
Let,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Given, distance between P1 and P2 is 1/3
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
And distance between P1 and P3 is 2/3
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.56. If the line Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev intersects the plane 2x + 3y - z + 13 = 0 at a point P and the plane 3x + y + 4z = 16 at a point Q, then PQ is equal to:    (2019)
(1) 14
(2) √14
(3) 2√7
(4) 2√14
Ans.
(4)
Let points P (3λ + 2, 2λ - 1, - λ + 1) and Q(3μ + 2, 2μ - 1, - μ + 1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, P (- 1, -3, 2)
Similarly, Q lies on 3x + y + 4z = 16
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, Q is (5, 1, 0)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.57. A plane which bisects the angle between the two given planes 2x - y + 2z -4 = 0 and x + 2y + 2z - 2 = 0, passes through the point:    (2019)
(1) (-1, -4, 1)
(2) (1, 4, -1)
(3) (2, 4, 1)
(4) (2, -4, 1)
Ans.
(4)
The equations of angle bisectors are,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
⇒ x - 3y - 2 = 0
or 3x + y + 4z - 6 = 0
(2, -4, 1) lies on the second plane.

Q.58. The length of the perpendicular drawn from the point (2, 1,4) to the plane containing the lines Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is:    (2019)
(1) 3
(2) 1/3
(3) √3
(4) 1/√3
Ans.
(3)
The equation of plane containing two given lines is,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
On expanding, we get x - y - z = 0
Now, the length of perpendicular from (2, 1, 4) to this plane
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev 

Q.59. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevbe a vector coplanar with the vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev perpendicular to Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevis equal to:    (2018)
(1) 336
(2) 315
(3) 256
(4) 84
Ans.
(1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.60. If L1 is the line of intersect ion of the planes  2x - 2y + 3z = 0, x - y + z = 0 and L2 is the line of intersection of the planes x + 2y - z - 3 = 0, 3x - y + 2z = 0 then the distance of the origin fro m the plane, containing the lane L1 and L2 , is:    (2018)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(2)
Vectors along the given lines L1, L2 are
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Putting y = 0 in 1st two equation
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Point on the plane is (–5, 0, 4) and normal vector of required plane is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Hence, equation of plane is - 7x + 7y - 8z - 3 = 0
Perpendicular distance is Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.61. The length of the project ion of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, x + y + z = 7 is:    (2018)
(1) 2/√3
(2) 2/3
(3) 1/3
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(4)
D.R’s of AB = (1, 0, 1)
D.R’s of normal to plane = (1, 1, 1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
AB = √2
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Length of projection = Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.62. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevbe a vector such that Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and the angle between Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevbe 30°. Then Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevis equal to    (2017)
(1) 1/8
(2) 25/8
(3) 2
(4) 5
Ans.
 (3)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev 

Q.63. If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line, x/1 = y/4 = z/5 is Q, then PQ is equal to    (2017)
(1) 6√5
(2) 3√5
(3) 2√42
(4) √42
Ans.
(3)
Equation of PQ,
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Let M be (λ+1, 4λ- 2, 5λ+ 3)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
As it lies on 2x + 3y – 4z + 22 = 0
λ = 1
For Q, λ = 2
Distance PQ
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev 

Q.64. The distance of the point (1, 3, –7) from the plane passing through the point (1, –1, –1), having normal perpendicular to both the lines Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is     (2017)
(1) 10/√74
(2) 20/√74
(3) 10/√83
(4) 5/√83
Ans. 
(3)
Let the plane be
a(x - 1)+ b( y + 1) + c(z +1) = 0
It is perpendicular to the given lines
a – 2b + 3c = 0
Solving, a : b : c = 5 : 7 : 3
∴ The plane is 5x + 7y + 3z + 5 = 0
Distance of (1, 3, –7) from this plane = 10/√83 

Q.65. The coodinates of the foot of the perpendicular from the point (1, -2,1) on the plane containing the lines, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is:    (2017)
(1) (2, -4, 2)
(2) (1, 1, 1)
(3) (0, 0, 0)
(4) (-1, 2, -`= 1)
Ans.
(3)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.66. The line of intersection of the planes Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is:    (2017)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(3)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
13 x = 6
x = 6/13y
y = 5/13
.... is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.67. The area (in sq. units) of the parallelogram whosed diagonals are along the vectors Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is:    (2017)
(1) 20
(2) 65
(3) 52
(4) 26
Ans.
(2)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.68. If the line, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev lies in the plane, 2x - 4y + 3z = 2, then the shortest distance between this line and the line, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is:    (2017)
(1) 1
(2) 2
(3) 3
(4) 0
Ans.
(4)
pt (3, -2, λ) on pline 2x - 4y +3z - 2 = 0
= 6 + 8 - 3λ - 2 = 0
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ...(2)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
p (1, 0, 0)
gives are ditersech - thortest distance = 0

Q.69. If x = a,  y = b, z = c is a solution of the system of linear equations
x + 8y + 7z = 0
9x + 2y +3z = 0
y + y + z = 0
such that the point (a, b, c) lies on the plane x + 2y + z = 6, then 2a + b + c equals:    (2017)
(1) 2
(2) -1
(3) 1
(4) 0
Ans.
(1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.70. If the vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevis written as the sum of a vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev parallel to Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and a vector Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev perpendicular to Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev is equal to:    (2017)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.71. If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the locus of the centroid of ΔABC is.    (2017)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans.
(1)
Let Centroid be (h, k, l)
∴ x - intp = 3h Y - intp = 3k, 3 - int = 3l
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.72. The distance of the point (1, -5, 9) from the plane x - y + z = 5 measured along the line x = y = z is:    (2016)
(1) 3√10
(2) 10√3
(3) 10/√3
(4) 20/3
Ans.
(2)
Let Q(1, -5, 9)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Line is Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev(say)
Any pt on line we can take P(r + 1, r - 5, r + 9)
So, Pt satisfy Plane
=>(r + 1) - (r - 5) + (r + 9) = 5 r = -10 So, Point P = (-9, -15, -1)
Distance is PQ = Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.73. If the line, Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev lies in the plane, lx + my - z = 9, then l2 + m2 is equal to:    (2016)
(1) 26
(2) 18
(3) 5
(4) 24
Ans. (4) 
Point on line is P = (3, -2, -4)
'P’ lies on lx + my - z = 9
=> 3l - 2m + 4  = 9
3l - 2m = 5 ....(1)
As line lies on plane
=> 2 x l + m x (-1) + 3 x (-1) = 0
2l - m = 3 .....(3) 
Solving l = 1, m = -1
So, l+ m= 2 

Q.74. Let Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevandPrevious year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevbe three unit vectors such that Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevIf Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevis not parallel to c , then the angle betweenPrevious year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevandPrevious year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevis:    (2016)
(1) 3π/4
(2) π/2
(3) 2π/3
(4) 5π/6
Ans.
(4)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Equate,Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev as Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRevunit vectors
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.75. The shortest distance between the lines Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev lies in the interval:    (2016)
(1) (2, 3]
(2) [0, 1)
(3) (3, 4]
(4) [1, 2)
Ans.
(1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.76. The distance of the point (1, -2, 4) from the plane passing through the point (1, 2, 2) and perpendicular to the planes x - y + 2z = 3 and 2x - 2y + z + 12 = 0, is   (2016)
(1) 1/√2
(2) 2
(3) √2
(4) 2√2
Ans.
(4)
Equation of plane ⊥ to the planes.
x - y + 2z = 3 & 2x - 2y + z + 12 = 0
and passes through (1, 2, 2) is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
3(x - 1) + 3(y - 2) = 0
x + y = 3 ..... (1)
distance of plane x + y - 3 = 0 from (1, - 2, 4) is
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.77. In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev then the point (p, q) lies on a line    (2016)
(1) parallel to y-axis
(2) making an acute angle with the positive direction of x-axis
(3) parallel to x-axis
(4) making an obtuse angle with the position direction of x-axis.
Ans.
(2)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev ⇒ - 8 + 2(9 - 1) - 3 (p + 1) = 0 ⇒ - 3p + 2q - 13 = 0
⇒ (p, q) lies on line
3x - 2y + 13 = 0
slope = 3/2

Q.78. ABC is a triangle in a plane with vertices A(2, 3, 5), B(-1, 3, 2) and C(λ, 5, µ). If the median through A is equally inclined to the coordinate axes, then the value of (λ3 + µ3  + 5) is    (2016)
(1) 676
(2) 1130
(3) 1348
(4) 1077
Ans.
(3)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
direction cosine of AD = Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.79. Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev and Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev respectively, then the position vector of the orthocentre of this triangle, is    (2016)
(1) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(2) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(3) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
(4) Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Ans. 
(4)
Position vector of the centroid of Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Now we known that centroid divides the line joining orthocentre to circum centre divided by centriod divided by centroid in the ratio in 2 : 1
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev

Q.80. The number of distinct real values of λ for which the lines Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev are coplanar is    (2016)
(1) 3
(2) 2
(3) 1
(4) 4
Ans.
(1)
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev
Previous year Questions (2016-20) - Vector Algebra and Three Dimensional Geometry Notes | EduRev 

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