JEE Main Previous year questions (2021-23): Vector Algebra and Three Dimensional Geometry - 1 PDF Download

Q.1. Let  and  is equal to       (JEE Main 2023)
(a) 2
(b) 3/2
(c) 1
(d)

Ans. c

Q.2. Let  and  be parallel to  be perpendicular to   then the value of          (JEE Main 2023)
(a) 7
(b) 9
(c) 6
(d) 11

Ans. a

Q.3. Let  is equal to       (JEE Main 2023)

Ans. 8

Q.4. The distance of the point (−1,9, −16) from the plane 2x + 3y − z = 5 measured parallel to the line
    (JEE Main 2023)
(a) 31
(b) 13√2
(c) 20√2
(d) 26

Ans. d

Q.5. The distance of the point (7, −3, −4) from the plane passing through the points (2, −3,1), (−1,1, −2) and (3, −4,2) is :    (JEE Main 2023)
(a) 5
(b) 4
(c) 5√2
(d) 4√2

Ans. c

Q.6. The shortest distance between the lines  is equal to    (JEE Main 2023)

Ans. 14

Q.7. Let the plane containing the line of intersection of the planes P1: x + (λ + 4)y + z = 1 and P2: 2x + y + z = 2 pass through the points (0,1,0) and (1,0,1). Then the distance of the point (2λ, λ, −λ) from the plane P2 is    (JEE Main 2023)
(a) 4√6
(b) 3√6
(c) 5√6
(d) 2√6

Ans. b
[x + (λ+4)y + z-1] + μ[2x+y + z- 2] =0
(0,1,0)
(i) (λ + 4 – 1) + μ[-1] = 0
λ - m = -3
(1,0,1) (ii) 1 + μ[1] = 0 ⇒ m = -1, λ = -4
∴ point (-8,-4,4); 2x + y + z - 2 = 0

Q.8. If the foot of the perpendicular drawn from (1,9,7) to the line passing through the point (3,2,1) and parallel to the planes x + 2y + z = 0 and 3y − z = 3 is (α, β, γ), then α + β + γ is equal to    (JEE Main 2023)
(a) 3
(b) 1
(c) −1
(d) 5

Ans. d

Q.9. If the shortest distance between the lines   then the square of sum of all possible values of λ is      (JEE Main 2023)

Ans. 384

Q.10. Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that
          (JEE Main 2023)
(a) 4
(b) 3
(c) 2
(d) 5

Ans. b

Q.11. Letbe the unit vectors along the three positive coordinate axes. Let 

be three vectors such that b2b3>0,and

Then, which of the following is/are TRUE?       (JEE Advanced 2022)
(a)
(b)
(c)
(d)

Ans. b, c, d

Q.12. Let S be the reflection of a point Q with respect to the plane given by

where t, p are real parameters andare the unit vectors along the three positive coordinate axes. If the position vectors of Q and S arerespectively, then which of the following is/are TRUE?       (JEE Advanced 2022) 
(a) 3(α + β) = −101
(b) 3(β + γ) = −71
(c) 3(γ + α) = −86
(d) 3(α + β + γ) = −121

Ans. a, b, c

Q.13. Let P₁ and P₂ be two planes given by
P₁: 10x + 15y + 12z − 60 = 0
P₂: −2x + 5y + 4z − 20 = 0
Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P₁ and P₂?        (JEE Advanced 2022) 
(a)
(b)
(c)
(d)

Ans. a, b, d

Q.14. Letbe two vectors such thatThenis equal to __________.        (JEE Main 2022)

Ans. 14

Q.15. Let a line with direction ratios a, −4a, −7 be perpendicular to the lines with direction ratios 3, −1, 2b and b, a, −2. If the point of intersection of the lineand the plane x − y + z = 0 is (α, β, γ), then α + β + γ is equal to _________.        (JEE Main 2022)

Ans. 10
Given a.3 + (−4a)(−1) + (−7)2b = 0 ...... (1)

and ab − 4a² + 14 = 0 ....... (2)

⇒ a² = 4 and b² = 1

⇒ General point on line is (5λ − 1, 3λ + 2, λ) for finding point of intersection with x − y + z = 0 we get (5λ − 1) − (3λ + 2) + (λ) = 0

⇒ 3λ − 3 = 0 ⇒ λ = 1

∴ Point at intersection (4, 5, 1)

∴ α + β + γ = 4 + 5 + 1 = 10

Q.16. Let P(−2,−1,1) and Q(56/17, 43/17, 111/17) be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are α, −1, β, where both α and β are integers of minimum absolute values, then α² + β² is equal to ____________.        (JEE Main 2022) 

Ans. 450

d.r's of RS = <α, −1, β>

as PQ and RS are diagonals of rhombus

α(45) + 30(−1) + 47(β) = 0

⇒ 45α + 47β = 30

for minimum integral value α = −15 and β = 15

⇒ α² + β² = 450.

Q.17. Letthree non-coplanar vectors such thatIfthen α is equal to __________.        (JEE Main 2022) 

Ans. 36
Given, 


Hence, 

Multiplying (iv), (v) and (vi) 

Q.18. Let the lineintersect the plane containing the lines and 4ax − y + 5z − 7a = 0 = 2x − 5y − z − 3, a ∈ R at the point P(α, β, γ). Then the value of α + β + γ equals _____________.        (JEE Main 2022)

Ans. 12

Equation of plane containing the line 4ax − y + 5z − 7a = 0 = 2x − 5y − z − 3 can be written as

4ax − y + 5z − 7a + λ(2x − 5y − z − 3) = 0

(4a + 2λ)x − (1 + 5λ)y + (5 − λ)z − (7z + 3λ) = 0

Which is coplanar with the line

4(4a + 2λ) + (1 + 5λ) − (7a + 3λ) = 0

9a + 10λ + 1 = 0 ..... (1)

(4a + 2λ)1 + (1 + 5λ)2 + 5 − λ = 0

4a + 11λ + 7 = 0 ...... (2)

a = 1, λ = −1

Equation of plane is x + 2y + 3z − 2 = 0

Intersection with the line

(7t + 3) + 2(−t + 2) + 3(−4t + 3) − 2 = 0

−7t + 14 = 0

t = 2

So, the required point is (17, 0, −5)
α + β + γ = 12

Q.19. The plane passing through the line L: lx − y + 3(1 − l)z = 1, x + 2y − z = 2 and perpendicular to the plane 3x + 2y + z = 6 is 3x − 8y + 7z = 4. If θ is the acute angle between the line L and the y-axis, then 415cos²⁡θ is equal to _____________.        (JEE Main 2022)

Ans. 125
L: lx − y + 3(1 − l)z = 1, x + 2y − z = 2 and plane containing the line p: 3x − 8y + 7z = 4

Letbe the vector parallel to L.

∵ R containing L

3(6l − 5) − 8(3 − 2l) + 7(2l + 1) = 0

18l + 16l + 14l − 15 − 24 + 7 = 0

∴ l = 32/48 = 2/3

Let θ be the acute angle between L and y-axis

∴ 415cos²θ = 125 

Q.20. The largest value of a, for which the perpendicular distance of the plane containing the linesfrom the point (2, 1, 4) is √3, is _________.        (JEE Main 2022)

Ans. 20

∴ Plane (1 − a)(x − 1) + (y − 1) + z = 0

Distance from (2, 1, 4) is 3 i.e.

⇒ 25 + a² − 10a = 3a² − 6a + 9

⇒ 2a² + 4a − 16 = 0

⇒ a² + 2a − 8 = 0

a = 2 or −4

∴ a_max = 2  

Q.21. Let Q and R be two points on the lineat a distance √26 from the point P(4, 2, 7). Then the square of the area of the triangle PQR is ___________.        (JEE Main 2022)

Ans. 153

Let T(2t − 1, 3t − 2, 2t + 1)

∵ PT ⊥^r QR

∴ 2(2t − 5) + 3(3t − 4) + 2(2t − 6) = 0

17t = 34

∴ t = 2

So T(3, 4, 5)

∴ Square of ar(ΔPQR) = 153.

Q.22. The line of shortest distance between the lines makes an angle ofwith the plane P: ax − y − z = 0, (a > 0). If the image of the point (1, 1, −5) in the plane P is (α, β, γ), then α + β − γ is equal to _________________.        (JEE Main 2022)

Ans. 3
Wrong Question. 

Q.23. Consider a triangle ABC whose vertices are A(0, α, α), B(α, 0, α) and C(α, α, 0), α > 0. Let D be a point moving on the line x + z − 3 = 0 = y and G be the centroid of ΔABC. If the minimum length of GD isthen α is equal to ____________.        (JEE Main 2022)

Ans. 6

Given, G is the centroid of ΔABC 

Also given, D is a point moving on the line x + z − 3 = 0 = y

Let D = (h, 0, k)

x + z − 3 = 0 ⇒ h + k − 3 = 0 ⇒ h = 3 − k

∴ D = (3 − k, 0, k)

Now, length of GD = d

Differentiating both side with respect to k, we get 

For maximum or minimum value of k, d' = 0

⇒ −3 + k = −k

⇒ k = 3/2

⇒ (4α − 9)² + 16α²  + (4α − 9)²  = 57 × 18

⇒ 16α²  − 72α + 81 + 16α²  + 16α²  − 72α + 81 = 57 × 18

⇒ 48α²  − 144α + 162 = 1026

⇒ 24α²  − 72α + 81 − 513 = 0

⇒ 24α² − 72α − 432 = 0

⇒ α² − 3α − 18 = 0

⇒ α²  − 6α + 3α − 18 = 0

⇒ α(α − 6) + 3(α − 6) = 0

⇒ (α − 6)(α + 3) = 0

⇒ α = 6, −3

Given, α > 0

∴ Possible value of α = 6.

Q.24. Letbe a vector such thatThen the value ofis equal to _________.        (JEE Main 2022)

Ans. 10

Q.25. Letbe a plane. Let P₂ be another plane which passes through the points (2, −3, 2), (2, −2, −3) and (1, −4, 2). If the direction ratios of the line of intersection of P₁ and P₂ be 16, α, β, then the value of α + β is equal to ________________.       (JEE Main 2022)

Ans. 28

Direction ratio of normal to P1 ≡ <2, 1, −3>

i.e. <−5, 5, 1>

d.r's of line of intersection are along vector

i.e. <16, 13, 15>

∴ α + β = 13 + 15 = 28

Q.26. Let d be the distance between the foot of perpendiculars of the points P(1, 2, −1) and Q(2, −1, 3) on the plane −x + y + z = 1. Then d² is equal to ___________.       (JEE Main 2022)

Ans. 26
Foot of perpendicular from P 

and foot of perpendicular from Q 

⇒ d² = 26

Q.27. Let the image of the point P(1, 2, 3) in the linebe Q. Let R (α, β, γ) be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the value of 22 (α + β + γ) is equal to __________.       (JEE Main 2022)

Ans. 125

The point dividing PQ in the ratio 1 : 3 will be mid-point of P & foot of perpendicular from P on the line.

∴ Let a point on line be λ

⇒ P′(3λ + 6, 2λ + 1, 3λ + 2)

as P' is foot of perpendicular

(3λ + 5)3 + (2λ − 1)2 + (3λ − 1)3 = 0

⇒ 22λ + 15 − 2 − 3 = 0

⇒ λ = −511

⇒ 22(α, β, γ) = 62 + 23 + 40 = 125

Q.28. Ifare coplanar vectors andthen 122(c₁ + c₂ + c₃) is equal to ___________.       (JEE Main 2022)

Ans. 150

2C₁ + C₂ + 3C₃ = 5 ...... (i)

3C₁ + 3C₂ + C₃ = 0 ...... (ii)

= 2(3C₃ − C₂) − 1(3C₃ − C₁) + 3(3C₂ − 3C₁)

= 3C₃ + 7C₂ − 8C₁

⇒ 8C₁ − 7C₂ − 3C₃ = 0 ...... (iii)

So 122(C₁ + C₂ + C₃) = 150

Q.29. Let the mirror image of the point (a, b, c) with respect to the plane 3x − 4y + 12z + 19 = 0 be (a − 6, β, γ). If a + b + c = 5, then 7β − 9γ is equal to ______________.       (JEE Main 2022)

Ans. 137

(x, y, z) ≡ (a − 6, β, γ)

⇒ 3a − 4b + 12c = 150 ..... (1)

a + b + c = 5

3a + 3b + 3c = 15 ...... (2)

Applying (1) - (2)

−7b + 9c = 135

7b − 9c = −135

7β − 9γ = 7(8 + b) − 9(−24 + c)

= 56 + 216 + 7b − 9c
= 56 + 216 − 135 = 137

Q.30. Let l1 be the line in xy-plane with x and y intercepts 1/8 and 1/4√2 respectively, and l₂ be the line in zx-plane with x and z interceptsrespectively. If d is the shortest distance between the line l₁ and l₂, then d⁻² is equal to _______________.       (JEE Main 2022)

Ans. 51

Equation of L₂ 


d⁻² = 51

Q.31. Letis a vector such thatthenis equal to _____________.       (JEE Main 2022)

Ans. 14

⇒ λy − z = 13, z − λx = −1, x − y = −4

and x + y + λz = −21

⇒ Clearly, λ = 3, x = −2, y = 2 and z = −7

 

Q.32. Let the lines

intersect at the point S. If a plane ax + by − z + d = 0 passes through S and is parallel to both the lines L₁ and L₂, then the value of a + b + d is equal to ____________.       (JEE Main 2022)

Ans. 5

As plane is parallel to both the lines we have d.r's of normal to the plane as <7, −2, −1>

Also point of intersection of lines is

∴ Equation of plane is

7(x − 2) − 2(y − 4) − 1(z − 6) = 0

⇒ 7x − 2y − z = 0
a + b + d = 7 − 2 + 0 = 5

Q.33. Let θ be the angle between the vectorsand θ ∈ (π/4, π/3). Thenis equal to __________.        (JEE Main 2022)

Ans. 576

Q.34. If the shortest distance between the linesandthen the integral value of a is equal to ___________.        (JEE Main 2022)

Ans. 2


⇒ 6(a² − 2a + 1) = 2a² − 2a + 2
⇒ (a − 2)(2a − 1) = 0 ⇒ a = 2 because a ∈ z.

Q.35. Let a line having direction ratios, 1, −4, 2 intersect the linesat the points A and B. Then (AB)² is equal to ___________.        (JEE Main 2022)

Ans. 84
Let A(3λ + 7, −λ + 1, λ − 2) and B(2μ, 3μ + 7, μ)
So, DR's of AB ∝ 3λ − 2μ + 7, −(λ + 3μ + 6), λ − μ − 2

⇒ 5λ − 3μ = −16
And λ − 5μ = 10
From (i) and (ii) we get λ = −5, μ = −3
So, A is (−8, 6, −7) and B is (−6, −2, −3)

Q.36. Letbe three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 andthenis equal to:        (JEE Main 2022)
(a) 10
(b) 14
(c) 16
(d) 18

Ans. c

Q.37. If (2, 3, 9), (5, 2, 1), (1, λ, 8) and (λ, 2, 3) are coplanar, then the product of all possible values of λ is:        (JEE Main 2022)
(a) 21/2
(b) 59/8
(c) 57/8
(d) 95/8

Ans. d
∵ A(2, 3, 9), B(5, 2, 1), C(1, λ, 8) and D(λ, 2, 3) are coplanar.

⇒ [−6(λ − 3) − 1] − 8(1 − (λ − 3)(λ − 2)) + (6 + (λ − 2) = 0
⇒ 3(−6λ + 17) − 8(−λ² + 5λ − 5) + (λ + 4) = 8
⇒ 8λ² − 57λ + 95 = 0
∴ λ₁λ₂ = 95/8

Q.38. Let Q be the foot of perpendicular drawn from the point P(1, 2, 3) to the plane x + 2y + z = 14. If R is a point on the plane such that ∠PRQ = 60^∘, then the area of △PQR is equal to:        (JEE Main 2022)
(a) √3/2
(b) √3
(c) 2√3
(d) 3

Ans. b

Q.39. Letbe two unit vectors such that the angle between them is π/4. If θ is the angle between the vectorsthen the value of 164 cos²⁡θ is equal to:        (JEE Main 2022)
(a) 90 + 27√2
(b) 45 + 18√2
(c) 90 + 3√2
(d) 54 + 90√2

Ans. a

Q.40. Letbe a vector satisfying are non-parallel, then the value of λ is:        (JEE Main 2022)
(a) −5
(b) 5
(c) 1
(d) −1

Ans. a

Q.41. If the foot of the perpendicular from the point A(−1, 4, 3) on the plane P: 2x + my + nz = 4, is (−2, 7/2, 3/2), then the distance of the point A from the plane P, measured parallel to a line with direction ratios 3, −1, −4, is equal to:        (JEE Main 2022)
(a) 1
(b) √26
(c) 2√2
(d) √14

Ans. b
(−2, 7/2, 3/2) satisfies the plane P: 2x + my + nz = 4

Line joining A(−1, 4, 3) and (−2, 7/2, 3/2) is perpendicular to P: 2x + my + nz = 4

Plane P: 2x + y + 3z = 4

Distance of P from A(−1, 4, 3) parallel to the line

for point of intersection of P & L

2(3r − 1) + (−r + 4) + 3(−4r + 3) = 4 ⇒ r = 1

Point of intersection: (2, 3, −1)

Required distance

Q.42. Let S be the set of all a ∈ R for which the angle between the vectors andis acute. Then S is equal to:        (JEE Main 2022)
(a)
(b) Φ
(c)
(d) (12/7, ∞)

Ans. b

⇒ a(log_eb)² − 12 + 6a(log_eb) > 0
∵ b > 1
Let log_eb = t ⇒ t > 0 as b > 1
at² + 6at − 12 > 0∀t > 0
⇒ a ∈ ϕ

Q.43. A plane P is parallel to two lines whose direction ratios are −2, 1, −3 and −1, 2, −2 and it contains the point (2, 2, −2). Let P intersect the co-ordinate axes at the points A, B, C making the intercepts α, β, γ. If V is the volume of the tetrahedron OABC, where O is the origin, and p = α + β + γ, then the ordered pair (V, p) is equal to:        (JEE Main 2022)
(a) (48, −13)
(b) (24, −13)
(c) (48, 11)
(d) (24, −5)

Ans. b

Vector normal to plane

= (4, −1, −3)

Plane through (2, 2, −2) and normal to

(x − 2, y − 2, z + 2) . (4, −1, −3) = 0

⇒ 4x − y − 3z = 12

Intercepts α, β, γ are 3, −12, −4

P = α + β + γ = −13

Q.44. Let the linesbe coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?        (JEE Main 2022)
(a) (0,−2,−2)
(b) (−5,0,−1)
(c) (3,−1,0)
(d) (0,4,5)

Ans. d



Equation of plane

≡ ((x − 1), (y − 2), (z − 3)) . (−3, −13, 11) = 0

⇒ 3x + 13y − 11z + 4 = 0

Checking the option gives (0, 4, 5) does not lie on the plane.

Q.45. Let a vectorhas magnitude 9. Let a vectorbe such that for every (x, y) ∈ R × R − {(0, 0)}, the vectoris perpendicular to the vectorThen the value of is equal to:        (JEE Main 2022)
(a) 9√3
(b) 27√3
(c) 9
(d) 81

Ans. b

As given equation is identity

Coefficient of x² = coefficient of y² = coefficient of xy = 0

Q.46. Let the vectorst ∈ R be such that for α, β, γ ∈ R,α = β = γ = 0. Then, the set of all values of t is:        (JEE Main 2022)
(a) a non-empty finite set
(b) equal to N
(c) equal to R−{0}
(d) equal to R

Ans. c

⇒ (1 + t)(1 + t + 2t) − (1 − t)(1 − t − 2t) + 1(t² − t − t − t²) ≠ 0

⇒ (3t² + 4t + 1) − (1 − t)(1 − 3t) − 2t ≠ 0
⇒ (3t² + 4t + 1) − (3t² − 4t + 1) − 2t ≠ 0
⇒ t ≠ 0

Q.47. If the line of intersection of the planes ax + by = 3 and ax + by + cz = 0, a > 0 makes an angle 30^∘ with the plane y − z + 2 = 0, then the direction cosines of the line are:        (JEE Main 2022)

(a)
(b)
(c)
(d)

Ans. b
P₁: ax + by + 0z = 3, normal vector:= (a, b, 0)

P₂: ax + by + cz = 0, normal vector:= (a, b, c)

Vector parallel to the line of intersection =

= (bc, −ac, 0)

Vector normal to 0 . x + y − z + 2 = 0 is= (0, 1, −1)

Angle between line and plane is 30^∘

⇒ a² = b²

Hence,= (ac, −ac, 0)

Direction ratios =

Q.48. If the length of the perpendicular drawn from the point P(a, 4, 2), a > 0 on the line  is 2√6 units and Q(α₁, α₂, α₃) is the image of the point P in this line, then is equal to:        (JEE Main 2022)
(a) 7
(b) 8
(c) 12
(d) 14

Ans. b
∵ PR is perpendicular to given line, so 

2(2λ − 1 − a) + 3(3λ − 1) − 1(−λ − 1) = 0

⇒ a = 7λ − 2

Now,

∵ PR = 2√6

⇒ (−5λ + 1)² + (3λ − 1)² + (λ + 1)² = 24

⇒ 5λ² − 2λ − 3 = 0 ⇒ λ = 1 or

∵ a > 0 so λ = 1 and a = 5

Now(Sum of co-ordinate of R) − (Sum of coordinates of P)

= 2(7) − 11 = 3

Q.49. If the plane P passes through the intersection of two mutually perpendicular planes 2x + ky − 5z = 1 and 3kx − ky + z = 5, k < 3 and intercepts a unit length on positive x-axis, then the intercept made by the plane P on the y-axis is        (JEE Main 2022)
(a) 1/11
(b) 5/11
(c) 6
(d) 7

Ans. d
P₁: 2x + ky − 5z = 1

P₂: 3kx − ky + z = 5

∵ P₁ ⊥ P₂ ⇒ 6k − k² + 5 = 0

⇒ k = 1, 5

∵ k < 3

∴ k = 1

P₁: 2x + y − 5z = 1

P₂: 3x − y + z = 5

P: (2x + y − 5z − 1) + λ(3x − y + z − 5) = 0

Positive x-axis intercept = 1

⇒ λ = 12

∴ P: 7x + y − 4z = 7

y intercept = 7.

Q.50. Letis equal to        (JEE Main 2022)
(a) 4
(b) 5
(c) 21
(d) 17

Ans. b

Q.51. Letbe two vectors, such that Then the projection ofis equal to        (JEE Main 2022)
(a) 2
(b) 39/5
(c) 9
(d) 46/5

Ans. d

4 + 5β = −1 ⇒ β = −1

−5α − 3 = 12 ⇒ α = −3

Q.52. A vectoris parallel to the line of intersection of the plane determined by the vectors and the plane determined by the vectorsThe obtuse angle between and the vectoris        (JEE Main 2022)
(a) 3π/4
(b) 2π/3
(c) 4π/5
(d) 5π/6

Ans. a
Ifis a vector normal to the plane determined bythen

Ifis a vector normal to the plane determined bythen

Cosine of acute angle between

Obtuse angle between

Q.53. LetIf the projection ofon the vectoris 30, then α is equal to:        (JEE Main 2022)
(a) 15/2
(b) 8
(c) 13/2
(a) 7

Ans. d


On solving α = −13/2 (Rejected as α > 0)
and α = 7

Q.54. The length of the perpendicular from the point (1, −2, 5) on the line passing through (1, 2, 4) and parallel to the line x + y − z = 0 = x − 2y + 3z − 5 is:        (JEE Main 2022)
(a)
(b)
(c)
(d) 1

Ans. a

The line x + y − z = 0 = x − 2y + 3z − 5 is parallel to the vector

Equation of the line through P(1, 2, 4) and parallel to

Let N ≡ (λ + 1, −4λ + 2, −3λ + 4)

= (λ, −4λ + 4, −3λ − 1)

 is perpendicular to

⇒ (λ, −4λ + 4, −3λ − 1) . (1, 4, −3) = 0

⇒ λ = 1/2

Q.55. Letbe a vector such that Then the projection ofon the vector  is:        (JEE Main 2022)
(a)
(b)
(c)
(d) 2/3

Ans. a

Q.56. The shortest distance between the linesis        (JEE Main 2022)
(a)
(b) 1
(c)
(d)

Ans. a


Shortest distance between L₁ and L₂ 

Q.57. A plane E is perpendicular to the two planes 2x − 2y + z = 0 and x − y + 2z = 4, and passes through the point P(1, −1, 1). If the distance of the plane E from the point Q(a, a, 2) is 3√2, then (PQ)² is equal to        (JEE Main 2022)
(a) 9
(b) 12
(c) 21
(d) 33

Ans. c
First plane, P₁ = 2x − 2y + z = 0, normal vector ≡=(2,−2,1)

Second plane, P₂ ≡ x − y + 2z = 4, normal vector ≡=(1,−1,2)

Plane perpendicular to P₁ and P₂ will have normal vector

Equation of plane E through P(1, −1, 1) andas normal vector

(x − 1, y + 1, z − 1) . (−3, −3, 0) = 0

⇒ x + y = 0 ≡ E

Distance of PQ(a, a, 2) from E =

Hence, Q ≡ (±3, ±3, 2)

Q.58. Let ABC be a triangle such thatConsider the statements:
 
Then        (JEE Main 2022)
(a) both (S1) and (S2) are true
(b) only (S1) is true
(c) only (S2) is true
(d) both (S1) and (S2) are false

Ans. c



Hence (S1) is not correct 

Q.59. Let P be the plane containing the straight lineand perpendicular to the plane containing the straight linesIf d is the distance of P from the point (2, −5, 11), then d² is equal to:        (JEE Main 2022)
(a) 147/2
(b) 96
(c) 32/3
(d) 54

Ans. c
Let ⟨a, b, c⟩ be direction ratios of plane containing

∴ 2a + 3b + 5c = 0… (i)  
and 3a + 7b + 8c = 0… (ii)
from eq. (i) and (ii):
∴ D.R^S. of plane are <11, 1, −5>
Let D.R^S of plane P be <a₁, b₁, c₁> then.
11a₁ + b₁ − 5c₁ = 0
and 9a₁ − b₁ − 5c₁ = 0
From eq. (iii) and (iv):

∴ D.A ⁵. of plane P are <1, −1, 2>
Equation plane P is: 1(x − 3) − 1(y + 4) + 2(z − 7) = 0
⇒ x − y + 2z − 21 = 0
Distance from point (2, −5, 11) is d =

Q.60. Let a vectorbe coplanar with the vectorsIf the vectoralso satisfies the conditionsthen the value of is equal to:        (JEE Main 2022)
(a) 24
(b) 29
(c) 35
(d) 42

Ans. c
Given,


⇒ x(−2) − y(−1) + z(−3) = 0

⇒ −2x + y − 3z = 0 ..... (1)

Given, 

⇒ 3y = 3

⇒ y = 1

Putting value of y = 1 in equation (1) and (2) we get,

−2x + 1 − 3z = 0 ..... (3)

and −6x + 3 + 5z = −42

⇒ −6x + 5z = −45 ..... (4)

Solving (3) and (4), we get

x = 5 and z = -3

Q.61. The distance of the point (3, 2, −1) from the plane 3x − y + 4z + 1 = 0 along the line is equal to:        (JEE Main 2022)
(a) 9
(b) 6
(c) 3
(d) 2

Ans. c

Line PQ is parallel to line

∴ DR of PQ = DR of line = <−2, 2, 1>

∴ Equation of line PQ passing through P(3, 2, −1) and DR = <−2, 2, 1> is

Any General point on line PQ = (x₁, y₁, z₁)

⇒ x₁ = −2λ + 3

y₁ = 2λ + 2

z₁ = λ − 1

∴ Point Q = (−2λ + 3, 2λ + 2, λ − 1)

Point Q lies on the plane 3x − y + 4z + 1 = 0. So point Q satisfy the equation.

3(−2λ + 3) − (2λ + 2) + 4(λ − 1) + 1 = 0

⇒ −6λ + 9 − 2λ − 2 + 4λ − 4 + 1 = 0

⇒ −4λ + 4 = 0

⇒ λ = 1

∴ Point Q = (−2 × 1 + 3, 2 × 1 + 2, 1 − 1)

= (1, 4, 0)

∴ Distance of the point P(3, 2, −1) from the plane = Length of PQ

= √9
= 3

Q.62. Let A, B, C be three points whose position vectors respectively are

If α is the smallest positive integer for whichare noncollinear, then the length of the median, in ΔABC, through A is:        (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. a

Q.63. Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line Then, which of the following points lies on T?        (JEE Main 2022)
(a) (2, 1, 0)
(b) (1, 2, 1)
(c) (1, 2, 2)
(d) (1, 3, 2) 

Ans. b

Q.64. Letlie on the plane px − qy + z = 5, for some p, q ∈ R. The shortest distance of the plane from the origin is:        (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. b

Q.65. Let where α, β ∈ R, be three vectors. If the projection ofthen the value of α + β is equal to:        (JEE Main 2022)
(a) 3
(b) 4
(c) 5
(d) 6

Ans. a



2β − 8 = −6 & 6 + β = 7
∴ β = 1
α + β = 2 + 1 = 3

Q.66. If the mirror image of the point (2, 4, 7) in the plane 3x − y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to:        (JEE Main 2022)
(a) 54
(b) 50
(c) −6
(d) −42

Ans. c
We know mirror image of point (x₁, y₁, z₁) in the plane ax + by + cz = d 

Here given point (2, 4, 7) and plane 3x − y + 4z = 2 then mirror image is 


∴  2a + b + 2c

Q.67. Letbe a vector which is perpendicular to the vectorthen the projection of the vectoron the vectoris:        (JEE Main 2022)
(a) 1/3
(b) 1
(c) 5/3
(d) 7/3

Ans. c


∴ a₂ = 2 ..... (ii)

and a₁ − 2a₃ = 13 ..... (iii)

From eq. (i) and (iii): a₁ = 3 and a₃ = −5

 

Q.68. Let the plane ax + by + cz = d pass through (2, 3, −5) and is perpendicular to the planes 2x + y − 5z = 10 and 3x + 5y − 7z = 12. If a, b, c, d are integers d > 0 and gcd (|a|, |b|, |c|, d) = 1, then the value of a + 7b + c + 20d is equal to:        (JEE Main 2022)
(a) 18
(b) 20
(c) 24
(d) 22

Ans. d
Equation of pane through point (2, 3, −5) and perpendicular to planes 2x + y − 5z = 10 and 3x + 5y − 7z = 12 is

∴ Equation of plane is (x − 2)(−7 + 25) − (y − 3)

(−14 + 15) + (z + 5) . 7 = 0

∴ 18x − y + 7z + 2 = 0

⇒ 18x − y + 7z = −2

∴ −18x + y − 7z = 2

On comparing with ax + by + cz = d where d > 0 is a = − 18, b = 1, c = − 7, d = 2

∴ a + 7b + c + 20d = 22

Q.69. Letwhere α ∈ R. If the area of the parallelogram whose adjacent sides are represented by the vectorsthen the value ofis equal to:        (JEE Main 2022)
(a) 10
(b) 7
(c) 9
(d) 14

Ans. d


⇒ (2 + α)² + (α − 2)² + (α² + 4)² = 15(α² + 4)

⇒ α⁴ − 5α² − 36 = 0

∴ α = ±3

Q.70. Let the planecontain the line of intersection of two planesandIf the plane P passes through the point (2, 3, 1/2), then the value ofis equal to        (JEE Main 2022)
(a) 90
(b) 93
(c) 95
(d) 97

Ans. b

P₁: x + 3y − z = 6

P₂: −6x + 5y − z = 7

Family of planes passing through line of intersection of P₁ and P₂ is given by
x(1 − 6λ) + y(3 + 5λ) + z(−1 − λ) − (6 + 7λ) = 0

It passes through (2, 3, 1/2)

Required plane is

−5x + 8y − 2z − 13 = 0

Q.71. The acute angle between the planes P₁ and P₂, when P₁ and P₂ are the planes passing through the intersection of the planes 5x + 8y + 13z − 29 = 0 and 8x − 7y + z − 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is        (JEE Main 2022)
(a) π/3
(b) π/4
(c) π/6
(d) π/12

Ans. a
Family of Plane's equation can be given by

(5 + 8λ)x + (8 − 7λ)y + (13 + λ)z − (29 + 20λ) = 0

P₁ passes through (2, 1, 3)

⇒ (10 + 16λ) + (8 − 7λ) + (39 + 3λ) − (29 + 20λ) = 0

⇒ −8λ + 28 = 0 ⇒ λ = 7/2

d.r, s of normal to P₁

P₂ passes through (0, 1, 2)

⇒ 8 − 7λ + 26 + 2λ − (29 + 20λ) = 0

⇒ 5 − 25λ = 0

⇒ λ = 1/5

d.r, s of normal to P₂

Angle between normals

Q.72. If two distinct point Q, R lie on the line of intersection of the planes −x + 2y − z = 0 and 3x − 5y + 2z = 0 and PQ = PR = √18 where the point P is (1, −2, 3), then the area of the triangle PQR is equal to        (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. b

Line L is x = y = z

⇒ (α − 3) + α + 2 + α − 1 = 0

Q.73. Letbe the vectors along the diagonals of a parallelogram having area 2√2. Let the angle betweenbe acute, then an angle betweenis        (JEE Main 2022)
(a) π/4
(b)
(c) 5π/6
(d) 3π/4

Ans. d
∵ be the vectors along the diagonals of a parallelogram having area 2√2. 



From (ii) and (iii) 

Q.74. The shortest distance between the lines
        (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. a

Q.75. Let the foot of the perpendicular from the point (1, 2, 4) on the linebe P. Then the distance of P from the plane 3x + 4y + 12z + 23 = 0 is        (JEE Main 2022)
(a) 5
(b) 50/13
(c) 4
(d) 63/13

Ans. a

Let P = (4t − 2, 2t + 1, 3t − 1)

∵ P is the foot of perpendicular of (1, 2, 4)

∴ 4(4t − 3) + 2(2t − 1) + 3(3t − 5) = 0

⇒ 29t = 29 ⇒ t = 1

∴ P = (2, 3, 2)

Now, distance of P from the plane

3x + 4y + 12z + 23 = 0, is

Q.76. LetThen the number of vectors∈ {1, 2, ........, 10} is:        (JEE Main 2022)
(a) 0
(b) 1
(c) 2
(d) 3

Ans. a


= 2 − 3 − 2 = 0

⇒ −3 = 0 (Not possible)

⇒ No possible value ofis possible.

Q.77. Letbe three given vectors. Let a vector in the plane ofwhose projection onis equal to:        (JEE Main 2022)
(a) 6
(b) 7
(c) 8
(d) 9

Ans. d


from equation (i) and (ii) 

Q.78. If the linesare co-planar, then the distance of the plane containing these two lines from the point (α, 0, 0) is:        (JEE Main 2022)
(a) 2/9
(b) 2/11
(c) 4/11
(d) 2

Ans. b
∵ Both lines are coplanar, so

⇒ α = 5/3

Equation of plane containing both lines

⇒ 9x + 2y + 6z = 13

So, distance of (5/3, 0, 0) from this plane

Q.79. If the plane 2x + y − 5z = 0 is rotated about its line of intersection with the plane 3x − y + 4z − 7 = 0 by an angle of π/2, then the plane after the rotation passes through the point:        (JEE Main 2022)
(a) (2, −2, 0)
(b) (−2, 2, 0)
(c) (1, 0, 2)
(d) (−1, 0, −2)

Ans. c
P₁: 2x + y − 52 = 0, P₂: 3x − y + 4z − 7 = 0

Family of planes P₁ and P₂

P: P₁ + λP₂

∴ P: (2 + 3λ)x + (1 − λ)y + (−5 + 4λ)z − 7λ = 0

∵ P ⊥ P₁

∴ 4 + 6λ + 1 − λ + 25 − 20λ = 0

λ = 2

∴ P: 8x − y + 32 − 14 = 0

It passes through the point (1, 0, 2)

Q.80. Ifthen the value of is:        (JEE Main 2022)
(a) 0
(b)
(c)
(d)

Ans. a


So, vectorsare coplanar, hence their Scalar triple product will be zero. 

Q.81. Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x − 3y + 5z = 8. If the mirror image of the point (2,2) in the rotated plane is B(a, b, c), then:        (JEE Main 2022)
(a)
(b)

(c)
(d)

Ans. a
Consider the equation of plane,

P: (2x + 3y + z + 20) + λ(x − 3y + 5z − 8) = 0

P: (2 + λ)x + 3(3 − 3λ)y + 1(1 + 5λ)z + (20 − 8λ) = 0

∵ Plane P is perpendicular to 2x + 3y + z + 20 = 0

So, 4 + 2λ + 9 − 9λ + 1 + 5λ = 0

⇒ λ = 7

P: 9x − 18y + 36z − 36 = 0

or P: x − 2y + 4z = 4

If image of (2, 2) in plane P is (a, b, c) then

So, a : b : c = 8 : 5 : −4

Q.82. If the two linesare perpendicular, then an angle between the linesis:       (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. b

∵ L₁ and L₂ are perpendicular, so

⇒ α = 3

Now angle between l₂ and l₃,

Q.83. Let p be the plane passing through the intersection of the planesandand the point (2, 1, −2). Let the position vectors of the points X and Y beandrespectively. Then the points       (JEE Main 2022)
(a) X and X + Y are on the same side of P
(b) Y and Y − X are on the opposite sides of P
(c) X and Y are on the opposite sides of P
(d) X + Y and X − Y are on the same side of P

Ans. c
Let the equation of required plane

π: (x + 3y − z − 5) + λ(2x − y + z − 3) = 0

∵ (2, 1, −2) lies on it so, 2 + λ(−2) = 0

⇒ λ = 1

Hence, π: 3x + 2y − 8 = 0

∵ π_x = −9, π_y = 5, π_x+y = 4

π_x−y = −22 and π_y−x = 6

Clearly X and Y are on opposite sides of plane π

Q.84. Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y + z = 5. If a line L passing through (1, −1, −1), parallel to the line PQ meets the plane S at R, then QR² is equal to:       (JEE Main 2022) 
(a) 2
(b) 5
(c) 7
(d) 11

Ans. b
As L is parallel to PQ d.r.s of S is <1, 1, 1> 

Point of intersection of L and S be λ

⇒ (λ + 1) + (λ − 1) + (λ − 1) = S

⇒ λ = 2

∴ R ≡ (3, 1, 1)

Let Q(α, β, γ)

⇒ α = 3, β = 2, γ = 3

⇒ Q ≡ (3, 2, 3)
(QR)² = 0² + (1)² + (2)² = 5

Q.85. Leti = 1, 2, 3 be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection ofon the vectorbe 7. Letbe a vector obtained by rotatingwith 90^∘. Ifand x-axis are coplanar, then projection of a vectoronis equal to:       (JEE Main 2022) 
(a) √7
(b) √2
(c) 2
(d) 7

Ans. b

Q.86. Letbe two unit vectors such that. If θ ∈ (0, π) is the angle between,then among the statements:
       (JEE Main 2022) 
 (a) Only (S1) is true.
(b) Only (S2) is true.
(c) Both (S1) and (S2) are true.
(d) Both (S1) and (S2) are false.

Ans. c

∴ cos⁡θ = cos⁡2θ

∴ θ = 2π/3

where θ is angle between .

(S1) is correct.

And projection of

(S2) is correct.

Q.87. Let the points on the plane P be equidistant from the points (−4, 2, 1) and (2, −2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is       (JEE Main 2022) 
(a) π/6
(b) π/4
(c) π/3
(d) 5π/12

Ans. c
Let P(x, y, z) be any point on plane P₁

Then (x + 4)² + (y − 2)² + (z − 1)² = (x − 2)² + (y + 2)² + (z − 3)²

⇒ 12x − 8y + 4z + 4 = 0

⇒ 3x − 2y + z + 1 = 0

And P₂: 2x + y + 3z = 0

∴ angle between P₁ and P₂

Q.88. If the shortest distance between the lines then the sum of all possible value of λ is:       (JEE Main 2022)
(a) 16
(b) 6
(c) 12
(d) 15

Ans. a


∴ Shortest distance 

⇒ 3(5 − 2λ)² = (15 − 4λ)² + (10 − λ)² + 25

⇒ 5λ² − 80λ + 275 = 0

∴ Sum of values of λ = 80/5 = 16

Q.89. Letbe unit vectors. Ifbe a vector such that the angle betweenthenis equal to:       (JEE Main 2022)
(a) 6(3 − √3)
(b) 3 + √3
(c) 6(3 + √3)
(d) 6(√3 + 1)

Ans. c