Practice Questions

Q87.Let →a = ˆi +ˆj + √2ˆk, b = b1ˆi + b2ˆj + √2ˆk and →c= 5ˆi +ˆj + √2ˆk be three vectors such that the projection → → → vector of b on →a is →a . If →a+ b is perpendicular to →c, then b is equal to: (1) √22 (2) √32 (3) 6 (4) 4

Q87.Let α = (λ −2) →a+ b and β = (4λ −2) →a+ 3 b, be two given vectors where vectors →a and b are non-collinear. → → The value of λ for which vectors α and β are collinear, is: (1) −4 (2) −3 (3) 4 (4) 3

Q87.Let →a = ^i + 2^j + 4^k,→b = ^i + λ^j + 4^k and →c = 2^i + 4^j + (λ2 −1)^k be coplanar vectors. Then the non-zero vector →a × →c is: (1) −10^i −5^j (2) −14^i −5^j (3) −14^i + 5^j (4) −10^i + 5^j

Q87.The magnitude of the projection of the vector 2^𝑖+ 3^𝑗+ ^𝑘 on the vector perpendicular to the plane containing the vectors ^𝑖+ ^𝑗+ ^𝑘 and ^𝑖+ 2^𝑗+ 3^𝑘, is: (1) 3√6 (2) √ 32 (3) √6 (4) √32

Q87.Let →𝑎= ^𝑖- ^𝑗, →𝑏= ^𝑖+ ^𝑗+ ^𝑘 and →𝑐 be a vector such that →𝑎× →𝑐+ →𝑏= →0 and →𝑎. →𝑐= 4, then |→𝑐| is equal to: 19 (1) (2) 9 2 (3) 17 (4) 8 2

Q87.The distance of the point having position vector -^𝑖+ 2^𝑗+ 6^𝑘 from the straight line passing through the point 2, 3, - 4 and parallel to the vector, 6^𝑖+ 3^𝑗- 4^𝑘 is (1) 4√3 (2) 6 (3) 2√13 (4) 7

Q87.If the volume of parallelepiped formed by the vectors ^𝑖+ 𝜆^𝑗+ ^𝑘, ^𝑗+ 𝜆^𝑘 and 𝜆^𝑖+ ^𝑘 is minimum, then 𝜆 is equal to: 1 (1) - (2) -√3 √3 1 (3) √3 (4) √3

Q88.The plane containing the line x−3 2 = y+2−1 = z−13 and also containing its projection on the plane 2x + 3y −z = 5 , contains which one of the following points? (1) (2,2,0) (2) (-2,2,2) (3) (0,-2,2) (4) (2,0,-2)

Q88.If the plane 2𝑥- 𝑦+ 2𝑧+ 3 = 0 has the distances 1 and 2 units from the planes 4𝑥- 2𝑦+ 4𝑧+ 𝜆= 0 and 3 3 2𝑥- 𝑦+ 2𝑧+ 𝜇= 0 , respectively, then the maximum value of 𝜆+ 𝜇 is equal to: (1) 9 (2) 15 (3) 13 (4) 5 𝑥- 1 𝑦+ 1 𝑧

Q88.Let A be a point on the line →r= (1 −3μ)ˆi + (μ −1)ˆj + (2 + 5μ)ˆk and B(3, 2, 6) be a point in the space. −→ Then the value of μ for which the vector AB is parallel to the plane x −4y + 3z = 1 is (1) 1 (2) 1 2 4 (3) −14 (4) 81

Q88.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 : (1) {√3} (2) {3, −3} (3) {1, −1} (4) {√3, −√3}

Q88.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? (1) (2, 1, 3) (2) (4, 1, −2) (3) (4, −1, 7) (4) (−2, 3, 5) y−5

Q88.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 : (1) 12 (2) 7 (3) 5 (4) 17

Q88.Let →𝑎= 3^𝑖+ 2^𝑗+ 2^𝑘 and →𝑏= ^𝑖+ 2^𝑗- 2^𝑘 be two vectors. If a vector perpendicular to both the vectors →𝑎+ →𝑏 and →𝑎- →𝑏 has the magnitude 12 then one such vector is: (1) 4(2^𝑖+ 2^𝑗+ ^𝑘) (2) 4(2^𝑖- 2^𝑗- ^𝑘) (3) 4( - 2^𝑖- 2^𝑗+ ^𝑘) (4) 4(2^𝑖+ 2^𝑗- ^𝑘)

Q88.The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines is + + + + →r= (ˆi ˆj) λ(ˆi + 2ˆj −ˆk) and →r= (ˆi ˆj) μ(−ˆi + ˆj −2ˆk) (1) 1 (2) 3 3 (3) √3 (4) 1 √3

Q88.A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(−1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR is JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper (1) cos−1( 317 ) (2) cos−1( 3117 ) (3) cos−1( 3519 ) (4) cos−1( 359 )

Q88.The length of the perpendicular from the point ( 2, - 1, 4 ) on the straight line 𝑥+ 3 = 𝑦- 2 = 𝑧 is 10 -7 1 (1) greater than 3 but less (2) greater than 4 (3) less than 2 (4) greater than 2 but less than 4 than 3

Q88.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 (1) →𝑟× ^𝑖+ ^𝑘+ 2 = 0 (2) →𝑟⋅^𝑖- ^𝑘- 2 = 0 (3) →𝑟× ^𝑖- ^𝑘+ 2 = 0 (4) →𝑟⋅(^𝑖- ^𝑘) + 2 = 0

Q88.A plane passing though the points (0, −1, 0) and (0, 0, 1) and making an angle π4 with the plane y–z + 5 = 0, also passes through the point −1, 1, (1) (√2, 4) (2) (√2, 4) −1, 1, (3) (−√2, −4) (4) (−√2, −4)

Q88.The vertices B and C of a ΔABC lie on the line, x+2 3 = y−10 = 4z such that BC = 5 units. Then the area (in sq. units) of this triangle, given the point A(1, −1, 2), is (1) 6 (2) 2√34 (3) √34 (4) 5√17

Q88.The plane through the intersection of the planes 𝑥+ 𝑦+ 𝑧= 1 and 2𝑥+ 3𝑦- 𝑧+ 4 = 0 and parallel to 𝑦- axis also passes through the point (1) 3, 3, - 1 (2) -3, 1, 1 (3) 3, 2, 1 (4) -3, 0, - 1

Q88.If the lines x = ay + b, z = cy + d and x = a′z + b′, y = c′ z + d′ are perpendicular, then (1) cc’ + a + a’ = 0 (2) aa’ + c + c’ = 0 (3) bb’ + cc’ + 1 = 0 (4) ab’ + bc’ + 1 = 0

Q88.If the length of the perpendicular from the point (β, 0, β), (β ≠0) to the line, x1 = y−10 = z+1−1 is √32 is equal to (1) 2 (2) −1 (3) −2 (4) 1

Q89.The equation of the plane containing the straight line x 2 = 3y = 4z and perpendicular to the plane containing the straight lines x 3 = 4y = 2z and x4 = 2y = 3z is: (1) 3x + 2y −3z = 0 (2) x + 2y −2z = 0 (3) x −2y + z = 0 (4) 5x + 2y −4z = 0

Q89.The equation of the line passing through -4, 3, 1, parallel to the plane 𝑥+ 2𝑦- 𝑧- 5 = 0 and intersecting the 𝑥 + 1 𝑦- 3 𝑧- 2 line = = is -3 2 -1 𝑥+ 4 𝑦- 3 𝑧- 1 𝑥+ 4 𝑦- 3 𝑧- 1 (1) = = (2) = = 3 -1 1 1 1 3 (3) 𝑥+ 4 = 𝑦- 3 = 𝑧- 1 (4) 𝑥- 4 = 𝑦+ 3 = 𝑧+ 1 -1 1 1 2 1 4