Practice Questions

Q87.Let →𝑎= 3^𝑖+ 2^𝑗+ 𝑥^𝑘 and →𝑏= ^𝑖- ^𝑗+ ^𝑘, for some real 𝑥. Then the condition for →𝑎× →𝑏 = 𝑟 to follow (1) 0 < 𝑟≤ 3 (2) 𝑟≥ 3 √ 2 5√ 2 (3) 3 < 3 (4) 3 3 < r < 3 √ 2 𝑟≤3√ 2 √ 2 5√ 2

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.The sum of the distinct real values of μ for which the vectors μˆi + ˆj + ˆk, ˆi + μˆj + ˆk, ˆi + ˆj + μˆk are co- planar, is (1) 0 (2) −1 (3) 1 (4) 2

Q87.If an angle between the line, x+1 , then a value 2 = 1 = z−3−2 and the plane, x −2y −kz = 3 is cos−1( 2√23 ) of k is (1) √53 (2) √35 (3) −35 (4) −53

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.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 (1) (2, 4, 1) (2) (1, −4, 1) (3) (1, 4, −1) (4) (2, −4, 1)

Q87.Let A(3, 0, −1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the mid-point of AC . If G divides BM in the ratio, 2 : 1 , then cos(∠GOA) ( O being the origin) is equal to (1) 1 (2) 1 √30 6√10 (3) 1 (4) 1 √15 2√15 , then β

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

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.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

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.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.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.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.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.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.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 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 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

Q89.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 (1) √91 (2) √91 4 2 (3) 2√13 (4) √65 2 JEE Main 2019 (10 Apr Shift 1) JEE Main Previous Year Paper

Q89.On which of the following lines lies the point of intersection of the line, x−4 2 = 2 = z−31 and the plane, x + y + z = 2? y−5 (1) x−4 1 = 1 = z−5−1 (2) x−11 = y−32 = z+4−5 (3) x−2 2 = y−32 = z+33 (4) x+33 = 4−y3 = z+1−2

Q89.A perpendicular is drawn from a point on the line = = to the plane 𝑥+ 𝑦+ 𝑧= 3 such that the 2 -1 1 foot of the perpendicular 𝑄 also lies on the plane 𝑥- 𝑦+ 𝑧= 3. Then the coordinates of 𝑄 are (1) 2, 0, 1 (2) – 1, 0, 4 (3) 4, 0, – 1 (4) 1, 0, 2

Q89.If the line, x−1 2 = y+13 = z−24 meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the origin is, (1) 2√5 (2) 92 (3) √5 (4) 7 2 2

Q89.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 JEE Main 2019 (08 Apr Shift 1) JEE Main Previous Year Paper (1) 𝑥- 3𝑦- 2𝑧= - 2 (2) 𝑥+ 3𝑦+ 𝑧= 4 (3) 𝑥- 𝑦- 𝑧= 0 (4) 2𝑥- 𝑧= 2