Practice Questions

Q85.If the vectors →a = λˆi + μˆj + 4ˆk, b = −2ˆi + 4ˆj −2ˆk and →c= 2ˆi + 3ˆj + ˆk are coplanar and the projection of →a → on the vector b is √54 units, then the sum of all possible values of λ + μ is equal to (1) 0 (2) 6 (3) 24 (4) 18 →

Q85.If →a = ˆi + 2ˆk, →b= ˆi + ˆj + ˆk, →c= 7ˆi −3ˆj + 4ˆk, →r×→b+→b×→c=→0 and →r⋅→a = 0 then →r.→cis equal to: (1) 34 (2) 12 (3) 36 (4) 30 + ˆj + × = 4

Q85.Let the vectors →a, b, →crepresent three coterminous edges of a parallelopiped of volume V . Then the volume of → → the parallelopiped, whose coterminous edges are represented by →a, b +→cand →a+ 2 b + 3→cis equal to (1) 2V (2) 6V (3) V (4) 3V

Q85.If the points with position vectors αˆi + 10ˆj + 13ˆk, 6ˆi + 11ˆj + 11ˆk, 92ˆi + βˆj −8ˆk are collinear, then (19α −6β)2 is equal to (1) 36 (2) 25 (3) 49 (4) 16 → →

Q85.Let the vectors u1→ = ˆi + ˆj + aˆk, u2→ = ˆi + bˆj + ˆk, and u3→ = cˆi + ˆj + ˆk be coplanar. If the vectors −−→ → v1 = (a + b)ˆi + cˆj + cˆk, v2 = aˆi + (b + c)ˆj + aˆk and →v3 = bˆi + bˆj + (c + a)ˆk are also coplanar, then 6(a + b + c) is equal to (1) 0 (2) 4 (3) 12 (4) 6

Q85.Let →a = ˆi + 4ˆj + 2ˆk, b = 3ˆi −2ˆj + 7ˆk and →c= 2ˆi −ˆj + 4ˆk. If a vector d satisfies d × b =→c× b and d ⋅→a = 24, →2 then d is equal to (1) 323 (2) 423 (3) 313 (4) 413 → → → 2

Q85.Let λ ∈Z, →a = λˆi + ˆj −ˆk and b = 3ˆi −ˆj + 2ˆk. Let →c be a vector such that + b = 0, →a⋅→c= −17 and b ⋅→c= −20. Then →c× (λˆi + ˆj + ˆk) is equal to (→a → → → 2 +→c) ×→c (1) 46 (2) 53 (3) 62 (4) 49 JEE Main 2023 (12 Apr Shift 1) JEE Main Previous Year Paper

Q85.If four distinct points with position vectors →a,→b,→cand →d are coplanar, then [→a→b→c] + + + + (1) [→d →b →a] [→a →c →d ] [→d→b →c] (2) [→a →d →b] [→d →c →a] [→d →b →c] (3) [→d →c →a] + [→b →d →a] + [→c →d →b ] (4) [→b →c →d ] + [→d →a →c] + [→d →b →a] → → → = 27 and b ⋅→c=

Q85.Let λ ∈R,→a = λˆi + 2ˆj −3ˆk,→b = ˆi −λˆj + 2ˆk, If ((→a →b) (→a →b)) (→a →b) → → + × − 2 is equal to λ(→a b) (→a b) (1) 140 (2) 132 (3) 144 (4) 136 → → b, then the value of × −3 b ⋅→cis

Q86.Let →a = 2ˆi −7ˆj + 5ˆk , b = ˆi + ˆk and→c= ˆi + 2ˆj −3ˆk be three given vectors. If→ris a vector such that →r×→a =→c×→a and→r⋅→b = 0 , then →r is equal to: (1) 11 7 √2 (2) 117 (3) 11 5 √2 (4) √9147

Q86.The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(−2, −3, 5) and D(1, −6, −7) is equal to (1) 48 (2) 8√38 (3) 54 (4) 9√38

Q86.Let →a = 4ˆi + 3ˆj and→b = 3ˆi −4ˆj + 5ˆk and→cis a vector such that →c⋅(→a → b) + 25 = 0,→c⋅(ˆi ˆk) → and projection of →con →a is 1 , then the projection of →con b equals: (1) 5 (2) 1 √2 5 (3) 1 (4) 3 √2 √2

Q86.The sum of all values of α, for which the points whose position vectors are ˆi −2ˆj + 3ˆk, 2ˆi −3ˆj + 4ˆk, (α + 1)ˆi + 2ˆk and 9ˆi + (α −8)ˆj + 6ˆk are coplanar, is equal to (1) −2 (2) 2 (3) 6 (4) 4

Q86.The vector →a = −ˆi + 2ˆj + ˆk is rotated through a right angle, passing through the y-axis in its way and the → → resulting vector is b. Then the projection of 3→a+ √2 b on →c= 5ˆi + 4ˆj + 3ˆk is (1) 3√2 (2) 1 (3) √6 (4) 2√3

Q86.Let →a = ˆi + 2ˆj + λˆk, b = 3ˆi −5ˆj −λˆk, →a⋅→c= 7 , 2( ⋅→c)

Q86.The shortest distance between the lines x + 1 = 2 y = −12z and x = y + 2 = 6z −6 is (1) 2 (2) 3 (3) 5 (4) 3 2 2

Q86.Let →aand→b be two vectors. Let →a = 1, →b = 4 and →a⋅→b = 2 . If →c= (2→a →b) (1) −24 (2) −48 (3) −84 (4) −60

Q87.Let the plane P pass through the intersection of the planes 2x + 3y −z = 2 and x + 2y + 3z = 6, and be perpendicular to the plane 2x + y −z + 1 = 0. If d is the distance of P from the point (−7, 1, 1), then d2 is equal to : (1) 250 (2) 15 83 53 (3) 25 (4) 250 83 82

Q87.Let the equation of plane passing through the line of intersection of the planes x + 2y + az = 2 and x −y + z = 3 be 5x −11y + bz = 6a −1. For c ∈Z, if the distance of this plane from the point (a, −c, c) is 2 , then a+bc is equal to √a (1) 2 (2) 4 (3) −4 (4) −2 = 10 parallel to the line of the shortest

Q87.The foot of perpendicular of the point (2, 0, 5) on the line x+12 = y−15 = z+1−1 is (α, β, γ). Then. Which of the following is NOT correct? (1) αβ γ = 154 (2) αβ = −8 (3) β γ = −5 (4) αγ = 85

Q87.Let the lines L1 : x+53 = y+41 = z−α−2 and L2 : 3x + 2y + z −2 = 0 = x −3y + 2z −13 be coplanar. If the point P(a, b, c) on L1 is nearest to the point Q(−4, −3, 2), then |a| + |b| + |c| is equal to (1) 12 (2) 14 (3) 8 (4) 10

Q87.The shortest distance between the lines x−4 4 = y+25 = z+33 and x−13 = y−34 = z−42 is (1) 6√3 (2) 2√6 (3) 6√2 (4) 3√6

Q87.Shortest distance between the lines x−1 2 = y+8−7 = z−45 and x−12 = y−21 = z−6−3 is JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper (1) 2√3 (2) 4√3 (3) 3√3 (4) 5√3

Q87.Let P be the plane passing through the points (5, 3, 0), (13, 3, −2) and (1, 6, 2). For α ∈N, if the distance of the points A(3, 4, α) and B(2, α, a) from the plane P are 2 and 3 respectively, then the positive value of a is (1) 6 (2) 3 (3) 5 (4) 4

Q87.A vector →vin the first octant is inclined to the x axis at 60° , to the y-axis at 45° and to the z-axis at an acute −1, (a, b, c), is normal to →v, then 1) and angle. If a plane passing through the points (√2, (1) √2a + b + c = 1 (2) a + b + √2c = 1 (3) a + √2b + c = 1 (4) √2a −b + c = 1