Practice Questions

Q77.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 (1) 2 3 √38 (2) 43 √38 (3) 8 3 √38 (4) √1523

Q77.Let →a and b be the vectors along the diagonal of a parallelogram having area 2√2. Let the angle between →a and → → → → → → × −2b, then an angle between b and →cis b be acute. →a = 1 and →a. b = →a× b . If →c= 2√2(→a b) (1) −π (2) 5π 4 6 (3) π (4) 3π 3 4 P . Then the

Q77.If dy + ex(x2 −2)y = (x2 −2x)(x2 −2)e2x and y(0) = 0 , then the value of y(2) is dx (1) −1 (2) 1 (3) 0 (4) e →

Q77.Let 𝐴𝐵𝐶 be a triangle such that 𝐵𝐶= →𝑎, 𝐶𝐴= 𝑏, 𝐴𝐵= →𝑐, →𝑎= 6√2, 𝑏= 2√3 and 𝑏· →𝑐= 12 Consider the statements : 𝑆1: →𝑎× →𝑏+ →𝑐× →𝑏- →𝑐= 62√2 - 1 𝑆2: ∠𝐴𝐵𝐶= cos-1√ 23. Then (1) both 𝑆1 and 𝑆2are true (2) only 𝑆1 is true (3) only 𝑆2 is true (4) both 𝑆1 and 𝑆2 are false 𝑥- 3 𝑦+ 4 𝑧- 7

Q77.Let →a = ˆi + ˆj −ˆk and →c= 2ˆi −3ˆj + 2ˆk. Then the number of vectors b such that b ×→c=→a and → b ∈{1, 2, … , 10} is (1) 0 (2) 1 (3) 2 (4) 3

Q77.Let S be the set of all a ∈R for which the angle between the vectors u = a(loge b)ˆi −6ˆj + 3ˆk and →v= (loge b)ˆi + 2ˆj + 2a(loge b)ˆk, (b > 1) is acute. Then S is equal to (1) (−∞, −43 ) (2) Φ (3) (−43 , 0) (4) ( 127 , ∞) JEE Main 2022 (28 Jul Shift 2) JEE Main Previous Year Paper

Q77.If the length of the perpendicular drawn from the point P(a, 4, 2), a > 0 on the line x+12 = y−33 = z−1−1 is 2√6 units and Q(α1, α2, α3) is the image of the point P in this line, then a + ∑3i=1 αi is equal to (1) 7 (2) 8 (3) 12 (4) 14

Q78.If two straight lines whose direction cosines are given by the relations l + m −n = 0, 3l2 + m2 + cnl = 0 are parallel, then the positive value of c is (1) 6 (2) 4 (3) 3 (4) 2

Q78.Let the solution curve of the differential equation x dxdy −y = √y2 + 16x2, y(1) = 3 be y = y(x). Then y(2) is equal to (1) 15 (2) 11 (3) 14 (4) 17 →

Q78.If the two lines l1 : x−23 = y+1−2 , z = 2 and l2 : x−11 = 2y+3α = z+52 are perpendicular, then an angle between the lines l2 and l3 : 1−x3 = 2y−1−4 = 4z is (1) cos−1( 294 ) (2) sec−1( 294 ) (3) cos−1( 292 ) (4) cos−1( √292 )

Q78.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: (1) √212 (2) √92 (3) √732 (4) 1

Q78.If the shortest distance between the lines x−1 2 = y−23 = z−3λ and x−21 = y−44 = z−55 is √31 , then the sum of all possible values of λ is: (1) 16 (2) 6 (3) 12 (4) 15

Q78.Let →a = ˆi + ˆj + 2ˆk, b = 2ˆi −3ˆj + ˆk and →c= ˆi −ˆj + ˆk be the three given vectors. Let →vbe a vector in the → plane of →a and b whose projection on →cis 2 . If →v,ˆj = 7 , then →v + is equal to √3 ⋅(ˆi ˆk) (1) 6 (2) 7 (3) 8 (4) 9

Q78.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 (1) 9 (2) 12 (3) 21 (4) 33 y−6

Q78.Let the foot of the perpendicular from the point (1, 2, 4) on the line x+24 = y−12 = z+13 be distance of P from the plane 3x + 4y + 12z + 23 = 0 is (1) 50 (2) 63 13 13 (3) 65 (4) 4 13

Q78.Let ˆa,ˆb be unit vectors. If →cbe a vector such that the angle between ˆa and →cis 12 π , and ˆb =→c+ 2(→c ˆa), then 6→c 2 is equal to: + (1) 6(3 −√3) (2) 6(3 √3) + (3) 3 + √3 (4) 6(√3 1)

Q78.Let →a = 2ˆi −ˆj + 5ˆk and b = αˆi + βˆj + 2ˆk. If ((→a b) ׈i) (1) 4 (2) 5 (3) √21 (4) √17

Q78.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 (1) 1 , 1 , 0 (2) 1 , −1 , 0 √2 √2 √2 √2 (3) 1 , −2 , 0 (4) 1 2 , −√32 , 0 √5 √5

Q78.Let x−2 3 = −2 = z+3−1 lie on the plane px −qy + z = 5, for some p, q ∈R. The shortest distance of the plane from the origin is: (1) √ 1093 (2) √ 1425 (3) √571 (4) √ 1421

Q78.Let ˆa and ˆb be two unit vectors such that the angle between them is π4 . If and + × then the value of 164 cos2 θ is equal to (ˆa ˆb) (ˆa + 2ˆb + 2(ˆa ˆb)) (1) 90 + 27√2 (2) 45 + 18√2 (3) 90 + 3√2 (4) 54 + 90√2

Q78.Let a vector →𝑎 has a magnitude 9. Let a vector →𝑏 be such that for every 𝑥, 𝑦𝑅× 𝑅- 0, 0, the vector 𝑥→𝑎+ 𝑦 →𝑏 is → → perpendicular to the vector 6𝑦 →𝑎- 18𝑥 𝑏. Then the value of →𝑎× 𝑏 is equal to (1) 9√3 (2) 27√3 (3) 9 (4) 81

Q78.If 𝑦= 𝑦𝑥 is the solution of the differential equation 2𝑥2𝑑𝑦 2𝑥𝑦+ 3𝑦2 = 0 such that 𝑦𝑒= 𝑒 then 𝑦1 is equal 𝑑𝑥- 3, to (1) 1 (2) 2 3 3 3 (3) (4) 3 2

Q78.Let →𝑎= 𝑎1 ^𝑖+ 𝑎2 ^𝑗+ 𝑎3 ^𝑘, 𝑎𝑖> 0, 𝑖= 1, 2, 3 be a vector which makes equal angles with the coordinate axes 𝑂𝑋, 𝑂𝑌 and 𝑂𝑍. Also, let the projection of →𝑎 on the vector 3 ^𝑖+ 4 ^𝑗 be 7 . Let →𝑏 be a vector obtained by rotating →𝑎 with 90°. If →𝑎, →𝑏 and 𝑥-axis are coplanar, then projection of a vector →𝑏 on 3 ^𝑖+ 4 ^𝑗 is equal to (1) √7 (2) √2 (3) 2 (4) 7

Q78.Let →a = αˆi + 2ˆj −ˆk and b = −2ˆi + αˆj + ˆk, where α ∈R. If the area of the parallelogram whose adjacent → 2 → → 2 b is equal to ⋅ sides are represented by the vectors →a and b is √15(α2 + 4), then the value of 2→a + (→a b) (1) 10 (2) 7 (3) 9 (4) 14 + = 2ˆi −13ˆj −4ˆk, then

Q78.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 (1) π (2) π 3 4 (3) π (4) π 6 12 JEE Main 2022 (28 Jun Shift 1) JEE Main Previous Year Paper = 6 and