Practice Questions

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

Q77.The area bounded by the curves 𝑦= 𝑥2 - 1 and 𝑦= 1 is (1) 2 + 1 (2) 4 - 1 3√2 3√2 8 (3) 2√2 - 1 (4) 3√2 - 1

Q77.If →a⋅ b = 1, b ⋅→c= 2 and →c⋅→a = 3 , then the value of [→a ( ×→c) ( ×→a)] b (1) 0 (2) −6→a⋅(→ ×→c) → −12b ⋅(→c×→a) (3) 12→c⋅(→a×→b) (4)

Q77.Let A, B, C be three points whose position vectors respectively are: →a = ˆi + 4ˆj + 3ˆk → b = 2ˆi + αˆj + 4ˆk, α ∈R →c= 3ˆi −2ˆj + 5ˆk → If α is the smallest positive integer for which →a, b, →care non-collinear, then the length of the median, △ABC , through A is: (1) √82 (2) √62 2 2 (3) √69 (4) √66 2 2 y+1

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.The area of the region enclosed between the parabolas 𝑦2 = 2𝑥- 1 and 𝑦2 = 4𝑥- 3 is. 1 1 (1) (2) 3 6 2 3 (3) (4) 3 4

Q77.Let →a = αˆi + ˆj −ˆk and b = 2ˆi + ˆj −αˆk, α > 0 . If the projection of →a× b on the vector −ˆi + 2ˆj −2ˆk is 30 , then α is equal to (1) 15 (2) 8 2 (3) 13 (4) 7 2

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

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 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.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 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 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.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 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.Let the lines x−1 λ = y−21 = z−32 and x+26−2 = y+183 = z+28λ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lies on P ? (1) (0, −2, −2) (2) (−5, 0, −1) (3) (3, −1, 0) (4) (0, 4, 5)

Q79.Let 𝑄 be the foot of perpendicular drawn from the point 𝑃1, 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 (1) √3 (2) √3 2 (3) 2√3 (4) 3

Q79.A vector →𝑎 is parallel to the line of intersection of the plane determined by the vectors ^𝑖, ^𝑖+ ^𝑗 and the plane determined by the vectors ^𝑖- ^𝑗, ^𝑖+ ^𝑘. The obtuse angle between →𝑎 and the vector →𝑏= ^𝑖- 2 ^𝑗+ 2 ^𝑘 is (1) 3𝜋 (2) 2𝜋 4 3 4𝜋 5𝜋 (3) (4) 5 6 4

Q79.Five numbers x1, x2, x3, x4, x5 are randomly selected from the numbers 1, 2, 3, … … , 18 and are arranged in the increasing order (x1 < x2 < x1 < x4 < x2). The probability that x2 = 7 and x4 = 11 is JEE Main 2022 (27 Jun Shift 1) JEE Main Previous Year Paper (1) 1 (2) 1 136 68 (3) 7 (4) 5 68 68

Q79.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 (1) π (2) π 6 4 (3) π (4) 5π 3 12