Practice Questions

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 the vectors →𝑎= 1 + 𝑡 ^𝑖+ 1 - 𝑡 ^𝑗+ ^𝑘, →𝑏= 1 - 𝑡 ^𝑖+ 1 + t ^𝑗+ 2 ^𝑘 and →𝑐= 𝑡 ^𝑖- 𝑡 ^𝑗+ ^𝑘, 𝑡∈𝑅 be such that for 𝛼, 𝛽, 𝛾∈𝑅, 𝛼 →𝑎+ 𝛽 →𝑏+ 𝛾 →𝑐= →0 ⇒𝛼= 𝛽= 𝛾= 0. Then, the set of all values of 𝑡 is (1) a non-empty finite set (2) equal to 𝑁 (3) equal to 𝑅- 0 (4) equal to 𝑅

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 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 and b be two unit vectors such that |(a + b) + 2(a × b)| = 2. If θ ∈(0, π) is the angle between ˆa and ˆb , then among the statements: (S1) : 2 ˆa × ˆb = ˆa −ˆb is 1 + (S2) : The projection of ˆa on 2 (ˆa ˆb) (1) Only (S1) is true. (2) Only (S2) is true. (3) Both (S1) and (S2) are true. (4) Both (S1) and (S2) are false. JEE Main 2022 (24 Jun Shift 2) JEE Main Previous Year Paper

Q77.Let →a = ˆi −ˆj + 2ˆk and let b be a vector such that →a× b = 2ˆi −ˆk and →a⋅ b = 3 . Then the projection of b on the → vector →a− b is: (1) 2 (2) √21 2√37 (3) 2 (4) 2 3 3 √73

Q77.If the solution curve 𝑦= 𝑦𝑥 of the differential equation 𝑦2 d𝑥+ 𝑥2 - 𝑥𝑦+ 𝑦2d𝑦= 0, which passes through the point 1, 1 and intersects the line 𝑦= √3𝑥 at the point 𝛼, √3𝛼, then value of log𝑒√3𝛼 is equal to 𝜋 𝜋 (1) (2) 2 4 (3) 𝜋 (4) 𝜋 6 12 JEE Main 2022 (25 Jun Shift 1) JEE Main Previous Year Paper

Q77.If 2, 3, 9, 5, 2, 1, 1, 𝜆, 8 and 𝜆, 2, 3 are coplanar, then the product of all possible values of 𝜆 is (1) 21 (2) 59 2 8 57 95 (3) (4) 8 8

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 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 = αˆi + ˆj + βˆk and b = 3ˆi −5ˆj + 4ˆk be two vectors, such that →a× b = −ˆi + 9ˆi + 12ˆk. Then the → → projection of b −2→a on b +→a is equal to (1) 2 (2) 395 (3) 9 (4) 465 → → → 23 × b × 2ˆj is equal to ⋅ˆk = 2 , then

Q77.Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tan x(cos x −y). if the curve passes π through the point ( π4 , 0), then the value of ∫ 0 2 ydx is equal to (1) (2 −√2) + √2π (2) 2 − √2π (3) (2 + √2) + √2π (4) 2 + √2π →

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

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

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.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 𝑦= 𝑓𝑥 of the differential equation 𝑑𝑦 𝑥𝑦 = 𝑥4 + 2𝑥 , 𝑥∈-1, 1 pass through the 𝑑𝑥+ 𝑥2 - 1 √1 - 𝑥2 √3 origin. Then ∫ 2 𝑓𝑥𝑑𝑥 is equal to -√3 2 𝜋 1 𝜋 √3 (1) - (2) - 3 4 3 4 (3) 𝜋 - √3 (4) 𝜋 - √3 6 4 6 2

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)

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 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.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= 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.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.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.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