Practice Questions

Q77.Let y = y(x) be the solution of the differential equation (x2 + 4)2dy + (2x3y + 8xy −2)dx = 0. If y(0) = 0, then y(2) is equal to (1) π (2) 2π 32 (3) π (4) π 8 16

Q77.Let 𝑦= 𝑦𝑥 be the solution of the differential equation 𝑑𝑦 2𝑥𝑥+ 𝑦3 −𝑥𝑥+ 𝑦−1, 𝑦0 = 1. Then, 1 + 𝑦1 𝑑𝑥= √2 √2 equals: (1) 4 (2) 3 4 + √𝑒 3 −√𝑒 2 1 (3) (4) 1 + √𝑒 2 −√𝑒

Q77.Let →𝑎= 3 ^𝑖+ ^𝑗−2 ^𝑘, 𝑏= 4 ^𝑖+ ^𝑗+ 7 ^𝑘 and →𝑐= ^𝑖−3 ^𝑗+ 4 ^𝑘 be three vectors. If a vectors →𝑝 satisfies →𝑝× →𝑏= →𝑐× →𝑏 and →𝑝⋅→𝑎= 0, then →𝑝⋅ ^𝑖− ^𝑗− ^𝑘 is equal to (1) 24 (2) 36 (3) 28 (4) 32

Q77.If the solution y = y(x) of the differential equation (x4 + 2x3 + 3x2 + 2x + 2)dy −(2x2 + 2x + 3)dx = 0 satisfies y(−1) = −π4 , then y(0) is equal to : (1) π 2 (2) −π2 (3) 0 (4) π 4 →

Q77.Consider three vectors →a,→b, →c. Let |→a| = 2, |→b| = 3 and →a = →b × →c. If α ∈[0, 3 ] is the angle between the vectors →b and →c, then the minimum value of 27|→c −→a|2 is equal to: (1) 110 (2) 124 (3) 121 (4) 105

Q77.Let OA→ =→a, OB→ = 12→a+ 4→b and OC→ = →b, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then area of the areaquadrilateralof S OABC is equal to _____ (1) 6 (2) 10 (3) 7 (4) 8

Q78.If the line 2−x 3 = 4λ+13y−2 = 4 −z makes a right angle with the line x+33μ = 1−2y6 = 5−z7 , then 4λ + 9μ is equal to : (1) 4 (2) 13 (3) 5 (4) 6

Q78.The distance of the point 𝑄( 0, 2, – 2 ) form the line passing through the point 𝑃( 5, – 4, 3 ) and perpendicular to the lines →𝑟= −3 ^𝑖+ 2 ^𝑘+ 𝜆2 ^𝑖+ 3 ^𝑗+ 5 ^𝑘, 𝜆∈ℝ and →𝑟= ^𝑖−2 ^𝑗+ ^𝑘+ 𝜇− ^𝑖+ 3 ^𝑗+ 2 ^𝑘, 𝜇∈ℝ is (1) √86 (2) √20 (3) √54 (4) √74

Q78.Let OA→ = 2→a, OB = 6→a + 5→b and OC = 3→b, where O is the origin. If the area of the parallelogram with −−→ → adjacent sides OA and OC is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to : (1) 32 (2) 40 (3) 38 (4) 35

Q78.Let a unit vector which makes an angle of 60∘ with 2^i + 2^j −^k and angle 45∘ with ^i −^k be C. Then → is : C + + (−12^i 1 ^j −√23 ^k) 3√2 (1) √2 + − + 3 + 21 )^i 1 )^j + √23 )^k ^i −12 ^k (2) ( √31 ( √31 3√2 ( √31 2√2 (3) √2 ^i + + 3 3√2 1 ^j −12 ^k (4) −√23 ^i + √23 ^j + ( 21 3 )^k

Q78.Let →a = ^i + 2^j + 3^k, b = 2^i + 3^j −5^k and→c= 3^i −^j + λ^k be three vectors. Let→rbe anit vector along →b + →c. If →r ⋅→a = 3, then 3λ is equal to: (1) 21 (2) 30 (3) 25 (4) 27

Q78.Let O be the origin and the position vector of A and B be 2ˆi + 2ˆj + ˆk and 2ˆi + 4ˆj + 4ˆk respectively. If the internal bisector of ∠AOB meets the line AB at C , then the length of OC is (1) 3 2 √31 (2) 32 √34 (3) 3 4 √34 (4) 23 √31

Q78.Let a unit vector ˆu = xˆi + yˆj + zˆk make angles π2 , π3 and 2π3 with the vectors √2ˆi1 + √21 ˆk, √21 ˆj + √21 ˆk and 1 + 1 ˆj respectively. If →v= 1 + 1 ˆj + 1 ˆk, then |^u −→v|2 is equal to √2ˆi √2 √2ˆi √2 √2 (1) 11 (2) 5 2 2 (3) 9 (4) 7

Q78.Let →𝑎= −5 ^𝑖+ ^𝑗−3 ^𝑘, →𝑏= ^𝑖+ 2 ^𝑗−4 ^𝑘 and →𝑐= →𝑎× →𝑏× ^𝑖× ^𝑖× ^𝑖. Then →𝑐⋅− ^𝑖+ ^𝑗+ ^𝑘 is equal to (1) -12 (2) -10 (3) -13 (4) -15

Q78.Let y = y(x) be the solution of the differential equation (2x loge x) dxdy + 2y = x3 loge x, x > 0 and y (e−1) = 0. Then, y(e) is equal to (1) −3e (2) −32e (3) −23e (4) −2e

Q78.Let 𝛼, 𝛽, 𝛾 be mirror image of the point 2, 3, 5 in the line 𝑥−1 = 𝑦−2 = 𝑧−3 . Then 2𝛼+ 3𝛽+ 4𝛾 is equal to 2 3 4 (1) 32 (2) 33 (3) 31 (4) 34 𝑥−1 𝑦+ 1 𝑧+ 4

Q78.Let →𝑎 and →𝑏 be two vectors such that | →𝑏| = 1 and | →𝑏× →𝑎| = 2 Then |( →𝑏× →𝑎) - →𝑏| (1) 3 (2) 5 (3) 1 (4) 4

Q78.If →a = ˆi + 2ˆj + ˆk, b = 3(ˆi −ˆj + ˆk) is equal to × − b →a⋅((→c → → b) −→c) (1) 32 (2) 24 (3) 20 (4) 36

Q78.Let →a = ^i + ^j + ^k,→b = 2^i + 4^j −5^k and →c = x^i + 2^j + 3^k, x ∈R. If →d is the unit vector in the direction of →b + →c such that →a ⋅→d = 1, then (→a × →b) ⋅→c is equal to (1) 11 (2) 3 (3) 9 (4) 6

Q78.If the mirror image of the point 𝑃( 3, 4, 9 ) in the line 𝑥−1 = 𝑦+ 1 = 𝑧−2 is 𝛼, 𝛽, 𝛾, then 14𝛼+ 𝛽+ 𝛾 is: 3 2 1 (1) 102 (2) 138 (3) 108 (4) 132 𝑥+ 3 𝑦−4 𝑧+ 1

Q78.Let →a = aiˆi + a2ˆj + a3ˆk and b = b1ˆi + b2ˆj + b3ˆk be two vectors such that →a = 1;→a⋅ b = 2 and b = 4. If × −3b, then the angle between b and →cis equal to : →c= 2(→a → → → b) JEE Main 2024 (30 Jan Shift 1) JEE Main Previous Year Paper (1) cos−1( √32 ) (2) cos−1(−1√3 ) 2 ) 3 (3) cos−1(−√32 ) (4) cos−1(

Q78.If the shortest distance between the lines is √n L2 : →r = 2(1 + μ)^i + 3(1 + μ)^j + (5 + μ)^k, μ ∈R , where gcd(m, n) = 1, then the value of m + n equals (1) 390 (2) 384 (3) 377 (4) 387

Q78.Let →a = 6^i + ^j −^k and b = ^i + ^j. If→cis a is vector such that |→c| ≥6,→a⋅→c= 6|→c|, |→c−→a| = 2√2 and the angle between →a × →b and →c is 60∘ , then |(→a × →b) × →c| is equal to: (1) 9 2 (6 −√6) (2) 23 √6 (3) 9 2 (6 + √6) (4) 23 √3

Q78.Let →a = 2^i + 5^j −^k,→b = 2^i −2^j + 2^k and→cbe three vectors such that (→c +^i) × (→a + →b +^i) = →a × (→c +^i). If →a ⋅→c = −29, then →c ⋅(−2^i + ^j + ^k) is equal to: (1) 15 (2) 12 (3) 10 (4) 5

Q79.Let the point, on the line passing through the points P(1, −2, 3) and Q(5, −4, 7), farther from the origin and at distance of 9 units from the point P, be (α, β, γ). Then α2 + β2 + γ 2 is equal to : (1) 165 (2) 160 (3) 155 (4) 150