Practice Questions

Q73.Let g : R →R be a non constant twice differentiable such that g′( 21 ) = g′( 23 ). If a real valued function f is defined as f(x) = 12 [ g(x) + g(2 −x)], then (1) f ′′(x) = 0 for atleast two x in (0, 2) (2) f ′′(x) = 0 for exactly one x in (0, 1) (3) f ′′(x) = 0 for no x in (0, 1) (4) f ′( 23 ) + f ′( 21 ) = 1

Q73.Let 𝑓: 𝑅→𝑅 be defined as 𝑎−𝑏cos2𝑥 ; 𝑥< 0 𝑥2 𝑓𝑥= 𝑥2 + 𝑐𝑥+ 2; 0 ≤𝑥≤1 2𝑥+ 1; 𝑥> 1 If 𝑓 is continuous everywhere in 𝑅 and 𝑚 is the number of points where 𝑓 is NOT differential then 𝑚 + 𝑎 + 𝑏 + 𝑐 equals: JEE Main 2024 (01 Feb Shift 1) JEE Main Previous Year Paper (1) 1 (2) 4 (3) 3 (4) 2 1

Q74.If the value of the integral ∫ −π2 2 ( x21+πxcos x 1+sin2 x π 1+e(sin x)2023 )dx (1) 3 (2) −32 (3) 2 (4) 32 JEE Main 2024 (29 Jan Shift 1) JEE Main Previous Year Paper

Q74.If 5𝑓𝑥+ 4𝑓 𝑥= 𝑥2 −2, ∀𝑥≠0 and 𝑦= 9𝑥2𝑓𝑥, then 𝑦 is strictly increasing in: (1) 0, 1 ∪1 ∞ (2) −1 0 ∪1 ∞ √5 √5, √5, √5, (3) −1 0 ∪0, 1 (4) −∞, 1 ∪0, 1 √5, √5 √5 √5 𝜋 Q75. 4 𝑥𝑑𝑥 The value of the integral ∫ equals: 0 sin42𝑥+ cos42𝑥 (1) √2𝜋2 (2) √2𝜋2 8 16 (3) √2𝜋2 (4) √2𝜋2 32 64

Q75.The solution curve, of the differential equation 2y dydx + 3 = 5 dydx , passing through the point (0, 1) is a conic, whose vertex lies on the line: JEE Main 2024 (09 Apr Shift 1) JEE Main Previous Year Paper (1) 2x + 3y = 9 (2) 2x + 3y = −9 (3) 2x + 3y = −6 (4) 2x + 3y = 6

Q75.If ∫ 3 3 √sin3 x cos3 x sin(x−θ) constant, then AB is equal to (1) 4 cosec (2θ) (2) 4 sec θ (3) 2 sec θ (4) 8 cosec (2θ) JEE Main 2024 (29 Jan Shift 2) JEE Main Previous Year Paper

Q75.For x ∈(−π2 , π2 ), if y(x) = ∫ cosecxcosecx+sinsec x+tan xx sin2 x dx and limπ = 0 then y( π4 ) is equal to x→( 2 )−y(x) (1) tan−1( √21 ) (2) 21 tan−1( √21 ) (3) −1 2 ) √2 tan−1( √21 ) (4) √21 tan−1(−1

Q75.Let f(x) be a positive function such that the area bounded by y = f(x), y = 0 from x = 0 to x = a > 0 is e−a + 4a2 + a −1. Then the differential equation, whose general solution is y = c1f(x) + c2 , where c1 and c2 are arbitrary constants, is d2y dy (1) (8ex −1) = 0 (2) (8ex −1) + dx d2y −dydx = 0 dx2 dx2 (3) (8ex + 1) + dxdy = 0 dx2 d2y −dydx = 0 (4) (8ex + 1) dx2d2y

Q76.Suppose the solution of the differential equation (2+α)x−βy+2 represents a circle passing through dx = βx−2αy−(βγ−4α) origin. Then the radius of this circle is : (1) 2 (2) √17 (3) 1 (4) √17 2 2 → →

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.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 →a, b and→cbe three non-zero vectors such that b and→care non-collinear if →a+ 5b is collinear with →c,→b + 6→cis collinear with →a and →a+ α→b + β→c= →0, then α + β is equal to (1) 35 (2) 30 (3) −30 (4) −25

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 = 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= −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

Q79.The shortest distance between lines 𝐿1 and 𝐿2, where 𝐿1: 2 = −3 = 2 and 𝐿2 is the line passing through 𝑥−3 𝑦 𝑧−1 the points 𝐴−4, 4, 3, 𝐵−1, 6, 3 and perpendicular to the line = = , is −2 3 1 (1) 121 (2) 24 √221 √117 (3) 141 (4) 42 √221 √117

Q79.Let P(α, β, γ) be the image of the point Q(3, −3, 1) in the line x−01 = y−31 = z−1−1 and R be the point (2, 5, −1). If the area of the triangle PQR is λ and λ2 = 14K , then K is equal to : (1) 36 (2) 81 (3) 72 (4) 18

Q80.The shortest distance between the lines x−34 = −11y+7 = z−15 and x−53 = y−9−6 = z+21 is: (1) 178 (2) 187 √563 √563 (3) 185 (4) 179 √563 √563

Q80.Let P be the point of intersection of the lines x−21 = y−45 = z−21 and x−32 = y−23 = z−32 . Then, the shortest distance of P from the line 4x = 2y = z is (1) 5√14 (2) 3√14 7 7 (3) √14 (4) 6√14 7 7

Q81.Let 𝑃= 𝑧∈ℂ: 𝑧+ 2 −3𝑖≤1 and 𝑄= 𝑧∈ℂ: 𝑧1 + 𝑖+ ¯𝑧1 −𝑖≤−8. Let in 𝑃∩𝑄, 𝑧−3 + 2𝑖 be maximum and minimum at 𝑧1 and 𝑧2 respectively. If 𝑧12 + 2𝑧2 = 𝛼+ 𝛽√2, where 𝛼, 𝛽 are integers, then 𝛼+ 𝛽 equals __________

Q81.Let α, β be the roots of the equation x2 −x + 2 = 0 with Im (α) >Im (β). Then α6 + α4 + β4 −5α2 is equal to

Q81.The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7 , when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to__________ JEE Main 2024 (08 Apr Shift 1) JEE Main Previous Year Paper

Q4. A small particle of mass m moves in such a way that its potential energy U = 12 mω2r2 where ω is constant and r is the distance of the particle from origin. Assuming Bohr’s quantization of momentum and circular orbit, the radius of nth orbit will be proportional to (1) √n (2) n1 (3) n2 (4) n

Q15.In a cuboid of dimension 2L × 2L × L , a charge q is placed at the centre of the surface S having area of 4L2 . The flux through the opposite surface to S is given by (1) q (2) q 12∈0 3∈0 (3) q (4) q 2∈0 6∈0

Q15.A dipole comprises of two charged particles of identical magnitude q and opposite in nature. The mass m of the positive charged particle is half of the mass of the negative charged particle. The two charges are separated → by a distance l . If the dipole is placed in a uniform electric field E ; in such a way that dipole axis makes a JEE Main 2023 (06 Apr Shift 2) JEE Main Previous Year Paper → very small angle with the electric field, E . The angular frequency of the oscillations of the dipole when released is given by: (1) √3qE2ml (2) √8qEml (3) √4qEml (4) √8qE3ml