Practice Questions

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

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.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.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.Let 𝑦= 𝑓( 𝑥) be a thrice differentiable function in ( - 5, 5 ) . Let the tangents to the curve 𝑦= 𝑓( 𝑥) at ( 1, f ( 1 ) ) and ( 3, f ( 3 ) ) make angles 𝜋 and 𝜋 respectively with positive x-axis. If 6 4, 3 2 27 ∫1 𝑓'𝑡 + 1𝑓"𝑡𝑑𝑡= 𝛼+ 𝛽√3 where 𝛼, 𝛽 are integers, then the value of 𝛼+ 𝛽 equals JEE Main 2024 (30 Jan Shift 2) JEE Main Previous Year Paper (1) -14 (2) 26 (3) -16 (4) 36 39 , then 𝑓𝑥- 𝑐𝑥𝑑𝑥=

Q75.The value of the integral ∫2−1 loge (x + √x2 + 1)dx JEE Main 2024 (09 Apr Shift 2) JEE Main Previous Year Paper (1) √5 −√2 + loge ( 7+4√51+√2 ) (2) √5 −√2 + loge ( 9+4√51+√2 ) + loge (3) √2 −√5 + loge ( 7+4√51+√2 ) (4) √2 −√5 ( 9+4√51+√2 )

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 → →

Q76.If (a, b) be the orthocentre of the triangle whose vertices are (1, 2), (2, 3) and (3, 1), and I1 = ∫ba xsin(4x −x2) dx, I2 = ∫ba sin(4x −x2) dx , then 36 I1I2 is equal to : (1) 72 (2) 88 (3) 80 (4) 66

Q77.Between the following two statements: Statement I : Let →a = ^i + 2^j −3^k and →b = 2^i + ^j −^k. Then the vector →r satisfying →a × →r = →a × →b and →a ⋅→r = 0 is of magnitude √10. Statement II : In a triangle ABC, cos 2A + cos 2B + cos 2C ≥−32 . (1) Statement I is incorrect but Statement II is (2) Both Statement I and Statement II are correct. correct. (3) Statement I is correct but Statement II is (4) Both Statement I and Statement II are incorrect. incorrect.

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.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 →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 = 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 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 the position vectors of the vertices A, B and C of a triangle be 2 ^i + 2 ^j + ^k, ^i + 2 ^j + 2 ^k and 2 ^i + ^j + 2 ^k respectively. Let l1, l2 and l3 be the lengths of perpendiculars drawn from the ortho centre of the triangle on the sides AB, BC and CA respectively, then l12 + l22 + l32 equals : 1 1 (1) (2) 5 2 (3) 1 (4) 1 4 3 x y - 1 z - 2

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.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

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 the image of the point ( 1, 0, 7 ) in the line = = be the point ( α, β, γ ) . Then which one of the 1 2 3 2π 3π following points lies on the line passing through ( α, β, γ ) and making angles and with y - axis and z - 3 4 axis respectively and an acute angle with x - axis? (1) ( 1, - 2, 1 + √2 ) (2) ( 1, 2, 1 - √2 ) (3) ( 3, 4, 3 - 2√2 ) (4) ( 3, - 4, 3 + 2√2 )

Q80.If an unbiased dice is rolled thrice, then the probability of getting a greater number in the ith roll than the number obtained in the (i −1)th roll, i = 2, 3, is equal to (1) 3/54 (2) 2/54 (3) 1/54 (4) 5/54

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 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.Let the set C = {(x, y) ∣x2 −2y = 2023, x, y ∈N}. Then ∑(x,y)∈C(x y)