Practice Questions

Q66.Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies: (1) r = 0 (2) 2r2 −4r + 1 = 0 (3) 2r2 −8r + 7 = 0 (4) r2 −8r + 8 = 0 JEE Main 2024 (05 Apr Shift 2) JEE Main Previous Year Paper

Q66.Let 𝐴𝑎, 𝑏, 𝐵3, 4 and −6, −8 respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point 𝑃2𝑎+ 3, 7𝑏+ 5 from the line 2𝑥+ 3𝑦−4 = 0 measured parallel to the line 𝑥−2𝑦−1 = 0 is (1) 15√5 (2) 17√5 7 6 (3) 17√5 (4) √5 7 17

Q66.Let 𝐴( 𝛼, 0 ) and 𝐵( 0, 𝛽) be the points on the line 5𝑥+ 7𝑦= 50. Let the point 𝑃 divide the line segment 𝐴𝐵 𝑥2 𝑦2 internally in the ratio 7: 3. Let 3𝑥- 25 = 0 be a directrix of the ellipse 𝐸: + = 1 and the corresponding 𝑎2 𝑏2 focus be 𝑆. If from 𝑆, the perpendicular on the 𝑥- axis passes through 𝑃, then the length of the latus rectum of 𝐸 is equal to 25 32 (1) (2) 3 9 (3) 25 (4) 32 9 5

Q67.Consider a hyperbola H having centre at the origin and foci on the x-axis. Let C1 be the circle touching the hyperbola H and having the centre at the origin. Let C2 be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of C1 and C2 are 36π and 4π, respectively, then the length (in units) of latus rectum of H is (1) 14 (2) 28 3 3 (3) 11 (4) 10 3 3

Q67.Let 𝑃 be a point on the ellipse 𝑥2 + 𝑦2 = 1. Let the line passing through 𝑃 and parallel to 𝑦- axis meet the 9 4 circle 𝑥2 + 𝑦2 = 9 at point 𝑄 such that 𝑃 and 𝑄 are on the same side of the 𝑥- axis. Then, the eccentricity of JEE Main 2024 (01 Feb Shift 2) JEE Main Previous Year Paper the locus of the point 𝑅 on 𝑃𝑄 such that 𝑃𝑅: 𝑅𝑄= 4: 3 as 𝑃 moves on the ellipse, is: 11 13 (1) (2) 19 21 (3) √139 (4) √13 23 7 𝑥

Q67.Let C be the circle of minimum area touching the parabola y = 6 −x2 and the lines y = √3|x|. Then, which one of the following points lies on the circle C ? (1) (1, 2) (2) (1, 1) (3) (2, 2) (4) (2, 4)

Q68.Let A and B be two square matrices of order 3 such that |A| = 3 and |B| = 2. Then ATA(adj(2 A))−1(adj(4 B))(adj(AB))−1AAT is equal to : (1) 108 (2) 32 (3) 81 (4) 64 Q69. 11x + y + λz = −5 If the system of equations 2x + 3y + 5z = 3 has infinitely many solutions, then λ4 −μ is equal to : 8x −19y −39z = μ (1) 51 (2) 45 (3) 47 (4) 49

Q68.Let f(x) = ∫x0 (t + sin (1 −e′))dt, x ∈R. Then, limx→0 f(x)x3 is equal to (1) −16 (2) 32 (3) −23 (4) 61

Q68.Let 𝑃 be a parabola with vertex 2, 3 and directrix 2𝑥+ 𝑦= 6. Let an ellipse 𝐸: 𝑥2 + 𝑦2 = 1, 𝑎> 𝑏 𝑎2 𝑏2 1 of eccentricity pass through the focus of the parabola 𝑃. Then the square of the length of the latus rectum √2 of 𝐸, is (1) 385 (2) 347 8 8 512 656 (3) (4) 25 25

Q68.Let f : (−∞, ∞) −{0} →R be a differentiable function such that f ′(1) = lima→∞a2f ( a1 ). Then a(a+1) lima→∞ 2 tan−1 ( a1 ) + a2 −2 loge a is equal to (1) 2 3 + π4 (2) 34 + π8 (3) 3 8 + π4 (4) 52 + π8

Q68.Let 𝑎 be the sum of all coefficients in the expansion of ( 1 – 2𝑥+ 2𝑥2 ) 2023 ( 3 - 4𝑥2 + 2𝑥3 ) 2024 and 𝑥log1 + 𝑡 ∫0 𝑑𝑡 𝑏= lim 𝑡2024 + 1 . If the equations 𝑐𝑥2 + 𝑑𝑥+ 𝑒= 0 and 2𝑏𝑥2 + 𝑎𝑥+ 4 = 0 have a common root, where 𝑥→0 𝑥2 𝑐, 𝑑, 𝑒∈𝑅, then 𝑑 : 𝑐 : 𝑒 equals (1) 2 : 1 : 4 (2) 4 : 1 : 4 (3) 1 : 2 : 4 (4) 1 : 1 : 4 Q69. 𝑥3 2𝑥2 + 1 1 + 3𝑥 If 𝑓𝑥= 3𝑥2 + 2 2𝑥 𝑥3 + 6 for all 𝑥∈ℝ, then 2𝑓0 + 𝑓'0 is equal to 𝑥3 −𝑥 4 𝑥2 −2 (1) 48 (2) 24 (3) 42 (4) 18

Q69.Let [t] be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and . The number of one-to-one functions from A to the + f : A →Z be the function f(x) = [log2 (x2 [ x35 ])] range of f is (1) 25 (2) 24 (3) 20 (4) 120

Q70.If A is a square matrix of order 3 such that det(A) = 3 and det (adj (−4 adj (−3 adj (3 adj ((2 A)−1))))) = 2m3n , then m + 2n is equal to : (1) 2 (2) 3 (3) 6 (4) 4 JEE Main 2024 (06 Apr Shift 2) JEE Main Previous Year Paper

Q71.The integral ∫3/41/4 cos (2 (1) 1/2 (2) −1/2 (3) −1/4 (4) 1/4

Q71.Let A = [ 10 21 ] and B = I + adj(A) + (adj A)2 + … + (adj A)10 . Then, the sum of all the elements of the matrix B is: (1) -124 (2) 22 (3) -88 (4) -110

Q72.If the domain of the function 𝑓𝑥= √𝑥2 −25 + + 2𝑥−15 is −∞, 𝛼∪𝛽, ∞, then 𝛼2 + 𝛽3 is equal to: 4 −𝑥2 log10𝑥2 (1) 140 (2) 175 (3) 150 (4) 125

Q72.Consider the function f: ( 0, 2 ) →R defined by f(x) = x + 2 and the function g ( x ) defined by 2 x min{f ( t ) }, 0 < t ≤x and 0 < x ≤1 gx = 3 . Then + x, 1 < x < 2 2 (1) g is continuous but not differentiable at x = 1 (2) g is not continuous for all x ∈( 0, 2 ) (3) g is neither continuous nor differentiable at x = 1(4) g is continuous and differentiable for all x ∈( 0, 2 )

Q72.If 𝑓𝑥= 4𝑥+ 3 , 𝑥≠2 and ( 𝑓𝑜𝑓) ( 𝑥) = 𝑔( 𝑥) , where 𝑔: 𝑅- 2 →𝑅- 2 then ( 𝑔𝑜𝑔𝑜𝑔) ( 4 ) is equal to 6𝑥- 4 3 3 3, 19 19 (1) - (2) 20 20 (3) -4 (4) 4 Q73. 𝑔𝑥, 𝑥≤0 Let 𝑔𝑥 be a linear function and 𝑓𝑥= 1 , is continuous at 𝑥= 0. If 𝑓'1 = 𝑓−1, then the value of 1 + 𝑥 𝑥, 𝑥> 0 2 + 𝑥 𝑔3 is JEE Main 2024 (31 Jan Shift 1) JEE Main Previous Year Paper 1 4 1 4 (1) 3log𝑒 1 (2) 3log𝑒 9 + 1 9𝑒 3 4 4 (3) log𝑒 9 −1 (4) log𝑒 1 9𝑒 3 𝑥𝑦𝑥- 1𝑥- 2

Q72.The function f: R->R, f(x) = x2+2x−15 , x ∈R is x2−4x+9 (1) one-one but not onto. (2) both one-one and onto. (3) onto but not one-one. (4) neither one-one nor onto. JEE Main 2024 (06 Apr Shift 1) JEE Main Previous Year Paper 1 ), x ≠0 x then

Q73.Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS . Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a + b)2 is equal to : (1) 72 (2) 60 (3) 64 (4) 80

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

Q73.If the function 𝑓: −∞, −1 →𝑎, 𝑏 defined by 𝑓𝑥= 𝑒𝑥3 −3𝑥+ 1 is one-one and onto, then the distance of the point 𝑃2𝑏+ 4, 𝑎+ 2 from the line 𝑥+ 𝑒−3𝑦= 4 is: (1) 2√1 + 𝑒6 (2) 4√1 + 𝑒6 (3) 3√1 + 𝑒6 (4) √1 + 𝑒6 JEE Main 2024 (31 Jan Shift 2) JEE Main Previous Year Paper

Q73.If f(x) = {x30 sin, x (= 0 (1) f ′′ ( π2 ) = 24−π22π (2) f ′′ ( π2 ) = 12−π22π (3) f ′′(0) = 1 (4) f ′′(0) = 0

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