Practice Questions

Q64.Let two straight lines drawn from the origin O intersect the line 3x + 4y = 12 at the points P and Q such that △OPQ is an isosceles triangle and ∠POQ = 90∘ . If l = OP2 + PQ2 + QO2 , then the greatest integer less than or equal to l is : (1) 42 (2) 46 (3) 44 (4) 48

Q64.Let a variable line of slope m > 0 passing through the point (4, −9) intersect the coordinate axes at the points A and B. The minimum value of the sum of the distances of A and B from the origin is JEE Main 2024 (06 Apr Shift 1) JEE Main Previous Year Paper (1) 30 (2) 25 (3) 15 (4) 10

Q65.The sum of the solutions x ∈R of the equation 3 cos 2x+cos3 2x = x3 −x2 + 6 is cos6 x−sin6 x (1) 0 (2) 1 (3) −1 (4) 3

Q65.If the value of 3 is a√5−b , where a, b, c are natural numbers and gcd(a, c) = 1, then a + b + c is c 5 cos 36∘−3 sin 18∘ equal to : (1) 40 (2) 52 (3) 50 (4) 54

Q65.Let C be a circle with radius √10 units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope -1. Then, a distance (in units) between the chord PQ and the chord MN is (1) 3 −√2 (2) √2 + 1 (3) √2 −1 (4) 2 −√3

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.The vertices of a triangle are A(−1, 3), B(−2, 2) and C(3, −1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is : (1) x + y + (2 −√2) = 0 (2) −x + y −(2 −√2) = 0 (3) x + y −(2 −√2) = 0 (4) x −y −(2 + √2) = 0

Q66.Let a circle passing through (2, 0) have its centre at the point (h, k). Let (xc, yc) be the point of intersection of the lines 3x + 5y = 1 and (2 + c)x + 5c2y = 1. If h = limc→1 xc and k = limc→1 yc , then the equation of the circle is : (1) 25x2 + 25y2 −2x + 2y −60 = 0 (2) 5x2 + 5y2 −4x + 2y −12 = 0 (3) 5x2 + 5y2 −4x −2y −12 = 0 (4) 25x2 + 25y2 −20x + 2y −60 = 0 JEE Main 2024 (09 Apr Shift 1) JEE Main Previous Year Paper

Q66.In a Δ ABC , suppose y = x is the equation of the bisector of the angle B and the equation of the side AC is 2x −y = 2. If 2 AB = BC and the point A and B are respectively (4, 6) and (α, β), then α + 2β is equal to (1) −4 (2) 42 (3) 2 (4) −1 Q67. 1 ( π2 )3 1 lim ∫ x3 cos( t3 is equal to (x−π2 )2 )dt) x→π2 ( (1) 3π (2) 3π2 8 4 (3) 3π2 (4) 3π 8 4

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

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 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)

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

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 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 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

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.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.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.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

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.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