Practice Questions

Q62.Let 𝑆= 𝑧∈𝐶: 𝑧−1 = 1 and √2 −1𝑧+ ¯𝑧- 𝑖𝑧- ¯𝑧= 2√2. Let 𝑧1, 𝑧2 ∈𝑆 be such that 𝑧1 = max𝑧∈𝑠𝑧 and 2 𝑧2 = min𝑧∈𝑠𝑧. Then √2𝑧1 −𝑧2 equals: (1) 1 (2) 4 (3) 3 (4) 2

Q62.Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to (1) 18 (2) 16 (3) 12 (4) 15

Q62.In an A.P., the sixth term a6 = 2. If the a1a4a5 is the greatest, then the common difference of the A.P., is equal to (1) 3 (2) 8 2 5 (3) 2 (4) 5 3 8

Q62.For 0 < 𝑐< 𝑏< 𝑎, let ( 𝑎+ 𝑏– 2𝑐) 𝑥2 + ( 𝑏+ 𝑐– 2𝑎) 𝑥+ ( 𝑐+ 𝑎– 2𝑏) = 0 and 𝛼≠1 be one of its root. Then, among the two statements (I) If 𝛼∈-1, 0, then 𝑏 cannot be the geometric mean of 𝑎 and 𝑐. (II) If 𝛼∈0, 1, then 𝑏 may be the geometric mean of 𝑎 and 𝑐. (1) Both (I) and (II) are true (2) Neither (I) nor (II) is true (3) Only (II) is true (4) Only (I) is true 1 2 3

Q62.Let 𝑎 and 𝑏 be two distinct positive real numbers. Let 11th term of a GP, whose first term is 𝑎 and third term is 𝑏, is equal to 𝑝th term of another GP, whose first term is 𝑎 and fifth term is 𝑏. Then 𝑝 is equal to (1) 20 (2) 25 (3) 21 (4) 24

Q63.The coefficient of x70 in x2(1 + x)98 + x3(1 + x)97 + x4(1 + x)96 + … + x54(1 + x)46 is 99Cp −46Cq . Then a possible value of p + q is : (1) 55 (2) 83 (3) 61 (4) 68

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.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.Two vertices of a triangle ABC are A(3, −1) and B(−2, 3), and its orthocentre is P(1, 1). If the coordinates of the point C are (α, β) and the centre of the of the circle circumscribing the triangle PAB is (h, k), then the value of (α + β) + 2( h + k) equals (1) 5 (2) 81 (3) 15 (4) 51 and the eccentricity

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.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.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

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 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.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

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

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.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

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 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

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