Practice Questions

Q65.The portion of the line 4x + 5y = 20 in the first quadrant is trisected by the lines L1 and L2 passing through the origin. The tangent of an angle between the lines L1 and L2 is : (1) 8 (2) 25 5 41 (3) 2 (4) 30 5 41

Q65.The sum of all rational terms in the expansion of 1 1 15 is equal to : 5 + 5 3 (2 ) (1) 3133 (2) 931 (3) 6131 (4) 633 JEE Main 2024 (04 Apr Shift 1) JEE Main Previous Year Paper

Q65.If for some 𝑚, 𝑛; 6 𝐶𝑚+ 26𝐶𝑚+ 1+6𝐶𝑚+ 2 >8 𝐶3 and 𝑛−1𝑃3:𝑛𝑃4 = 1: 8, then 𝑛𝑃𝑚+ 1+𝑛+ 1𝐶𝑚 is equal to (1) 380 (2) 376 (3) 384 (4) 372 JEE Main 2024 (31 Jan Shift 2) JEE Main Previous Year Paper

Q65.If tan𝐴= 1 tan𝐵= and tan𝐶= 𝑥−3 + 𝑥−2 + 𝑥−1 2, 0 < 𝐴, 𝐵, 𝐶< 𝜋 then 𝐴+ 𝐵 is equal √𝑥𝑥2 + 𝑥+ 1, √𝑥2 + 𝑥+ 1 2, to: (1) 𝐶 (2) 𝜋−𝐶 (3) 2𝜋−𝐶 (4) 𝜋 −𝐶 2 JEE Main 2024 (01 Feb Shift 1) JEE Main Previous Year Paper

Q65.The equations of two sides AB and AC of a triangle ABC are 4x + y = 14 and 3x −2y = 5, respectively. The point (2, −43 ) divides the third side BC internally in the ratio 2 : 1. the equation of the side BC is (1) x + 3y + 2 = 0 (2) x −6y −10 = 0 (3) x −3y −6 = 0 (4) x + 6y + 6 = 0 touch each other

Q65.A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of m is equal to: (1) 150 (2) 180 (3) 160 (4) 125

Q65.A ray of light coming from the point P(1, 2) gets reflected from the point Q on the x-axis and then passes through the point R(4, 3). If the point S(h, k) is such that PQRS is a parallelogram, then hk2 is equal to : (1) 70 (2) 80 (3) 60 (4) 90

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.The number of solutions of the equation 4sin2𝑥−4cos3𝑥+ 9 −4cos𝑥= 0; 𝑥∈−2𝜋, 2𝜋 is: (1) 1 (2) 3 (3) 2 (4) 0

Q65.If A(1, −1, 2), B(5, 7, −6), C(3, 4, −10) and D(−1, −4, −2) are the vertices of a quadrilateral ABCD , then its area is : (1) 48√7 (2) 12√29 (3) 24√7 (4) 24√29

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 one of the diameters of the circle 𝑥2 + 𝑦2 - 10𝑥+ 4𝑦+ 13 = 0 is a chord of another circle 𝐶, whose center is the point of intersection of the lines 2𝑥+ 3𝑦= 12 and 3𝑥- 2𝑦= 5, then the radius of the circle 𝐶 is (1) √20 (2) 4 (3) 6 (4) 3√2

Q65.Let (5, a4 ), be the circumcenter of a triangle with vertices A(a, −2), B(a, 6) and C( a4 , −2). Let α denote the circumradius, β denote the area and γ denote the perimeter of the triangle. Then α + β + γ is (1) 60 (2) 53 (3) 62 (4) 30 JEE Main 2024 (29 Jan Shift 1) JEE Main Previous Year Paper

Q65.If the circles (x + 1)2 + (y + 2)2 = r2 and x2 + y2 −4x −4y + 4 = 0 intersect at exactly two distinct points, then (1) 5 < r < 9 (2) 0 < r < 7 (3) 3 < r < 7 (4) 21 < r < 7

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

Q65.If A(3, 1, −1), B ( 35 , 37 , 13 ), C(2, 2, 1) and D ( 103 , 23 , −13 ) are the vertices of a quadrilateral ABCD, then its area is (1) 2√2 (2) 5√2 3 3 (3) 2√2 (4) 4√2 3

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.Four distinct points (2k, 3k), (1, 0), (0, 1) and (0, 0) lie on a circle for k equal to : (1) 2 (2) 3 13 13 (3) 5 (4) 1 13 13

Q66.Let R be the interior region between the lines 3x - y + 1 = 0 and x + 2y - 5 = 0 containing the origin. The set of all values of 𝑎, for which the points a2, a + 1 lie in R, is : (1) ( - 3, - 1) ∪- 1 1 (2) ( - 3, 0) ∪ 1 1 3, 3, (3) ( - 3, 0) ∪ 2 1 (4) ( - 3, - 1) ∪ 1 1 3, 3,

Q66.Let the foci of a hyperbola H coincide with the foci of the ellipse E : (x−1)2100 + (y−1)275 = 1 of the hyperbola H be the reciprocal of the eccentricity of the ellipse E . If the length of the transverse axis of JEE Main 2024 (09 Apr Shift 2) JEE Main Previous Year Paper H is α and the length of its conjugate axis is β , then 3α2 + 2β2 is equal to (1) 237 (2) 242 (3) 205 (4) 225 Q67. ∫(π/2)3x3 (sin(2t1/3)+cos(t1/3))dt limx→π2 is equal to (x−π2 )2 ( ) (1) 5π2 (2) 9π2 9 8 (3) 11π2 (4) 3π2 10 2

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 the circles C1 : (x −α)2 + (y −β)2 = r21 and C2 : (x −8)2 + (y −152 ) 2 = r22 externally at the point (6, 6). If the point (6, 6) divides the line segment joining the centres of the circles C1 and C2 internally in the ratio 2 : 1, then (α + β) + 4 (r21 + r22) equals (1) 125 (2) 130 (3) 110 (4) 145

Q66.Let 𝐶: 𝑥2 + 𝑦2 = 4 and 𝐶': 𝑥2 + 𝑦2 −4𝜆𝑥+ 9 = 0 be two circles. If the set of all values of 𝜆 so that the circles 𝐶 and 𝐶' intersect at two distinct points, is 𝑅−𝑎, 𝑏, then the point 8𝑎+ 12, 16𝑏−20 lies on the curve: (1) 𝑥2 + 2𝑦2 −5𝑥+ 6𝑦= 3 (2) 5𝑥2 −𝑦= −11 (3) 𝑥2 −4𝑦2 = 7 (4) 6𝑥2 + 𝑦2 = 42 𝑥2 𝑦2

Q66.The maximum area of a triangle whose one vertex is at (0, 0) and the other two vertices lie on the curve y = −2x2 + 54 at points (x, y) and (−x, y) where y > 0 is : (1) 88 (2) 122 (3) 92 (4) 108

Q66.If the image of the point (−4, 5) in the line x + 2y = 2 lies on the circle (x + 4)2 + (y −3)2 = r2 , then r is equal to: (1) 2 (2) 3 (3) 1 (4) 4