Practice Questions

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 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 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.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.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.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.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 𝑥2 - 𝑦2 + 2ℎ𝑥𝑦+ 2𝑔𝑥+ 2𝑓𝑦+ 𝑐= 0 is the locus of a point, which moves such that it is always equidistant from the lines 𝑥+ 2𝑦+ 7 = 0 and 2𝑥- 𝑦+ 8 = 0, then the value of 𝑔+ 𝑐+ ℎ- 𝑓 equals (1) 14 (2) 6 (3) 8 (4) 29

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.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.Let 𝐴 and 𝐵 be two finite sets with 𝑚 and 𝑛 elements respectively. The total number of subsets of the set 𝐴 is 56 more than the total number of subsets of 𝐵. Then the distance of the point P ( m, n ) from the point Q ( - 2, - 3 ) is (1) 10 (2) 6 (3) 4 (4) 8

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.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 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.If the foci of a hyperbola are same as that of the ellipse 𝑥2 + 𝑦2 = 1 and the eccentricity of the hyperbola is 15 9 25 8 14 2 times the eccentricity of the ellipse, then the smaller focal distance of the point √2, 3 √ 5 on the hyperbola, JEE Main 2024 (31 Jan Shift 1) JEE Main Previous Year Paper is equal to 2 8 2 4 (1) (2) - - 7√ 14√ 5 3 5 3 2 16 2 8 (3) (4) - + 14√ 7√ 5 3 5 3

Q66.If P(6, 1) be the orthocentre of the triangle whose vertices are A(5, −2), B(8, 3) and C(h, k), then the point C lies on the circle: (1) x2 + y2 −61 = 0 (2) x2 + y2 −52 = 0 (3) x2 + y2 −65 = 0 (4) x2 + y2 −74 = 0

Q66.A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are m and n, respectively, then m + n2 is equal to (1) 408 (2) 414 (3) 396 (4) 312

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 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( 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.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 PQ be a chord of the parabola y2 = 12x and the midpoint of PQ be at (4, 1). Then, which of the following point lies on the line passing through the points P and Q? (1) (3, −3) (2) (2, −9) (3) ( 23 , −16) (4) ( 12 , −20)

Q66.Let A be the point of intersection of the lines 3x + 2 y = 14, 5 x −y = 6 and B be the point of intersection of the lines 4 x + 3 y = 8, 6 x + y = 5. The distance of the point P(5, −2) from the line AB is (1) 13 (2) 8 2 (3) 5 (4) 6 2

Q66.Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5, 5) is : (1) 2√2 (2) 4√2 (3) 4 (4) 5