Practice Questions

Q64.Let the first three terms 2, p and q , with q ≠2, of a G.P. be respectively the 7th , 8th and 13th terms of an A.P. If the 5th term of the G.P. is the nth term of the A.P., then n is equal to: (1) 163 (2) 151 (3) 177 (4) 169

Q64.If α, −π2 < α < π2 is the solution of 4 cos θ + 5 sin θ = 1, then the value of tan α is (1) 10−√10 (2) 10−√10 6 12 (3) √10−10 (4) √10−10 12 6

Q64.If the term independent of x in the expansion of (√ax2 + 2x31 )10 is 105 , then a2 is equal to : (1) 2 (2) 4 (3) 6 (4) 9 JEE Main 2024 (08 Apr Shift 2) JEE Main Previous Year Paper cos 36∘+5 sin 18∘

Q64.If 2tan2𝜃- 5sec𝜃= 1 has exactly 7 solutions in the interval 0, n𝜋 , for the least value of n ∈N then n k is 2 ∑k = 1 2k equal to : - 15 (1) 2152141 - 14 (2) 2142151 15 1 (3) 1 - (4) - 15 213 213214

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.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.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.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.Let A(−1, 1) and B(2, 3) be two points and P be a variable point above the line AB such that the area of △PAB is 10 . If the locus of P is ax + by = 15, then 5a + 2 b is : (1) 6 (2) −65 (3) 4 (4) −125

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.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

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 (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 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 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.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 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 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 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

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.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.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.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.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,