Practice Questions

Q64.Let 𝛼, 𝛽, 𝛾, 𝛿∈𝑍 and let 𝐴𝛼, 𝛽, 𝐵1, 0, 𝐶𝛾, 𝛿 and 𝐷1, 2 be the vertices of a parallelogram 𝐴𝐵𝐶𝐷. If 𝐴𝐵= √10 and the points 𝐴 and 𝐶 lie on the line 3𝑦= 2𝑥+ 1, then 2𝛼+ 𝛽+ 𝛾+ 𝛿 is equal to (1) 10 (2) 5 (3) 12 (4) 8

Q64.A line passing through the point A(9, 0) makes an angle of 30° with the positive direction of x-axis. If this line is rotated about A through an angle of 15° in the clockwise direction, then its equation in the new position is (1) y + x = 9 (2) x + y = 9 √3−2 √3−2 (3) x + y = 9 (4) y + x = 9 √3+2 √3+2

Q64.Let 2nd, 8th and 44th, terms of a non-constant 𝐴. 𝑃. be respectively the 1st, 2nd and 3rd terms of 𝐺. 𝑃. If the first term of A.P. is 1 then the sum of first 20 terms is equal to- (1) 980 (2) 960 (3) 990 (4) 970

Q64.If the constant term in the expansion of 12 + , x ≠0, is α × 28 × 5√3, then 25α is equal to : ( 5√3x 2x ) 3√5 (1) 724 (2) 742 (3) 639 (4) 693

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 ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then : (1) P2 = 6√3Q (2) P2 = 36√3Q (3) P = 36√3Q2 (4) P2 = 72√3Q

Q64.For 𝛼, 𝛽∈0, let 3sin ( 𝛼+ 𝛽) = 2sin ( 𝛼- 𝛽) and a real number 𝑘 be such that tan𝛼= tan𝛽. Then the 2 value of 𝑘 is equal to (1) -5 (2) 5 (3) 2 (4) -2 3 3

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 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 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.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.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.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 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.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 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 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 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.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 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 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