Practice Questions

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 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

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.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

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 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 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.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.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 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.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.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

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.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.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.Let the locus of the mid points of the chords of circle 𝑥2 + 𝑦−12 = 1 drawn from the origin intersect the line 𝑥+ 𝑦= 1 at 𝑃 and 𝑄. Then, the length of 𝑃𝑄 is: 1 (1) (2) √2 √2 1 (3) (4) 1 2

Q67.If the shortest distance of the parabola y2 = 4x from the centre of the circle x2 + y2 −4x −16y + 64 = 0 is d , then d2 is equal to : (1) 16 (2) 24 (3) 20 (4) 36 y2 x2

Q67.A square is inscribed in the circle x2 + y2 −10x −6y + 30 = 0. One side of this square is parallel to y = x + 3. If (xi, yi) are the vertices of the square, then Σ (x2i + y2i ) is equal to: (1) 148 (2) 152 (3) 160 (4) 156

Q67.Let e1 be the eccentricity of the hyperbola x2 - y2 = 1 and e2 be the eccentricity of the ellipse 16 9 x2 y2 + = 1, a > b, which passes through the foci of the hyperbola. If e1e2 = 1, then the length of the chord a2 b2 of the ellipse parallel to the x-axis and passing through ( 0, 2 ) is : (1) 4√5 (2) 8√5 3 (3) 10√5 (4) 3√5 3 JEE Main 2024 (27 Jan Shift 2) JEE Main Previous Year Paper

Q67.If the line segment joining the points (5, 2) and (2, a) subtends an angle π4 at the origin, then the absolute value of the product of all possible values of a is : (1) 6 (2) 8 (3) 2 (4) -4

Q67.Let 𝑃 be a point on the hyperbola H: 𝑥2 - 𝑦2 = 1, in the first quadrant such that the area of triangle formed by 𝑃 9 4 and the two foci of H is 2√13. Then, the square of the distance of 𝑃 from the origin is JEE Main 2024 (30 Jan Shift 2) JEE Main Previous Year Paper (1) 18 (2) 26 (3) 22 (4) 20 Q68. 𝑥 0 0 2𝜋 4𝜋 Let 𝑅= 0 𝑦0 be a non-zero 3 × 3 matrix, where 𝑥sin𝜃= 𝑦sin𝜃+ = 𝑧sin𝜃+ ≠0, 𝜃∈( 0, 2𝜋) . 3 3 0 0 𝑧 For a square matrix 𝑀, let Trace𝑀 denote the sum of all the diagonal entries of 𝑀. Then, among the statements: I Trace ( 𝑅) = 0 ( II ) If Trace ( adj ( adj ( 𝑅) ) = 0, then 𝑅 has exactly one non-zero entry. (1) Both ( I ) and ( II ) are true (2) Only ( II ) is true (3) Neither ( I ) nor ( II ) is true (4) Only ( I ) is true

Q67.Let f(x) = x2 + 9, g(x) = x−9x and a = f ∘g(10), b = g ∘f(3). If e and l denote the eccentricity and the x2 y2 length of the latus rectum of the ellipse a + b = 1, then 8e2 + l2 is equal to. (1) 8 (2) 16 (3) 6 (4) 12

Q67.Let the circle C1 : x2 + y2 −2(x + y) + 1 = 0 and C2 be a circle having centre at (−1, 0) and radius 2 . If the line of the common chord of C1 and C2 intersects the y-axis at the point P, then the square of the distance of P from the centre of C1 is : (1) 2 (2) 1 (3) 4 (4) 6