Practice Questions

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.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 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 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.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.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 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 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 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.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.Let H : −x2 + y2 = 1 be the hyperbola, whose eccentricity is √3 and the length of the latus rectum is 4√3. a2 b2 Suppose the point (α, 6), α > 0 lies on H . If β is the product of the focal distances of the point (α, 6), then α2 + β is equal to (1) 172 (2) 171 (3) 169 (4) 170 Q68. ⎡ 2 a 0 ⎤ Let A = 1 3 1 . If A3 = 4A2 −A −21I , where I is the identity matrix of order 3 × 3, then 2a + 3b is ⎣ 0 5 b ⎦ equal to (1) -9 (2) -13 (3) -10 (4) -12

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.Let + = 1, 𝑎> 𝑏 be an ellipse, whose eccentricity is 1 and the length of the latus rectum is √14. Then 𝑎2 √2 𝑏2 𝑥2 𝑦2 the square of the eccentricity of − = 1 is: 𝑎2 𝑏2 7 (1) 3 (2) 2 3 5 (3) (4) 2 2

Q67.Let C be the circle of minimum area touching the parabola y = 6 −x2 and the lines y = √3|x|. Then, which one of the following points lies on the circle C ? (1) (1, 2) (2) (1, 1) (3) (2, 2) (4) (2, 4)

Q67.The distance of the point (2, 3) from the line 2x −3y + 28 = 0, measured parallel to the line √3x −y + 1 = 0, is equal to JEE Main 2024 (29 Jan Shift 2) JEE Main Previous Year Paper (1) 4√2 (2) 6√3 (3) 3 + 4√2 (4) 4 + 6√3

Q67.Let the line 2x + 3y −k = 0, k > 0 , intersect the x -axis and y -axis at the points A and B , respectively. If the equation of the circle having the line segment AB as a diameter is x2 + y2 −3x −2y = 0 and the length of the latus rectum of the ellipse x2 + 9y2 = k2 is mn , where m and n are coprime, then 2 m + n is equal to (1) 11 (2) 10 (3) 12 (4) 13 JEE Main 2024 (05 Apr Shift 1) JEE Main Previous Year Paper

Q67.Let a variable line passing through the centre of the circle 𝑥2 + 𝑦2 −16𝑥−4𝑦= 0, meet the positive co- ordinate axes at the point 𝐴 and 𝐵. Then the minimum value of 𝑂𝐴+ 𝑂𝐵, where 𝑂 is the origin, is equal to (1) 12 (2) 18 (3) 20 (4) 24

Q67.If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is : (1) √5 (2) √3 3 2 (3) 1 (4) 2 √3 √5 π 1 x ∫x0 f(t)dt lim = α, then 8α2 is equal

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 𝑃 be a point on the ellipse 𝑥2 + 𝑦2 = 1. Let the line passing through 𝑃 and parallel to 𝑦- axis meet the 9 4 circle 𝑥2 + 𝑦2 = 9 at point 𝑄 such that 𝑃 and 𝑄 are on the same side of the 𝑥- axis. Then, the eccentricity of JEE Main 2024 (01 Feb Shift 2) JEE Main Previous Year Paper the locus of the point 𝑅 on 𝑃𝑄 such that 𝑃𝑅: 𝑅𝑄= 4: 3 as 𝑃 moves on the ellipse, is: 11 13 (1) (2) 19 21 (3) √139 (4) √13 23 7 𝑥

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.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.Consider a hyperbola H having centre at the origin and foci on the x-axis. Let C1 be the circle touching the hyperbola H and having the centre at the origin. Let C2 be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of C1 and C2 are 36π and 4π, respectively, then the length (in units) of latus rectum of H is (1) 14 (2) 28 3 3 (3) 11 (4) 10 3 3

Q67. lim 𝑒2sin𝑥- 2sin𝑥- 1 𝑥→0 𝑥2 (1) is equal to -1 (2) does not exist (3) is equal to 1 (4) is equal to 2