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.In a Δ ABC , suppose y = x is the equation of the bisector of the angle B and the equation of the side AC is 2x −y = 2. If 2 AB = BC and the point A and B are respectively (4, 6) and (α, β), then α + 2β is equal to (1) −4 (2) 42 (3) 2 (4) −1 Q67. 1 ( π2 )3 1 lim ∫ x3 cos( t3 is equal to (x−π2 )2 )dt) x→π2 ( (1) 3π (2) 3π2 8 4 (3) 3π2 (4) 3π 8 4

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.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 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

Q66.Let a circle passing through (2, 0) have its centre at the point (h, k). Let (xc, yc) be the point of intersection of the lines 3x + 5y = 1 and (2 + c)x + 5c2y = 1. If h = limc→1 xc and k = limc→1 yc , then the equation of the circle is : (1) 25x2 + 25y2 −2x + 2y −60 = 0 (2) 5x2 + 5y2 −4x + 2y −12 = 0 (3) 5x2 + 5y2 −4x −2y −12 = 0 (4) 25x2 + 25y2 −20x + 2y −60 = 0 JEE Main 2024 (09 Apr Shift 1) JEE Main Previous Year Paper

Q66.Let 𝐴𝑎, 𝑏, 𝐵3, 4 and −6, −8 respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point 𝑃2𝑎+ 3, 7𝑏+ 5 from the line 2𝑥+ 3𝑦−4 = 0 measured parallel to the line 𝑥−2𝑦−1 = 0 is (1) 15√5 (2) 17√5 7 6 (3) 17√5 (4) √5 7 17

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

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 + = 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 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.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 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.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 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.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.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 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

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. 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

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 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.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.If the locus of the point, whose distances from the point (2, 1) and (1, 3) are in the ratio 5 : 4, is ax2 + by2 + cxy + dx + ey + 170 = 0, then the value of a2 + 2b + 3c + 4d + e is equal to : (1) 37 (2) 437 (3) -27 (4) 5 (12−1)(n−1)+(22−2)(n−2)+⋯+((n−1)2−(n−1))⋅1

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