Practice Questions

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.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.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.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.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 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.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 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 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.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 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.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 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 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.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

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 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 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.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

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)