Practice Questions

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 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.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 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.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 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 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.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 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

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 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.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 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 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.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.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.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 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 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

Q68.If the mean and variance of five observations are 24 and 194 respectively and the mean of first four 5 25 observations is 7 , then the variance of the first four observations in equal to 2 (1) 4 (2) 77 5 12 (3) 5 (4) 105 4 4

Q68.Let 𝑓𝑥= 𝑥−1, 𝑥 is even, 𝑥∈𝑁. If for some 𝑎∈𝑁, 𝑓𝑓𝑓𝑎= 21, then lim 𝑥3 where 𝑡 denotes the 2𝑥, 𝑥 is odd, 𝑥→𝑎− 𝑎− 𝑎, greatest integer less than or equal to 𝑡, is equal to: (1) 121 (2) 144 (3) 169 (4) 225

Q68. e−(1+2x) 2x1 limx→0 x is equal to (1) 0 (2) −2 e (3) e (4) e −e2