Practice Questions

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.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 + = 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.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. 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.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 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 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 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.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.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

Q68.Let f(x) = ∫x0 (t + sin (1 −e′))dt, x ∈R. Then, limx→0 f(x)x3 is equal to (1) −16 (2) 32 (3) −23 (4) 61

Q68.Let f : (−∞, ∞) −{0} →R be a differentiable function such that f ′(1) = lima→∞a2f ( a1 ). Then a(a+1) lima→∞ 2 tan−1 ( a1 ) + a2 −2 loge a is equal to (1) 2 3 + π4 (2) 34 + π8 (3) 3 8 + π4 (4) 52 + π8

Q68.For 0 < 𝜃< 𝜋/ 2, if the eccentricity of the hyperbola 𝑥2 −𝑦2cosec2𝜃= 5 is √7 times eccentricity of the ellipse 𝑥2cosec2𝜃+ 𝑦2 = 5, then the value of 𝜃 is: (1) 𝜋 (2) 5𝜋 6 12 𝜋 𝜋 (3) (4) 3 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.The frequency distribution of the age of students in a class of 40 students is given below. Age 15 16 17 18 19 20 If the mean deviation about the median is 1.25, then 4x + 5y No of Students 5 8 5 12 x y is equal to : (1) 46 (2) 43 (3) 44 (4) 47 Q69. 3x + 5y + λz = 3 Let λ, μ ∈R. If the system of equations 7x + 11y −9z = 2 has infinitely many solutions, then μ + 2λ is 97x + 155y −189z = μ equal to : (1) 24 (2) 25 (3) 22 (4) 27

Q68.Let the set S = {2, 4, 8, 16, … , 512} be partitioned into 3 sets A, B, C with equal number of elements such that A ∪B ∪C = S and A ∩B = B ∩C = A ∩C = ϕ. The maximum number of such possible partitions of S is equal to: (1) 1680 (2) 1640 (3) 1520 (4) 1710 Q69. ⎡ β α 3 ⎤ ⎡ 3α −9 3α ⎤ Let αβ ≠0 and A = α α β . If B = −α 7 −2α is the matrix of cofactors of the elements ⎣−β α 2α ⎦ ⎣ −2α 5 −2β ⎦ of A , then det(AB) is equal to : (1) 64 (2) 216 (3) 343 (4) 125

Q68.Let f : [−π2 , 2 ] →R be a differentiable function such that f(0) = 2 , If ex2−1 x→0 to : (1) 16 (2) 2 (3) 1 (4) 4

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.If lim 3 + 𝛼sin𝑥+ 𝛽cos𝑥+ log𝑒( 1 - 𝑥) = 1 then 2𝛼- 𝛽 is equal to : 𝑥→0 3tan2𝑥 3, (1) 2 (2) 7 (3) 5 (4) 1 Q69. 1 3 𝛼+ 3 2 2 The values of 𝛼, for which 1 1 = 0, lie in the interval 1 𝛼+ 3 3 2𝛼+ 3 3𝛼+ 1 0 (1) ( - 2, 1 ) (2) ( - 3, 0 ) (3) -3 3 (4) ( 0, 3 ) 2, 2

Q68.Let R be a relation on Z × Z defined by (a, b)R(c, d) if and only if ad −bc is divisible by 5 . Then R is (1) Reflexive and symmetric but not transitive (2) Reflexive but neither symmetric not transitive (3) Reflexive, symmetric and transitive (4) Reflexive and transitive but not symmetric Q69. ⎡ 1 0 0 ⎤ 3 Let A = 0 α β and 2A = 221 where α, β ∈Z , Then a value of α is ⎣ 0 β α⎦ (1) 3 (2) 5 (3) 17 (4) 9 is equal to

Q68.Let α, β ∈R. Let the mean and the variance of 6 observations −3, 4, 7, −6, α, β be 2 and 23 , respectively. The mean deviation about the mean of these 6 observations is : (1) 13 (2) 16 3 3 (3) 11 (4) 14 3 3 Q69. ⎡ 1 2 α⎤ Let α ∈(0, ∞) and A = 1 0 1 . If det (adj (2A −AT) ⋅adj (A −2AT)) = 28 , then (det(A))2 is equal ⎣ 0 1 2 ⎦ to: (1) 36 (2) 16 (3) 1 (4) 49

Q68.Let 𝑃 be a parabola with vertex 2, 3 and directrix 2𝑥+ 𝑦= 6. Let an ellipse 𝐸: 𝑥2 + 𝑦2 = 1, 𝑎> 𝑏 𝑎2 𝑏2 1 of eccentricity pass through the focus of the parabola 𝑃. Then the square of the length of the latus rectum √2 of 𝐸, is (1) 385 (2) 347 8 8 512 656 (3) (4) 25 25

Q68.Let A = {2, 3, 6, 8, 9, 11} and B = {1, 4, 5, 10, 15}. Let R be a relation on A × B defined by (a, b)R(c, d) if and only if 3ad −7bc is an even integer. Then the relation R is (1) an equivalence relation. (2) reflexive and symmetric but not transitive. (3) transitive but not symmetric. (4) reflexive but not symmetric. Q69. α b c If α ≠a, β ≠b, γ ≠c and a β c = 0, then α−aa + β−bb + γ−cγ is equal to: a b γ (1) 3 (2) 0 (3) 1 (4) 2