Practice Questions

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 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

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.The length of the chord of the ellipse 25 + 16 = 1, whose mid point is (1, 52 ), is equal to: (1) √1691 (2) √2009 5 5 (3) √1741 (4) √1541 5 5

Q68.Let A and B be two square matrices of order 3 such that |A| = 3 and |B| = 2. Then ATA(adj(2 A))−1(adj(4 B))(adj(AB))−1AAT is equal to : (1) 108 (2) 32 (3) 81 (4) 64 Q69. 11x + y + λz = −5 If the system of equations 2x + 3y + 5z = 3 has infinitely many solutions, then λ4 −μ is equal to : 8x −19y −39z = μ (1) 51 (2) 45 (3) 47 (4) 49

Q68. is equal to : limn→∞ (13+23+⋯⋯+n3)−(12+22+⋯⋯+n2) (1) 2 (2) 1 3 3 (3) 3 (4) 1 4 2

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

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 𝑎 be the sum of all coefficients in the expansion of ( 1 – 2𝑥+ 2𝑥2 ) 2023 ( 3 - 4𝑥2 + 2𝑥3 ) 2024 and 𝑥log1 + 𝑡 ∫0 𝑑𝑡 𝑏= lim 𝑡2024 + 1 . If the equations 𝑐𝑥2 + 𝑑𝑥+ 𝑒= 0 and 2𝑏𝑥2 + 𝑎𝑥+ 4 = 0 have a common root, where 𝑥→0 𝑥2 𝑐, 𝑑, 𝑒∈𝑅, then 𝑑 : 𝑐 : 𝑒 equals (1) 2 : 1 : 4 (2) 4 : 1 : 4 (3) 1 : 2 : 4 (4) 1 : 1 : 4 Q69. 𝑥3 2𝑥2 + 1 1 + 3𝑥 If 𝑓𝑥= 3𝑥2 + 2 2𝑥 𝑥3 + 6 for all 𝑥∈ℝ, then 2𝑓0 + 𝑓'0 is equal to 𝑥3 −𝑥 4 𝑥2 −2 (1) 48 (2) 24 (3) 42 (4) 18

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.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 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 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

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

Q69.If a = lim √1+√1+x4−√2 and b = lim sin2 x , then the value of ab3 is : x→0 x4 x→0 √2−√1+cos x (1) 36 (2) 32 (3) 25 (4) 30

Q69.Let M denote the median of the following frequency distribution. Class 0 −4 4 −8 8 −12 12 −16 16 −20 Frequency 3 9 10 8 6 Then 20M is equal to : (1) 416 (2) 104 (3) 52 (4) 208 Q70. 2 cos4 x 2 sin4 x 3 + sin2 2x If f(x) = 3 + 2 cos4 x 2 sin4 x sin2 2x then 15 f ′(0) is equal to ________. 2 cos4 x 3 + 2 sin4 x sin2 2x JEE Main 2024 (30 Jan Shift 1) JEE Main Previous Year Paper (1) 0 (2) 1 (3) 2 (4) 6

Q69.Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, 𝑎, 𝑏 be 170 205 and respectively. Then the mean deviation about the mean of these 7 observations is: 7 (1) 31 (2) 28 (3) 30 (4) 32 0