Practice Questions

Q79.Let a curve y = f(x), x ∈(0, ∞) pass through the points P(1, 32 ) and Q(a, 12 ). If the tangent at any point R(b, f(b)) to the given curve cuts the y-axis at the point S(0, c) such that bc = 3, then (PQ)2 is equal to JEE Main 2023 (06 Apr Shift 2) JEE Main Previous Year Paper _____.

Q79.If the equation of the plane passing through the line of intersection of the planes 𝑥+ 1 𝑦+ 3 𝑧- 2 2𝑥- 𝑦+ 𝑧= 3, 4𝑥- 3𝑦+ 5𝑧+ 9 = 0 and parallel to the line = = is 𝑎𝑥+ 𝑏𝑦+ 𝑐𝑧+ 6 = 0, -2 4 5 then 𝑎+ 𝑏+ 𝑐 is equal to (1) 12 (2) 14 (3) 16 (4) 13

Q79.Let 𝑃 be the point of intersection of the line = = and the plane 𝑥+ 𝑦+ 𝑧= 2. If the distance of 3 1 2 the point 𝑃 from the plane 3𝑥- 4𝑦+ 12𝑧= 32 is 𝑞, then 𝑞 and 2𝑞 are the roots of the equation (1) 𝑥2 - 18𝑥- 72 = 0 (2) 𝑥2 - 18𝑥+ 72 = 0 (3) 𝑥2 + 18𝑥+ 72 = 0 (4) 𝑥2 + 18𝑥- 72 = 0 𝑚

Q79.Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6} . Then the number of functions f : A →B satisfying f(1) + f(2) = f(4) −1 is equal to........ .Then and g(x) =

Q79.Let the shortest distance between the lines L: 𝑥- = = , 𝜆≥0 and L1: 𝑥+ 1 = 𝑦- 1 = 4 - 𝑧 be 2√6. -2 0 1 If ( 𝛼, 𝛽, 𝛾) lies on L, then which of the following is NOT possible? (1) 𝛼+ 2𝛾= 24 (2) 2𝛼+ 𝛾= 7 (3) 2𝛼- 𝛾= 9 (4) 𝛼- 2𝛾= 19

Q79.Let a unit vector →𝑂𝑃 make angle 𝛼, 𝛽, 𝛾 with the positive directions of the co-ordinate axes OX, OY, OZ 𝜋 respectively, where 𝛽∈0, →𝑂𝑃 is perpendicular to the plane through points 1, 2, 3, 2, 3, 4 and 1, 5, 7, then 2. which one of the following is true ? (1) 𝛼∈𝜋 𝜋 and 𝛾∈𝜋 𝜋 (2) 𝛼∈0, 𝜋 and 𝛾∈0, 𝜋 2, 2, 2 2 𝜋 𝜋 𝜋 𝜋 (3) 𝛼∈ 2, 𝜋 and 𝛾∈0, 2 (4) 𝛼∈0, 2 and 𝛾∈ 2, 𝜋

Q79.Let f : R −{2, 6} →R be real valued function defined as f(x) = x+2x+1 . Then range of f is x2−8x+12 (1) (−∞, −214 ] ∪[ 214 , ∞) (2) (−∞, −214 ] ∪[0, ∞) (3) (−∞, −214 ) ∪(0, ∞) (4) (−∞, −214 ] ∪[1, ∞)

Q79.If y(x) = xx, x > 0 , then y′′(2) −2y′(2) is equal to : (1) 8 loge 2 −2 (2) 4 loge 2 + 2 (3) 4(loge 2)2 −2 (4) 4(loge 2)2 + 2

Q79.The distance of the point -1, 9, - 16 from the plane 2𝑥+ 3𝑦- 𝑧= 5 measure parallel to the line 𝑥+ 4 2 - 𝑦 𝑧- 3 = = is 3 4 12 (1) 13√2 (2) 31 (3) 26 (4) 20√3

Q79.Let S be the set of all values of λ, for which the shortest distance between the lines x−λ0 = y−34 = z+61 and x+λ 3 = −4y = z−60 is 13. Then 8 ∑λ∈S λ is equal to (1) 306 (2) 304 (3) 308 (4) 302

Q79.If the functions f(x) = x33 + 2bx + ax22 and g(x) = x33 + then a + 2b + 7 is equal to (1) 4 (2) 32 (3) 3 (4) 6 1 + constant, then β −α is equal to + cos β x)

Q79.Let f(x) be a function such that f(x + y) = f(x) ⋅f(y) for all x, y ∈N , If f(1) = 3 and ∑nk=1 f(k) = 3279 , then the value of n is (1) 6 (2) 8 (3) 7 (4) 9

Q79.Let the line = = intersect the lines = = and = = at the points A and B 1 2 5 4 3 1 6 3 1 respectively. Then the distance of the mid-point of the line segment 𝐴𝐵 from the plane 2𝑥- 2𝑦+ 𝑧= 14 is (1) 3 (2) 11 3 10 (3) 4 (4) 3

Q80.The number of points, where the curve y = x5 −20x3 + 50x + 2 crosses the x-axis, is _____. x dx is equal to

Q80.In a binomial distribution B ( 𝑛, 𝑝) , the sum and product of the mean & variance are 5 and 6 respectively, then find 6 ( 𝑛+ 𝑝- 𝑞) is equal to :- (1) 51 (2) 52 (3) 53 (4) 50

Q80.Let I(x) = ∫√x+7x dx and I(9) = 12 + 7 loge 7. If I(1) = α + 7 loge(1 2√2), then α4 is equal to _____. dx = 3000k , then k is equal to _____.

Q80.Let 𝑁 denote the sum of the numbers obtained when two dice are rolled. If the probability that 2𝑁< 𝑁! is 𝑛 where 𝑚 and 𝑛 are coprime, then 4𝑚- 3𝑛 is equal to (1) 6 (2) 12 (3) 10 (4) 8

Q80.Let 𝑆= 𝑀= 𝑎𝑖𝑗, 𝑎𝑖𝑗∈0, 1, 2, 1 ≤𝑖, 𝑗≤2 be a sample space and 𝐴𝑀∈𝑆: 𝑀 is invertible be an even. Then 𝑃𝐴 is equal to 16 47 (1) (2) 27 81 49 50 (3) (4) 81 81 + 𝑎17 + 𝑏17 is equal to

Q80.If f(x) = x3 −x2f ′(1) + xf ′′(2) −f ′′′(3), x ∈R, then (1) 3f(1) + f(2) = f(3) (2) f(3) −f(2) = f(1) (3) 2f(0) −f(1) + f(3) = f(2) (4) f(1) + f(2) + f(3) = f(0) Q81. 3√34 48 ∫ 3√2 dx is equal to 4 √9−4x2 JEE Main 2023 (24 Jan Shift 2) JEE Main Previous Year Paper (1) π (2) π 3 2 (3) π (4) 2π 6 such that f(x) > 0 and

Q80.The absolute minimum value, of the function f(x) = x2 −x + 1 + [x2 −x + 1], where [t] denotes the greatest integer function, in the interval [−1, 2], is (1) 3 (2) 1 2 4 (3) 5 (4) 3 4 4 dx = 16+20√215 then α is equal to :

Q80.Let x = 2 be a local minima of the function f(x) = 2x4 −18x2 + 8x + 12, x ∈(−4, 4). If M is local maximum value of the function f in (−4, 4), then M = (1) 12√6 −332 (2) 12√6 −312 (3) 18√6 −332 (4) 18√6 −312

Q80.Let k and m be positive real numbers such that the function f(x) = {3x2mx2+ k√x+ k2,+ 1, 0 <x ≥1x < 1 8f ′(8) is differentiable for all x > 0 . Then 1 is equal to f ′( 8 ) x dx is equal to

Q80.Let a die be rolled n times. Let the probability of getting odd numbers seven times be equal to the probability 𝑘 of getting odd numbers nine times. If the probability of getting even numbers twice is 215, then 𝑘 is equal to (1) 60 (2) 15 (3) 90 (4) 30

Q80.If aα is the greatest term in the sequence an = n3 , n = 1, 2, 3. . . . , then α is equal to ______ n4+147

Q80.The random variable 𝑋 follows binomial distribution 𝐵( 𝑛, 𝑝) , for which the difference of the mean and the variance is 1. If 2 𝑃( 𝑋= 2 ) = 3 𝑃( 𝑋= 1 ) , then 𝑛2𝑃( 𝑋> 1 ) is equal to (1) 15 (2) 11 (3) 12 (4) 16