Practice Questions

Q66.If (α, β) is the orthocenter of the triangle ABC with vertices A(3, –7), B(–1, 2) and C(4, 5), then 9α −6β + 60 is equal to (1) 25 (2) 35 (3) 30 (4) 40

Q66.Let x = 13 9 13) and (7√2 9) . If (8√3 (1) [x] + [y] is even (2) [x] is odd but [y] is even (3) [x] is even but [y] is odd (4) [x] and [y] are both odd Q67. 50th root of a number x is 12 and 50th root of another number y is 18 . Then the remainder obtained on dividing (x + y) by 25 is ________. O be the origin

Q66.Let the ellipse 𝐸: 𝑥2 + 9𝑦2 = 9 intersect the positive 𝑥- and 𝑦-axes at the points 𝐴 and 𝐵 respectively. Let the major axis of 𝐸 be a diameter of the circle 𝐶. Let the line passing through 𝐴 and 𝐵 meet the circle 𝐶 at the 𝑚 point 𝑃. If the area of the triangle with vertices 𝐴, 𝑃 and the origin 𝑂 is 𝑛, where 𝑚 and 𝑛 are coprime, then 𝑚- 𝑛 is equal to (1) 16 (2) 15 (3) 17 (4) 18

Q66.If the orthocentre of the triangle, whose vertices are 1, 2, 2, 3 and 3, 1 is 𝛼, 𝛽, then the quadratic equation whose roots are 𝛼+ 4𝛽 and 4𝛼+ 𝛽, is (1) 𝑥2 - 19𝑥+ 90 = 0 (2) 𝑥2 - 18𝑥+ 80 = 0 (3) 𝑥2 - 22𝑥+ 120 = 0 (4) 𝑥2 - 20𝑥+ 99 = 0

Q66.The compound statement ( ~ ( 𝑃∧𝑄) ) ∨( ( ~𝑃) ∧𝑄) ⇒( ( ~𝑃) ∧( ~𝑄) ) is equivalent to (1) ( ( ~𝑃) ∨𝑄) ∧( ( ~𝑄) ∨𝑃) (2) ( ~𝑄) ∨𝑃 (3) ( ( ~𝑃) ∨𝑄) ∧( ~𝑄) (4) ( ~𝑃) ∨𝑄

Q66.Let 𝑆= 𝑥∈- 𝜋 (𝛽- 14 ) 2 is equal to 2, 2: 91 - tan2𝑥+ 9tan2𝑥= 10 and 𝛽= ∑𝑥∈𝑆tan2 3,𝑥 6 (1) 16 (2) 8 (3) 64 (4) 32

Q66.Let α be the constant term in the binomial expansion of (√x − x 32 ) , n ≤15. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of x−n is λα, then λ is equal to ________.

Q66.The sum of the common terms of the following three arithmetic progressions. 3, 7, 11, 15, … … … … , 399 2, 5, 8, 11, . . . . . . . . . 359 and 2, 7, 12, 17, … … , 197 , is equal to _____ .

Q67.The relation 𝑅= 𝑎, 𝑏: 𝑔𝑐𝑑𝑎, 𝑏= 1, 2𝑎≠𝑏, 𝑎, 𝑏∈ℤ is: (1) transitive but not reflexive (2) symmetric but not transitive (3) reflexive but not symmetric (4) neither symmetric nor transitive

Q67.The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48 , is (1) 472 (2) 432 (3) 507 (4) 400 JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper

Q67.Let 𝑅 be a relation on 𝑁× 𝑁 defined by 𝑎, 𝑏𝑅𝑐, 𝑑 if and only if 𝑎𝑑𝑏- 𝑐= 𝑏𝑐𝑎- 𝑑. Then 𝑅 is (1) symmetric but neither reflexive nor transitive (2) transitive but neither reflexive nor symmetric (3) reflexive and symmetric but not transitive (4) symmetric and transitive but not reflexive Q68. 1 0 0 Let 𝐴= 0 4 -1 . Then the sum of the diagonal elements of the matrix 𝐴+ 𝐼11 is equal to: 0 12 -3 (1) 6144 (2) 4094 (3) 4097 (4) 2050

Q67.Let 𝐴 be the point 1, 2 and 𝐵 be any point on the curve 𝑥2 + 𝑦2 = 16. If the centre of the locus of the point 𝑃, which divides the line segment 𝐴 𝐵 in the ratio 3: 2 is the point 𝐶𝛼, 𝛽, then the length of the line segment 𝐴𝐶 is (1) 3√5 (2) 4√5 5 5 (3) 2√5 (4) 6√5 5 5

Q67.If the term without x in the expansion of 23 + 22 (x x3α ) is 7315 , then |α| is equal to _____ . m 21 . + 5√2(x−2) log2 3) powers of 2(x−2) log2 3 , be

Q67.If the 1011th term from the end in the binomial expansion of ( 4x5 − 2x5 ) 2022 the beginning, then 32|x| is equal to (1) 15 (2) 10 (3) 12 (4) 8

Q67.The sum, of the coefficients of the first 50 terms in the binomial expansion of (1 −x)100, is equal to (1) 101C50 (2) 99C49 (3) −101C50 (4) −99C49

Q67.If the coefficients of x7 in (ax2 + 2bx1 ) 11 3bx2 and x−7 in (ax 1 ) (1) 729ab = 32 (2) 32ab = 729 (3) 64ab = 243 (4) 243ab = 64

Q67.The number of common tangents, to the circles x2 + y2 −18x −15y + 131 = 0 and x2 + y2 −6x −6y −7 = 0 , is (1) 3 (2) 1 (3) 4 (4) 2

Q67.If the co-efficient of x9 in 11 11 − βx3 1 ) are equal, then (αβ)2 is + βx1 ) and the co-efficient of x−9 in (αx (αx3 equal to : f

Q67.Let R be a rectangle given by the lines 𝑥= 0, 𝑥= 2, 𝑦= 0 and 𝑦= 5. Let A𝛼, 0 and B0, 𝛽, 𝛼∈0, 2 and 𝛽∈0, 5, be such that the line segment 𝐴𝐵 divides the area of the rectangle 𝑅 in the ratio 4: 1. Then, the mid- point of 𝐴𝐵 lies on a (1) straight line (2) parabola (3) hyperbola (4) circle

Q67.The constant term in the expansion of 5 + x71 + 3x2) is _____ . (2x

Q67.if the coefficients of three consecutive terms in the expansion of (1 + x)n are the ratio 1 : 5 : 20 then the coefficient of the fourth term is (1) 2436 (2) 5481 (3) 1827 (4) 3654 is α then [α] is

Q68.The equations of the sides AB, BC & CA of a triangle ABC are 2x + y = 0 , x + py = 21a (a ≠0) and x −y = 3 respectively. Let P(2, a) be the centroid of the triangle ABC , then (BC)2 is equal to

Q68.The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is (1) 1072 (2) 1792 (3) 1216 (4) 1456 JEE Main 2023 (01 Feb Shift 1) JEE Main Previous Year Paper

Q68.If 𝑃( ℎ, 𝑘) be point on the parabola 𝑥= 4𝑦2, which is nearest to the point 𝑄( 0, 33 ) , then the distance of 𝑃 from the directrix of the parabola 𝑦2 = 4 ( 𝑥+ 𝑦) is equal to: (1) 2 (2) 4 (3) 8 (4) 6

Q68.If lim = 17, then 5𝑎2 + 𝑏2 is equal to 𝑥→0 1 - cos ( 2𝑥) (1) 64 (2) 72 (3) 68 (4) 76