Practice Questions

Q64.Let a, b, c > 1, a3, b3 and c3 be in A. P. and loga b, logc a and logb c be in G. P. If the sum of first 20 terms of an A. P., whose first term is a+4b+c3 and the common difference is a−8b+c10 is −444, then abc is equal to (1) 343 (2) 216 (3) 343 (4) 125 8 8

Q64.If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is (1) 103 (2) 102 (3) 101 (4) 104

Q64.The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together is (1) 720 (2) 126(5!)2 (3) 7(360)2 (4) 7(720)2

Q64.Let a circle 𝐶1 be obtained on rolling the circle 𝑥2 + 𝑦2 - 4𝑥- 6𝑦+ 11 = 0 upwards 4 units on the tangent T to it at the point 3, 2. Let 𝐶2 be the image of 𝐶1 in 𝑇. Let 𝐴 and 𝐵 be the centers of circles 𝐶1 and 𝐶2 respectively, and 𝑀 and 𝑁 be respectively the feet of perpendiculars drawn from 𝐴 and 𝐵 on the 𝑥-axis. Then the area of the trapezium AMNB is: (1) 22 + √2 (2) 41 + √2 (3) 3 + 2√2 (4) 21 + √2

Q64.Let A1 and A2 be two arithmetic means and G1, G2 and G3 be three geometric means of two distinct positive numbers. Then G41 + G42 + G43 + G21G23 is equal to (1) (A1 + A2)2G1G3 (2) 2(A1 + A2)G1G3 (3) (A1 + A2)G21G23 (4) 2(A1 + A2)G21G23

Q64.Let a tangent to the curve 𝑦2 = 24𝑥 meet the curve 𝑥𝑦 = 2 at the points 𝐴 and 𝐵. Then the mid- points of such line segments 𝐴𝐵 lie on a parabola with the (1) directrix 4𝑥= 3 (2) directrix 4𝑥= - 3 3 (3) Length of latus rectum (4) Length of latus rectum 2 2 Q65. 1 1 1 1 sin2𝑡 𝑡→01lim sin 2𝑡+ 2 sin 2𝑡+ 3 sin 2𝑡. . . . . . 𝑛 sin 2𝑡 is equal to (1) 𝑛2 + 𝑛 (2) 𝑛 𝑛𝑛+ 1 (3) (4) 𝑛2 2

Q64.Let an be nth term of the series 5 + 8 + 14 + 23 + 35 + 50+. . . . . . .and Sn = ∑nk=1 ak . Then S30 −a40 is equal to (1) 11310 (2) 11260 (3) 11290 (4) 11280

Q64.Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is _____ .

Q64.Let the number ( 22 2022 + ( 2022 22 leave the remainder α when divided by 3 and β when divided by 7 ) ) . Then (α2 + β2 ) is equal to (1) 20 (2) 13 (3) 5 (4) 10

Q65.The combined equation of the two lines 𝑎𝑥+ 𝑏𝑦+ 𝑐= 0 and 𝑎'𝑥+ 𝑏'𝑦+ 𝑐' = 0 can be written as 𝑎𝑥+ 𝑏𝑦+ 𝑐𝑎'𝑥+ 𝑏'𝑦+ 𝑐' = 0. The equation of the angle bisectors of the lines represented by the equation 2𝑥2 + 𝑥𝑦- 3𝑦2 = 0 is (1) 3𝑥2 + 5𝑥𝑦+ 2𝑦2 = 0 (2) 𝑥2 - 𝑦2 + 10𝑥𝑦= 0 (3) 3𝑥2 + 𝑥𝑦- 2𝑦2 = 0 (4) 𝑥2 - 𝑦2 - 10𝑥𝑦= 0

Q65.If the maximum distance of normal to the ellipse 𝑥2 + 𝑦2 = 1, 𝑏< 2, from the origin is 1 , then the eccentricity 4 𝑏2 of the ellipse is: (1) 1 (2) √3 √2 2 (3) 1 (4) √3 2 4

Q65.If 2𝑛𝐶3: 𝑛𝐶3 = 10: 1, then the ratio 𝑛2 + 3𝑛: 𝑛2 - 3𝑛+ 4 is (1) 35: 16 (2) 27: 11 (3) 65: 37 (4) 2: 1

Q65.If gcd(m, n) = 1 and 12 −22 + 32 −42+. . . . +(2021)2 −(2022)2 + (2023)2 = 1012m2n then m2 −n2 is equal to (1) 240 (2) 200 (3) 220 (4) 180

Q65.A line segment 𝐴𝐵 of length 𝜆 moves such that the points 𝐴 and 𝐵 remain on the periphery of a circle of radius 𝜆. Then the locus of the point, that divides the line segment 𝐴𝐵 in the ratio 2: 3, is a circle of radius (1) 3 (2) 2 5𝜆 3𝜆 (3) √19 𝜆 (4) √19 𝜆 5 7 JEE Main 2023 (10 Apr Shift 1) JEE Main Previous Year Paper

Q65.If tan15° + + + tan195° = 2a, then the value of 𝑎+ is : tan75° tan105° 𝑎 (1) 4 (2) 4 - 2√3 (3) 2 (4) 5 - 3 2√3

Q65.Let (a + bx + cx2)10 = ∑20i=10 pixi, a, b, c ∈N. If p1 = 20 and p2 = 210, then 2(a + b + c) is equal to (1) 6 (2) 15 (3) 12 (4) 8 JEE Main 2023 (15 Apr Shift 1) JEE Main Previous Year Paper

Q65.Let f(x) = 2xn + λ, λ ∈R, n ∈N, and f(4) = 133 , f(5) = 255 . Then the sum of all the positive integer divisors of (f(3) −f(2)) is (1) 61 (2) 60 (3) 58 (4) 59

Q65.Let < an > be a sequence such that a1 + a2+. . . +an = (n+1)(n+2)n2+3n . If 28 ∑10k=1 ak1 p1, p2, . . . pm are the first m prime numbers, then m is equal to JEE Main 2023 (12 Apr Shift 1) JEE Main Previous Year Paper (1) 5 (2) 8 (3) 6 (4) 7

Q65.If (30C1)2 + 2(30C2)2 + 3(30C3)2. . . . . . . . . . 30(30C30)2 = (30!)2α60! , then (1) 30 (2) 60 (3) 15 (4) 10

Q65.Let a, b, c and d be positive real numbers such that a + b + c + d = 11 . If the maximum value of a5b3c2d is 3750β, then the value of β is (1) 90 (2) 110 (3) 55 (4) 108

Q65.Fractional part of the number 42022 is equal to 15 (1) 8 (2) 4 15 15 (3) 14 (4) 1 15 15 n 6

Q65.The coefficient of 𝑥5 in the expansion of 2𝑥3 - 1 5 is 3𝑥2 (1) 80 (2) 9 9 (3) 8 (4) 26 3

Q65.The number of elements in the set 𝑆= 𝜃∈[0, 2𝜋]: 3cos4𝜃- 5cos2𝜃- 2sin6𝜃+ 2 = 0 is (1) 10 (2) 8 (3) 12 (4) 9

Q65.Let a1, a2, a3, … . be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24 , then a1a9 + a2a4a9 + a5 + a7 is equal to

Q65.The sum ∑∞n=1 2n2+3n+4(2n)! is equal to : (1) 11e 2 + 2e7 (2) 13e4 + 4e5 −4 (3) 11e 2 + 2e7 −4 (4) 13e4 + 4e5