Practice Questions

Q60.In an oligopeptide named Alanylglycylphenyl alanyl isoleucine, the number of sp2 hybridised carbons is _____. is equal to

Q60.Compound A, C5H10O5 , given a tetraacetate with AC2 O and oxidation of A with Br2 −H2O gives an acid, C5H10O6 . Reduction of A with HI gives isopentane. The possible structure of A is : JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper (1) (2) (3) (4)

Q60.How many of the transformation given below would result in aromatic amines? JEE Main 2023 (31 Jan Shift 1) JEE Main Previous Year Paper

Q60.A trisubstituted compound 'A', C10H12O2 gives neutral FeCl3 test positive. Treatment of compound 'A' with NaOH and CH3Br gives C11H14O2, with hydroiodic acid gives methyl iodide and with hot conc. NaOH gives a compound B, C10H12O2. Compound 'A' also decolorises alkaline KMnO4. The number of 𝜋 bond/s present in the compound 'A' is _____ .

Q61.Let a ∈R and let α, β be the roots of the equation x2 + 60 41 x + a = 0. If α4 + β4 = −30, then the product of all possible values of a is _____ .

Q61.Let α, β be the roots of the quadratic equation x2 + √6x + 3 = 0. Then α15+β15+α10+β10α23+β23+α14+β14 (1) 81 (2) 9 (3) 72 (4) 729

Q61.If the value of real number α > 0 for which x2 −5αx + 1 = 0 and x2 −αx −5 = 0 have a common real roots is 3 then β is equal to ________ √2β

Q61.The number of points, where the curve f(x) = e8x −e6x −3e4x −e2x + 1, x ∈R cuts x-axis, is equal to............ ¯¯¯¯

Q61.If the solution of the equation 1, 𝑥∈0, 𝜋 is sin-1𝛼+ √𝛽 , where 𝛼, 𝛽 are integers, logcos𝑥cot𝑥+ 4logsin𝑥tan𝑥= 2 2 then 𝛼+ 𝛽 is equal to: (1) 3 (2) 5 (3) 6 (4) 4 -2

Q61.The sum of all the roots of the equation 𝑥2 - 8𝑥+ 15 - 2𝑥+ 7 = 0 is (1) 9 - √3 (2) 9 + √3 (3) 11 - √3 (4) 11 + √3

Q61.The number of real solutions of the equation 3(x2 + x21 ) −2(x + x1 ) + 5 = 0 , is (1) 4 (2) 0 (3) 3 (4) 2 2π 2π 3 1+sin 9 +i cos 9

Q61.Let 𝑝, 𝑞∈ℝ and (1 - √3𝑖) 200 = 2199 (𝑝+ 𝑖𝑞), 𝑖= √-1. Then, 𝑝+ 𝑞+ 𝑞2 and 𝑝- 𝑞+ 𝑞2 are roots of the equation. (1) 𝑥2 + 4𝑥- 1 = 0 (2) 𝑥2 - 4𝑥+ 1 = 0 (3) 𝑥2 + 4𝑥+ 1 = 0 (4) 𝑥2 - 4𝑥- 1 = 0

Q61.The number of integral solution 𝑥 of 7 ≥0 is log𝑥+ 2𝑥- 3 2 (1) 7 (2) 8 (3) 6 (4) 5

Q61.Let α1, α2, … , α7α1, α2, … , α7 be the roots of the equation x7 + 3x5 −13x3 −15x = 0 and |α1| ≥|α2| ≥… ≥|α7|. Then, α1α2 −α3α4 + α5α6 is equal to _______ ¯

Q61.Let w = zz + k1z + k2iz + λ(1 + i), k1, k2 ∈R. . Let Re(w) = 0 be the circle C of radius 1 in the first quadrant touching the line y = 1 and the y−axis. If the curve Im(w) = 0 intersects C at A and B, then 30(AB)2 is equal to _______. JEE Main 2023 (13 Apr Shift 1) JEE Main Previous Year Paper

Q61.Let a ≠b be two non-zero real numbers. Then the number of elements in the set X = {z ∈C : Re(az2 + bz) = a and Re(bz2 + az) = b} is equal to (1) 0 (2) 1 (3) 3 (4) 2

Q61.Let 𝛼, 𝛽 be the roots of the equation 𝑥2 - √2𝑥+ 2 = 0 Then 𝛼14 + 𝛽14 is equal to (1) -64 (2) -64√2 (3) -128 (4) -128√2

Q61.The number of integral values of k, for which one root of the equation 2x2 −8x + k = 0 lies in the interval (1, 2) and its other root lies in the interval (2, 3), is : JEE Main 2023 (01 Feb Shift 2) JEE Main Previous Year Paper (1) 2 (2) 0 (3) 1 (4) 3

Q61.The number of real roots of the equation √𝑥2 - 4𝑥+ 3 + √𝑥2 - 9 = √4𝑥2 - 14𝑥+ 6, is: (1) 0 (2) 1 (3) 3 (4) 2

Q61.The equation e4x + 8e3x + 13e2x −8ex + 1 = 0, x ∈R has : (1) four solutions two of which are negative (2) two solutions and both are negative (3) no solution (4) two solutions and only one of them is negative

Q61.The number of real roots of the equation x|x| −5|x + 2| + 6 = 0 , is (1) 5 (2) 4 (3) 6 (4) 3 ¯ ¯

Q61.Let λ ≠0 be a real number. Let α, β be the roots of the equation 14x2 −31x + 3λ = 0 and α, γ be the roots of the equation 35x2 −53x + 4λ = 0. Then 3αβ and 4αγ are the roots of the equation : (1) 7x2 + 245x −250 = 0 (2) 7x2 −245x + 250 = 0 (3) 49x2 −245x + 250 = 0 (4) 49x2 + 245x + 250 = 0

Q61.Let the complex number 𝑧= 𝑥+ 𝑖𝑦 be such that is purely imaginary. If 𝑥+ 𝑦2 = 0, then 𝑦4 + 𝑦2 - 𝑦 is 2𝑧+ 𝑖 equal to (1) 2 (2) 3 3 2 3 4 (3) (4) 4 3

Q61.Let m and n be the numbers of real roots of the quadratic equations x2 −12x + [x] + 31 = 0 and x2 −5 x + 2 −4 = 0 respectively, where [x] denotes the greatest integer ≤x. Then m2 + mn + n2 is equal to

Q61.Let 𝑆= 𝑧= 𝑥+ 𝑖𝑦: is a real number }. Then which of the following is NOT correct? 4𝑧+ 2𝑖 (1) 𝑦+ 𝑥2 + 𝑦2 ≠- 1 (2) (𝑥, 𝑦) = 0, - 1 4 2 (3) 𝑥= 0 (4) 𝑦∈- ∞, - 1 ∪-1 ∞ 2 2,