JEE Main Guess Paper 2027

gamma_A is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. gamma_B is the specific heat ratio of polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If gamma_A/gamma_B = (1 + 1/n), then the value of n is ______.

Let B₁ be the magnitude of magnetic field at center of a circular coil of radius R carrying current I. Let B₂ be the magnitude of magnetic field at an axial distance 'x' from the center. For x : R = 3 : 4, B₂/B₁ is : (1) 25 : 16 (2) 16 : 25 (3) 4 : 5 (4) 64 : 125

In an adiabatic process, which of the following statements is true ? (1) Work done by the gas equals the increase in internal energy (2) The internal energy of the gas decreases as the temperature increases (3) The molar heat capacity is zero (4) The molar heat capacity is infinite

A zener diode with 5 V zener voltage is used to regulate an unregulated dc voltage input of 25 V. For a 400 ohm resistor connected in series, the zener current is found to be 4 times load current. The load current (I_L) and load resistance (R_L) are : (1) I_L = 0.02 mA; R_L = 250 ohm (2) I_L = 20 mA; R_L = 250 ohm (3) I_L = 10 mA; R_L = 500 ohm (4) I_L = 10 A; R_L = 0.5 ohm

For a hydrogen atom, the ratio of the largest wavelength of Lyman series to that of the Balmer series is (1) 3 : 4 (2) 5 : 27 (3) 27 : 5 (4) 5 : 36

Moment of inertia of a rod of mass 'M' and length 'L' about an axis passing through its center and normal to its length is 'alpha'. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is : (1) alpha (2) alpha/2 (3) alpha/8 (4) alpha/4

Choose the correct logic circuit for the given truth table having inputs A and B. Truth table (A, B -> Y): 0,0 -> 0 ; 0,1 -> 0 ; 1,0 -> 1 ; 1,1 -> 1 Options: (logic gate circuit diagrams) (1) A and B into an OR gate, its output inverted (NOT) then fed with A into an AND gate producing Y (2) A and B into an OR gate whose output feeds an AND gate producing Y (3) A and B into an OR gate whose output is inverted at the AND gate input, producing Y (4) A and B into an OR gate whose output feeds an AND gate (with A) producing Y

Four particles A, B, C, D of mass m/2, m, 2m, 4m, have same momentum, respectively. The particle with maximum kinetic energy is: (1) A (2) B (3) C (4) D

A parallel plate capacitor is filled equally (half) with two dielectrics of dielectric constants epsilon₁ and epsilon₂, as shown in figures. The distance between the plates is d and area of each plate is A. If capacitance in first configuration and second configuration are C₁ and C₂ respectively, then C₁/C₂ is: (First Configuration: the two dielectric slabs stacked horizontally one above the other; Second Configuration: the two dielectric slabs placed side by side vertically) Options: (1) epsilon₀(epsilon₁ + epsilon₂)/2 (2) 4 epsilon₁ epsilon₂/(epsilon₁ + epsilon₂)² (3) epsilon₁ epsilon₂/(epsilon₁ + epsilon₂) (4) epsilon₁ epsilon₂²/(epsilon₁ + epsilon₂)²

A small bob of mass 100 mg and charge +10 microC is connected to an insulating string of length 1 m. It is brought near to an infinitely long non-conducting sheet of charge density 'sigma' as shown in figure. If string subtends an angle of 45 degrees with the sheet at equilibrium the charge density of sheet will be. (Given, epsilon₀ = 8.85*10^-12 F/m and acceleration due to gravity, g = 10 m/s²) (1) 1.77 nC/m² (2) 0.885 nC/m² (3) 17.7 nC/m² (4) 885 nC/m²

The relationship between the magnetic susceptibility (chi) and the magnetic permeability (mu) is given by : (mu₀ is the permeability of free space and mu_r is relative permeability) (1) chi = mu_r + 1 (2) chi = 1 - mu/mu₀ (3) chi = mu/mu₀ - 1 (4) chi = mu_r/mu₀ + 1

Match List - I with List - II. List - I: (A) Coefficient of viscosity (B) Intensity of wave (C) Pressure gradient (D) Compressibility List - II: (I) [M L⁰ T^-3] (II) [M L^-2 T^-2] (III) [M^-1 L T²] (IV) [M L^-1 T^-1] Choose the correct answer from the options given below : (1) (A)-(IV), (B)-(II), (C)-(I), (D)-(III) (2) (A)-(II), (B)-(III), (C)-(IV), (D)-(I) (3) (A)-(I), (B)-(IV), (C)-(III), (D)-(II) (4) (A)-(IV), (B)-(I), (C)-(II), (D)-(III)

A wire of resistance R is bent into a triangular pyramid as shown in figure with each segment having same length. The resistance between points A and B is R/n. The value of n is: (1) 10 (2) 12 (3) 14 (4) 16

For a given series LCR circuit it is found that maximum current is drawn when value of variable capacitance is 2.5 nF. If resistance of 200 Ω and 100 mH inductor is being used in the given circuit. The frequency of ac source is _______ x 10³ Hz. (given π² = 10)

The work function of a metal is 3 eV. The color of the visible light that is required to cause emission of photoelectrons is. Options: (1) Yellow (2) Green (3) Red (4) Blue

To find the spring constant (k) of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of k is: (1) 3% (2) 5% (3) 1% (4) 4%

A total of 48 J heat is given to one mole of helium kept in a cylinder. The temperature of helium increases by 2°C. The work done by the gas is : Given, R = 8.3 J K^-1 mol^-1. (1) 24.9 J (2) 48 J (3) 72.9 J (4) 23.1 J

Two coherent monochromatic light beams of intensities 4I and 9I are superimposed. The difference between the maximum and minimum intensities in the resulting interference pattern is xI. The value of x is ______.

Pressure inside a soap bubble is greater than the pressure outside by an amount : (given : R = Radius of bubble, S = Surface tension of bubble) (1) 2S/R (2) 4S/R (3) S/R (4) 4R/S

A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample, is: (1) 2 sqrt(2) / 1 (2) 1 / (2 sqrt(2)) (3) 1/32 (4) 1/4

The electrostatic potential on the surface of uniformly charged spherical shell of radius R = 10 cm is 120 V. The potential at the centre of shell, at a distance r = 5 cm from centre, and at a distance r = 15 cm from the centre of the shell respectively, are: Options: (1) 40V, 40V, 80V (2) 0V, 120V, 40V (3) 120V, 120V, 80V (4) 0V, 0V, 80V

An electric field, E = (2 i + 6 j + 8 k)/sqrt(6) passes through the surface of 4 m² area having unit vector n = ((2 i + j + k)/sqrt(6)). The electric flux for that surface is __________ V m.

A block of a mass 2 kg is attached with two identical springs of spring constant 20 N/m each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is pi/sqrt(x) in SI unit. The value of x is ______.

A 100 m long wire having cross-sectional area 6.25 x 10^-4 m² and Young's modulus is 10¹⁰ Nm^-2 is subjected to a load of 250 N, then the elongation in the wire will be: Options: 1) 4 x 10^-4 m 2) 4 x 10^-3 m 3) 6.25 x 10^-6 m 4) 6.25 x 10^-3 m

Considering Bohr's atomic model for hydrogen atom : (A) the energy of H atom in ground state is same as energy of He^+ ion in its first excited state. (B) the energy of H atom in ground state is same as that for Li^++ ion in its second excited state. (C) the energy of H atom in its ground state is same as that of He^+ ion for its ground state. (D) the energy of He^+ ion in its first excited state is same as that for Li^++ ion in its ground state. Choose the correct answer from the options given below : (1) (A), (B) only (2) (A), (C) only (3) (B), (D) only (4) (A), (D) only

The metal ions that have the calculated spin-only magnetic moment value of 4.9 B.M. are: A. Cr²⁺ B. Fe²⁺ C. Fe³⁺ D. Co²⁺ E. Mn³⁺ Choose the correct answer from the options given below: Options: (1) A, B and E Only (2) A, D and E Only (3) B and E Only (4) A, C and E Only

Choose the correct tests with respective observations. (A) CuSO₄ (acidified with acetic acid) + K₄[Fe(CN)₆] -> Chocolate brown precipitate. (B) FeCl₃ + K₄[Fe(CN)₆] -> Prussian blue precipitate. (C) ZnCl₂ + K₄[Fe(CN)₆], neutralised with NH₄OH -> White or bluish white precipitate. (D) MgCl₂ + K₄[Fe(CN)₆] -> Blue precipitate. (E) BaCl₂ + K₄[Fe(CN)₆], neutralised with NaOH -> White precipitate. Choose the correct answer from the options given below : (1) A, B and C only (2) B, D and E only (3) C, D and E only (4) A, D and E only

Given below are two statements : Statement (I) : In octahedral complexes, when Delta₀ < P high spin complexes are formed. When Delta₀ > P low spin complexes are formed. Statement (II) : In tetrahedral complexes because of Delta_t < P, low spin complexes are rarely formed. In the light of the above statements, choose the most appropriate answer from the options given below : (1) Both Statement I and Statement II are correct (2) Both Statement I and Statement II are incorrect (3) Statement I is correct but Statement II is incorrect (4) Statement I is incorrect but Statement II is correct

The number of paramagnetic complexes among [FeF₆]^3-, [Fe(CN)₆]^3-, [Mn(CN)₆]^3-, [Co(C₂O₄)₃]^3-, [MnCl₆]^3-, and [CoF₆]^3-, which involved d²sp³ hybridization is ______

The density of 'x' M solution ('x' molar) of NaOH is 1.12 g mL^-1, while in molality, the concentration of the solution is 3 m (3 molal). Then x is (Given: Molar mass of NaOH is 40 g/mol) (1) 3.5 (2) 3.8 (3) 3.0 (4) 2.8

Among SO₂, NF₃, NH₃, XeF₂, ClF₃ and SF₄, the hybridization of the molecule with non-zero dipole moment and highest number of lone-pairs of electrons on the central atom is : (1) dsp² (2) sp³d (3) sp³d² (4) sp³

On complete combustion 1.0 g of an organic compound (X) gave 1.46 g of CO₂ and 0.567 g of H₂O. The empirical formula mass of compound (X) is ______ g. (Given molar mass in g mol^-1 C : 12, H : 1, O : 16) (1) 15 (2) 30 (3) 45 (4) 60

Which of the following compounds is least likely to give effervescence of CO₂ in presence of aq. NaHCO₃? (1) 4-nitrobenzoic acid (p-NO_2-C₆H_4-COOH) (2) anilinium chloride (Ph-NH₃^+ Cl^-) (3) 3-nitrophenol (m-NO_2-C₆H_4-OH) (4) 2,4,6-trinitrophenol (picric acid)

Consider the following equilibrium, CO(g) + 2H₂(g) <=> CH₃OH(g) 0.1 mol of CO along with a catalyst is present in a 2 dm³ flask maintained at 500 K. Hydrogen is introduced into the flask until the pressure is 5 bar and 0.04 mol of CH₃OH is formed. The K_p^theta is ______ *10^-3 (nearest integer). Given : R = 0.08 dm³ bar K^-1 mol^-1 Assume only methanol is formed as the product and the system follows ideal gas behaviour.

Match the LIST-I with LIST-II LIST-I (Molecule / ion): A. ICl₂^-; B. H₂O; C. SO₂; D. XeF₄ LIST-II (Bond pair : lone pair on the central atom): I. 4 : 2; II. 4 : 1; III. 2 : 3; IV. 2 : 2 Choose the correct answer from the options given below: (1) A-III, B-IV, C-I, D-II (2) A-IV, B-III, C-II, D-I (3) A-III, B-IV, C-II, D-I (4) A-II, B-I, C-IV, D-III

Consider the following electrochemical cell at standard condition. Au(s) | QH₂, Q | NH₄X (0.01 M) || Ag^+ (1 M) | Ag(s) E_cell = +0.4 V The couple QH₂/Q represents quinhydrone electrode, the half cell reaction is given below : Q + 2e^- + 2H^+ -> QH₂, E°_Q/QH₂ = +0.7 V (Given : E°_Ag+/Ag = +0.8 V and 2.303*R*T/F = 0.06 V) The pK_b value of the ammonium halide salt (NH₄X) used here is ______. (nearest integer)

Given below are two statements: Statement I: Ozonolysis followed by treatment with Zn, H₂O of cis-2-butene gives ethanal. Statement II: The product obtained by ozonolysis followed by treatment with Zn, H₂O of 3,6-dimethyloct-4-ene has no chiral carbon atom. In the light of the above statements, choose the correct answer from the options given below (1) Both Statement I and Statement II are true (2) Both Statement I and Statement II are False (3) Statement I is true but Statement II is false (4) Statement I is false but Statement II are true

In a reaction A + B -> C, initial concentrations of A and B are related as [A]₀ = 8[B]₀. The half lives of A and B are 10 min and 40 min, respectively. If they start to disappear at the same time, both following first order kinetics, after how much time will the concentration of both the reactants be same? Options: (1) 20 min (2) 40 min (3) 60 min (4) 80 min

Given below are two statements: Statement (I): Fusion of MnO₂ with KOH and an oxidising agent gives dark green K₂MnO₄. Statement (II): Manganate ion on electrolytic oxidation in alkaline medium gives permanganate ion. In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are true (2) Both Statement I and Statement II are false (3) Statement I is true but Statement II is false (4) Statement I is false but Statement II is true

When 'x' ×10^-2 mL methanol (molar mass = 32 g; density = 0.792 g/cm³) is added to 100 mL water (density = 1 g/cm³), the following diagram is obtained. [Vapour pressure vs Temperature/K plot for Water, Methanolic solution and Frozen system, with values 270.65 and 273.15 marked on the temperature axis.] x = ________ (nearest integer). [Given : Molal freezing point depression constant of water at 273.15 K is 1.86 K kg mol^-1]

Given below are two statements: Statement I: Mohr's salt is composed of only three types of ions-ferrous, ammonium and sulfate. Statement II: If the molar conductance at infinite dilution for ferrous, ammonium and sulfate ions are x₁, x₂ and x₃ S cm² mol^-1, respectively then the molar conductance for Mohr's salt solution at infinite dilution would be given by x₁ + x₂ + 2x₃. In the light of the given statements, choose the correct answer from the options given below: (1) Both Statements I and Statement II are true (2) Both Statements I and Statement II are false (3) Statement I is true but Statement II are false (4) Statement I is false but Statement II is true

Identify [A], [B] and [C], respectively in the following reaction sequence: [A] --NaNO₂, HCl (273-278 K)--> C₆H_5-N₂^+Cl^- (benzenediazonium chloride) --KI--> [B] --2Na / Dry ether--> [C] Options: (1) [A] aniline (C₆H₅NH₂), [B] chlorobenzene (C₆H₅Cl), [C] benzene (C₆H₆) (2) [A] aniline (C₆H₅NH₂), [B] chlorobenzene (C₆H₅Cl), [C] iodobenzene (C₆H₅I) (3) [A] nitrobenzene (C₆H₅NO₂), [B] iodobenzene (C₆H₅I), [C] biphenyl (C₆H_5-C₆H₅) (4) [A] aniline (C₆H₅NH₂), [B] iodobenzene (C₆H₅I), [C] biphenyl (C₆H_5-C₆H₅)

Given: DeltaH^o_sub [C (graphite)] = 710 kJ mol^-1 Delta_(C-H) H^o = 414 kJ mol^-1 Delta_(H-H) H^o = 436 kJ mol^-1 Delta_(C=C) H^o = 611 kJ mol^-1 The DeltaH_f^o for CH₂=CH₂ is ______ kJ mol^-1 (nearest integer value)

An aqueous solution of HCl with pH 1.0 is diluted by adding equal volume of water (ignoring dissociation of water). The pH of HCl solution would (Given log 2 = 0.30) (1) remain same (2) reduce to 0.5 (3) increase to 2 (4) increase to 1.3

CaCO₃(s) + 2HCl(aq) -> CaCl₂(aq) + CO₂(g) + H₂O(l) Consider the above reaction, what mass of CaCl₂ will be formed if 250 mL of 0.76 M HCl reacts with 1000 g of CaCO₃ ? (Given : Molar mass of Ca, C, O, H and Cl are 40, 12, 16, 1 and 35.5 g mol^-1, respectively) (1) 5.272 g (2) 2.636 g (3) 3.908 g (4) 10.545 g

For a reaction taking place in three steps at same temperature, overall rate constant K = (K₁ K₂)/K₃. If Ea₁, Ea₂ and Ea₃ are 40, 50 and 60 kJ/mol respectively, the overall Ea is ______ kJ/mol.

Number of compounds with one lone pair of electrons on central atom amongst following is ______ . O₃, H₂O, SF₄, ClF₃, NH₃, BrF₅, XeF₄

Number of molecules having bond order 2 from the following molecules is _______. C₂, O₂, Be₂, Li₂, Ne₂, N₂, He₂

0.1 mol of the given antiviral compound (P) will weigh ______ *10^-1 g (nearest integer). Compound P is an iodinated, fluorinated pyrimidine nucleoside (a uracil base bearing an iodine, N-linked to a deoxyribose-type sugar ring carrying two OH groups and a ring F substituent) as shown. (Given : molar mass in g mol^-1 H : 1, C : 12, N : 14, O : 16, F : 19, I : 127)

Designate whether each of the following compounds is aromatic or not aromatic. The structures shown are: (a) cyclopentadienyl anion, (b) cyclopentadienyl cation, (c) cyclobutadiene dication, (d) cyclobutadiene dianion, (e) cycloheptatrienyl (tropylium) cation, (f) cyclooctatetraene, (g) cyclobutadiene, (h) cyclopropenyl cation. (1) b, e, f, g aromatic and a, c, d, h not aromatic (2) a, c, d, e, h aromatic and b, f, g not aromatic (3) a, b, c, d aromatic and e, f, g, h not aromatic (4) e, g aromatic and a, b, c, d, f, h not aromatic

Let the shortest distance between the lines (x - 3)/3 = (y - alpha)/(-1) = (z - 3)/1 and (x + 3)/(-3) = (y + 7)/2 = (z - beta)/4 be 3 sqrt(30). Then the positive value of 5 alpha + beta is (1) 40 (2) 42 (3) 46 (4) 48

If the system of linear equations 3x + y + beta*z = 3 2x + alpha*y - z = -3 x + 2y + z = 4 has infinitely many solutions, then the value of 22*beta - 9*alpha is : (1) 49 (2) 43 (3) 37 (4) 31

Let a belongs to R and A be a matrix of order 3x3 such that det(A) = -4 and A + I = [[1, a, 1], [2, 1, 0], [a, 1, 2]], where I is the identity matrix of order 3x3. If det((a + 1)*adj((a - 1)*A)) is 2^m * 3ⁿ, m, n belongs to {0, 1, 2, ..., 20}, then m + n is equal to : (1) 14 (2) 15 (3) 16 (4) 17

Let y = y(x) be the solution curve of the differential equation x(x² + e^x) dy + (e^x (x - 2) y - x³) dx = 0, x > 0, passing through the point (1, 0). Then y(2) is equal to (1) 4/(4 - e²) (2) 2/(2 - e²) (3) 4/(4 + e²) (4) 2/(2 + e²)

If the area of the region bounded by the curves y = 4 - x²/4 and y = (x - 4)/2 is equal to alpha, then 6 alpha equals (1) 210 (2) 220 (3) 240 (4) 250

Let the vertices Q and R of the triangle PQR lie on the line (x + 3)/5 = (y - 1)/2 = (z + 4)/3, QR = 5 and the coordinates of the point P be (0, 2, 3). If the area of the triangle PQR is m/n then : (1) 5*m - 21*sqrt(2)*n = 0 (2) 5*m - 2*sqrt(21)*n = 0 (3) m - 5*sqrt(21)*n = 0 (4) 2*m - 5*sqrt(21)*n = 0

Let z be a complex number such that |z| = 1. If (2 + k²*z)/(k + z_bar) = k*z, k belongs to R, then the maximum distance of k + i*k² from the circle |z - (1 + 2i)| = 1 is : (1) sqrt(3) + 1 (2) 2 (3) sqrt(5) + 1 (4) 3

Let A be the set of all functions f: Z -> Z and R be a relation on A such that R = {(f, g) : f(0) = g(1) and f(1) = g(0)}. Then R is : (1) Reflexive but neither symmetric nor transitive (2) Symmetric but neither reflexive nor transitive (3) Transitive but neither reflexive nor symmetric (4) Symmetric and transitive but not reflexive

The value of integral from -1 to 1 of [ (1 + sqrt(|x| - x)) e^x + (sqrt(|x| - x)) e^(-x) ] / (e^x + e^(-x)) dx is equal to (1) 2 + 2 sqrt(2)/3 (2) 3 - 2 sqrt(2)/3 (3) 1 - 2 sqrt(2)/3 (4) 1 + 2 sqrt(2)/3

Let f, g : (1, infinity) -> R be defined as f(x) = (2x + 3)/(5x + 2) and g(x) = (2 - 3x)/(1 - x). If the range of the function fog : [2, 4] -> R is [alpha, beta], then 1/(beta - alpha) is equal to (1) 2 (2) 29 (3) 56 (4) 68

Let a = 2 i - 3 j + 4 k, b = 3 i + 4 j - 5 k and a vector c be such that a x (b + c) + b x c = i + 8 j + 13 k. If a . c = 13, then (24 - b . c) is equal to ________

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is 70/3 and the product of the third and fifth terms is 49. Then the sum of the 4th, 6th and 8th terms is equal to : (1) 78 (2) 84 (3) 91 (4) 96

Let P_n = alphaⁿ + betaⁿ, n belongs to N. If P₁₀ = 123, P₉ = 76, P₈ = 47 and P₁ = 1, then the quadratic equation having roots 1/alpha and 1/beta is : (1) x² - x + 1 = 0 (2) x² + x - 1 = 0 (3) x² - x - 1 = 0 (4) x² + x + 1 = 0

lim_x->0^+ [tan(5 (x)^(1/3)) log_e(1 + 3x²)] / [(tan^-1(3 sqrt(x)))² (e^(5 (x)^(4/3)) - 1)] is equal to (1) 1 (2) 1/3 (3) 5/3 (4) 1/15

If the solution y(x) of the given differential equation (e^y + 1) cos x dx + e^y sin x dy = 0 passes through the point (π/2, 0), then the value of e^y(π/6) is equal to _______.

If the function f(x) = 2*x³ - 9*a*x² + 12*a²*x + 1, where a > 0, attains its local maximum and local minimum values at p and q, respectively, such that p² = q, then f(3) is equal to : (1) 37 (2) 55 (3) 10 (4) 23

The term independent of x in the expansion of ((x + 1)/(x^2/3 + 1 - x^1/3) - (x - 1)/(x - x^1/2))¹⁰, x > 1, is : (1) 150 (2) 210 (3) 120 (4) 240

Considering the principal values of the inverse trigonometric functions, sin^-1( (sqrt(3)/2) x + (1/2) sqrt(1 - x²) ), -1/2 < x < 1/sqrt(2), is equal to (1) pi/4 + sin^-1 x (2) pi/6 + sin^-1 x (3) 5pi/6 - sin^-1 x (4) -5pi/6 - sin^-1 x

A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the remaining are manufactured at plant B. 80% of the motorcycles manufactured at plant A are rated of the standard quality, while 90% of the motorcycles manufactured at plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If p is the probability that it was manufactured at plant B, then 126p is (1) 56 (2) 64 (3) 66 (4) 54

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at 315th position in this arrangement is : (1) NRAGPU (2) NRAPGU (3) NRAGUP (4) NRAPUG

Let m and n be the number of points at which the function f(x) = max{ x, x³, x⁵, ..., x²¹ }, x in R, is not differentiable and not continuous, respectively. Then m + n is equal to ______

Let A = [[alpha, -1], [6, beta]], alpha > 0, such that det(A) = 0 and alpha + beta = 1. If I denotes 2x2 identity matrix, then the matrix (I + A)⁸ is : (1) [[4, -1], [6, -1]] (2) [[257, -64], [514, -127]] (3) [[766, -255], [1530, -509]] (4) [[1025, -511], [2024, -1024]]

If the area of the region {(x, y): |4 - x²| <= y <= x², y <= 4, x >= 0} is (80*sqrt(2)/alpha - beta), alpha, beta belongs to N, then alpha + beta is equal to ______.

Let [.] denote the greatest integer function. If integral from 0 to e³ of [1/e^x-1] dx = alpha - log_e(2), then alpha³ is equal to ______.

The number of relations on the set A = {1, 2, 3}, containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is ______