JEE Main Guess Paper 2027

Match the LIST-I with LIST-II LIST-I: A. Triatomic rigid gas; B. Diatomic non-rigid gas; C. Monoatomic gas; D. Diatomic rigid gas LIST-II: I. Cp/Cv = 5/3; II. Cp/Cv = 7/5; III. Cp/Cv = 4/3; IV. Cp/Cv = 9/7 Choose the correct answer from the options given below: (1) A-III, B-II, C-IV, D-I (2) A-III, B-IV, C-I, D-II (3) A-II, B-IV, C-I, D-III (4) A-IV, B-II, C-III, D-I

An element delta l = delta x i-hat is placed at the origin and carries a large current I = 10 A. The magnetic field on the y-axis at a distance of 0.5 m from the element (delta x of 1 cm length) is: (1) 4x10^-8 T (2) 8x10^-8 T (3) 10x10^-8 T (4) 12x10^-8 T

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P-V diagram. The relation between the ratio V_a/V_d and the ratio V_b/V_c is: (1) V_a/V_d = V_b/V_c (2) V_a/V_d != V_b/V_c (3) V_a/V_d = (V_b/V_c)^(-1) (4) V_a/V_d = (V_b/V_c)²

In the following circuit, the reading of the ammeter will be (Take Zener breakdown voltage = 4 V) (1) 24 mA (2) 80 mA (3) 10 mA (4) 60 mA

The energy levels of an hydrogen atom are shown below (arrows A, B, C, D between levels n = 1, 2, 3, 4). The transition corresponding to emission of shortest wavelength is (1) A (2) C (3) B (4) D

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

Match List-I with List-II : List-I (Y vs X) (A) Y = magnetic susceptibility, X = magnetising field (B) Y = magnetic field, X = distance from centre of a current carrying wire for x < a (where a = radius of wire) (C) Y = magnetic field, X = distance from centre of a current carrying wire for x > a (where a = radius of wire) (D) Y = magnetic field inside solenoid, X = distance from centre List-II (Shape of Graph) (I) straight line through origin with positive slope (II) constant then falling curve (III) horizontal constant line (IV) decreasing (inverse) curve Choose the correct answer from the options given below : (1) (A)-(IV), (B)-(I), (C)-(III), (D)-(II) (2) (A)-(III), (B)-(I), (C)-(IV), (D)-(II) (3) (A)-(I), (B)-(III), (C)-(II), (D)-(IV) (4) (A)-(III), (B)-(IV), (C)-(I), (D)-(II)

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

In series RLC resonator, if the self inductance and capacitance become double, the new resonant frequency (f₂) and new quality factor (Q₂) will be: (f₁ = original resonant frequency, Q₁ = original quality factor) (A) f₂ = f₁/2 and Q₂ = Q₁ (B) f₂ = f₁ and Q₂ = Q₁/2 (C) f₂ = 2f₁ and Q₂ = Q₁ (D) f₂ = f₁ and Q₂ = 2Q₁

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%

The pressure of a gas changes linearly with volume from A to B as shown in figure. If no heat is supplied to or extracted from the gas then change in the internal energy of the gas will be (in the P-V graph, B is at (50 cc, 50 kPa) and A is at (200 cc, 10 kPa)). Options: (1) zero (2) 4.5 J (3) 6 J (4) -4.5 J

Two coherent sources of light interfere. The intensity ratio of two sources is 1 : 4. For this interference pattern if the value of (I_max + I_min)/(I_max - I_min) is equal to (2 alpha + 1)/(beta + 3), then alpha/beta will be : Option A: 1.5 Option B: 2 Option C: 0.5 Option D: 1

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

For a uniformly charged thin spherical shell, the electric potential (V) radially away from the centre (O) of shell can be graphically represented as - (options are four V vs r graphs) (1) V constant for r < R, then decreasing linearly to zero (2) V rising (concave) to a cusp at r = R, then decaying (3) V rising to a peak at r = R, then decaying on both sides (4) V constant for r < R, then decaying as 1/r for r > R

Expression for an electric field is given by E = 4000 x² i V/m. The electric flux through the cube of side 20 cm when placed in electric field (as shown in the figure) is ______ V cm. (Cube of side 20 cm with one corner at origin (0,0,0).)

A mass m is attached to two strings as shown in figure. The spring constants of two springs are K₁ and K₂. For the frictionless surface, the time period of oscillation of mass m is (1) (1/2 pi) sqrt((K₁ + K₂)/m) (2) (1/2 pi) sqrt((K₁ - K₂)/m) (3) 2 pi sqrt(m/(K₁ - K₂)) (4) 2 pi sqrt(m/(K₁ + K₂))

Two wires each of radius 0.2 cm and negligible mass, one made of steel and the other made of brass are loaded as shown in the figure. The elongation of the steel wire is ____ × 10^-6 m. [Young's modulus for steel = 2 × 10¹¹ N m^-2 and g = 10 m s^-2] (figure: 1.6 m steel wire with 2 kg load, 1 m brass wire with 1.14 kg load)

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

Match List - I with List - II. List - I (Complex ion) — List - II (Spin only magnetic moment in B.M.) (A) [Cr(NH₃)₆]³⁺ (I) 4.90 (B) [NiCl₄]^2- (II) 3.87 (C) [CoF₆]^3- (III) 0.0 (D) [Ni(CN)₄]^2- (IV) 2.83 Choose the correct answer from the options given below: (1) (A)-(IV), (B)-(III), (C)-(I), (D)-(II) (2) (A)-(II), (B)-(IV), (C)-(I), (D)-(III) (3) (A)-(I), (B)-(IV), (C)-(II), (D)-(III) (4) (A)-(II), (B)-(III), (C)-(I), (D)-(IV)

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

Match the LIST-I with LIST-II LIST-I (Molecules/ion) | LIST-II (Hybridisation of central atom) A. PF₅ | I. dsp² B. SF₆ | II. sp³d C. Ni(CO)₄ | III. sp³d² D. [PtCl₄]^2- | IV. sp³ Choose the correct answer from the options given below: Options: (1) A-I, B-II, C-III, D-IV (2) A-II, B-III, C-IV, D-I (3) A-III, B-I, C-IV, D-II (4) A-IV, B-I, C-II, D-III

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

The least acidic compound, among the following is: (A) 3-ethoxyphenol (EtO-C₆H_4-OH) (B) benzenesulfonic acid (C₆H_5-SO₃H) (C) butanoic acid (CH₃CH₂CH_2-COOH) (D) ethyl propiolate (EtO₂C-C(triple bond)C-H, a terminal alkyne ester) Options: (1) B (2) D (3) C (4) A

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

At -20 degrees C and 1 atm pressure, a cylinder is filled with equal number of H₂, I₂ and HI molecules for the reaction H₂(g) + I₂(g) <=> 2HI(g), the K_p for the process is x x 10^-1. x = ________. [Given: R = 0.082 L atm K^-1 mol^-1] (1) 1 (2) 10 (3) 0.01 (4) 2

The emf of cell Tl | Tl^+ (0.001M) || Cu²⁺ (0.01M) | Cu is 0.83 V at 298 K. It could be increased by: (1) increasing concentration of Tl^+ ions (2) increasing concentration of Cu²⁺ ions (3) increasing concentration of both Tl^+ and Cu²⁺ ions (4) decreasing concentration of both Tl^+ and Cu²⁺ ions

Which compound would give 3-methyl-6-oxoheptanal upon ozonolysis? Options: (four cyclohexene structures bearing methyl substituents at different ring positions; select the alkene whose reductive ozonolysis of the ring C=C yields 3-methyl-6-oxoheptanal) (1) 1,6-dimethylcyclohexene (2) 1-methyl-4-methylcyclohexene (1-methyl, methyl at position 4/5) (3) 1,4-dimethylcyclohexene (4) 1,3-dimethylcyclohexene

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

In the depression of freezing point experiment A. Vapour pressure of the solution is less than that of pure solvent B. Vapour pressure of the solution is more than that of pure solvent C. Only solute molecules solidify at the freezing point D. Only solvent molecules solidify at the freezing point. Choose the most appropriate answer from the options given below: Options: 1) B and C only 2) A and D only 3) A and C only 4) A only

Correct order of limiting molar conductivity for cations in water at 298 K is: Options: (1) Mg²⁺ > H^+ > Ca²⁺ > K^+ > Na^+ (2) H^+ > Na^+ > K^+ > Ca²⁺ > Mg²⁺ (3) H^+ > Na^+ > Ca²⁺ > Mg²⁺ > K^+ (4) H^+ > Ca²⁺ > Mg²⁺ > K^+ > Na^+

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 (A) 2CO(g) + O₂(g) -> 2CO₂(g), ΔH₁° = -x kJ mol^-1 (B) C(graphite) + O₂(g) -> CO₂(g), ΔH₂° = -y kJ mol^-1 The ΔH° for the reaction C(graphite) + 1/2 O₂(g) -> CO(g) is (1) (2x - y)/2 (2) (x + 2y)/2 (3) (x - 2y)/2 (4) 2y - x

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

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

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 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 vector a = 4i - j + k, vector b = 11i - j + k and vector c be a vector such that (a + b) x c = c x (-2a + 3b). If (2a + 3b) . c = 1670, then |c|² is equal to : (1) 1609 (2) 1600 (3) 1618 (4) 1627

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

Let y = y(x) be the solution of the differential equation (1 + y²) e^(tan x) dx + cos² x (1 + e^(2 tan x)) dy = 0, y(0) = 1. Then y(pi/4) is equal to (1) 1/e² (2) 1/e (3) 2/e (4) 2/e²

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

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

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

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 ______