Question 1:

If f(x) =
f(x) = { x³ sin(1/x), x ≠ 0; 0, x = 0 }, then:

1. f''(0) = 1
2. f''(2π) = (24 - π²) / (2π)
3. f''(2π) = (12 - π²) / (2π)
4. f''(0) = ?

Question 2:

The area of a quadrilateral ABCD with vertices A(3, 1, −1), B(5/3, 7/3, 1/3), C(2, 2, 1), D(10/3, 2/3, −1/3) is:

1. 4√2 / 3
2. 5√2 / 3
3. 2√2
4. 2√2 / 3

Question 3:

The integral ∫₀^π/4 (cos²x sin²x) / (cos³x + sin³x)² dx equals:

1. 1/12
2. 1/9
3. 1/6
4. 1/3

Question 4:

The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, an observation was mistakenly taken as 8 instead of 12. The correct standard deviation is:

1. √3.86
2. 1.8
3. √3.96
4. 1.94

Question 5:

The function f(x) = (x² + 2x − 15) / (x² − 4x + 9) is:

1. both one-one and onto
2. onto but not one-one
3. neither one-one nor onto
4. one-one but not onto

Question 6:

Let A = {n ∈ [100, 700] ∩ N : n is neither a multiple of 3 nor a multiple of 4}. Then the number of elements in A is:

1. 300
2. 280
3. 310
4. 290

Question 7:

Let C be the circle of minimum area touching the parabola y = 6 − x² and the lines y = √3|x|. Then, which one of the following points lies on the circle C?

1. (2, 4)
2. (1, 2)
3. (2, 2)
4. (1, 1)

Question 8:

For α, β ∈ ℝ and a natural number n, let Ar = | r 1 n² + α
2r 2 n² − β
3r − 2 3 n(3n − 1)/2 |. Then 2A₁₀ − A₅ is:

1. 4α + 2β
2. 2α + 4β
3. 2n
4. 0

Question 9:

The shortest distance between the lines:
(x − 3)/2 = (y + 15)/−7 = (z − 9)/5 and (x + 1)/2 = (y − 1)/1 = (z − 9)/−3, is:

1. 6√3
2. 4√3
3. 5√3
4. 8√3

Question 10:

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. 54
2. 64
3. 66
4. 56

Question 11:

Let α, β be the distinct roots of the equation:
x² − (t² − 5t + 6)x + 1 = 0, t ∈ ℝ, and a_n = αⁿ + βⁿ. Then the minimum value of (a₂₀₂₃ + a₂₀₂₅) / a₂₀₂₄ is:

1. 1/4
2. −1/2
3. −1/4
4. 1/2

Question 12:

Let the relations R₁ and R₂ on the set X = {1, 2, 3, ..., 20} be given by:
R₁ = {(x, y) : 2x − 3y = 2} and R₂ = {(x, y) : −5x + 4y = 0}. If M and N be the minimum number of elements required to be added in R₁ and R₂, respectively, in order to make the relations symmetric, then M + N equals:

1. 8
2. 16
3. 12
4. 10

Question 13:

A variable line of slope m > 0 passing through the point (4, −9) intersects the coordinate axes at the points A and B. The minimum value of the sum of the distances of A and B from the origin is:

1. 25
2. 30
3. 15
4. 10

Question 14:

The interval in which the function f(x) = x^x is strictly increasing is:

1. (0, 1/e]
2. [1/e², 1)
3. (0, ∞)
4. [1/e, ∞)

Question 15:

A circle is inscribed in an equilateral triangle of side 12. If the area and perimeter of any square inscribed in this circle are m and n, respectively, then m + n² is equal to:

1. 396
2. 408
3. 312
4. 414

Question 16:

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is:

1. 24
2. 56
3. 16
4. 48

Question 17:

Let y = y(x) be the solution of the differential equation:
(1 + x²) dy/dx + y = e^tan⁻¹(x), y(1) = 0. Then y(0) is:

1. 1/4(e^π/2 − 1)
2. 1/2(1 − e^−π/2)
3. 1/4(1 − e^π/2)
4. 1/2(e^π/2 − 1)

Question 18:

Let y = y(x) be the solution of the differential equation:
(2x ln x) dy/dx + 2y = 3/x ln x, x > 0, and y(e⁻¹) = 0. Then, y(e) is equal to:

1. −3/2e
2. −2/3e
3. −3/e
4. −2/e

Question 19:

Let the area of the region enclosed by the curves y = 3x, 2y = 27 − 3x, and y = 3x − x√x be A. Then 10A is equal to:

1. 184
2. 154
3. 172
4. 162

Question 20:

Let f : (−∞, ∞) → ℝ \ {0} be a differentiable function such that:
f'(1) = lim_a→∞ a²f(1/a). Then:

1. 3/2 + π/4
2. 3/8 + π/4
3. 5/2 + π/8
4. 3/4 + π/8

Question 21:

Let αβγ = 45; α, β, γ ∈ ℝ. If:
x(α, 1, 2) + y(1, β, 2) + z(2, 3, γ) = (0, 0, 0),
for some x, y, z ∈ ℝ, xyz ≠ 0, then 6α + 4β + γ is equal to:

1. 55
2. 60
3. 50
4. 65

Question 22:

A conic C passes through the point (4, −2) and P(x, y), x ≥ 3, is any point on C. Let the slope of the line touching the conic C only at a single point P be half the slope of the line joining the points P and (3, −5). If the focal distance of the point (7, 1) on C is d, then 12d equals:

1. 70
2. 75
3. 80
4. 85

Question 23:

Let:
Iₖ = ∫₀¹ (1 − x)ᵏ dx, k ∈ ℕ.
Then the value of:
Σ_k=1¹⁰ (1/7)(Iₖ − 1) is equal to:

1. 60
2. 62
3. 65
4. 68

Question 24:

Let x₁, x₂, x₃, x₄ be the solution of the equation 4x⁴ + 8x³ − 17x² − 12x + 9 = 0 and (4 + x²₁)(4 + x²₂)(4 + x²₃)(4 + x²₄) = 125/16m. Then the value of m is:

1. 210
2. 215
3. 221
4. 225

Question 25:

Let L₁, L₂ be the lines passing through the point P(0, 1) and touching the parabola 9x² + 12x + 18y − 14 = 0. Let Q and R be the points on the lines L₁ and L₂ such that the ΔPQR is an isosceles triangle with base QR. If the slopes of the lines QR are m₁ and m₂, then 16(m₁² + m₂²) is equal to:

1. 60
2. 64
3. 68
4. 72

Question 26:

If the second, third, and fourth terms in the expansion of (x + y)ⁿ are 135, 30, and 10³, respectively, then 6(n³ + x² + y) is equal to:

1. 800
2. 806
3. 812
4. 820

Question 27:

Let the first term of a series be T₁ = 6 and its r-th term Tᵣ = 3Tᵣ₋₁ + 6r, r = 2, 3, ..., n. If the sum of the first n terms of this series is 1/5 (n² − 12n + 39)(4.6n − 5.3n + 1), then n is equal to:

1. 5
2. 6
3. 7
4. 8

Question 28:

For n ∈ ℕ, if cot⁻¹ 3 + cot⁻¹ 4 + cot⁻¹ 5 + cot⁻¹ n = π/4, then n is equal to:

1. 45
2. 46
3. 47
4. 48

Question 29:

Let P(10, −2, −1) and Q be the foot of the perpendicular drawn from the point R(1, 7, 6) on the line passing through the points (2, −5, 11) and (−6, 7, −5). Then the length of the line segment PQ is equal to:

1. 12
2. 13
3. 14
4. 15

Question 30:

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

1. 44
2. 45
3. 46
4. 47

Question 31:

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

1. 3%
2. 1%
3. 4%
4. 5%

Question 32:

A bullet of mass 50 g is fired with a speed of 100 m/s on a plywood and emerges with 40 m/s. The percentage loss of kinetic energy is:

1. 32%
2. 44%
3. 16%
4. 84%

Question 33:

The ratio of the shortest wavelength of the Balmer series to the shortest wavelength of the Lyman series for hydrogen atom is:

1. 4:1
2. 1:2
3. 1:4
4. 2:1

Question 34:

To project a body of mass m from Earth’s surface to infinity, the required kinetic energy is (assume the radius of Earth is Re, g = acceleration due to gravity on the surface of Earth):

1. 2mgRe
2. mgRe
3. 0.5mgRe
4. 4mgRe

Question 35:

Electromagnetic waves travel in a medium with speed 1.5 × 10⁸ m/s. The relative permeability of the medium is 2.0. The relative permittivity will be:

1. 5
2. 1
3. 4
4. 2

Question 36:

Which of the following phenomena is not explained by the wave nature of light?

(A) Reflection
(B) Diffraction
(C) Photoelectric effect
(D) Interference
(E) Polarization

Choose the most appropriate answer from the options below:

1. E only
2. C only
3. B, D only
4. A, C only

Question 37:

While measuring the diameter of a wire using a screw gauge, the following readings were noted. The main scale reading is 1 mm, and the circular scale reading is equal to 42 divisions. The pitch of the screw gauge is 1 mm, and it has 100 divisions on the circular scale. The diameter of the wire is x/50 mm. The value of x is:

1. 142
2. 71
3. 42
4. 21

Question 38:

σ is the uniform surface charge density of a thin spherical shell of radius R. The electric field at any point on the surface of the spherical shell is:

1. σ/ε₀
2. σ/2ε₀
3. σ/ε₀R
4. σ/4ε₀

Question 39:

The value of unknown resistance (x) for which the potential difference between B and D will be zero in the arrangement shown, is:

1. 3 Ω
2. 9 Ω
3. 6 Ω
4. 42 Ω

Question 40:

The specific heat at constant pressure of a real gas obeying PV² = RT equation is:

1. Cv + R
2. R/3 + Cv
3. R
4. Cv + R/2V

Question 41:

Match List I with List II

List I (Quantity) | List II (Dimension)

• A. Torque | I. [ML²T⁻²]
• B. Magnetic field | II. [MA⁻¹T⁻²]
• C. Magnetic moment | III. [AL²]
• D. Permeability of free space | IV. [MLT⁻²A⁻²]

Choose the correct answer from the options below:

1. A-I, B-III, C-II, D-IV
2. A-IV, B-III, C-II, D-I
3. A-III, B-I, C-II, D-IV
4. A-IV, B-II, C-III, D-I

Question 42:

Given below are two statements:

Statement I: In an LCR series circuit, current is maximum at resonance.
Statement II: Current in a purely resistive circuit can never be less than that in a series LCR circuit when connected to the same voltage source.

In the light of the above statements, choose the correct option from the given below:

1. Statement I is true but Statement II is false.
2. Statement I is false but Statement II is true.
3. Both Statement I and Statement II are true.
4. Both Statement I and Statement II are false.

Question 43:

The correct truth table for the following logic circuit is:

The circuit consists of an AND gate, a NOT gate, and an OR gate. The output Y is determined as follows:

• The output of the AND gate is A ⋅ B.
• The output of the NOT gate is A ⋅ B.
• The output Y is the OR of A ⋅ B and B: Y = A ⋅ B + B.

Question 44:

A sample contains a mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample is:

1. 1/32
2. 2√2
3. 1
4. 1/(2√2)

Question 45:

A light string passing over a smooth light pulley connects two blocks of masses m₁ and m₂ (where m₂ > m₁). If the acceleration of the system is g√2, then the ratio of the masses m₁/m₂ is:

1. (√2−1)/(√2+1)
2. (1+√5)/(√5−1)
3. (1+√5)/(√2−1)
4. (√3+1)/(√2−1)

Question 46:

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

1. D
2. C
3. A
4. B

Question 47:

A train starting from rest first accelerates uniformly up to a speed of 80 km/h for time t, then it moves with a constant speed for time 3t. The average speed of the train for this duration of the journey will be (in km/h):

1. 80
2. 70
3. 30
4. 40

Question 48:

An element Δl = Δxî is placed at the origin and carries a large current I = 10A. The magnetic field on the y-axis at a distance of 0.5m from the element Δx of 1 cm length is:

1. 4 × 10⁻⁸ T
2. 8 × 10⁻⁸ T
3. 12 × 10⁻⁸ T
4. 10 × 10⁻⁸ T

Question 49:

A small ball of mass m and density ρ is dropped in a viscous liquid of density ρ₀. After some time, the ball falls with constant velocity. The viscous force on the ball is:

1. mg(ρ₀/ρ − 1)
2. mg(1 + ρ/ρ₀)
3. mg(1 − ρ/ρ₀)
4. mg(1 − ρ₀/ρ)

Question 50:

In the photoelectric experiment, energy of 2.48 eV irradiates a photo-sensitive material. The stopping potential was measured to be 0.5 V. The work function of the photo-sensitive material is:

1. 0.5 eV
2. 1.68 eV
3. 2.48 eV
4. 1.98 eV

Question 51:

If the radius of Earth is reduced to three-fourths of its present value without change in its mass, then the value of the duration of the day of Earth will be ____ hours 30 minutes.

1. 13 hours 30 minutes
2. 12 hours 30 minutes
3. 11 hours 30 minutes
4. 14 hours 30 minutes

Question 52:

Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point P is xσ/ε₀. The value of x is ______.

1. 2
2. 3
3. 4
4. 5

Question 53:

A big drop is formed by coalescing 1000 small droplets of water. The ratio of surface energy of 1000 droplets to that of the big drop is 10^x. The value of x is ______.

1. 0
2. 1
3. 2
4. 3

Question 54:

When a DC voltage of 100V is applied to an inductor, a DC current of 5A flows through it. When an AC voltage of 200V peak value is connected to the inductor, its inductive reactance is found to be 20√3 Ω. The power dissipated in the circuit is ____ W.

1. 250 W
2. 300 W
3. 350 W
4. 400 W

Question 55:

The refractive index of a prism is μ = √3 and the ratio of the angle of minimum deviation to the angle of prism is one. The value of the angle of the prism is ____°.

1. 45°
2. 50°
3. 60°
4. 75°

Question 56:

A wire of resistance R and radius r is stretched till its radius becomes r/2. If the new resistance of the stretched wire is xR, then the value of x is ____.

1. 4R
2. 8R
3. 12R
4. 16R

Question 57:

The radius of a certain orbit of the hydrogen atom is 8.48 Å. If the energy of the electron in this orbit is E/x, then x = ____ (Given a₀ = 0.529 Å, E = energy of electron in ground state).

1. 8
2. 12
3. 16
4. 20

Question 58:

A circular coil having 200 turns, 2.5 × 10⁻⁴ m² area and carrying 100 μA current is placed in a uniform magnetic field of 1 T. Initially, the magnetic dipole moment (M) was directed along B. Amount of work required to rotate the coil through 90° from its initial orientation such that M becomes perpendicular to B is ____ μJ.

1. 2 μJ
2. 5 μJ
3. 8 μJ
4. 10 μJ

Question 59:

A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s. The maximum velocity of the particle is ____ cm/s.

1. 8 cm/s
2. 10 cm/s
3. 12 cm/s
4. 14 cm/s

Question 60:

For three vectors A = (−x î − 6 ĵ − 2 k̂), B = (−î + 4 ĵ + 3 k̂) and C = (−8 î − ĵ + 3 k̂), if A · (B × C) = 0, then the value of x is ____.

1. 3
2. 5
3. 6
4. 4

Question 61:

Functional group present in sulphonic acid is:

1. SO₃H
2. SO₃H
3. −S−OH
4. −SO₂

Question 62:

Match List I with List II:

Choose the correct answer from the options given below:

1. A-IV, B-I, C-III, D-II
2. A-III, B-I, C-II, D-IV
3. A-III, B-III, C-I, D-IV
4. A-III, B-IV, C-II, D-I

Question 63:

Given below are two statements:

• Statement I: Picric acid is 2,4,6-trinitrotoluene.
• Statement II: Phenol-2,4-disulphuric acid is treated with conc. HNO₃ to get picric acid.

Choose the most appropriate answer from the options given below:

1. Statement I is incorrect but Statement II is correct.
2. Both Statement I and Statement II are incorrect.
3. Statement I is correct but Statement II is incorrect.
4. Both Statement I and Statement II are correct.

Question 64:

Which of the following is a metamer of the given compound (X)?

1. Option 1
2. Option 2
3. Option 3
4. Option 4

Question 65:

DNA molecule contains 4 bases whose structures are shown below. One of the structures is not correct. Identify the incorrect base structure.

1. Option 1
2. Option 2
3. Option 3
4. Option 4

Question 66:

Match List I with List II:

Choose the correct answer from the options below:

1. A-III, B-I, C-IV, D-II
2. A-III, B-I, C-II, D-IV
3. A-IV, B-III, C-I, D-II
4. A-III, B-IV, C-I, D-II

Question 67:

Given below are two statements:

• Statement I: Gallium is used in the manufacturing of thermometers.
• Statement II: A thermometer containing gallium is useful for measuring the freezing point (256 K) of brine solution.

Choose the correct answer from the options below:

1. Both Statement I and Statement II are false.
2. Statement I is false but Statement II is true.
3. Both Statement I and Statement II are true.
4. Statement I is true but Statement II is false.

Question 68:

Which of the following statements are correct?

• A. Glycerol is purified by vacuum distillation because it decomposes at its normal boiling point.
• B. Aniline can be purified by steam distillation because it decomposes at its normal boiling point.
• C. Ethanol can be separated from ethanol-water mixture by azeotropic distillation because it forms azeotrope.
• D. An organic compound is pure if mixed melting point remains unchanged.

Choose the correct answer from the options below:

1. A, B, C only
2. A, C, D only
3. B, C, D only
4. A, B, D only

Question 69:

Match List I with List II:

Choose the correct answer from the options below:

1. A-II, B-III, C-I, D-IV
2. A-III, B-IV, C-II, D-I
3. A-II, B-IV, C-III, D-I
4. A-III, B-II, C-IV, D-I

Question 70:

In Reimer-Tiemann reaction, phenol is converted into salicylaldehyde through an intermediate. The structure of the intermediate is:

1. Option 1
2. Option 2
3. Option 3
4. Option 4

Question 71:

Which of the following material is not a semiconductor?

1. Germanium
2. Graphite
3. Silicon
4. Copper oxide

Question 72:

Consider the following complexes:

• (A) [CoCl(NH₃)₅]²⁺
• (B) [Co(CN)₆]^3-
• (C) [Co(NH₃)₅(H₂O)]³⁺
• (D) [Cu(H₂O)₄]²⁺

The correct order of A, B, C, and D in terms of wavenumber of light absorbed is:

1. C < D < A < B
2. D < A < C < B
3. A < C < B < D
4. B < C < A < D

Question 73:

Match List I with List II:

Choose the correct answer from the options below:

1. A-IV, B-III, C-II, D-I
2. A-IV, B-II, C-I, D-II
3. A-III, B-IV, C-II, D-I
4. A-III, B-IV, C-II, D-I

Question 74:

The electron affinity values are negative for:

• A. Be → Be^-
• B. N → N^-
• C. O → O^-
• D. Na → Na^-
• E. Al → Al^-

Choose the most appropriate answer from the options below:

1. D and E only
2. A, B, D and E only
3. A and D only
4. A, B and C only

Question 75:

The number of elements from the following that do not belong to lanthanoids is:

Eu, Cm, Er, Tb, Yb, and Lu

1. 3
2. 4
3. 1
4. 5

Question 76:

The density of 'x' M solution ('x' molar) of NaOH is 1.12 g/mL, while in molality, the concentration of the solution is 3m (3 molal). Then x is:

1. 3.5
2. 3.0
3. 3.8
4. 2.8

Question 77:

Which among the following aldehydes is most reactive towards nucleophilic addition reactions?

1. HCHO
2. C₂H₅CHO
3. CH₃CHO
4. C₃H₇CHO

Question 78:

At −20°C and 1 atm pressure, a cylinder is filled with an equal number of H₂, I₂, and HI molecules for the reaction:

H₂(g) + I₂(g) ⇌ 2HI(g)

K_p for the process is x × 10⁻¹. x =?

1. 2
2. 1
3. 10
4. 0.01

Question 79:

Match List I with List II:

Choose the correct answer from the options below:

1. A-I, B-II, C-III, D-IV
2. A-III, B-II, C-IV, D-I
3. A-III, B-I, C-IV, D-II
4. A-II, B-IV, C-I, D-III

Question 80:

A conductivity cell with two electrodes (dark side) is half filled with an infinitely dilute aqueous solution of a weak electrolyte. If volume is doubled by adding more water at constant temperature, the molar conductivity of the cell will:

1. Increase sharply
2. Remain same or cannot be measured accurately
3. Decrease sharply
4. Depend upon type of electrolyte

Question 81:

Consider the dissociation of the weak acid HX as given below:

HX(aq) ⇌ H⁺(aq) + X⁻(aq), K_a = 1.2×10⁻⁵

The osmotic pressure of 0.03 M aqueous solution of HX at 300 K is ___ ×10⁻² bar (nearest integer).

Given: R = 0.083 L·bar·mol⁻¹·K⁻¹

Solution: The osmotic pressure is calculated using the formula:

Π = iCRT, where i is the van 't Hoff factor considering dissociation.

Question 82:

The difference in the ‘spin-only’ magnetic moment values of KMnO₄ and the manganese product formed during titration of KMnO₄ against oxalic acid in acidic medium is ___ BM (nearest integer).

Solution: In KMnO₄, Mn is in the +7 oxidation state (no unpaired electrons, μ = 0 BM). In the reduced product (Mn²⁺), there are 5 unpaired electrons, giving a spin-only magnetic moment of 6 BM.

Question 83:

Time required for 99.9% completion of a first-order reaction is ___ times the time required for completion of 90% reaction (nearest integer).

Solution: The time for a given percentage completion is proportional to ln(remaining concentration). For 99.9% completion:

t_99.9% = (ln(1000) / ln(10)) × t_90%, which simplifies to 3 times t_90%.

Question 84:

Number of molecules from the following which can exhibit hydrogen bonding is ___ (nearest integer):

CH₃OH, H₂O, C₆H₆, C₆H₅NO₂, HF, NH₃

Solution: Hydrogen bonding is exhibited by CH₃OH, H₂O, HF, NH₃, and C₆H₅NO₂, due to the presence of N-H, O-H, or F-H bonds.

Question 85:

9.3 g of pure aniline upon diazotization followed by coupling with phenol gives an orange dye. The mass of orange dye produced (assume 100% yield/conversion) is _____ g (nearest integer).

Solution: The molecular weight of aniline is 93 g/mol. Given 9.3 g of aniline, this corresponds to 0.1 mol. Each mole of aniline produces 1 mole of dye, and the molecular weight of the dye is 200 g/mol, so:

Mass of dye = 0.1 × 200 = 20 g.

Question 86:

The major product of the following reaction is P:

CH₃C≡C-CH₃
(i) Na/liq. NH₃ → (ii) dil. KMnO₄, 273 K → P
The number of oxygen atoms present in product P is _____ (nearest integer).

Solution: The reaction produces a vicinal diol as the final product, with two hydroxyl groups attached to the adjacent carbon atoms, contributing two oxygen atoms.

Question 87:

The frequency of the de-Broglie wave of an electron in Bohr’s first orbit of the hydrogen atom is ___ ×10¹³ Hz (nearest integer).

Given: RH (Rydberg constant) = 2.18×10⁻¹⁸ J, h (Planck’s constant) = 6.6×10⁻³⁴ J·s.

Solution: The frequency is calculated using the relation ν = E/h, where E is the kinetic energy of the electron in the first orbit.

Question 88:

The major products from the following reaction sequence are product A and product B:

B (i) Br₂ → (ii) alc. KOH (3 eq.) → A
B (i) Br₂ → (ii) Na⁺/O⁻ (1.0 eq.) → B
The total sum of π electrons in product A and product B are ____ (nearest integer).

Solution: Product A is benzene (6 π electrons), and product B is an alkene (2 π electrons). The total sum is 8 π electrons.

Question 89:

Among CrO, Cr₂O₃, and CrO₃, the sum of spin-only magnetic moment values of basic and amphoteric oxides is ___ ×10⁻² BM (nearest integer).

Given: Atomic number of Cr is 24.

Solution: CrO (Cr²⁺) has 4 unpaired electrons, Cr₂O₃ (Cr³⁺) has 3 unpaired electrons, and CrO₃ (Cr⁶⁺) has no unpaired electrons.

Question 90:

An ideal gas, C_V = 5/2 R, is expanded adiabatically against a constant pressure of 1 atm until it doubles in volume. If the initial temperature and pressure is 298 K and 5 atm, respectively, then the final temperature is ____ K (nearest integer).

Solution: Using the adiabatic relation PV^γ = constant, where γ = C_P/C_V, the final temperature is calculated.