Doubtly
@Doubtly · Joined Feb 2021
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- If ω is a complex cube root of unity, prove that (1 – ω + ω2)6 + (1 + ω – ω2)6 = 128. ω is the complex cube root of unity. ∴ ω³ = 1 and 1 + ω + ω² = 0 Also, 1 + ω² = -ω, 1 + ω = -ω² ∴ L.H.S. = (1 – ω + ω²)⁶ + (1 + ω – ω²)⁶ = [... 1
- 18i complex number Let √18i = a + bi, where a, b ∈ R. Squaring on both sides, we get 18i = a² + b² i² + 2abi 0 + 18i = a² – b² + 2abi …..[∵ i² = -1] Equating r... 1
- i131 + i49 complex number i¹³¹ + i⁴⁹ = (i⁴)³² . i³ + (i⁴)¹² . i = (1)³² (-i) + (1)¹² . i = -i + i = 0 2
- 2x + i9 y(2 + i) = x i7 + 10 i16 2x + i⁹ y(2 + i) = x i⁷ + 10 i¹⁶ 2x + (i⁴)² . i . y(2 + i) = x(i²)³ . i + 10 . (i⁴)⁴ 2x + (1)² . iy(2 + i) = x(-1)³ . i + 10(1)⁴ ……..[∵ i² =... 1
- A 0.1 m solution of K2SO4 in water has a freezing point of – 4.3 °C. Given: Molality of K₂SO₄ solution = m = 0.1 m Freezing point of solution = T_f = – 4.3 °C K_f of water = 1.86 K kg mol^(–1)To find: van’t Ho... 1
- Write the equation relating boiling point elevation to the concentration of the solution. The boiling point elevation is directly proportional to the molality of the solution. Thus, Δ Tb ∝ m ∴ Δ Tb ∝ Kb m where, m is the molality... 1
- The distance between two consecutive bright fringes in a biprism experiment using light of wavelength 6000 Å Given : λ₁ = 6000 Å = 6 x 10⁻⁷ m, λ_(2 )= 4800 Å = 4.8 x 10⁻⁷ m, W₁ = 0.32 mm =3.2 x 10⁻⁴ m Distance between consecutive bright fringes, W =... 1
- A star is emitting light at the wavelength of 5000 Å. Determine the limit of resolution of a telescope Given : λ = 5000 Å = 5 x 10⁻⁷ m, D = 200 x 2.54 cm = 5.08 m θ = 1.22λ/D = frac(1.22)(5×10⁻⁷)5.08 = 1.2×10⁻⁷ rad 1
- Monochromatic electromagnetic radiation from a distant source passes through a slit. Given : 2W = 6mm W = 3mm = 3 x 10⁻^(3 )m, y=2.5 m, (a) λ₁ = 500 nm = 5 x 10⁻^(7 )m (b) λ₂ = 50 μm = 5 x 10⁻^(5 )m (c) λ_(3 )= 0.500 nm = 5 x... 1
- What must be the ratio of the slit width to the wavelength for a single slit to have the first diffraction Given : θ = 45°, m = 1 a sin θ = mλ for (m = 1, 2, 3... minima) Here, m = 1 (First minimum) a sin 45° = (1) λ ∴ a/λ = 1/sin 45° = 1.414 This... 1
- A parallel beam of green light of wavelength 546 nm passes through a slit of width 0.4 mm. Given : λ = 546 nm = 546 x 10⁻⁹ m, a = 0.4 mm = 4 x 10⁻⁴ m, D = 40 cm = 40 x 10⁻² m, y_md = mλD/a ∴ y₁d = 1λD/a And 2y₁d = 2λD/a =(frac2×546... 1
- The intensity of the light coming from one of the slits in Young’s experiment is twice the intensity of the light coming Given : I₁:I₂ = 2:1 If E₁₀ and E₂₀ are the amplitudes of the interfering waves, the ratio of the maximum intensity to the minimum intensity... 1
- In Fraunhoffer diffraction by a narrow slit, a screen is placed at a distance of 2 m from the lens Given: D = 2 m, y₁d = 5 mm = 5 x 10⁻³ m, a = 0.2 mm = 0.2 x 10⁻³ m = 2 x 10⁻⁴ m y_md = mλD/a ∴ λ = fracy_1daD=frac5×10⁻³×0.2×10⁻³2 ∴ λ = 5 x... 1
- In a biprism experiment, the fringes are observed in the focal plane of the eyepiece at a distance of 1.2 m Given : D = 1.2 m The distance between the central bright band and the 20th bright band is 0.4 cm. ∴ y₂₀ = 0.4 cm = 0.4 x 10⁻² m W = y₂₀/20... 1
- Unpolarized light with intensity I0 is incident on two polaroids. The axis of the first polaroid makes According to Malus’ law, when the unpolarized light with intensity I₀ is incident on the first polarizer, the polarizer polarizes this incide... 1
- In a double-slit arrangement the slits are separated by a distance equal to 100 times the wavelength Data : d = 100 λ, D = 50.0 cm a) The condition for maximum intensity in Young's experiment is, d sin θ = nλ , n = 0, 1, 2..., The angle betw... 2
- A double-slit arrangement produces interference fringes for sodium light (λ =589 nm) that are 0.20° apart. Given : θ₁ = 0.20°, n_w = 1.33 In the first approximation, D sin θ_(1 )= y₁ and D sin θ_(2 )= y₂ sin θ₂/sin θ₁ = y₂/y₁ …..(1) Now, y ∝ λD/d... 1
- The optical path of a ray of light of a given wavelength travelling a distance of 3 cm in flint Let d_fg and d_m be the distances by the ray of light in the flint glass and the medium respectively. Also, let n_fg and n_m be the refractiv... 1