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22 Aug 2023: 3, 2). 10. Basic identities. Starting from the identity sin(A B) = sin A cos B cos A sin B, use the basic properties of the trigono-metric functions (such as ... i) sin(A B) = sin A cos B cos A sin B ;. (ii) cos(A B) = cos A cos B sin A sin B ;. (iii) tan

30 Aug 2019: cos a = cos b cos c sin b sin c cos A. ... sin a cos B = cos b sin c sin b cos c cos A.

17 Jun 2019: π2. 8=. n=1. 1(2n 1)2. [17]. 3B Consider the matrix. A =(. cos θ sin θsin θ cos θ. ). (a) Determine the eigenvalues of A. ... r = (a cos φ sin θ, b sin φ sin θ, c cos θ).

31 Jul 2023: y1(t) = A cos t , y0(t) = B sin t C sin 3t (18). ... for suitable values of the constants A, B and C [recall that cos3 θ = 34 cos θ 14 cos 3θ, and.

4 Aug 2022: Consider the vectors a = (0, 0, 1),b = (sin β, 0, cos β),a′ = (1, 0, 0),b′ =(sin β′, 0, cos β′). ... Show that. |E(a,b) E(a,b′)| |E(a′,b) E(a′,b′)| = |cos β cos β′| 2c0c1|sin β sin β′|.

17 Jun 2019: 1]. (b). J = π/2. 0. sin x cos x dx. ... 4]. (d) The surface S of an ellipsoid is defined parametrically byx = (b sin θ cos φ, b sin θ sin φ, a cos θ), where 0 6 θ 6 π

17 Jun 2019: cos α =(a b) (a c)|a b||a c|. Using identities (1) and (2), prove that. ... cos δ(B, C) = cos δ(A, B) cos δ(A, C) sin δ(A, B) sin δ(A, C) cos α.

17 Jun 2019: b)dy. dxsin x y cos x = 12 sin 2x [5]. ... 2yx=y. (2x2 y)dxdy. Evaluate the integral. [10]. (b) The equation r = (1 cos θ) defines a closed surface in three dimensions.

30 Aug 2019: 6|1〉B, sin. π. 6|0〉B cos. π. 6|1〉B. ). ,(. sinπ. 6|0〉B cos. ... π. 6|1〉B, cos. π. 6|0〉B sin. π. 6|1〉B. ). determine the condition, which α and β must satisfy in order for the measurementstatistics to violate the

17 Jun 2019: 2]. 3 Differentiate(a) ln(x3);. [1]. (b) x cos(2x). [1]. 4 Integrate. ... a) y = (ln x)2 for 0 < x < ;. [1]. (b) y = sin(x) cos(x) for π 6 x 6 π.

17 Jun 2019: Prove the Pythagorean theorem for spherical geometry in the form. cos c = cos a cos b. ... E = (E0 cos(kz ωt), 0, 0) B = (0,B0 cos(kz ωt), 0),.

30 Aug 2019: χ =(A cos αr. B sin αrr. cr2). sin2 θ. Find conditions on A,B,C and α for which B = 0 on the sphere r = a, and givean expression for ... πc(τ) sin 2πz,. χ =1k. [πd(τ). ( 8p. )1/2sin kx cos πz e(τ) sin 2kx. ],.

17 Jun 2019: φ(r,θ) = A B log r. n=1. [rn(An cos nθ Bn sin nθ) rn(Cn cos nθ Dn sin nθ)]. ... where An, Bn, Cn and Dn are constants. [10]. Obtain the solution for φ(r,θ) given the boundary conditions φ(a,θ) = U andφ(b,θ) = V cos θ.

17 Jun 2019: σ(u,v) = ((a b cos u) cos v, (a b cos u) sin v,b sin u),. ... ψ(x) =15a. cos(πx. 2a. ). 25a. sin(πxa. ),. in the region a x a, and zero otherwise.

9 Nov 2023:  , G =. y cos(xy) by (b c)z. x cos(xy) bx bz(c b)x by. ... a) Find the general solution of the differential equation. d2y. dx2+ y = cos(kx) ,. where k is a positive integer. [10]. (b) (i) Rewrite the set of simultaneous differential equations

17 Jun 2019: sationσ(u, v) =. ((a b cos u) cos v, (a b cos u) sin v, b sin u. ),. ... 2π. 0. sin2 θdθ. a b cos θ=. 2π. b2. [a.

17 Jun 2019: 1]. B C. A. D. 2. 5. Solve for x in the interval 0 6 x 6 π: cos 2x = sinx. ... 1]. Natural Sciences IA, Paper 1. 5. SECTION B. 11Z. (a) Use De Moivre’s theorem to express 8 cos(6θ) 15 sin(4θ) sin(2θ) in terms of cos θand

17 Jun 2019: u(x,y,0) =. m=1. n=1. Amne β. 2x sin(mπx) cos. (. 2n12 πy. ). ,. ... exp(M) =. . . . . cosha sinha 0 0sinha cosh a 0 00 0 cos b sinb0 0 sinb cos b.

17 Jun 2019: Φ(r, θ) =. n=0. (Anr. n Bnrn1)Pn(cos θ). for constants An, Bn. ... The temperatureon the boundaries of the shell is. T (a, θ) = T0 T1 cos θ, T (b, θ) = T2 T3 cos θ ,. for constants T0, T1, T2, T3. Determine the

30 Aug 2019: X(θ) = b(cos θ, sin θ, θ tan φ), 0 6 θ 6 L cos φ/b, (). where φ is the pitch of the helix (i.e. the constant angle ... between dX/dθ and the xy-plane)and L cos φ b.

17 Jun 2019: Thecoordinates are related by the transformation,. x = r cos 2θ, y = r sin 2θ,. ... 1]. (b) B =. (. cos θ sin θsin θ cos θ. ).

17 Jun 2019: By settingthe former equal to cos θ|a|b̂, where b̂ is a unit vector along b, show that. ... cos θ =a b|a||b|. Explain why this is a sensible definition of the angle θ between a and b.

30 Aug 2019: 1 < p <. Letf (x) = a cos nx b sin nx,. ... n. b) Let. f (x) =. k=0. ak cos 5kx, ak > 0,.

30 Aug 2019: b) By considering f (x) = cos kx show that we cannot have a small-o estimate. ... q(x)| 6 n1 x2. at the n zeros of Tn ,. then|q(x)| 6 |T ′. n(x)| , |x| > cos π. 2n.

17 Jun 2019: A = cos(. θ. 2. )I sin. (θ. 2. )E1E2,. where θ is a scalar. ... 6]. (b) A curve is given parametrically by. x = cos θ, y = cos(4θ), 0 < θ < π.

17 Jun 2019: y(0) = y′(0) = 0 , f (x) = 2k cos kx ,. [10]. and. y(0) = y( π. 6k. )= 0 , f (x) =. k2. 2. [10]. [You may use the identity sin(A B) = ... sin A cos B cos A sin B.].

17 Jun 2019: dx(sin x). 12 (cos x). 72. [8]. [Hint: substitute tan x = u.]. ... and v. [6]. (c) Suppose thatx = u cos v ,y = u sin v.

4 Aug 2022: ex(sin x cos x) dx and. ex(sin x cos x) dx. ... 2]. (b) By performing the integration, show that. enx(sin x cos x)dx = e.

17 Jun 2019: F = (sin x, sin y, sin z), G = (cos x, cos y, cos z). ... i) (2xy2 y)dx (2x x2y)dy = 0 , [5]. (ii) (2y sin x 3y4 sin x cos x)dx (4y3cos2x cos x)dy = 0.

17 Jun 2019: a)2. |x 3| for < x < , [1]. (b)12. (sin x cos x) for π/4 6 x 6 π/4. ... B(p, q) = 2. π/2. 0. (sin θ)2p1(cos θ)2q1 dθ. [4].

17 Jun 2019: find all the solutions of this equation. [2]. 2. Differentiate. a) cos 5x [1]. ... i; ln(e); 2i; (ii)i. [8]. (b) Write down expressions for sin x and cos x in terms of eix and eix.

17 Jun 2019: 1]. (b) sin(cos x). [1]. 9. One side of a rectangle has fixed length a = 10 cm while the length of the other side, b,increases at a rate of ... 7]. (c) Find the general solution of. dy. dx+. 2. xy = x2y2 cos x.

17 Jun 2019: The inner circle, definedby τ = a, is held at Φ = 0 and the outer circle, defined by τ = b, is held atΦ = cos(2σ). ... Hint: consider α = cos θ.] [5]. (b) What restriction must be placed on p for Bp to have a cyclic subgroup C = {I,c}?Give the

17 Jun 2019: Calculate T̂ and κ for the circular helix. x(t) = (a cos t , a sin t , bt) ,. where a and b are constants. (ii) Find the normal vector and the equation ... x = a cos θ , y = b sin θ , 0 θ 2π.

17 Jun 2019: y = b cos θ,. for 0 < θ < 2π. ... 0. R. 0. r2 cos θ sin θdr dθ. [4]. (b) Find the area of the finite region of the plane bounded by the line y = 2x 3and the curve y =

17 Jun 2019: θ. ]. ]. (i) Using separation of variables, find an expression for B as an expansion in Legendrepolynomials Pl(cos θ) when r > R. ... B(r, θ) =. l=0. Bl fl(s1/2 r) Pl(cos θ) ,. where Bl is a constant and fl(s1/2 r) satisfies the equation.

17 Jun 2019: i) For B = R1A with. R1 =. 1 0 00 cos θ1 sin θ10 sin θ1 cos θ1. ... ii) λ = 1 and S = 32 in the case where x y = cos 2π3.

17 Jun 2019: 9E. (a) Show thatsinh(ix) = i sin x. and thatcosh(ix) = cos x. ... 4]. (c) Evaluate π/20. cosh x cos x dx. [10]. Paper 1 [TURN OVER.

17 Jun 2019: 4. 4A. The waveform φ(t) transmitted by an analogue radio is produced by modulating acarrier wave c(t) = cos(t) of frequency by the signal s(t) to be broadcast ... 4]. (b) For the case. s(t) =1 cos t. t2,. compute the Fourier transform to show that.

30 Aug 2019: α =|k3|. (k21 k22)1/2. |σ13 cos φ σ23 sin φ|a2 b2. ... as t. What is the physical interpretation of α? Now assume that a and b are independent Gaussian random variables with zeromean and variance Γ2 and that σ13 cos φ σ23

4 Aug 2022: V (x) in part (a) orV (cos θ) in part (b). ... I. d4k. (2π)4f(k2) =. 1. 16π2. dk2 k2f(k2). II. B. A.

20 Jul 2021: M =. (f(θ). 2 i sinθ2f(θ) sinθ2f(θ)sinθ2f(θ). f(θ)2. i sinθ2f(θ). ), f(θ) =. 1 cos θ,. where θ R. (ii) Now consider an element of ... Rab = cos θδab (1 cos θ)nanb sin θabcnc,. where a,b,c = 1, 2, 3, nana = 1 and abc is the 3-dimensional

17 Jun 2019: 3 cos θ sin θ =. 2. in the range 0 6 θ < 2π. ... i). cos xdydx. (cos x sin x)y = 2 sin 2x.

30 Aug 2019: ψA〉 = sin(θ) |a〉 cos(θ) |ω〉. for 0 6 θ 6 π2. ... G = cos(2θ) (|a〉 〈a| |ω〉 〈ω|) sin(2θ) (|a〉 〈ω| |ω〉 〈a|).

3 Aug 2022: b) Determine the Wronskian of the equation (†) as a function of x.Let λ = 1. ... 6A Differential Equations. (a) Let f(x,y) be a real-valued function depending smoothly on real variables x andy, and g(t) = f( a t cos γ, b t sin

30 Aug 2019: b) Let X be a continuous local martingale such that Xt X almost surely. ... dΘ = cos(Θ)dZ. (b) Hence, show that. P(Xt ) =1. 2+.

30 Aug 2019: δ′b k2θb = 0 , θ′b. a′a θb c. 2sbδb = RaneσT(θb θγ)/k2 ,. where csb is the baryon sound speed and R is given in terms of the relative ... δγ (k, τ ) [A(k) cos(kτ /. 3) B(k) sin(kτ /. 3)). ]exp(k2/k2D) ,. where kD is a time-dependent

20 Jul 2021: b). (i) Consider two states ρ1 = |0〉〈0|, ρ2 = |θ〉〈θ|, where |θ〉 = cos(θ)|0〉 sin(θ)|1〉,θ (0,π/2). ... c,d (0, 1), a < b < c < d and〈φ1|φ1〉 = 〈φ2|φ2〉 = 1.

30 Aug 2019: In longitudinal Newtonian gauge we take instead E = B = 0. ... δC = Aτ 2 =34δR , for τ 2π/k ,. δC B ln τ , δR C cos(kτ /. 3) D sin(kτ /. 3) , for τ 2π/k ,. where A, B, C, D are

30 Aug 2019: real k axis at k = k0 cos θ and k = k0 sec θ). ... φi = cos(nπy/h) exp(iωt ikx) ,. where k =k20 n2π2/h2 is diffracted by the two edges.