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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: For u = ex cos y, find v and hence f(z). ... Let the energy eigenstates of even parity be given by. ψeven(x) =. . . . A cos λx B sin λx a < x < 0A cos λx B sin λx 0 < x <

30 Aug 2019: 3)(f) Assume f(t) = A1(t) cos(ω1t) and ω = ω2 ω1. ... Show that exp(iθA) = cos(θ)I i sin(θ)A for θ R. (b) Use the result from part (a) to show that.

30 Aug 2019: η2 = 2cos θ1 sin θ1. cos θ2 sin(θ1 θ2)h0,. where θ1 and θ2 are the angles between lines of constant phase and the vertical inthe two regions. ... You may assume thath = h0 at x = 0 and that sin θ >. 12E cos θ. Find expressions for h(x) and

19 Jun 2019: Xt = cos Bt and Yt = sin Bt. (c) Find the stochastic differential equation for the process (X, Y ) and write it inits standard form, i.e. ... Here σ1,. , σm, b are boundedmeasurable vector fields on Rd and B = (B1,. ,

3 Aug 2022: b) Show that. B =. . B1B2B3.  cos(kz ωt). satisfies the Maxwell equations given appropriate conditions on the constants B1, B2, B3,ω and k that you should find. ... State the hyperbolic sine formula. The hyperbolic cosineformula is cosh a =

12 Jul 2023: i) Λ = R(φ) :=. . 1 0 0 00 cos φ sin φ 00 sin φ cos φ 00 0 0 1. ...  , ii) Λ = B(β) :=. . cosh β sinh β 0 0sinh β cosh β 0 0.

30 Aug 2019: ln λ)2. ). (b) The integral I(σ) is defined by. I(σ) = 2π. ... 0. exp (σx(1 cos x)) dx. Find the the first two terms of the asymptotic expansion for I(σ) as σ.

17 Jun 2019: i) dydx =y1x21 ; [6]. (ii) cos x dydx y = sec x tan x. ... 6]. (b) Write down formulae for cos θ and sin θ in terms of complex exponentials.

30 Aug 2019: B = B0η. sin θ cos θr3. r0. x4ω(x)dx. (iii) Estimate the time it would take to establish this steady solution in a fluid ofvery high conductivity, starting from an ... Explain the significance of the variousterms in these equations and obtain the

3 Jul 2024: T̂ =. (a b. b a. ),. where a(t) and b(t) are functions of time and a, b denote the complexconjugates, such that. ( ... Assume that w(x) = cos(πx). Show that in this case, the eigenvalues of B havenon-positive real part.

30 Aug 2019: The region y < 0 is insulating. A line current J cos ωt flows in a wire parallel to thez-axis and located at x = 0, y = b (b > 0). ... b) Suppose that the frequency ω is large, so that (η/ω)1/2 b.

3 Jul 2024: b) Consider spatial perturbations of the form cos(kx) about the solution found inpart (a), and set λ = D1/D2. ... Ai(t) π1/2 (t)1/4 cos[23(t)3/2 14π. ]as t. ]. (c) We seek approximations to the smallest eigenvalues λ, which are close to 1.Using the

17 Jun 2019: 7]. (c)e2x 2ex 1. cos(3x) 2 cos(2x) cos x. [7]. [You may assume, if you wish, the standard Taylor series expansions for exponen-tial, trigonometric and logarithmic functions.]. ... 21. 21. dx dy dz. x2y3z4. [3]. (b) Evaluate using polar coordinates 0.

17 Jun 2019: The coordinates u and v aredefined by. u = x cos θ y sin θ,v = x sin θ y cos θ,. ... y(x) = (a bx) sin x (c dx) cos x,. where the constants a, b, c, and d are to be determined.

17 Jun 2019: 1. c22E. t2 2E = 0 and E = 0 where c =. 1. ǫ0µ0. (b) Suppose that E = E02(1,1,0) cos(kz ωt) where E0, k and ω are real constants. ... f(u). (b) Now take f(s) = cos s 2, g(s) = sins, 0 6 s < 2π.

19 Jun 2019: f (λ) = π |λ|,. find the autocovariance function. [5]. (c) Let Xt = A cos(ω0t) B sin(ω0t), where A and B are uncorrelated randomvariables A and B, each with mean ... f (x)g(x). ]. [3]. (b) Consider generating random numbers from a distribution with

17 Jun 2019: a) Show that (r/x)y = cos φ and find a similar formula for (r/y)x. ... K(r, θ) =1. 2+. n=1. rn cos nθ,. and for an arbitrary 2π-periodic function f, let.

30 Aug 2019: b) Let Tn(x) = cos n arccos x be the Chebyshev polynomial of degree n, and let. ... x ti). Prove also that. (b) b. a. Mi(t)dt = 1, (c)n.

30 Aug 2019: i) Xt = t0. sgn(Bs)dBs,. (ii) Yt = eWt cos Bt,. ... t0. φ(Bs)ds}). where B is a Brownian motion starting from x.

30 Aug 2019: B = (0, 0, f (t) b(q) cos mφ) , q =r. ... It is not requiredto solve the equation for b(q) when m 6= 0.

17 Jun 2019: π. n=1. cos[(2n 1)x] cos[(2n 1)ct]. (2n 1)2. (b) Determine a periodic function P(x) such that this solution may be expressed as. ... d) Suppose that E = E0(0,1,0) and B =E0c (0,cos θ,sinθ) where 0 6 θ 6 π/2,.

12 Jul 2023: g(θ) =. . cos θ sin θ 0sin θ cos θ 0. ... Aµ 7 A′µ = gAµg1 (µg)g1. (b) Given that Aµ(x) g, show that A′µ(x) g.

17 Jun 2019: The inner cylinder is held at φ = 0 and the outer at φ = V cos 2θ. ... 12]. Find φ if the inner cylinder is held at φ = V and the outer at φ = V cos 2θ.

17 Jun 2019: Show that Amaps Π to itself and induces a rotation of Π by angle θ, where cos θ = 12 (trace (A) 1).Show that A is a reflection in Π if ... Prove that det(A I) = 4(cos θ 1). Paper 3. 5.

30 Aug 2019: 2 LetTn(x) = cos n arccos x, x [1, 1], n = 0, 1,. ... f (x) = α cos x β sin x,. from Tn1 in L1(T) is identically zero.

30 Aug 2019: Tt=1. xt cos(ωt) and B =1πT. Tt=1. xt sin(ωt). Show that. ... t=k1. cos(ωt) sin(ω(t k)). )]. Deduce that. |EAB| 6 1πT. T1k=1.

17 Jun 2019: a) (cosh θ sinh θ cos φ)dθ (cosh θ sin φ cos φ)dφ,. ... 6. 11A Vector Calculus. Prove that. (a b) = a b b a (b )a (a )b.

30 Aug 2019: b) From general principles we know that the set of eigenfunctions of such an operatordefine a complete set of basis functions. ... Show that these can be written as. W (n)(x) = A cos(k(n)x) B sin(k(n)x) D cosh(k(n)x) E sinh(k(n)x) ,. and

30 Aug 2019: A. t= α(z)B η2A. B. t=. dV. dz. A. x+ η2B. ... p, u ẑ, b ẑ cos πz eilximyst. (i) Derive an expression for the growth rate s in terms of the other parameters.

4 Aug 2022: m are orthogonal. with respect to w1(x). [6]. (c) Using the substitution x = cos θ, with θ [0,π], verify that. ... as. u(x,t) = c0etex. n=1. cneπ2n2tVn(x),. whereVn(x) = πn cos(πnx) sin(πnx),. and c0 and cn are real constants.

17 Jun 2019: n! nn. ;. (b). n>1. 1. n (log n)2;. (c). n>1. ... f (x) = x2 sign(x). cos. π. x. , x 6= 0.

17 Jun 2019: x(u) = aebu cos u , y(u) = aebu sin u ,. where the constants a, b > 0. Sketch the curve segment corresponding to the range0 6 u 6 3π. ... Show that if g G stabilizes the point y X , then there exists an h G withhgh1 B.

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: Show that these can be written as. W (n)(x) = A cos(k(n)x) B sin(k(n)x) D cosh(k(n)x) E sinh(k(n)x) ,. and ... ut = Duuxx a (b 1)u u2v , (1). vt = Dvvxx bu u2v , (2).

17 Jun 2019: cos(A B) cos(A B)]. Paper 1 [TURN OVER. 6. 9C The functions p(x), q(x) and w(x) are defined on the interval 0 x 1. ... I = b. a. f(x,y,y′)dx ,. stationary, subject to appropriate boundary conditions.

17 Jun 2019: 4]. (b) Find the eigenvalues of the matrixcos 2θ sin 2θ 0sin 2θ cos 2θ 00 0 2. ... 11 cos 2θ. dθ. [6]. (c) Using your results from (b), or otherwise, evaluate the definite integral π/20.

17 Jun 2019: Give a reason for your answer. [8]. (b) For what real values of λ does the equation cos θ sin θ 0sin θ cos θ 00 0 5. ... a)d2ydx2. 4y = cos x ,[6]. (b)d2ydx2. 4y = cos2 x ,[6].

17 Jun 2019: Show that the following are. equivalent. (i) a1b1ab H for all a,b G. ... y. 0. Q(x,y′)dy′. or otherwise, show that V is conservative. (b) Now let V =(. 1 cos(2πx 2πy),2 cos(2πx 2πy)). Show that there existsa smooth function

17 Jun 2019: Ψ(r,θ) =. =0. (ar. br1. )P(cos θ). Here, P(cos θ) is the th Legendre polynomial, i.e., the solution of the differentialequation. ... d. dx. ((1 x2)dP. dx. ) ( 1)P = 0 ,. with x = cos θ, which is regular at x = 1.

17 Jun 2019: 3 5i. 5 4i. [2]. 6 Give the first three terms of the Taylor expansion of the function f (x) = cos xabout the point x = π/3. ... A B cos θ. where A > B > 0. By rewriting the equation of this curve in terms of cartesiancoordinates (x, y), or otherwise,

30 Aug 2019: Wn(x) = B sin knx D cos knx E sinh knx F cosh knx ,. where kn is the nth root of the transcendental equation cos kL cosh kL = 1. ... Wn(x) = N[(sin qn sinh qn) (cos knx cosh knx). (cos qn cosh qn) (sin knx sinh knx)] , (1).

30 Aug 2019: Find the order of ω(g, δ) for the Weierstrass function. g(x) :=. k=012k. cos 3kx. using the fact that, for 3m n < 3m1, the polynomial of best approximation of ... degreen to g is the partial sum tn(x) =. mk=0. 12k. cos 3kx. 4 Let (Ni,k)ni=0 be the

17 Jun 2019: 2 cos θ| > 1. [2]. 10 Show that the probability distribution. ... g(x) =1. 3+. 4. π2. n=1. (1)nn2. cos(πnx). [6]. (b) By considering.

30 Aug 2019: u = R (u1 cos t u2 sin t). where ui = ûieikix for distinct vectors k1, k2 with |k1| = |k2| = k. ... Show that b takes the form. b = R. j=1,2. eikjx(pj cos t qj sin t).

17 Jun 2019: d. dξ. (. (1 ξ2)dP. dξ. ). λP = 0,. where ξ = cos θ and λ = n(n 1), which is non-singular at ξ = 1.[8]. ... You may use the formulae P0 (cos(θ)) = 1, P2 (cos(θ)) = (3 cos2(θ) 1)/2.

30 Aug 2019: b) Let. f(x) =. k=0. ak cos 5kx, ak > 0,. ... tn(x) =m. k=0. ak cos 5kx. is the best approximant to f from Tn and find the value of En(f) in terms of ak.

17 Jun 2019: Show that. E′ =E(1 β cos θ). 1 β2, cos θ′ =. cos θ β1 β cos θ. ,. where β = v/c. ... x1/n. 1 x2dx =. π. 2 cos(π/2n),. where n > 1,.

17 Jun 2019: n=1. (. an rn. bnrn. )(. An cos(nθ) Bn sin(nθ). ). [4]. Find a solution Φ(r, θ) which is bounded inside the disc r 6 R and such that. ... I =. π. π. d θ. a b cos θ c sin θ.[5].

4 Jul 2024: Ai(t) π1/2 (t)1/4 cos[23(t)3/2 14π. ]as t. ]. (c) We seek approximations to the smallest eigenvalues λ, which are close to 1.Using the ... S. A S. a b A A. a b b b a b.