9502 - +a s g r >PI,sg+t a +t,z% /aL 1 6 R P / R [SS*+I...

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- ++ a sg r >PI ,sg - +t a +t,z% /aL 16 R P / R [SS*+I e+g 1. Suppose that f (x,y,t) = x2 - e2Y + t2 . Which of the following equations is correct? df - af 'ik dy (a) If x = x(t) and y = y(t), - - -- +-- dt dx dt ay dt (c) - = x2 -2eZY (d) - = t dt (e) none of the above 2. Suppose that x = x(u,v), y = y(u,v) . Which of the following equations is correct? (a) - . - = 1 du ax dxdu axav -.-+-.-=I auax avax axau (e) zs5+av.v= I 3. If xy -y3 =1, yl(x) atx =O is equal to xi2 sinx 4. The integral [ i,2 - kdy is equal to X n/2 di A (e) - = 0 dr (c) - = 6 do 5.The polar coordinates (r,e) and the Cartesian coordinates (x,y) are related by , A x = r cose, y = r sin 6 . Which of the following equations for unit vectors i, j, is INCORRECT? i=cosei+sin~'j d6 ,, "I) dB = -r
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- fi - *€l&.h,ql 1 / A *%& [ +a 321521),!5~9. - %%%+I @*% 6. The spherical coordinates @, 4, 19) of a point P are defined by that p is the distance form origin 0 to P 4 the angle between the z axis and OP, 0 x axis and the projection of OP on th~ x-y plane. The Jacobian YyZ) is equal to ~(PY 499) . (c) p2 cos@sinI9 (d) cos 4 cos I9 (e) p2 sin 4 cos 8 7. If 0 is a scalar field and A, B are vector fields, which of the following equations is INCORRECT? (a) V.(@A)=V@.A+<DV.A (b) Vx(@A)=VQxA+QVxA (c) V-(VxA)=O (d) VX(VXA)=V(V-A)-V'A (e) none of the above 8. A vector field A(*, y, z) = xy i' + yz ] - xy2 & , then V . A is equal to (a> X+Y (b) Y + Y+Z-~XY (d) xy + - xy2 (e) none of the above 9. Suppose that V(x, y) = 2xy + x2 j is a gradient field and a curve R(r) = (1 + cosr)i + sinr j i =n/2 defined by - n / 2 5 z S ~t / 2. The line integral v.a~ise~ualto =-n/2 0. A function f (x) defined in 0 < x < L has a Fourier half range cosine expansion a nm f (x) = a, + Can cos- . Which of the following equations is correct? n=l L nnx = 2 rf(x)cos--tic LO L .I. The Sturrn-Liouville equation the form [p(x) y']' + q(x) y + Aw(x) y = 0, a < x < b . For the differential equation y" - 2 y' + Ay = 0,0 < x < n , which of the following function, wllien multiplied into the above equation, can turn it into a Sturrn-Liouville equation. (a) x Co) -x (c) ex (dl e -X (el e -2x
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- *a 3g13p~,g&-- 44a+35 lwl- % 16 a% 3 FI **m C%K~I&+I fij+% 00 -io x 12. The Fourier transform of f (x) is defined by F(f (x)} = f (x) e h. If f ("'(x) is the nth -00 I derivative, F{ f '"' (x)) is equal to I (d) (-io)" f (x)) (e) none of the above 13. If H(x) is the Heaviside step hction and f = H(x + 1) - - . The Fourier transform I F( f is equal to (b) 2% 0 sin w (c) - W COS W (4 - W (e) none of the above 14. Consider the wave equation c2u, = u,,, u (0,t) = u(L, t) = 0, u(x, 0) = f (x), u,(x,O) = g(x) for I 0 5 x S L, 0 5 t. By the method of separation of variables, we have u(x, = X(x)T(t) and 1 115. Following the above problem, u(x,,t) is the superposition ofthe possible X(x)T(t) has the form CO u(x, t) = A,X, (x)T, (t) . Then X, (x) is equal to n=l (2n - 1)n (a) sin x L (2n - 1)n (b) cos x 2L 2nn (c) sin- x L 16. Which of the following complex hctions is NOT differentiable? nn (d) sin-x L I (e) cos-x (a) sinz
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$+ €I 3% I ;3!!).%32 2 4% / R % Y- R *3$& [%%%+I &t+% 17. Which of the following is a particular solution x,(Q for the differential equation x" + 2x' + 3x = cos 3t ? ei3t ei31 ei3~ ' (a) Re [- - 6(1- i) I (b) Re[- 6(1+ i) I (c)Ret-- 3(1- i) 1 @)Re[m 1 @]Re[-m 1 2-1 18.
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9502 - +a s g r &gt;PI,sg+t a +t,z% /aL 1 6 R P / R [SS*+I...

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