Not yet checked 7 of 7 questions have not yet been compared with the official paper.
Q. 1
(a)Evaluate the surface integral ∬ A ⃗ ⋅ n ⃗ d S \iint \vec{A} \cdot \vec{n} dS ∬ A ⋅ n d S where A ⃗ = z i ^ + x j ^ − 3 y 2 z k ^ \vec{A} = z\hat{i} + x\hat{j} - 3y^2z\hat{k} A = z i ^ + x j ^ − 3 y 2 z k ^ and S is the portion of the cylinder x2+y2=8 lying in the first octant between z = 0 z = 0 z = 0 and z = 4 z = 4 z = 4 . [10]
(b)Prove that ∇ ⋅ ( f ( r ) r ⃗ ) = f ′ ( r ) r r ⋅ f ⃗ \nabla \cdot (f(r)\vec{r}) = \frac{f'(r)}{r} r \cdot \vec{f} ∇ ⋅ ( f ( r ) r ) = r f ′ ( r ) r ⋅ f where r ⃗ = x i ^ + y j ^ + z k ^ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} r = x i ^ + y j ^ + z k ^ and r = ∣ r ⃗ ∣ r = |\vec{r}| r = ∣ r ∣ . [10]
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Q. 2
(a)The greatest resultant that two forces can have is of magnitude P and the least is of magnitude Q. Show that, when they act at an angle α , their resultant is of magnitude P 2 c o s 2 α 2 + Q 2 s i n 2 α 2 2 \sqrt{\frac{P^2cos^2 \frac{\alpha}{2} + Q^2sin^2 \frac{\alpha}{2}}{2}} 2 P 2 co s 2 2 α + Q 2 s i n 2 2 α . [10]
(b)A sphere of weight W and radius a is suspended by a string of length l from a point P and a weight w is also suspended from P by a string sufficiently long for the weight to hang below the sphere. Show that the inclination of the first string to the vertical is sin−1(W+w)(a+l)wa [10]
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Q. 3
(a)Show that the law of force towards the pole, of a particle describing the curve rn=ancosnθ is given by f = f=r2n+3(n+1)h2a2n [10]
(b)The maximum velocity that a particle executing simple harmonic motion of amplitude a attains, is v. If it is disturbed in such a way that its maximum velocity becomes nv. Find the change in the amplitude and the time-period of motion. [10]
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Q. 4
(a)Define ordinary and singular points of the differential equation a2(x)y′′+a1(x)y′+a0(x)y=0 . When a singular point is said to be regular and irregular? Find regular and irregular singular points of the differential equation (x2−4)2y′′+(x−2)y′+y=0 . [10]
(b)Show that I 3 / 2 π x = 2 π x [ sin x − cos x ] \frac{I_{3/2}}{\pi x} = \frac{2}{\pi x} [\sin x - \cos x] π x I 3/2 = π x 2 [ sin x − cos x ] [10]
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Q. 5
(a)Solve the equation by using method of undetermined coefficients y ′ ′ − y ′ + y = 2 cos 3 x y'' - y' + y = 2 \cos 3x y ′′ − y ′ + y = 2 cos 3 x . [10]
(b)Use the method of Frobenius to find two linear independent series solutions in powers of x of the DE. x2y′′–(x2+x)y′+y=0 . [10]
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Q. 6
(a)Classify general second order partial differential equation (PDE) into elliptic, parabolic and hyperbolic form. Discuss the nature of the PDE ( 1 – (1–x2)uxx−2xyuxy+(1–y2)uyy=0 at each ( x , y ) (x,y)∈R2 . [10]
(b)Use the method of separation of variables to find the solution u ( x , t ) : [ 0 , T ] u(x,t):[0,T]×[0,L]→R to the initial/boundary value problem u ( x , t ) = u(x,t)=uxx(x,t) for 0u(x,0) = f(x),for00 \leq x \leq L,u(0,t) = u(L,t) = 0,for0<t0 < t \leq T,wheref:[0,L]f: [0, L] \rightarrow R$ is a known function. [10]
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Q. 7
(a)Use Simpson's 3/8 rule to estimate the integral ∫13(x3−2x2+7x−5)dx . By comparing your answer with exact value, find the error. [10]
(b)Solve the system of equations by Jacobi iterative method. 10 x + 3 y + z = 19 , 3 x + 10 y + 2 z = 29 , x + 2 y + 10 z = 35 10x + 3y + z = 19, 3x + 10y + 2z = 29, x + 2y + 10z = 35 10 x + 3 y + z = 19 , 3 x + 10 y + 2 z = 29 , x + 2 y + 10 z = 35 [10]
The 2021 CSS Applied Mathematics paper set by the FPSC. Question wording only; questions marked “Not yet checked” have not been compared with the official paper yet.
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