Not yet checked 6 of 6 questions have not yet been compared with the official paper.
Q. 1
(a)Let u=[y, z, x] and v=[yz, zx, xy], f = xyz and g = x+ y+ z. Find div (grad (fg)).
(b)Evaluate ∫CF(r).dr counter clockwise around the boundary C of the region R by Green's theorem, where F = [ y , − x ] F = [y, -x] F = [ y , − x ] , C the circle x2+y2=1/4 Part (b):
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Q. 2
(a)Three forces P, Q, R, acting at a point, are in equilibrium, and the angle between P and Q is double of the angle between P and R. Prove that R 2 = Q ( Q – P ) .
(b)Find the centre of mass of a semi-circular lamina of radius a whose density varies as the square of the distance from the centre.
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(10)
Q. 4
(a)Solve the given initial-value problem.
(b)Find the general solution of the given higher-order differential equation. y ′ – 4 y ′ ′ – 5 y ′ = 0 y''' – 4y'' – 5y' = 0 y ′′′ –4 y ′′ –5 y ′ = 0 Part (b):
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(10)
Q. 5
(a)Find two power series solutions of the given differential equation about the ordinary point x=0. y′′–2xy′+y=0 .
(b)Find the general solution of the given Bessel's equation on (0, ∞). x2y′′+xy′+(9x2−4)y=0 Part (b):
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(10)
Q. 6
(a)Find the Fourier series of the given function f(x), which is assumed to have the period 2π.
(b)Find u(x,t) for the string of length L=1 and c²=1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is kx(1 – x).
Not yet checked
(10)
Q. 7
(a)Use the Bisection method to determine an approximation to the root of the given function in the interval [1,2] that is accurate to at least within 10−4 . f ( x ) = f(x)=x3+4x2–10=0 .
The 2022 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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