Not yet checked 1 of 8 questions have not yet been compared with the official paper.
Q. 1
Let F be a differentiable vector field, and f a differentiable scaler field, defined on an open subset U ⊂ Rⁿ.
(a)(i) Define the gradient ∇f. If n = 3 and f(x, y, z) = x²ycosz, calculate ∇f at the point (2, -1, 0) [3]
(b)Show that the following 3-dimensional vector field F is irrotational. F(x, y, z) = (2y − cosz, 2x − cosz, 2z + (x + y)sinz). Construct a scalar potential φ for F. [10]
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Q. 2
A particle is constrained to move in the x − y plane such that
(a)x(t) = A cos(wt) y(t) = A sin(wt) What is the speed of the particle? What is the acceleration of this particle? Find unit vectors that are tangent and normal to the curve that describes the particle's path. [10]
(b)Consider a driven plane pendulum. The pendulum is attached to an oscillating support that oscillates at frequency w with amplitude h. Find the equation of motion for the angle the pendulum makes with the vertical. [10]
(20)
Q. 3
Suppose we are given the following differential equation: y'' + (1/x)y' + (1 − 1/(4x²))y = 0
(a)What are the singular points of this equation? [5]
(b)Show that one solution to the equation is y = x^(-1/2) sin(x) [5]
(c)Find a second, linearly independent solution to this equation using the method of Reduction of Order. [10]
Not yet checked
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Q. 4
(a)Use Simpson's rule with n = 4 to estimate tan⁻¹(2). [5]
(b)Derive the following Trapezoidal rule for approximating ∫ₐᵇ f(x)dx. ∫ₐᵇ f(x)dx = (h/2)[f(a) + f(b)] − (h³/12)f''(ε), where h = b − a [5]
(c)Use Euler's method with step size 0.2 to estimate y(0.4) where y(x) is the solution to the initial value problem. y' = 10(x + y)², y(0) = 0 [10]
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Q. 5
(a)Let A = [[1, 2, -2], [1, 1, 1], [2, 2, 1]] decide whether Jacobi method converge to the solution of AX = b. [10]
(b)Prove that the Newton-Raphson method is quadratically convergent. [10]
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Q. 6
(a)Find the Fourier series of the Identity function f(x) = x, 0 < x < 2π. [10]
(b)Expand x(π − x) in half-range Sin series when 0 < x < π. [10]
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Q. 7
(a)Solve ydx − (y − 3x − 3)dy = 0 [10]
(b)Find the Orthogonal Trajectories of the family of rectangular hyperbolas xy = c. [10]
(20)
Q. 8
(a)A particle is moving along the parabola x² = 4ay, with constant speed v. Determine the tangential and the normal components of its acceleration when it reaches the point whose abscissa is √(5a). [10]
(b)Let A, B and C are three vectors. Then prove that: A × (B × C) = (A.C)B − (A.B)C. [10]
The 2021 CCE Sindh Applied Mathematics paper set by the Sindh Public Service Commission. Question wording only; questions marked “Not yet checked” have not been compared with the official paper yet. Open the official paper beside the questions to check any of them.
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