Not yet checked 8 of 8 questions have not yet been compared with the official paper.
Section A
Q. 1
(a)State and prove Tawlor's theorem with Cauchy's form of remainder. [8]
(b)Evaluate lim x → 0 tan x x \lim_{x \to 0} \frac{\tan x}{x} lim x → 0 x t a n x . [6]
(c)Evaluate the integral of e raised to the power ax, multiplied by sine of (bx + c), with respect to x. That is: ∫eaxsin(bx+c) dx\int e^{ax} \sin(bx + c)\, dx∫eaxsin(bx+c)dx [6]
(d)∫0π/2sinpxcosqxdx=2Γ(2p+q+1)Γ(2p+1)Γ(2q+1) Proof of the Integral Identity The given integral can be proven using the properties of the Beta and Gamma functions.
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Q. 2
(a)Sketch the graph of the curve r 2 = a 2 sin 2 θ r^2 = a^2 \sin 2\theta r 2 = a 2 sin 2 θ , a > 0 a > 0 a > 0 . Also write pedal equation for this curve. [8]
(b)Show that the parabola a2x2−b2y2=1 has asymptotes y = y=±abx and y = y=−abx . [6]
(c)Define extrema (local and global) of a function of two variables. Find three positive numbers whose sum is 48 and whose product is as large as possible. [6]
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Q. 3
(a)Find the volume of the tetrahedron bounded by the coordinate planes and the plane ax+by+cz=1 , a,b,c>0 . [8]
(b)Evaluate ∫ 0 π / 2 sin x d x \int_0^{\pi/2} \sin x dx ∫ 0 π /2 sin x d x . [6]
(c)Determine the values of x for which the power series ∑ n = 1 ∞ x n \sum_{n=1}^{\infty} \frac{x^n}{n^n} ∑ n = 1 ∞ n n x n converges absolutely, converges conditionally and diverges. [6]
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Q. 4
(a).Define a metric on a non-empty set X . If d is a metric on X , show that if d ( x , y ) = ∣ x − y ∣ ∣ 1 + x y ∣ d(x, y) = \frac{|x - y|}{|1 + xy|} d ( x , y ) = ∣1 + x y ∣ ∣ x − y ∣ then d ′ d' d ′ is also a metric on X . Also write open and closed balls (spheres) in the discrete metric space ( X , (X,d0) with radius 1 1 1 and 1.1 1.1 1.1 centered at some x x∈X . [10]
(b)Define limit point of a subset A of a metric space X. Show that an open sphere containing a limit point X of A contains infinitely many points of A other than X. [10]
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Q. 5
(a)Show that Rⁿ (n-dimensional real space) is a complete metric space under the metric defined by: d(x, y )= i=1∑n ∣ ξi − ηi ∣,x, y ∈R n Where: x=(ξ1,ξ2,…,ξn)andy=(η1,η2,…,ηn)x = (\xi_1, \xi_2, \ldots, \xi_n) \quad \text{and} \quad y = (\eta_1, \eta_2, \ldots, \eta_n)x=( ξ1 , ξ2 ,…, ξn )and y =( η1 , η2 ,…, ηn ) [8]
(b)Show that a function f : (X, d) → (Y, d) is continuous if and only if for every open subset V of Y, the preimage f⁻¹(V) is an open subset of X. [6]
(c)Find the radius of convergence and interval of convergence of the following power series: n=0∑∞ 5n+1(−1)n(2n+1) ⋅(x−1) 2n [6]
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Section B
Q. 6
(a)If C is a continuous curve and f(z) is defined on each point of C , then prove that ∣ ∫ C f ( z ) d z ∣ ≤ M L \left| \int_C f(z) dz \right| \le ML ∫ C f ( z ) d z ≤ M L : Where M = max ∣ f ( z ) ∣ M = \max |f(z)| M = max ∣ f ( z ) ∣ and L is length of curve C . [10]
(b)Suppose f(z)=u−iv(x,y) is differentiable at a point z = x + i y z = x + iy z = x + i y , then at z the first order partial derivatives of U an V exist and satisfy Cauchy-Reiman equations: ∂x∂U=∂y∂V , ∂y∂U=−∂x∂V . Verify Cauchy-Reiman equations for the function f ( z ) = e z cos y − i e z sin y f(z) = e^z \cos y - ie^z \sin y f ( z ) = e z cos y − i e z sin y . [10]
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Q. 7
(a)Define singularity of a function f ( z ) = f(z)=z2 . Investigate for the pole, singularities and zeros, the function f ( z ) = f(z)=z2 [8]
(b)Let D be simply connected domain and f(z) be analytic in D . Let f′(z) exist and is continuous at each point of D then prove that ∮Cf(z)dz=0 , where C is any closed contour in D . [6]
(c)State De Moivre's theorem and hence prove that cos 5 θ = 16 cos 5 θ − 20 cos 3 θ + 5 cos θ \cos 5\theta = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta cos 5 θ = 16 cos 5 θ − 20 cos 3 θ + 5 cos θ . sin 5 θ = ( − 1 ) n − 1 1 2 n − 1 [ sin n θ − sin ( n − 1 ) θ + n ( n − 1 ) 2 sin ( n − 2 ) θ − . ] \sin 5\theta = (-1)^{n-1} \frac{1}{2^{n-1}} [\sin n\theta - \sin(n-1)\theta + \frac{n(n-1)}{2} \sin(n-2)\theta - ...] sin 5 θ = ( − 1 ) n − 1 2 n − 1 1 [ sin n θ − sin ( n − 1 ) θ + 2 n ( n − 1 ) sin ( n − 2 ) θ − ... ] . [6]
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Q. 8
(a)Solve the equation x4−x2+1=0 and find which of its roots satisfy the equation x4−x2+1=0 . [6]
(b)Show that multiplication of a vector z by eiα where α is a real number, rotates the vector z counter clockwise through an angle of measure α . [6]
(c)Sum the series n n ! sin n θ + n ( n + 1 ) 2 ! sin 2 θ + n ( n + 1 ) ( n + 2 ) 3 ! sin 3 θ + . \frac{n}{n!}\sin n\theta + \frac{n(n+1)}{2!}\sin 2\theta + \frac{n(n+1)(n+2)}{3!}\sin 3\theta + ... n ! n sin n θ + 2 ! n ( n + 1 ) sin 2 θ + 3 ! n ( n + 1 ) ( n + 2 ) sin 3 θ + ... [8]
The 2012 CSS Pure 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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