C/18/BSc/3rd Sem/MTMH/GE3T
2018
CBCS
3rd Semester
MATHEMATICS
PAPER-GE3T
(Honours)
Full Marks: 60
Time: 3 Hours

The figures in the right-hand margin indicate full marks.
Candidates are required to give their answers in their own words as far as practicable.
Illustrate the answers wherever necessary.
Answer all questions.

Differential Equation and Vector Calculus

1. Answer any ten questions: 10 x 2
(a) Prove that $ [\vec{\alpha}+\vec{\beta}, \vec{\beta}+\vec{\gamma}, \vec{\gamma}+\vec{\alpha}] = 2[\vec{\alpha}, \vec{\beta}, \vec{\gamma}] $.
(b) If $ \vec{R} $ be a unit vector in the direction of $ \vec{r} $, then Prove that $ \vec{R} \times \frac{d\vec{R}}{dt} = \frac{1}{r^{2}} \vec{r} \times \frac{d\vec{r}}{dt} $ where $ r = |\vec{r}| $.
(c) State Picard's Theorem.
(d) Calculate the Wronskian of the set $ \{x, x^{2}, x^{3}\} $.
(e) State the Principle of Superposition for homogeneous equation.
(f) Find the total work done in moving a particle in a force field given by $ \vec{F} = 3xy\hat{i} - 5z\hat{j} + 10x\hat{k} $ along the curve $ x = t^{2}+1, y = 2t^{2}, z = t^{3} $ from $ t = 1 $ to $ t = 2 $.
(g) If $ \theta $ be the angle between two unit vectors $ \vec{u} $ and $ \vec{v} $, then prove that $ 2 \sin\frac{\theta}{2} = |\vec{u} - \vec{v}| $.
(h) Define equilibrium points.
(i) Solve the equation $ \frac{dx}{dt} = -wy $ and $ \frac{dy}{dt} = wx $ and show that the point (x, y) lies on a circle.
(j) Define basic theory of linear systems in normal form of two equations in two unknown functions.
(k) Find the value of the constant d, such that the vectors $ 2\hat{i}-\hat{j}+\hat{k}, \hat{i}+2\hat{j}-3\hat{k} $ and $ 3\hat{i}+d\hat{j}+5\hat{k} $ are coplanar.
(l) Show that the vector $ \vec{r} = (x+3y)\hat{i} + (y+az)\hat{j} + (x+az)\hat{k} $ is solenoidal, if $ a = -2 $.
(m) If S is any closed surface enclosing a volume V and $ \vec{F} = x\hat{i} + 2y\hat{j} + 3z\hat{k} $, Prove that $ \int_{S} \vec{F} \cdot n dS = 6V $.
(n) Solve: $ (xy^{2} - e^{\frac{y'}{x^{3}}})dx - x^{3}y dy = 0 $.
(o) Find the differential equation, whose Primitives are $ (x^{2}+1)(y^{2}+1) = C $ where C is arbitrary constant.

2. Answer any four questions: 4 x 5
(a) (i) Solve the equation $ \frac{d^{2}y}{dx^{2}} + (x-1)^{2}\frac{dy}{dx} - 4(x-1)y = 0 $ in series about the ordinary point $ x = 1 $.
(ii) Determine the singular point of the following differential equation: $ x^{2}(x-1)^{2}\frac{d^{2}y}{dx^{2}} + 2(x-2)\frac{dy}{dx} + (x+3)y = 0 $. Also find the indicial equation.
(iii) Evaluate $ \int_{S} \vec{F} \cdot \vec{n} ds $, where $ \vec{F} = 4x\hat{i} - 2y^{2}\hat{j} + z^{2}\hat{k} $ and S is the curved surface of the cylinder $ x^{2}+y^{2}=4 $ bounded by the planes $ z=0, z=3 $.
(b) (i) Find the directional derivative of $ f = xy^{2} + yz^{2} + zx^{2} $ at the point (1, 2, 5) in the direction of x-axis.
(ii) Find the unit vector in the direction of the tangent at any point on the curve given by $ \vec{r} = (a \cos t)\hat{i} + (a \sin t)\hat{j} + bt\hat{k} $.
(c) (i) Prove that $ \frac{d\vec{n}}{ds} = \tau\vec{b} - \kappa\vec{t} $, where $ \vec{t}, \vec{b}, \vec{n} $ are the unit tangent, binormal and principal normal vectors respectively and $ \tau, \kappa $ and s are torsion, curvature and arc length respectively.
(ii) Find the magnitude of the volume of the parallelopiped having the vectors $ \vec{a} = -3\hat{i}+7\hat{j}+5\hat{k}, \vec{b} = 5\hat{i}+7\hat{j}-3\hat{k} $ and $ \vec{c} = 7\hat{i}-5\hat{j}-3\hat{k} $ as the concurrent edges.
(d) Solve the differential equation $ \frac{d^{2}y}{dx^{2}} - y = \frac{2}{1+e^{x}} $ by the method of variation parameter.
(e) Verify Green's theorem in the plane for $ \oint_{c}(xy+y^{2})dx + x^{2}dy $ where c is the closed region bounded by $ y=x $ and $ y=x^{2} $.
(f) Solve by the method of undetermined coefficients: $ \frac{d^{2}y}{dx^{2}} - \frac{dy}{dx} - 2y = 8 $.

3. Answer any two questions: 2 x 10
(a) Consider the linear system $ \frac{dx}{dt} = 5x+3y, \frac{dy}{dt} = 4x+y $.
(i) Show that $ x=3e^{7t}, y=2e^{7t} $ and $ x=e^{-t}, y=-2e^{-t} $ are solutions of this system.
(ii) Show that the above two solutions are linearly independent on every interval $ a \le t \le b $ and write the general solution of the system.
(iii) Find the solution $ x=f(t), y=g(t) $ of the system such that $ f(0)=0 $ and $ g(0)=0 $.
(b) If $ \vec{F} = (3x^{2}+6y)\hat{i} - 14yz\hat{j} + 20xz^{2}\hat{k} $, evaluate $ \int_{c} \vec{F} \cdot d\vec{r} $ from (0, 0, 0) to (1, 1, 1) along the following paths C:
(i) $ x=t, y=t^{2}, z=t^{3} $.
(ii) the straight lines from (0, 0, 0) to (1, 0, 0) then to (1, 1, 0), and then to (1, 1, 1).
(iii) the straight line joining (0, 0, 0) and (1, 1, 1).
(c) (i) Solve the following differential equation $ \frac{d^{2}w}{dz^{2}} - 2z\frac{dw}{dz} + 2\gamma w = 0 $ in a series about the ordinary point $ z = 0 $.
(ii) Solve: $ x^{3}\frac{d^{3}y}{dx^{3}} + 2x^{2}\frac{d^{2}y}{dx^{2}} + 2y = 10(x + \frac{1}{x}) $.

Unit-I (Real Analysis)

(b) Show that $ f(x) = \frac{1}{x}, x > 0 $ is not uniformly continuous in (0, 1].
3. Answer any one question: 1 x 10
(a) (i) Prove that if a real valued function f is continuous on a closed interval $ I = [a, b] $, then it is bounded there.
(ii) Prove that if a function f is continuous on a closed interval $ I = [a, b] $ then f is uniformly continuous on I.
(b) (i) Let f be a real-valued continuous function in a closed interval [a, b]. Suppose $ f(a) \neq f(b) $. Then prove that f assumes every value between $ f(a) $ and $ f(b) $ at least once.
(ii) Apply $ \epsilon - \delta $ definition to show that the function $ f(x) = x \sin(\frac{1}{x}), x \neq 0 $ and $ f(x) = 0, x = 0 $ is continuous at $ x = 0 $.

Unit-II

4. Answer any two questions: 2 x 2
(a) State Rolle's theorem.
(b) Is Rolle's theorem applicable to $ f(x) = 1 - x^{2/3}, -1 \le x \le 1 $? Justify your answer.
(c) For what range of value of x, $ f(x) = 2x^{3} - 9x^{2} + 12x - 3 $ decreases as x increases?

5. Answer any two questions: 2 x 5
(a) Show that $ \frac{x}{1+x} < \log(1+x) < x $, if $ x > 0 $.
(b) In the Mean value theorem $ f(x+h) = f(x) + hf'(x+\theta h), 0 < \theta < 1 $. Show that the limiting value of $ \theta $ as $ h \to 0+ $ is $ \frac{1}{2} $ when $ f(x) = \sin x $.
(c) State and prove Lagrange's Mean value theorem.

Unit-III

6. Answer any two questions: 2 x 2
(a) Show that the function $ f(x) = x^{3} - 3x^{2} + 6x + 3 $ does not possess any maximum or minimum value.
(b) State Taylor's theorem with Lagrange's form of remainder.
(c) State Maclaurin's theorem with remainder.

7. Answer any one question: 1 x 10
(a) (i) Expand the function $ f(x) = \cos x $ in power of x in infinite series.
(ii) State and prove Cauchy's mean value theorem.
(b) (i) If in the Cauchy's MVT we take $ f(x) = \sqrt{x}, g(x) = \frac{1}{\sqrt{x}} $ then prove that c is the geometric mean between a and b.
(ii) Find Cauchy's Remainder after n terms in the expansions of $ (1+x)^{m} $ and $ \log(1+x) $ in power of x.

Unit-IV

8. Answer any two questions: 2 x 3
(a) If (X, d) be a metric space, then show that $ (X, \frac{d}{1+d}) $ is also a metric space.
(b) Define closure of a set in a metric space. Prove that $ \overline{A \cup B} = \overline{A} \cup \overline{B} $.
(c) Let (X, d) be a metric space and $ A, B \subseteq X $. Show that $ \text{diam}(A \cup B) \le \text{diam}(A) + \text{diam}(B) + d(A, B) $.

9. Answer any one question: 1 x 5
(a) Prove that any finite set has no limit point.
(b) In a metric space prove that any open sphere is an open set.

Group Theory-1

Unit-I

1. Answer any two questions: 2 x 2
(a) Prove that a group $ (G, \circ) $ is Abelian if $ (a \circ b)^{-1} = a^{-1} \circ b^{-1} $ for all $ a, b \in G $.
(b) Consider the group (D, *) where D is the set of all odd integers and $ a*b = a+b-1 $ for $ a, b \in D $. Find $ 3^{-1} $.
(c) Define dihedral group $ D_{3} $.

2. Answer any one question: 1 x 5
(a) Let $ G = \{(a, b) \in Q \times Q : b \neq 0\} $. Prove that $ (G, \circ) $ is an abelian group, where $ \circ $ is defined by $ (a, b) \circ (c, d) = (ad+bc, bd) $ for $ (a, b), (c, d) \in G $.
(b) Let X be a non-empty set and P(X) be the power set of X. Examine if P(X) is a group under the composition '*' defined by $ A * B = A \cap B $ for $ A, B \in P(X) $.

Unit-II

3. Answer any two questions: 2 x 2
(a) Let G be an abelian group. Prove that the subset $ H = \{g \in G : g = g^{-1}\} $ forms a subgroup of G.
(b) Let G be a group and $ a \in G $. Prove that Z(G), the centre of the group G, is a subgroup of C(a), the centralizer of a.
(c) Let G be a group and $ H_{1}, H_{2} $ be two subgroups of G. Then show that $ H_{1} \cap H_{2} $ is a subgroup of G.

4. Answer any two questions: 2 x 5
(a) State and prove the necessary and sufficient condition that a non-empty subset H of G to be a subgroup of G.
(b) Let G be a group on which $ (ab)^{3} = a^{3}b^{3} $ for all $ a, b \in G $. Show that $ H = \{x^{2} : x \in G\} $ is a subgroup of G.
(c) If H and K be two subgroups of a group G, then show that HK is a subgroup of G iff $ HK = KH $.

Unit-III

5. Answer any two questions: 2 x 2
(a) Prove that every cyclic group is Abelian.
(b) A cyclic group G has only one generator. Prove that either $ o(G) = 1 $ or $ o(G) = 2 $.
(c) Examine whether the permutation $ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 4 & 5 & 1 & 3 & 2 & 8 & 6 & 7 \end{pmatrix} $ is odd or even.

6. Answer any one question: 1 x 10
(a) (i) Define cyclic group. Prove that every subgroup of a cyclic group is cyclic.
(ii) Define alternating group. Show that every permutation on a finite set is either a cycle or it can be expressed as a product of disjoint cycles.
(b) (i) State and Prove Lagrange's theorem for a finite group.
(ii) Let $ S = \{1, w, w^{2}, -1, -w, -w^{2}\} $, where $ w = \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} $. Prove that S is a cyclic group under multiplication.

Unit-IV

7. Answer any two questions: 2 x 2
(a) Define external direct product of two groups.
(b) Show that the alternating group $ A_{3} $ is a normal subgroup of the symmetric group $ S_{3} $.
(c) State Cauchy's theorem for finite abelian group.

8. Answer any one question: 1 x 10
(a) (i) Let G be the group of all $ n \times n $ real non-singular matrices and H be the group of all $ n \times n $ real orthogonal matrices. Prove that H is a subgroup of G but H is not a normal subgroup of G.
(ii) Let M and N be normal subgroups of a group G such that $ M \cap N = \{e\} $. Prove that $ mn = nm $ for all $ m \in M $ and for all $ n \in N $.
(b) (i) Find the number of elements of order 5 in the group $ Z_{15} \times Z_{10} $.
(ii) Prove that the group $ Z \times Z $ is not cyclic.
(iii) Prove that the group $ Z_{3} \times Z_{4} $ is cyclic.

Unit-V

9. Answer any two questions: 2 x 2
(a) If $ f: G \to G' $ be a homomorphism then show that $ f(a^{-1}) = [f(a)]^{-1}, \forall a \in G $.
(b) Define image and Kernel of a homomorphism.
(c) Let $ \phi: (G, \circ) \to (G', *) $ be an isomorphism. Then show that $ \phi^{-1}: (G', *) \to (G, \circ) $ is also an isomorphism.

10. Answer any one question: 1 x 5
(a) State and prove first isomorphism theorem.
(b) State and prove Cayley theorem for finite group.

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