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B.A. / B.Sc. / B.C.A. 3rd Semester Examination

BS-IIIS/11

228406

Mathematics

Course No. : 301

Time Allowed: 3 hours                                                                        Maximum Marks: 80

Ψ Ψ Ψ

B.A. / B.Sc. / B.C.A. 3rd Semester Examination

BS-IIIS/11

225395

Mathematics

Course No. : 301

Time Allowed: 3 hours                                                                        Maximum Marks: 80

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B.C.A. 1st Semester Examination

BS-IS/12

230294

PROBLEM SOLVING USING C-LANGUAGE

Course No.: BCA - 102

Time Allowed: 3 hours                                                                        Maximum Marks: 80

Note: Attempt total five questions in all, selecting one question from each unit. Q1 is compulsory. Each question carries 16 marks.

1.     a) Explain what is meant by precedence and associativity.

b) Explain with the help of an example what in nesting of loops.

c) Explain four standard library functions used for string manipulation.

d) Discuss how unions are declared and accessed.

UNIT - I

2. Write an algorithm to find the largest element in a list of elements. Also, draw its flowchart.
3. What is an error? Explain the different types of errors? Give example for each one of them.

UNIT - II

4. Discuss the various decision control statements.
5. Differentiate between while loop, Do while loop and for loop.

UNIT - III

6. Write a program in C to sort an array of integers in descending order.
7. Discuss the features of various storage classes in C language.

UNIT - IV

8. Write a program in C to illustrate how two dimensional arrays are accessed using pointers.
9. Write a program in C to read and display an array of structures.

 

B.A. / B.Sc. 1st Semester Examination

BS-IS/12

229944 (S)

Mathematics

Course No. : 101

Time Allowed: 3 hours                                                                        Maximum Marks: 80

Note: Attempt five questions in all, selecting one question from each unit. All questions carry equal marks.

Unit - I

1. a) Discuss the continuity of the function f(x, y) defined by:
       f(x, y) = { xy sin(1/x), if x ≠ 0
                        0, if x = 0
       at origin.
   b) Determine a and b so that the curve y = ax³ + 3bx² has a point of inflexion at (-1,2).

2. a) If Z = log(x² + xy + y²), then prove that:
        x(∂Z/∂x) + y(∂Z/∂y) = 2.
   b) Find the position and nature of double point(s) on the curve:
       x³ - y² - 7x² + 4y + 15x - 13 = 0.

Unit - II

3. a) Trace the curve y = x²(x - 3a), a > 0.
   b) Evaluate lim(x→0) (sin x)^(tan/x).

4. a) Find the envelope of the family of lines (x/a) + (y/b) = 1 where a and b are connected by the relation ab = c².
   b) Show that x + y + a = 0 is the only asymptote of the curve:
       x³ + y³ - 3axy = 0.

Unit - III

5. a) Find the value of ȓ satisfying the equation d²ȓ/dt² = ȃ,

       where ȃ is a constant vector. Also, it is given that ȓ = ô and dȓ/dt = ȗ, at t = 0.

   b) Find the directional derivative of the function 2xy + z² at (1, -1, 3) in the direction of the vector î + 2ĵ + 2ќ.

6. a) If F^→ = (2x² + y²) î + (3y - 4x) ĵ, evaluate _Δ∫F^→.dȓ around the triangle ΔABC whose vertices are A(0,0), B(2,0), and C(2,1).

   b) Prove that div ȓ = 2/r.

Unit - IV

7. a) With usual notations, prove that tan φ = (r dθ/dr).
   b) Trace the curve r = a(1 + cos θ).

8. a) Trace the curve r = a sin 2θ.
   b) i) Transform xy³ + x³y = a² to polar form.

       ii) Transform r² = sin 2θ to Cartesian form.

Unit - V

9. a) Find the curved surface of the solid generated by the revolution about the x-axis of the area bounded by the parabola y² = 4ax, the ordinate x = 3a, and the x-axis.
   b) Obtain a reduction formula for ∫ secⁿ x dx, n being a positive integer. Hence evaluate ∫sec⁴ x dx.

10. a) Evaluate ₀∫^a (x⁴/√(a² - x²)) dx.
     b) A segment is cut off from a sphere of radius a by a plane at a distance (1/2)a from the center. Show that the volume of the segment is (5/32) of the volume of the sphere.

B.A. / B.Sc. VI - Semester Examination

BS-VIS/4

211728

MATHEMATICS

Course No. : MA - 601

Time Allowed: 3 hours                                                                           Maximum Marks: 80

Note: Attempt any five questions, selecting one from each unit. All questions carry equal marks.

B.A. / B.Sc. VI Semester Examination

BS-VI/7(B)

222247

MATHEMATICS

Course No. : 601

Time Allowed: 3 hours                                                                       Maximum Marks: 80

Note: Do any five questions, selecting one from each unit.

B.A. / B.Sc. VI Semester Examination

BS-VIS/5

26687

MATHEMATICS

Course NO.: 601

Time Allowed: 3 hours                                                                       Maximum Marks: 80

Note: Do any five questions, selecting one from each unit.

B.A. / B.Sc. IV - Semester Examination
BS-IVS/5
26966
MATHEMATICS
Course No. : 401

Time Allowed: 3 hours                                                                        Maximum Marks: 80

Note: Do any five questions, selecting one from each unit.

UNIT - I

1. a) If A is a non-empty subset of R that is bounded below, show that A has a greatest lower bound in R. Find g.l.b. of {π+1, π+1/2, π+1/3,.....}

b) When do we say that sequence {x_n} of real numbers has the limit L? Show that the limit of a convergent sequence of real numbers is unique.

2. a) i) If a sequence {x_n} of real numbers is convergent, show that {x_n} is bounded.

ii) Show that a non- decreasing sequence which is bounded above is convergent.

b) Define Cauchy sequence of real numbers and show that a Cauchy sequence of real numbers is convergent.

b) State D’Alembert’s ratio test and Raabe’s test and show by giving an example that Raabe’s test is stronger than D’Alembert’s ratio test.

UNIT – III

5. a) Prove that ƒ(x) = xⁿ, x ∈ R (n is a positive integer) is continuous at each x and deduce that easy polynomial function is continuous at each point of R. Give an example to show that not easy continuous function is differentiable.

b) State Rolle’s theorem and find a suitable point c of Rolle’s theorem for the function,
ƒ(x) = (x-a) (b-x), x ∈ [a,b]. Show that between two consecutive zeroes of ƒ'(x) there lies a zero of ƒ(x).

6. a) State Mean value theorem & use it to show that if ƒ'(x) = 0 ∀ x ∈ (a, b), then ƒ(x) is constant on [a, b]. What happens, if you replace ƒ'(x) = 0 ∀ x ∈ (a, b) by ƒ'(x) = 0 ∀ x ∈ (a,b)? 

b) State Taylor’s theorem with Lagrange’s form of the remainder and obtain Maclaurin's infinite series for ƒ(x) = log (1 + x).

UNIT – IV

7. a) Prove that if θ is real and n is rational, then (cos θ + isin θ)ⁿ = cos nθ + isin nθ.

b) Determine all roots of x⁹ – 1 = 0 by De - Moivre’s theorem and find out which of these roots satisfy x³ – 1 = 0.

B.A. / B.Sc. V Semester Examination

BS-VS/11

229006

MATHEMATICS

Course NO.: 501

Time Allowed: 3 hours                                                                       Maximum Marks: 80

Note: Attempt any five questions, selecting one from each unit. All questions carry equal marks.

Unit – I

1. a) Define the Cartesian product of two sets A and B. Show that (A×B) ∩ (C×D) = (A∩B) × (B∩D) for any sets A, B, C, and D.

b) Define a relation and inverse of a relation. If R^-1 and S^-1 are inverse of relations R and S, respectively, then show that (SoR)^-1 = R^-1oS^-1.

2. a) If R is an equivalence relation on a set X and [ɑ] denotes the equivalence class of ɑ ∈ X, the show that [ɑ] = [b] if and only if ɑ ∈ [b], ∀ ɑ, b ∈ X.

b) If ƒ: A ➝ B and g: B ➝ C are maps such that goƒ: A ➝ C is onto, then show that g is onto but ƒ need not be onto.

Unit – II

3. a) Give an example of the following:

i) A finite non-abelian group.

ii) An infinite non-abelian group.

b) Define the order of a group. If G is a group of even order, then prove that there is an element ɑ ≠ e in G such that ɑ² = e.

4. a) Define a subgroup of a group. Let G be a group and H = {a ∈ G; ab = ba, ∀ b ∈ G}. Prove that H is a subgroup of G.

b) Define a cyclic group. Prove that every group of prime order is cyclic.

Unit – III

5. a) Define left coset of a subgroup in a group. Prove that any two left cosets of a subgroup H in a group G are either identical or disjoint.

b) If H and K are subgroups of a group G, then prove that HK is a subgroup of G iff HK = KH.

6. a) Give an example of a non-abelian group G a normal subgroup H of G such that the quotient group G/H is abelian.

b) Prove that every proper subgroup of an infinite cyclic group is infinite.

Unit – IV

7. a) Let G be a group and ɑ be any fixed element of G. Prove that the mapping ƒ: G ➝ G defined by ƒ(x) = xɑx^-1, ∀ x ∈ G is an isomorphism of G onto G.

b) Show that any cyclic group of order n is homomorphic to the additive group of residue classes modulo n.

Unit - V

9. a) Give an example of a ring which contains elements a, b such that (a + b)² ≠ a²  + 2ab + b².

b) Show that a finite integral domain is a field.

10. a) Define left ideal of a ring. Show that any intersection of left ideals of a ring is also a left ideal, but the union of two left ideals need not be a left ideal.

b) Show that an ideal M of a commutative ring R with unity is a maximal ideal iff R/M is a field.

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