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MATHEMATICS - GENERAL
Bangabasi Evening College
B. Sc(General)
Fourth Paper
Part-III
Module- VII
Computer Science & Programming
1. Describe different generation of computers with reference to their electronic components.
2. What are the advantage and disadvantage of high-level and machine language?
3. Multiply (11001)2 and (1101) 2 by using binary arithmetic multiplication and divide (1001000)2 by (110)2 .
4. Using truth table, determine whether the following expression is identical xy  x y  ( xy  x y ) .
5. Point out the errors (if any) in the following program segments:
(a) DO 10 I=1, 10
DO 20 J=1, 20
X=I*J
WRITE (*,*) X
10 CONTINUE
20 CONTINUE
(b) IF (A.LT.5)
X=1
ENDIF
6. (a) What are the differences among I, F and E format.
(b) Explain the function of IMPLICIT declaration statement FORTRAN.
7. Convert the following numbers of different bases (35.3125) 10 to binary form and ( A8C )16 to decimal
form.
Module-VIII
Answer Any one of the following groups
Group A
A Course of Calculus
1. Find the general solution of the PDE:
( z  y ) p  ( x  z )q  y  x .
2. Find the Fourier series of the periodic function of period
2 .
f ( x)  x ,    x  
2
Hence, deduce that (i) 1 
1
1
1
2





6
2 2 32 4 2
1 1 1
2
.
1 2  2  2  
12
2 3 4
3. Find the Fourier cosine series for the function (  x) on [0,  ] .
(ii)
Hence, deduce the value of 1 
1
1
 2 .
2
3
5
4. Solve the following differential equation by method of variation of parameters (D 2  9) y  Sec3x
2
5. Solve: (1  x) 2 d 2y  (1  x) dy  y  4 cos log(1  x)
dx
dx
6. (a)Find L[ f (t )] , where f (t )  sin t , 0  t  
t 
0,
(b) Apply convolution theorem to find the inverse Laplace transform of
s
.
( s 2  4) 2
7. Solve the following differential equations, using Laplace and inverse Laplace transform:
where
y(0)  0, y ' (0)  1 .
8. Solve the simultaneous linear differential equations:
dx
 y ;
dt
dy
 x , where  is a constant.
dt
Group-B
Discrete Mathematics
y '' (t )  y(t )  sin 2t ,
1. Solve the system of linear congruence’s: x  2(mod3) , x  3(mod5) and x  1(mod7) .
2. Prove that there exists infinite number of prime integers.
3. (a) Suppose there are eight teachers in the BEC and their coded identification numbers 2733, 1396,
2724, 1522, 1952, 2088, 2155, 1850. Create a file for each teacher in this order in the computer if there
are fifteen memory locations indexed as 0, 1, 2, …, 14 available for this purpose, using the Hasing
function h(k )  k (mod 13) .
(b) Use principle of Mathematical Induction to show that 7 2n  16 n  1 is divisible by 64, where n is a
positive integer.
4. (a) Solve the following difference equation with given initial condition: a n  a n1  a n2 for n  3 ,
a1  1, a 2  3 .
(b) Find the solution of the recurrence relation an  7an1  10an2  0 with a0  3, a1  3
5. (a) If a | bc and gcd(a, b)  1 , then prove that a | c .
(b) Find the remainder when 1920 is divided by 181.
6. What is the remainder when 1!2!3!4!   100! is divided by 15?
7. (a) A student spends Rs. 321 to buy some books and pens. The cost of each book is Rs. 12 and that of
each pen is Rs. 7. Find the possible integral solution by forming a Diophntine equation.
(b). Construct a Round Robin Tournament Schedule for 7 teams using congruences of integers.
8.(i) Find  (72) where  is the Euler’s Phi function.
(ii) Show that a n   2 n  3n is a solution of the difference equation a n  5a n1  6a n2 , n  2 , ,  are
constants.
(iii) Find all integral solutions of 13x  4 y  8 .
(iv) Which of the following equations 3x  2 y  6 , 6x  4 y  91 has no integral solutions? Give reasons.
(v) Find the generating function of the numeric function {a r } , where ar  2.3r , r  0 .
9 (a) In a Boolean algebra ( B,,., ' ) prove that a  b  a  c and a.b  a.c imply b  c .
(b) Express the Boolean function ( x  y )( y  z )( x' y ' z ' ) as CNF and DNF in three variables.