# MCA_DCA6103_Foundation of Mathematics__InternalAssignments_Set 1&2

SESSION JUL/AUG 2021

PROGRAM MASTER OF COMPUTER APPLICATIONS (MCA)

SEMESTER I

COURSE CODE & NAME DCA6103 – FOUNDATION OF MATHEMATICS

Ques 1 a) Suppose that p and q are prime numbers and that n = pq. Use the principle of inclusion-exclusion to find the number of positive integers not exceeding n that are relatively prime to n

Since n is the product of two prime numbers p and q, any number which is relatively prime to n must be relatively prime to both p and q. Consider the following two sets of . Its Half solved only

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Ques1 b) Show that ~[p V(~p ^q)] and ~p ^~q are logically equivalent.

Ques 2 a) Find lim x~0 (27x-9x-3x+1)/(1-cos x)

Ques2 b)Find the derivative of f(x)=log e x ,x>0, using first principle.

Sol:

F(x)= Ques 3 a)If y = e ^ (a * sin-1(x)) , prove that (1-x2)yn+2 -(2n+1)xyn+1-(n² + a²)yn = 0.

Hence find the value of yn when x=0.

Sol:

Differentiate

Ques 3 b)Find all the asymptotes of the curve y ^ 3 – 6x * y ^ 2 + 11x ^ 2 * y – 6x ^ 3 + x + y = 0.

Sol:

Given :

Ques 4 a)Show that Sol:

Ques 4 b) Ifu = (3xy-y3)-(y²-2x)³/2 show that Sol:

Ques 5 Verify Green’s theorem for , where c is the region bounded by the parabola y = x and the line y=x.

Sol:

Boundary condition

x=0,1

Ques 6 a)Solve the system of equations x + y + z = 9; 2x + 5y + 7z = 52; 2x+y=z=0.

Sol:

x+y+z=9

Ques 6 b)Find the value of √-5+12i.

Sol:  Let z=x+iy