Maharashtra State Board of Technical Education 2008 Diploma Computer Engineering Applied Mathematics Computers Group - Question Paper

Sample Question Paper - I

9035

-    Computer Group

-    CO/CM/IF/CD

-    Third

-    Applied Mathematics

-    3 hours

Instructions:

   All Questions are compulsory

   Figures to the right indicate full marks

   Assume suitable data if necessary

f) Find the order and degree of differential equation k-

dx ²    dx

g)    Prove that E ^V-VE -A

_h)    Show that the second differences of the polynomial y=x² when x=1,3,5,7,9, are constant.

By using Simpsons 1/3 Rule Evaluate

5

| f (x)dx using following table.

x	1	2	3	4	5
f(x)	10	50	70	80	100

j) If X={1,2,3,4,5,6,...........,15}

A= {5,6,7,8,9,10,11}

B= {9,10,11,12,13,14}

Find A B'

Q2. Attempt Any Three    Marks-12

a) Verify that y=logx is a solution of differential equation

d 2 y dy _ xf + - = 0 dx dx

b)    Solve ^dy = cos(x + y)

dx

c)    Solve (2x+3cosy) dx +(2y-3xsiny) dy=0

dy

d)    Solve x logx + y = 2log x

dx

Q3. Attempt any Three a) Given

x 3 7 9 10 y 168 120 72 63

Estimate y when x=6 Using Lagranges interpolation formula

b) Using Newtons forward formula for interpolation find f(1.5) from the Following data_

x 1 2 3 4 5 f(x) 2.38 3.65 5.85 9.95 14.85

c) From the following table find the number of students who obtained marks more than 65[Use Newtons backward interpolation formula]

Marks obtained 30-40 40-50 50-60 60-70 70-80 No. of Students 30 41 52 36 31

d) The current I flowing in the circuit containing resistance R and inductance L

dl

in series with voltage source E at time t is given by L--+ RI = E. Show that

dt

I= ^E (1 - e ~^Rt/L)

R

Q4. Attempt any four    Marks-16

a)    Evaluate j x tan^-1 xdx

cos x

b)    Evaluate I-dx

(1 + sin x)(2 + sin x)

n / 2

sin x

c)    Evaluate j -^xdx

0 (1 + cos x)3

d)    Evaluate j , +⁵ dx

0 V x + 5 + V 9 x

I x + 5 + ' _

^e)    Using integration find the area of the ellipse ^x- + = 1

a b

f)    Find the area between parabola y=x and the line y=x

Q5. Attempt Any Three    Marks-12

a) Find y' (0) from the following data

x	0	1	2	3	4	5
y	4	8	15	7	6	2

f dx

b)    Evaluate I-_T by Trapezoidal Rule by taking h=4 and hence obtain

0¹ + x²

approximate value of n

c)    A curve is drawn to pass through the points given by the following table

x	1	1.5	2	2.5	3	3.5
y	2	2.4	2.7	2.8	3	2.6

Using Simpsons 1/3 rule estimate the area bounded by the curve y=f(x), the x-axis and x=1

d) By Using Range-Kutta method fourth order solve the differential equation

^y, y(1)=1obtain y when x=1.1 (Take h=0.1) dx x

Q6. Attempt any Three    Marks-12

^a) Using second order Runge-Kutta method solve differential equation y = - y ,y(0)=1 for x₁=0.2 and x₂=0.4

b)    Using Runge-Kutta method of Fourth order to find an approximate value of y

when x=0.02 given that =x²+y²with y(0)=1

dx

c)    If A={X/x²- 11x+28=0} B={ x/x²+8x-48=0} and C={x/x²+12x+35=0}

And the universal set X={-12,-10,-7,-6,-5,4,5,7} Verify that

i)    A U (B n C)=(A U B) n (A U C)

ii)    (a u C)' = A' n C'

d)    Find how many integers from 1 to 300 are not divisible by 3 nor by 5.

Attachment: maharashtra-state-board-of-technical-education-2008-diploma-computer-engineering-30895-1.pdf

Earning:   Approval pending.