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AUCET 2012

Testpapers of Andhra University BCA - Mathematics
Max Marks :75 Time :3 hrs.
Answer any FIVE questions and all questions carry equal marks

1.
a. Find the derivative of tan(2x+3)
b. A circular plate expands, when heated, from a radius of 5 cms.to 5.06 cms. Find the approximate increase in its area.
c. Trace the graph of the function y2=x2(a2-x2)
d. Find the equation of the circle passing through (0,0),(1,0) and directrix.

2.
a. Find the equation to the parabola whose focus is (1,-1) and directrix 3x+4y+2=0
b. Find the area of the circle x2+y2=a3 by the method of integration.
c. Integrate the following functions
(i) ex(tanx+sec2x)
(ii) 2x/1+x4

d. Evalute integral from -1 to 1 dx/1+x2

3.
a. Show that the vectors (1,2,1),(2,1,0) and (1,-1,2) form a basis of R3
b. Let F be the field of Complex numbers and
		T:F3 -> F3 be defined by
		T(x1,x2,x3) = (x1-x2+2x3,2x1+x2-x3,-x1-2x2)
		Show that T is a linear transformation. Describe the null space of T.

4.
a. Find the characteristic roots and characteristic vectors of the matrix
| 0 1 0 | A= | 0 0 1 | | -12 -20 -9 |

b.
Solve the system of equations:
2x+3y+z=2; x+2y-z=6; 3x+z=2 by Cramer's rule

5.
a. If A and B are similar linear transformations on a vector space V, show that A2 and B2 are similar
b.Let T be the linear operator on R3 defined by
T(x1,x2,x3) = (3x1+x2,-2x1+x2,-x1+2x2+4x3)

Prove that T is inevitable and find formula for T-1

6.
a. Calculate mean, median and mode of the following frequency distribution. Class 0-9 10-19 20-29 30-39 40-49 50-59 Frequency 13 38 67 76 22 4

b.
A Problem in mathematics is given to three students A,B and C whose chances of solving it are ½,1/3 and ¼ respectively. What is the probability that the problem is solved.

7.
a.Fit a straight line to the following data
X :	1	2	3	4	6	8
Y :	2.4	3	3.6	4	5	6

b.
The two regression equations of the variables x and y are x=19.13-0.87y and y=11.64-0.50x.Find (i) mean of x's ,(ii) mean of y's and (iii) Correlation coefficient between x and y.

8.
a. If A, B, C are sets prove that
(i) (A-B) U (B-A) = (AUB)- (AÇB)
(ii) A U (B Ç C)=(AUB) Ç (AUC)
b.
Define reflexive,symmetric and transitive relations.Give examples for
(i) A relation which is reflexive and symmetric but not transitive
(ii) A relation which is reflexive and transitive but not symmetric

9.
a.For any two elements a nd b of a Boolean algebra,show that a £ b if and only if b' £ a'.
b. If A={2,3,4},B={1,2} and C={4,5,6} find
( I ) A + B (ii) B + C and (iii) A+B+C
c. For any three elements a,b and c in a Boolean algebra,show that
(a Ç b Ç c) U (b Ç c) = (b Ç c)

10.
a.Define a distributive lattice and prove that every chain is a distributive lattice.
b.Show that, in a lattice, a £ b < c Þ a Å b = b-c
c.Show that in a complemented lattice with more than one element, no element is it's own compliment.
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