Bangalore University 2008 B.A MATHEMATICS - Question Paper
IV Semester B.A./B.Sc.Degree Examination, June 2008
(Semester Scheme)
MATHEMATICS(Paper-IV)
SM - 222
Time : 3 Hours
II Semester B.A./B.Sc. Examination, June 2008 (Semester Scheme) Paper - II: MATHEMATICS
Max. Marks : 90
Instructions : i) Answer all questions.
ii) Answers should be written completely either in English or in Kannada.
I. Answer any 15 of the following :
(15x2=30)
1) If A is any square matrix and A' is its transpose then prove that A and A' wave the same eigen values.
2) Find the value of X for which the system 7x + 4y + 3z = 0, x + 2y + Xz = 0, x + 3y + 2z = 0 has a non-trivial solution.
3) Show that O is the characteristic root of a matrix iff the matrix is singular.
/2 -T 0 1
4) Find the eigen values of
12 3 1 2 4 6 2
5) Find the rank of the matrix
12 3 2
/
71
6) For the curve r = a(l + cos0) at 0 = find the polar subtangent.
ds
7) For the curve x = acost, u = bsint, find .
dt
8) Find the angle between the radius vector and the tangent for the curve r2 = a2cos20 at 0 = .
-2-
2
r
9) For the curve r = a0, show that P - r .
Vr + a
10) Show that y = e\ x > 0, is concave upwards.
11) Find the envelope of the family of curves x2 + y2 - 2gx + g2 - c2 = 0 where g is a parameter.
12) Find the asymptotes parallel to the coordinate axes for the curve y2(x2 - a2) = x.
13) Define node and cusp of a given curve.
14) Find the area of the loop of the curve 3ay2 = x(a - x2).
x y
15) Find the volume generated by an ellipse + = 1 about y-axis.
a b
17) Solve (ax + hy + g) dx + (hx + by + f) dy = 0.
18) Solve p2 - 5p + 6 = 0 where p =
dx
19) Solve 1 + = ex+y dx
ap
20) Find the general solution of y - px + , .
V1 + P
(3x5=15)
II. Answer any three :
1) Find the inverse of matrix A by elementory transformations where
3) For what values of X and the equations x + y + z = 6, x + 2y + 3z = 10, x + 2y+ A, z = (a have 1 no solution 2 a unique solution 3 infinite number of solutions.
-3-
SM - 222
|
1 |
1 |
-1 |
1 |
|
1 |
3 |
2 |
1 |
|
2 |
0 |
3 |
2 |
|
3 |
3 |
3 |
3 |
2) Reduce the matrix A =
to normal form and hence find the rank.
4) State and prove cayley Hamilton theorem.
5') Show that the following system of equations x + 2y - z = 3, 3x - y + 2z = 1, 2x - 2y + 3z = 2 are consistent and solve.
III. Answer any two :
(2x5=10)
1) With usual notation, prove that tan(|) = r.
dr
2) Show that r = a(l + sin0) and r = b(l - sin0) intersect orthogonally.
3) Find the pedal equation of the curve rn = a" cosn0 .
4) Show that the evolute of parabola y2 = 4ax is 4(x - 2a)3 = 27ay2.
IV. Answer any two : (2x5=10)
1) Find the asymptotes of the curve 4x2(y - x) + y(y - 2) (x - y) - 4x - 4y + 7 = 0.
2) Find the position and nature of the double points on the curve x3 + x2 + y2 - x - 4y + 3 = 0.
X V
3) Find the envelope of family of lines + = 1 where a.b = c2.
a b
4) Trace the curve r = a(l + cos0).
r
1) Find the perimeter of the cardiod r = a(l + cos0).
2) Find the surface area of the solid generated by revolving the cycloid x = a(0 - sin0) y = a(l - cos0) about the axis.
3) Find the area of the Astroid x = acos30 y = asin30 .
VI. Answer any three : (3x5=15)
1) Solve (x2 + y2) dx = 2xydy.
dy
2) Solve = sin(x + y) + cos(x + y).
dx
3) Find the orthogonal trajectories of rn = an sinn0.
4) Solve (px - y) (py + x) = a2p using transformation x2 = u, y2 = v.
I. oijsradd 15 advert : (15x2=30)
1) A iQedroftdo,A' edd as.sPsSpeTF iQedrodd A sk&A' ode
CS J -o
soairaoQdbd <aodo
< -e *
2) 7x + 4y, + 3z = 0, x + 2y + Xz = 0, x + 3y + 2z = 0 toetfdrarttf srfoaoadrttfrfj sLaoQdd X tfodo&QcGoQ.
< am
3) 'O' oc> &rsed c&p&p5' dossd Add do&sidd &6>edDd?&
|
4) |
r2 0 V |
-1> 1 |
Irsedd so rto*1 odo&o2oo. erf | |
|
(1 |
2 |
3 1> | ||
|
5) |
2 |
4 |
6 2 |
loedd d2F0>?& odo&ccoo |
|
1 V |
2 |
3 2 | ||
|
Attachment: |
| Earning: Approval pending. |
