Bangalore University 2007 B.Sc Mathematics - 1 - Question Paper
Mathematics
Instructions: 1) Answer all questions.
I Semester B.A./B.Sc. Examination, Nov ./Dec. 2007 (Semester Scheme) MATHEMATICS-I
Time : 3 Hours
Max. Marks: 90
2) Answer should be written completely either in English or in Kannada.
(15x2*30)
1. Answer any IS of the following :
1) Write the negation of Vx L p (x) -> q (x)]. -
2) Find T [ p(x) ] where p(x): | x - 3 ] < 5 with R [p(x)3 = Z.
3) Define an equivalence relation.
4) Show that the function f : R - R defined by f(x) = 4x + 2 Vxe R, is a bijeciion.
5) Find the ndl derivative of e3x sin 5x. '
6) If Z = e2'sin 3y find .
dxdy
7) Find the n,h derivative of log (9x2 - 4).
(V + v0') da du
8) If u = log -1 , find x + y-
x + y ox dy
x + y
9) If u = x3 + y3 where x = 3t and y = t2 find .
3(u, v)
10) lfu = x2 -2y andv = x + y find .
n
11) Evaluate f
x COS* x dx.
12) If I = f tan" xdx, prove that 1 +In_2 =--
J n - I
13) Find the ratio in which the line joining the points (2, 4, 5) and (3. 5, -4) is
divided by XY-plane.
14) Find the direction ratios and direction cosines of the line joining the points
(4, 3, -5) and (-2, 1, -8).
. 15) Find the equation of the line passing through the point (3,4, 5) and is parallel to the vector 2i + 2 j - 3k.
16) Find the equation of the plane passing through the point (2,4, 3) and parallel to the plane 5x - 6y + 7z = 3.
17) Find the length of the perpendicular from the point (1,-1, 3) on to the plane 5x + 2y - Iz + 9 = 0.
i 8) Find the equation of the sphere whose centre is (2, -3,4) and radius 5.
19) Find the equation of ihe light circular cylinder of radius 3 and axis x-1 y-3 _ z-5
2 2 -1
20) State any two properties of the hyperboloid of one sheet-
11. Answer any two questions : (2x5=10)
1) With usual notations, prove that T[p(x) a q(x)]= l[p(x)]n T[q(x)J
2) Prove that any two equivalence classes are either equal or disjoint.
31 If f: X > Y is a function where A and B are non-empty subsets of X, then prove that
f(AuB)=f(A)uf(B)
4) If f: R - R defined by f(x) = x2 and g : R - R defined by g(x) = x + 3 find (gof) (4) and (fog) (4).
III. Answer any three of the following : (3x5=15)
x - 2
1) Find the n,h derivative of 5----
6x + x -1
2) If y = cos (m cos-1 x) prove that
(1 -x2)yn+2 -(2n + l)xy0.M -(n2-m2)y =0
3) State and prove Eulers theorem for a homogeneous function of two variables.
4) If z s sin (ax + y) + cos (ax - y), prove that
3x dy
 |
(2x5=10) |
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2) Evaluate ( |  |
3) Verify the Leibnitz rale of differentiation under the integral sign for
(3*5=15)
 |
a is a parameter. |
|
cc (I + cos x) 0 | |
V. Answer any three :
1) Find the volume of the tetrahedron formed by the points (1, 2, 3), (2, 3, 5), (-2, -1, 2) and (3, 0, -3).
2) Find the angle between the two lines whose direction cosines are given by the equations / + m + n = 0 and /2 + m2 - n2 = 0.
3) Find the vector and Cartesian equations of a line passing through two points.
4) Find the image of the point (1,2, 3) in the plane x + y + z = 9.
5) Find the shortest distance between the lines
x+3 y-6 z x+2 y 2-7 = =T=
VI. Answer any two : (2x5=10)
1) Find the equation of the sphere which passes through the points (3, 0, 2),
(-i, I, i), (2, - 5, 4) and having its centre on the plane 2x + 3y + 4z = 6.
2) Derive the equation of a right circular cone in its standard form x2 + y2 = z2 tan2a.
3) Find the right circular cylinder generated by revolving the line
x 1 y-3 z-5 x + t y + 3 z + 5 = ---- about the line -----.
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Attachment: |
| Earning: Approval pending. |
