University of Delhi 2010-2nd Year B.Sc PMCS (Physics, Mathematics, Computer Science) Prog MATHEMATICS-1 UNIVERSITY - Question Paper
This question paper contains 4+2 printed pages]
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Your Roll No
5186
B.Sc. ProgJII
. MA-201MATHEMATICSI
(Calculus and Geometry)
(For Physical Sciences/Applied Sciences)
(Admissions of 2008 and onwards)
This question paper contains 4+2 printed pages]
J
Your Roll No
5186
Maximum Marks . 112
Time 3 House
This question paper contains 4+2 printed pages]
J
Your Roll No
5186
(Write your Roll No on the top immediately on receipt of this question paper )
All questions are compulsory Attempt any two parts from each question except Question No 5 In questions No 5 attempt any one
Section I
1 (a) (0 Represent geometrically the set
{z \z - i\ > \z + 11}
5 &
(w) Find centre and radius of the circle whose
equation is
\z - i\ = 4|z + i[
Check if the origin lies inside the circle 7
x4 2x3 - 21x2 - 22x + 40 = 0,
*
the sum of the two roots being equal to sum of the other two 7
(u) If a, P and y are the roots of the equation x3 - ax2 + bx - c = 0, find the value of
(c) (i) Prove that :
(1 + t)n + (1 - if = 2n/2+1 cos(nn/4) 5VS (w) Solve the equation
zG-z5 + z4-z3 + z2-z+l = 0 7 Section II
2 (a) State intermediate value theorem and show that if the continuity hypothesis of the theorem is dropped, then its coticIusvmv fail to hold 10
(6) Show that the function f defined as
fix) = \x\ + \x - 1|
is continuous at x = 0, and x = 1 But is not derivable at these points 10
(c) Define uniform continuity of a function. Give an example of a continuous function which is not uniformly continuous 10
3 (a) Find the asymptotes of the curve
x3 + 2x2y + xy2 - x$ - xy + 2 = 0. 10
(
(b) (i) Determine the position and nature of double * points on the curve
x3 + x2 + y2 x - 4y + 3 = 0
(ii) Trace the polar curve
r =. a(l + cos 0) 5+5 (c) Trace the curve 10
>(1 - x2) = x2
4 (a) Obtain the reduction formula for
J sin" x dx,
n being a positive integer and hence evaluate
it/2
nn
J sin6 x dx
10
(ib) Find the area of the parabola y2 = 4ax
bounded by its latus rectum Also find the
volume of the solid obtained by rotating this
area about x-axis 10
(c) Find the arc length of the cycloid
x = a(Q + sm 8), = 0(1 + cos -u < ft < n 10
Section III
5 Identify and sketch the graph of the conic
4x2 - 4xy + y2 - 8x - 6y + 5 = 0 by rotating co-ordinate axes 14
Or
Find the nature of the following conic and trace it completely giving essential details
x2 + xy + y2 - x + Ay + 3 = 0 14
6 (a) A particle moves along the curve x = f3 + 1,
y - f2, z - 2f + 5, where t is the time Find the component of the velocity and acceleration at t - 1 in the direction i + j + Zk 614
(b) Show that
721
(c) For what value of the component a will the vector
A = (axy - z3)i + (a - 2) x2j + (1 - a) xz2k have its curl identically equal to zero 9 6
5186 6 2,000
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