University of Delhi 2009 B.A Mathematics -I set-II, ., SOL, - Question Paper
Your Roll No.
Paper I (Algebra and Calculus) (New Course : Admissions of 2004 and onwards)
Time : 3 Hours Maximum Marks: 75
, (Write your Roll No. on the top immediately on receipt of this question paper. )
Note: The maximum marks printed on the question paper are applicable for the students of the regular colleges (Cat. A). These marks will, however, be scaled up proportionately in respect of the students of NCWEB at the time of posting of awards for compilation of result.
All sections are compulsory and have equal marks.
Attempt any two parts from each section.
I. (a) Do the vectors (1, 2, 1), (1, 0, -1) and (0, -3, 2) form a basis of V =IR3 (TR). Give reasons. What
: is dim V ?
(b) Verify that the matrix :
A-
A4.
4x + 5y + 6z = 0
5x + 6y :+lz-. 0 Ix + 8y + 9z 0
2. (a) If a, P are roots of the equation x2-2x + 2 = Q, prove that:
nn
an + (V7 = 7> } cosj and hence evaluate a6 + (36.
(b) Using De Moivre s theorem, solve the equation z7 + z ~ 0
(c) If the sum of two roots of the equation
4x4 - 24a-3 + 3 ix2 + 6.x: - 8 = 0
is zero, find all the roots of the equation.
3. (a) Show that function/defined as f(x) = x when 0 <* c--
= 1 when x =
' ' 2 ,
= I- x when < * < 1 -
: V'""' ' l .
is discontinuous at jc = . Examine the type of
discontinuity.
(b) If x = (0 - sin 0), y = a (1 cos 0),
find at 0 = tl
. dx ' f V.. f'.
, use Eulers theorem to
du du 1 . _ prove x + y = sin zu
If u = tarr1
dx xy 4
4. (a) Show that the pedal equation of the curve
X= ae (sin 0 - cos 0) \
y = aee (sin 0+ cos 0) , . is r -
(b)
on the.curve:
- y2 - lx2 + 4y + 15x- 13 = 0
(c) Trace the curve :
; y2 (a2 + x2) = x2 (a2 -jt2)
5. (a) Explain why Roll * s Theorem is not applicable to the function -
f(x)= 1 -x2/3\n [-1, 1]
(b) Find the values of a and b such that lim x (1 + a cos x) ~ b sin x _
.*->0 . ; jP
(c) Show that .
; x5 - 5'4-Sx3 - 1;
has a maximum value when jc = 1, a minimum A value when jc = 3 and neither when jc = 0.
6. (a) Evaluate f - * dX 7.....>
q a cos x b sin x
(b) Find the surface of the solid generated by revolving the arc of the parabola )>2 = Aax bounded by its latus rectum about x-axis.
(c) Prove that the volume of the solid generated by the
- ' ' a3 ] """"
revolution of the curve y2 = ,: about its
--v' . d X ;
- ' _2. 3
. n a
asymptote is -
934 5 2000
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