University of Delhi 2009 B.A Mathematics set-III, ., - Question Paper
BA ProgJI 1-1
Your Roll No......
MATHEMATICSPaper I (Algebra and Calculus)
(NC : Admissions of 2006 onwards)
Time : 3 Hours Maximum Marks : 100
(Write your Roll No. on the top immediately on receipt of this question paper.)
All Sections are compulsory and have equal marks. Attempt any two parts from each Section.
Section I
1. (a) Prove that the following set of vectors in R3 S = K2, 0, -1), (5, 1, 0); (0, 1, 3)1 is linearly independent.
A -
compute A3, A4 and A2.
Or
(a) Verify that the matrix
(2 1 1 1 2 1
A =
1 1 2
satisfies its characteristic qtiatioh arid hence
find A'1.
(6) Solve by matrix method the following system of equations :
x + y + z = 7 x + 2y + 3z = 16 x + 3y + 4z = 22.
Section II
(a) Show that :
(1 + cos a + i sin a)p + (1 + cos a i sin a)p
(6) Find a necessary condition for the roots of the equation :
atffi + 3ajjc2 + Zac + a3 = 0
to be in G.P. lev Geometric Progression.
, Or '
Prove that
(a) 32 sin4 0 cos2 0 = cos 60 - 2 cos 40 - cos 20 + 2.
(b) Solve the equation
16jc4 64x3 + 56x2 + 16a: 15= 0.
Given that the roots are in Arithmetic Progression.
3. (a) Discuss the derivability of the function f defined
' . ' by
x for x < 1
2 - x for 1 < * S 2
/(*)=
-2 + 3* - x2 for x >2
at x - 1, 2.
(6) If u - show that
d - - = (l + 3xyz + x2y222)
Or
(a) Prove that the function
= x sin , if x *0
x
, if x ~ 0
0
is continuous at jc = 0 but not derivable at x - 0.
(b) If * s o sin3 6, y - b cos 6,
Section IV
(o) The tangents at two points P, Q of the cycloid :
x = a (0 - sin 0), y = a (1 - cos 0)
are at right angles. Show that if Px and P2 be the radii of curvatures at these points then :
Pf + if = 16a2-
(6) Find the asymptotes of the curve :
(#2 - jy2) (* + 2y) + + y2 j + x + y 0.
Or ;
(a) Trace the curve :
y2 (2a - x) - x3.
(b) Show that the pedal equation of the ellipse :
is
1 11 r2
2 ~ 2 + *2 2t2 p a b a b
Sectkm V
5. (a) State the prove Rolles Theorem. Show that there is no real number k for which the equation x2 - Zk +k = 0 Has two distinct roots iii [0, 1].
(b) Obtain Maclaurins series expansion of sin x for all x e R.
Or
(a) Separate the interval in which the function ;
fix) - 2x3 - I5x2 + 36* + 1 is increasing or decreasiiig.
(b) Evaluate any two of the following :
(j) lim (cot *)1/log *
1 + sin x - cos x + logfl - *)
(U) lim--5v
x -*Q x tan x
lim (l - sin x) tan x KUl} x k/2 7
Section VI
Ji/2
Im> = J sinm x coS*1 x dx
0
prove that
m + l Am,n-2;
where m and n are positive integers.
(6) Find the area enclosed by the ellipse :
*2 2
-+ = 1
a2 b2 *
Or
(a) Evaluate :
x dx
a2 cos2 x + b2 sin2 x '
(6) Find the volume of the solid obtained by the revolution of the loop of the curve y2(a + x) = x2(a - x) about x-axis.
1144 7 2,500
BA ProgJI 1-1
Your Roll No......
MATHEMATICSPaper I (Algebra and Calculus)
(NC : Admissions of 2006 onwards)
Time : 3 Hours Maximum Marks : 100
(Write your Roll No. on the top immediately on receipt of this question paper.)
All Sections are compulsory and have equal marks. Attempt any two parts from each Section.
Section I
1. (a) Prove that the following set of vectors in R3 S = K2, 0, -1), (5, 1, 0); (0, 1, 3)1 is linearly independent.
A -
compute A3, A4 and A2.
Or
(a) Verify that the matrix
(2 1 1 1 2 1
1 1 2
satisfies its characteristic qtiatioh arid hence
find A'1.
(6) Solve by matrix method the following system of equations :
x + y + z = 7 x + 2y + 3z = 16 x + 3y + 4z = 22.
Section II
(a) Show that :
(1 + cos a + i sin a)p + (1 + cos a i sin a)p
(6) Find a necessary condition for the roots of the equation :
atffi + 3ajjc2 + Zac + a3 = 0
to be in G.P. lev Geometric Progression.
, Or '
Prove that
(a) 32 sin4 0 cos2 0 = cos 60 - 2 cos 40 - cos 20 + 2.
(b) Solve the equation
16jc4 64x3 + 56x2 + 16a: 15= 0.
Given that the roots are in Arithmetic Progression.
3. (a) Discuss the derivability of the function f defined
' . ' by
x for x < 1
2 - x for 1 < * S 2
/(*)=
-2 + 3* - x2 for x >2
at x - 1, 2.
(6) If u - show that
d - - = (l + 3ryz + x2y222)
Or
(a) Prove that the function
= x sin , if x *0 x
, if x ~ 0
0
is continuous at jc = 0 but not derivable at x - 0.
(b) If * s o sin3 6, y - b cos 6,
Section IV
(o) The tangents at two points P, Q of the cycloid :
x = a (0 - sin 0), y = a (1 - cos 0)
are at right angles. Show that if Px and P2 be the radii of curvatures at these points then :
Pf + if = 16a2-
(6) Find the asymptotes of the curve :
(#2 -,y2)(* + 2y) + + y2 j + x + y 0.
Or ;
(a) Trace the curve :
y2 (2a - x) - x3.
(b) Show that the pedal equation of the ellipse :
2 2 2 V
is
1 1 1 r2
2 ~ 2 + *2 2t2 p a b a b
Sectkm V
5. (a) State the prove Rolles Theorem. Show that there is no real number k for which the equation x2 - Zk +k = 0 Has two distinct roots iii [0, 1].
(b) Obtain Maclaurins series expansion of sin x for all x e R.
Or
(a) Separate the interval in which the function ;
fix) - 2x3 - I5x2 + 36* + 1 is increasing or decreasing.
(b) Evaluate any two of the following :
(j) lim (cot *)1/log *
1 + sin x - cos x + logfl - *)
(U) lim--5v
x -*Q x tan x
lim (l - sin x) tan x KUl} x k/2 7
Section VI
Ji/2
Im> = J sinm x coS*1 x dx 0
prove that
m + l Am,n-2;
where m and n are positive integers.
(6) Find the area enclosed by the ellipse :
*2 2
-+ = 1
a2 +b2 *
Or
(a) Evaluate :
x dx
a2 cos2 x + b2 sin2 x '
(6) Find the volume of the solid obtained by the revolution of the loop of the curve y2(a + x) = x2(a - x) about x-axis.
1144 7 2,500
BA ProgJI 1-1
Your Roll No......
MATHEMATICSPaper I (Algebra and Calculus)
(NC : Admissions of 2006 onwards)
Time : 3 Hours Maximum Marks : 100
(Write your Roll No. on the top immediately on receipt of this question paper.)
All Sections are compulsory and have equal marks. Attempt any two parts from each Section.
Section I
1. (a) Prove that the following set of vectors in R3 S = K2, 0, -1), (5, 1, 0); (0, 1, 3)1 is linearly independent.
A -
compute A3, A4 and A2.
Or
(a) Verify that the matrix
(2 1 1 1 2 1
1 1 2
satisfies its characteristic qtiatioh arid hence
find A'1.
(6) Solve by matrix method the following system of equations :
x + y + z = 7 x + 2y + 3z = 16 x + 3y + 4z = 22.
Section II
(a) Show that :
(1 + cos a + i sin a)p + (1 + cos a i sin a)p
(6) Find a necessary condition for the roots of the equation :
atffi + 3ajjc2 + Zac + a3 = 0
to be in G.P. lev Geometric Progression.
, Or '
Prove that
(a) 32 sin4 0 cos2 0 = cos 60 - 2 cos 40 - cos 20 + 2.
(b) Solve the equation
16jc4 64x3 + 56x2 + 16a: 15= 0.
Given that the roots are in Arithmetic Progression.
3. (a) Discuss the derivability of the function f defined
' . ' by
x for x < 1
2 - x for 1 < * S 2
/(*)=
-2 + 3* - x2 for x >2
at x - 1, 2.
(6) If u - show that
d - - = (l + 3ryz + x2y222)
Or
(a) Prove that the function
= x sin , if x *0 x
, if x ~ 0
0
is continuous at jc = 0 but not derivable at x - 0.
(b) If * s o sin3 6, y - b cos 6,
Section IV
(o) The tangents at two points P, Q of the cycloid :
x = a (0 - sin 0), y = a (1 - cos 0)
are at right angles. Show that if Px and P2 be the radii of curvatures at these points then :
Pf + if = 16a2-
(6) Find the asymptotes of the curve :
(#2 -,y2)(* + 2y) + + y2 j + x + y 0.
Or ;
(a) Trace the curve :
y2 (2a - x) - x3.
(b) Show that the pedal equation of the ellipse :
2 2 2 V
is
1 1 1 r2
2 ~ 2 + *2 2t2 p a b a b
Sectkm V
5. (a) State the prove Rolles Theorem. Show that there is no real number k for which the equation x2 - Zk +k = 0 Has two distinct roots iii [0, 1].
(b) Obtain Maclaurins series expansion of sin x for all x e R.
Or
(a) Separate the interval in which the function ;
fix) - 2x3 - I5x2 + 36* + 1 is increasing or decreasing.
(b) Evaluate any two of the following :
(j) lim (cot *)1/log *
1 + sin x - cos x + logfl - *)
(U) lim--5v
x -*Q x tan x
lim (l - sin x) tan x KUl} x k/2 7
Section VI
Ji/2
Im> = J sinm x coS*1 x dx 0
prove that
m + l Am,n-2;
where m and n are positive integers.
(6) Find the area enclosed by the ellipse :
*2 2
-+ = 1
a2 +b2 *
Or
(a) Evaluate :
x dx
a2 cos2 x + b2 sin2 x '
(6) Find the volume of the solid obtained by the revolution of the loop of the curve y2(a + x) = x2(a - x) about x-axis.
1144 7 2,500
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