Download PTU B.Sc-Hons 2020 March 1st Sem 77312 Calculas I Question Paper

Download PTU (I.K. Gujral Punjab Technical University Jalandhar (IKGPTU) B.Sc Hons (Bachelor of Science Honours) 2020 March Previous Question Papers

1 | M-77312 (S1)-2338

Roll No. Total No. of Pages : 03
Total No. of Questions : 09
Bachelor of Science - Honours (Mathematics) (Sem.?1)
CALCULAS-I
Subject Code : UC-BSHM-101-19
M.Code : 77312
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks
each.
2. SECTION - B & C. have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.

SECTION-A
l. Solve the following :
a) Find the l.u.b. and g.l.b., if they exist for the set
2 1
;| 4 | 2
5
x
A x
x
? ? ?
? ? ?
? ?
?
? ?
.
b) Define the greatest integer function. Also write its domain and range.
c) Prove that (sinh ) cosh
d
x x
dx
? .
d) Differentiate cos
?1
(2x
2
?1) with respect to x if 0 < x < 1.
e) Discuss the applicability of Rolle?s Theorem for the function f (x) = | x | in the interval
[?3, 3].
f) Evaluate
1
log
lim
x
x
x x
?
?

g) Show that y = x + a is the only asymptote of the curve x
2
(x ? y) + ay
2
= 0.
h) Find the nth derivative of
1
( 2)( 3) x x ? ?
.
i) Using ? ? ? definition, prove that is continuous f (x) = 3x + 2 at x = 2.
j) State Cauchy?s Mean Value theorem.
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1 | M-77312 (S1)-2338

Roll No. Total No. of Pages : 03
Total No. of Questions : 09
Bachelor of Science - Honours (Mathematics) (Sem.?1)
CALCULAS-I
Subject Code : UC-BSHM-101-19
M.Code : 77312
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks
each.
2. SECTION - B & C. have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.

SECTION-A
l. Solve the following :
a) Find the l.u.b. and g.l.b., if they exist for the set
2 1
;| 4 | 2
5
x
A x
x
? ? ?
? ? ?
? ?
?
? ?
.
b) Define the greatest integer function. Also write its domain and range.
c) Prove that (sinh ) cosh
d
x x
dx
? .
d) Differentiate cos
?1
(2x
2
?1) with respect to x if 0 < x < 1.
e) Discuss the applicability of Rolle?s Theorem for the function f (x) = | x | in the interval
[?3, 3].
f) Evaluate
1
log
lim
x
x
x x
?
?

g) Show that y = x + a is the only asymptote of the curve x
2
(x ? y) + ay
2
= 0.
h) Find the nth derivative of
1
( 2)( 3) x x ? ?
.
i) Using ? ? ? definition, prove that is continuous f (x) = 3x + 2 at x = 2.
j) State Cauchy?s Mean Value theorem.
2 | M-77312 (S1)-2338

SECTION-B
2. a) State and prove Archimedean property of real numbers.
b) Express the function ( )
1
x
h x
x
?
?
as a composite of two ?simpler? functions, and
state necessary conditions on their domains.
3. a) Prove that the function f (x)
1
x
? is continuous in (0, 1) but is not uniformly
continuous.
b) Find all the asymptotes of the following curve :
x
3
? 4x
2
y + 5xy
2
?

2y
3
+ 3x
2
? 4xy + 2y
2
? 3x + 2y ? 1 = 0
4. a) If f (x) =
2
2
2
x a ab ac
ab x b bc
ac bc x c
?
?
?
, find f
/
(x).
b) If sin y = x sin (a + y), prove that
2
sin ( )
sin
dy a y
dx a
?
? .
5. a) If y = log
? ?
2 2
x x a ? ? , show that
? ?
2
2 2
2
0
d y dy
x a x
dx dx
? ? ? .
b) Find the derivative of x
tanh x
.

SECTION-C
6. a) Find the interval of concave upwards for the curve y = (cos x + sin x) e
x
in (0, 2 ?).
b) Show that the curve x = log
y
x
? ?
? ?
? ?
has a point of inflexion at (?2, ?2e
?2
).
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1 | M-77312 (S1)-2338

Roll No. Total No. of Pages : 03
Total No. of Questions : 09
Bachelor of Science - Honours (Mathematics) (Sem.?1)
CALCULAS-I
Subject Code : UC-BSHM-101-19
M.Code : 77312
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks
each.
2. SECTION - B & C. have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.

SECTION-A
l. Solve the following :
a) Find the l.u.b. and g.l.b., if they exist for the set
2 1
;| 4 | 2
5
x
A x
x
? ? ?
? ? ?
? ?
?
? ?
.
b) Define the greatest integer function. Also write its domain and range.
c) Prove that (sinh ) cosh
d
x x
dx
? .
d) Differentiate cos
?1
(2x
2
?1) with respect to x if 0 < x < 1.
e) Discuss the applicability of Rolle?s Theorem for the function f (x) = | x | in the interval
[?3, 3].
f) Evaluate
1
log
lim
x
x
x x
?
?

g) Show that y = x + a is the only asymptote of the curve x
2
(x ? y) + ay
2
= 0.
h) Find the nth derivative of
1
( 2)( 3) x x ? ?
.
i) Using ? ? ? definition, prove that is continuous f (x) = 3x + 2 at x = 2.
j) State Cauchy?s Mean Value theorem.
2 | M-77312 (S1)-2338

SECTION-B
2. a) State and prove Archimedean property of real numbers.
b) Express the function ( )
1
x
h x
x
?
?
as a composite of two ?simpler? functions, and
state necessary conditions on their domains.
3. a) Prove that the function f (x)
1
x
? is continuous in (0, 1) but is not uniformly
continuous.
b) Find all the asymptotes of the following curve :
x
3
? 4x
2
y + 5xy
2
?

2y
3
+ 3x
2
? 4xy + 2y
2
? 3x + 2y ? 1 = 0
4. a) If f (x) =
2
2
2
x a ab ac
ab x b bc
ac bc x c
?
?
?
, find f
/
(x).
b) If sin y = x sin (a + y), prove that
2
sin ( )
sin
dy a y
dx a
?
? .
5. a) If y = log
? ?
2 2
x x a ? ? , show that
? ?
2
2 2
2
0
d y dy
x a x
dx dx
? ? ? .
b) Find the derivative of x
tanh x
.

SECTION-C
6. a) Find the interval of concave upwards for the curve y = (cos x + sin x) e
x
in (0, 2 ?).
b) Show that the curve x = log
y
x
? ?
? ?
? ?
has a point of inflexion at (?2, ?2e
?2
).
3 | M-77312 (S1)-2338

7. a) Find the values of a and b, so that the
3
0
(1 cos ) sin
lim
x
x a x b x
x
?
? ?
exists and is equal to
1
3
.
b) Use Lagrange?s Mean Value theorem to prove that
1
2
sin
1
x
x x
x
?
? ?
?
for 0 < x < 1.
8. a) Find the nth derivative of sin x sin 2x sin 3x.
b) If y = (sin
?1
x)
2
, find y
n
(0)
9. a) If f (x) = tan x, then prove that
n
C
0
f
n
(0) ?
n
C
2
f
n?2
(0) +
n
C
4
f
n?4
(0) ? ?. = sin
2
n ?
.
b) Use Maclaurin?s Theorem (with Lagrange?s form of remainder) to expand sin x.








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