INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous)
Dundigal, Hyderabad - 500 043
AERONAUTICAL ENGINEERING
QUESTION BANK
Course Title
MATHEMATICAL TRANSFORM TECHNIQUES
Course Code
AHSC07
Program
B.Tech
Semester
II
Course Type
Foundation
Regulation
IARE - UG20
Course Structure
Course Coordinator
AE / ECE / EEE / ME / CE
Lecture
3
Theory
Tutorials
1
Credits
4
Practical
Laboratory
Credits
-
Mr.Satyanarayana.G, Assistant Professor
COURSE OBJECTIVES:
The students will try to learn:
I
The transformation of ordinary differential equations in Laplace field and its
applications.
II
The operation of non-periodic functions by Fourier transforms .
III
The concepts of multiple integration for find integration and volumes of physical
quantities.
IV
The Integration of several functions by transforming the co-ordinate system in scalar
and Vector fields.
COURSE OUTCOMES:
After successful completion of the course, students should be able to:
CO 1
CO 2
CO 3
Explain the properties of Laplace and inverse transform to various
functions such as continuous, piecewise continuous, step, impulsive
and complex variable functions.
Make use of the integral transforms which converts operations of
calculus to algebra in solving linear differential equations
Apply the Fourier transform as a mathematical function that
transforms a signal from the time domain to the frequency domain,
non-periodic function up to infinity
Understand
Apply
Apply
CO 4
CO 5
CO 6
Apply the definite integral calculus to a function of two or more
variables in calculating the area of solid bounded regions
Develop the differential calculus which transforms vector functions,
gradients. Divergence, curl, and integral theorems to different
bounded regions in calculating areas.
Solve Lagrange’s linear equation related to dependent and
independent variables the nonlinear partial differential equation by
the method of Charpit concern to the engineering field
Apply
Apply
Apply
QUESTIN BANK:
Q.No
1
2
3
4
QUESTION
Taxonomy
How does this subsume
the level
CO’s
MODULE I
LAPLACE TRANSFORM
PART A-PROBLEM SOLVING AND CRITICAL THINKING QUESTIONS
Find the inverse Laplace transform
Apply
Learner to recall inverse
CO
Laplace formulae, explain
s+3
their existence and
L( 2
)
s − 10s + 29
properties, and apply
suitable inverse Laplace
formula to the given
function
Find
Apply
Learner to recall Laplace
CO
e−3t sin2t
formulae,
explain
their
)
L(
t
existence and properties,
and apply suitable inverse
Laplace formula to the
given function
Find
Apply
Learner to recall Laplace
CO
Z t
1 − e− t
formulae,
explain
their
L{
dt}
t
existence and properties,
0
and apply suitable inverse
Laplace formula to the
given function
Find Laplace of
Apply
Learner to recall Laplace
CO
formulae, explain their
(a)e−3t cosh4tsin3t(b)(t + 1)2
existence and properties,
and apply suitable inverse
Laplace formula to the
given function
Page 2
2
2
2
2
5
Find the Laplace Transform of the
following functions
Apply
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply derivative
theorem to the given
function
CO 2
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply derivative
theorem to the given
function
Apply convolution theorem to
Apply
Learner to recall Laplace
evaluate
formulae, explain their
existence and properties,
1
and apply derivative
L−1
S(S 2 + 4)2
theorem to the given
function
Find the inverse Laplace transform
Apply
Learner to recall inverse
Laplace formulae, explain
s2 + s − 2
their existence and
s(s + 3)(s − 2)
properties, and apply
suitable inverse Laplace
formula to the given
function
Find the inverse Laplace transform
Apply
Learner to recall inverse
Laplace formulae, explain
s2 + 2s − 4
their existence and
(s2 + 9)(s − 5)
properties, and apply
suitable inverse Laplace
formula to the given
function
Find the L{f (t)} for the function
Apply
Learner to recall inverse
sint
−5t
(a) t (b)e sint
Laplace formulae, explain
their existence and
properties, and apply
suitable inverse Laplace
formula to the given
function
PART-B LONG ANSWER QUESTIONS
CO 2
a)eat
b)cosat
c)tsinat
6
7
8
9
10
Apply convolution theorem to
evaluate
S2
−1
L
(S 2 + a2 )(S 2 + b2 )
Apply
Page 3
CO 2
CO 2
CO 2
CO 2
1
Find the Laplace transform
e−t sint
t
Apply
2
Find the Laplace transform sin3 2t
Apply
3
Find the
Laplace transform 
 1
0<t<2 




 2
2<t<4 
f (t) =

3
4<t<6 






0
t>6
Apply
4
Find the Laplace transform of
e−t cos2 t
Apply
5
Find the Laplace transform of
t2 cos 3t
Apply
6
Find the Laplace transform of
Apply
sin 3t cos t
t
7
Find
Laplace transform of g)(t) =
(
cos(t − π3 )
if t > π3
π
0
t< 3
Apply
Page 4
Learner to recall inverse
Laplace formulae, explain
their existence and
properties, and apply
suitable inverse Laplace
formula to the given
function
Learner to recall inverse
Laplace formulae, explain
their existence and
properties, and apply
suitable inverse Laplace
formula to the given
function
Learner to recall inverse
Laplace formulae, explain
their existence and
properties, and apply
suitable inverse Laplace
formula to the given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
CO 2
CO 1
CO 1
CO 2
CO 2
CO 2
CO 2
8
Find the Laplace transform of
Rt 2 −4t
t e sin 2t dt
Apply
0
9
Find the Laplace transform of
et−3 u (t − 3)
Apply
10
Find the Laplace transform of
cos t cos 2t cos 3t
Apply
11
Find L−1
12
Find the inverse Laplace transform
2S 2 −6S+5
of S 3 −6S
2 +11S−6
Apply
13
Find the inverse Laplace transform
e−2s
of s2 +4s+5
Apply
14
Find the inverse Laplace transform
s
of (s+1)(s
2 +9)
Apply
s+1
(s2 +2s+2)
Apply
Page 5
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
CO 2
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
CO 2
CO 2
CO 1
CO 2
CO 2
CO 2
15
16
17
18
19
20
1
2
3
Find the
Laplace transform
2 inverse
s +4
of log s2 +9
Apply
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
Find the inverse Laplace transform
Apply
Learner to recall Laplace
2 +2s−4
of (s2s+16)(s−3)
formulae, explain their
existence and properties,
and apply first shifting
theorem to the given
function
By using Laplace transform
Apply
Learner to recall inverse
method
Laplace formulae, explain
(D2 + 2D + 5)t = e−t sin t, y(0) =
their existence and
′
0, y (0) = 1
properties, and apply
Laplace formula to linear
differential equation
By using Laplace transform
Apply
Learner to recall inverse
method
Laplace formulae, explain
y ′′ + 9y = cos 2t, y(0) = 1, y ′ (0) = 1
their existence and
y( π2 ) = −1
properties, and apply
Laplace formula to linear
differential equation
By using Laplace transform
Apply
Learner to recall inverse
method,solve
Laplace formulae, explain
′′′
′′
′
y − 2y + 5y = 0,
their existence and
y(0) = 1, y ′ (0) = 0, y ′′ (0) = 1
properties, and apply
Laplace formula to linear
differential equation
By using Laplace transform
Apply
Learner to recall inverse
method,solve
Laplace formulae, explain
their existence and
(D3 − D2 + 4D − 4)t = 68ex sin 2x,
2
properties, and apply
atx = 0,y = 1, Dy = −19, D y = −37
Laplace formula to linear
differential equation
PART-C SHORT ANSWER QUESTIONS
Define Laplace Transform
Remember —
Define Laplace Transform, and
Remember —
write the sufficient conditions for
the existence of Laplace Transform
State Laplace transform of
Remember —
derivatives and integrals
Page 6
CO 2
CO 2
CO 2
CO 2
CO 2
CO 2
CO 1
CO 2
CO 2
4
6
State Unit step function of Laplace
transform
State Laplace transform of
multiplied by t and division by t
Find L{sin3 t}
7
Find L[3 cos 3t. cos 4t]
Apply
8
Find the Laplace transform of
f (t) = t3
Apply
9
Find Laplace transform of
Apply
5
Remember
—
CO 1
Remember
—
CO 1
CO 2
Remember
Learner to recall Laplace
formulaeand explain
property of Laplace
Transformation in linearity
Learner to recall Laplace
formulae, explain their
existence and properties,
andsolve for the Laplace
transformation of given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
andsolve for the Laplace
transformation of given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
andsolve for the Laplace
transformation of given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
andsolve for the Laplace
transformation of given
function
—
Remember
—
CO 2
Learner to recall inverse
Laplace formulae, explain
their existence, properties,
and apply suitable Laplace
inverse formula to the given
function
CO 1
Apply
e−t (3 sin 2t − 5cosh2t)
10
Find L (t sin 3t cos 2t)
11
State First and second Shifting
theorem of inverse Laplace
transform
State inverse Laplace transform of
derivatives and integrals
Find the inverse Laplace transform
of (s−1)1 2 +4
12
13
Apply
Apply
Page 7
CO 1
CO 1
CO 2
CO 2
CO 2
14
Find the inverse Laplace transform
s
of s2 −a
2
Apply
15
Find the inverse Laplace transform
−2s
of se2 +16
Apply
16
Find
Laplace transform
o
n the inverse
2s−5
of 4s2 +25
Apply
17
Find L
t
R
e−t cos tdt
Learner to recall inverse
Laplace formulae, explain
their existence, properties,
and apply suitable Laplace
inverse formula to the given
function
Learner to recall inverse
Laplace formulae, explain
their existence, properties,
and apply suitable Laplace
inverse formula to the given
function
Learner to recall inverse
Laplace formulae, explain
their existence, properties,
and apply suitable Laplace
inverse formula to the given
function
CO 1
Learner to recall inverse
Laplace formulae, explain
their existence, properties,
and apply suitable Laplace
inverse formula to the given
function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply change of scale
property of Laplace to the
given function
Learner to recall Laplace
formulae, explain their
existence and properties,
and apply initial and final
value theorems of Laplace
to the given function
—–
CO 2
CO 2
CO 1
Apply
0
18
FindL e−3t sinh 3t
Apply
19
Find the Laplace of e−t (t + 1)2
Apply
20
State change of scale property of
Laplace Transforms and inverse
Laplace transform
Remember
CO 2
CO 2
CO 2
MODULE II
FOURIER TRANSFORMS
PART A-PROBLEM SOLVING AND CRITICAL THINKING QUESTIONS
Page 8
1
Find the Fourier cosine transform
of the function
( f(x) defined
cos x, 0 < x < a
byf (x) =
0,
x≥a
Apply
Learner to recall Orthogonal
transform, explain Fourier
Cosine integral and apply to
the given function
CO 3
2
Find the Fourier sine transform of
the function
( f(x) defined
sin x, 0 < x < a
byf (x) =
0,
x≥a
Apply
Learner to recall Orthogonal
transform, explain Fourier
Cosine integral and apply to
the given function
CO 3
3
Find the Fourier sine and cosine
transform of the function f(x)
defined by 3e−7x + 7e−3x
Apply
CO 3
4
Find the Fourier sine and cosine
transform of the function f(x)
defined
by f (x) =


f or 0 < x < 1
 x,
2 − x, f or 1 < x < 2

 0,
f or
x>2
Find the Fourier cosine transform
of the function
f(x) defined by


x
−
2,
0<x<1

f (x) =
2 + x, 1 < x < 2

 0,
x>2
Find the inverse finite sine
transform if FS (n) =
(1 − cosnπ)/(n2 π 2 )where0 < x < π
Apply
Learner to recall Orthogonal
transform, explain Fourier
Cosine integral and apply to
the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine integral and apply to
the given function
Apply
Learner to recall Orthogonal
transform, explain Fourier
Cosine integral and apply to
the given function
CO 3
Apply
CO 3
7
Find the inverse finite cosine
transform f(x),
cos2nπ/3
ifFc (n) = (2nπ+1)
2 where0 < x < 4
Apply
8
Using Fourier integral show that
R∞ 2
cosλxdλ
e−x cos x = π2 λλ2 +2
+4
Apply
9
Find the finite Fourier sine and
cosine transforms
off (x) = x (π − x) in (0, π) .
Apply
10
Find the inverse finite cosine
transform f(x),
iff (x) = cos axin (0, l) and (0, π)
Apply
Learner to recall Orthogonal
transform, explain Fourier
Cosine integral and apply to
the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine integral and apply to
the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine integral and apply to
the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine integral and apply to
the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine integral and apply to
the given function
5
6
0
Page 9
CO 3
CO 3
CO 3
CO 3
CO 3
1
2
3
PART-B LONG ANSWER QUESTIONS
Find the Fourier transform
of f(x)
Apply
Learner to recall Orthogonal
(
transform, explain Infinite
1, |x| < a
defined by f (x) =
Fourier transform and apply
0, |x| > a
to the given function
and
R ∞ sin ap. cos px
R ∞ sin p
dp
p dp. and −∞
p
0
Find the Fourier transform of f(x)
Apply
Learner to recall Orthogonal
defined by
transform, explain Infinite
(
Fourier transform and apply
1 − x2 , |x| ≤ 1
f (x) =
Hence
to the given function
0,
|x| > 1
R ∞ x cos x−sin x
x
(i) 0
cos 2 dx
x3
R∞
evaluate
x
(ii) 0 x cos x−sin
dx
x3
Find the Fourier Transform of f(x)
Apply
Learner to recall Orthogonal
defined by
transform, explain Infinite
−x2
Fourier transform and apply
f (x) = e 2 , −∞ < x < ∞ or,
to the given function
Show that the Fourier Transform of
CO 3
CO 3
CO 3
−x2
4
e 2 is reciprocal.
Find Fourier cosine and sine
transforms of f (x) = e−ax and
hence deduce the inversion formula
(or)
deduce the integrals
R∞
R ∞ p sin px
px
i. 0 acos
2 +p2 dp ii.
0 a2 +p2 dp
Apply
Learner to recall Orthogonal
transform, explain Infinite
Fourier transform and apply
to the given function
CO 3
Learner to recall Orthogonal
transform, explain Fourier
Sine transform and apply to
the given function.
Learner to recall Orthogonal
transform, explain Fourier
Sine transform and apply to
the given function.
CO 3
Learner to recall Orthogonal
transform, explain Fourier
Sine transform and apply to
the given function.
Learner to recall Orthogonal
transform, explain Fourier
Sine transform and apply to
the given function.
CO 3
5
Find the Fourier sine Transform of
e−|x| andR hence
∞ sin mx
evaluate 0 x 1+x
2 dx
Apply
6
Find the Fourier cosine Transform
of
Apply
(a) e−ax cos ax (b) e−ax sin ax
7
Find the Fourier sine and cosine
transform of xe−ax
Apply
8
Find the Fourier sine transform of
x
and Fourier cosine transform
a2 +x2
1
of a2 +x2
Apply
Page 10
CO 3
CO 3
9
Find the Fourier sine and cosine
transform and invesre fourier
transform of f (x) = e−ax
Apply
10
Find the finite Fourier sine and
cosine transform of f(x) defined by
2
f (x) = 1 − πx where0 < x < π
Apply
11
Find the finite Fourier sine and
cosine transform of f(x) defined by
f (x) = sin ax
in[0, π]
Apply
12
Find the finite Fourier sine
transform
( of f(x) defined by
x, 0 ≤ x ≤ π/2
f (x) is
π − x, π/2 ≤ x ≤ π
Apply
13
Find theFourier Sine transform of

 1for 0 < x < a
f (x) =
0 for x > a


Apply
14
Find the inverse Fourier transform
of f(x) defined by F (p) = e−|p|y
Apply
15
Find the Fourier transform of f(x)
defined (
by
a2 − x2 if |x| < a
f (x) =
0 if |x| > a
hence show that
R∞ sin x−cos x
dx = π4
x3
Apply
16
Find the finite Fourier sine and
cosine transforms of f (x) = cos ax
in [0, π].
Apply
Learner to recall Orthogonal
transform, explain Fourier
Sine transform and apply to
the given function.
Learner to recall Orthogonal
transform, explain Fourier
Sine and Cosine transform
and apply to the given
function
Learner to recall Orthogonal
transform, explain Fourier
Sine and Cosine transform
and apply to the given
function
Learner to recall Orthogonal
transform, explain Fourier
Sine and Cosine transform
and apply to the given
function
Learner to recall Orthogonal
transform, explain Fourier
Sine and Cosine transform
and apply to the given
function
Learner to recall Orthogonal
transform, explain Fourier
Sine and Cosine transform
and apply to the given
function
Learner to recall Orthogonal
transform, explain Fourier
Sine and Cosine transform
and apply to the given
function
CO 3
Learner to recall Orthogonal
transform, explain Fourier
Sine and Cosine transform
and apply to the given
function
CO 3
CO 3
CO 3
CO 3
CO 3
CO 3
CO 3
0
Page 11
17
18
19
20
1
2
3
4
5
6
Find the inverse Fourier cosine
−ap
transform f(x) of Fc (p) = e p
Apply
Learner to recall Orthogonal
transform, explain Fourier
Sine and Cosine transform
and apply to the given
function
Using Fourier integral show that
Apply
Learner to recall Orthogonal
∞
R
transform, explain Fourier
cos λx
e−ax = 2a
dλ
π
λ2 +a2
Sine and Cosine transform
0
( a > 0, x > 0)
and apply to the given
function
Using Fourier integral show
Apply
Learner to recall Orthogonal
thate−ax − e−bx =
transform, explain Fourier
2(b2 −a2 ) R ∞
λ sin λx
Sine and Cosine transform
π
0 (λ2 +a2 )(λ2 +b2 ) dλ, a >
and apply to the given
0, b > 0
function
Using
Fourier integral show that
Apply
Learner to recall Orthogonal
R ∞ 1−cos
λπ
transform, explain Fourier
.
sin
λx
dλ
λ
0
(
π
Sine and Cosine transform
2 if 0 < x < π
=
and apply to the given
0, if x > π
function
PART-C SHORT ANSWER QUESTIONS
Write the Fourier sine integral and
Remember cosine integral.
Write the infinite Fourier sine
Apply
Learner to recall Orthogonal
transform of f(x)
transform, explain Fourier
sine transform and apply to
the given function
Write the infinite Fourier transform Remember of f(x)
CO 3
Write the properties of Fourier
transform of f(x)
Find the Fourier sine transform of
f(x)?
Apply
-
CO 3
Apply
CO 3
Find the Fourier cosine transform
of f (x) = 2e−5x + 5e−2x
Apply
Learner to recall Orthogonal
transform, explain Fourier
sine transform, and apply to
the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
Page 12
CO 3
CO 3
CO 3
CO 3
CO 3
CO 3
CO 3
7
What is the value of FC {e−at } ?
8
9
10
State Fourier integral theorem.
Define Fourier transform.
Find the finite Fourier cosine
transform of
f (X) = 1f or0 < x < π
11
Find the Fourier cosine transform
of f (x) = 3e−2x + 2e−3x
Apply
12
State and prove Linear property of
Fourier Transform
Apply
13
State and prove change of scale
property
Apply
14
State and prove Shifting Property
Apply
15
State and prove Modulation
Theorem
Apply
16
solveFc {f (x) cos ax} =
1
2 [FC (p + a) − FC (p − a)]
Apply
Apply
Remember
Remember
Apply
Page 13
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and apply
to the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
CO 3
CO 3
CO 3
CO 3
CO 3
CO 3
CO 3
CO 3
CO 3
CO 3
17
Find the Fourier
Cosine transform



1for0 < x < a 






of f (x) = 0


forx > a






Apply
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
CO 3
18
Solve FS {f (x) cos ax} =
1
2 [FS (p + a) + FS (p − a)]
Apply
CO 3
19
solveFc {f (x) sin ax} =
1
2 [FS (p + a) − FS (p − a)]
Apply
20
solveFs {f (x) sin ax} =
1
2 [Fc (p − a) − Fc (p + a)]
Apply
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
Learner to recall Orthogonal
transform, explain Fourier
Cosine transform and
Linearity Property and
apply to the given function
1
2
CO 3
CO 3
MODULE III
MULTIPLE INTEGRALS
PART A-PROBLEM SOLVING AND CRITICAL THINKING QUESTIONS
Evaluate the double integral
Apply
Learner to recall basic
CO 4
integral
formulae,
explain
Z 2 Z x+2
double integration and
dydx
apply it to the given
−1 2
Cartesian form function.
Evaluate where R is the region
bounded by the parabolas y 2 = 4x
and x2 = 4y
ZZ
ydxdy
Apply
Learner to recall basic
integral formulae ,explain
double integration and
apply it to the given
function in polar form.
CO 4
Apply
Learner to recall basic
integral formulae, explain
double integration and
apply the double integration
to get the area inCartesian
form.
CO 4
R
3
..
Evaluate
ZZ
x2 dxdy
over the region bounded by
hyperbola xy=4,y=0,x=1,x=4
Page 14
4
Find the area bounded by curves
xy=2,4y = x2 and the line y=4.
Apply
5
Evaluate the double integral
Z 2Z x
e[x+y] dydx
Apply
0
Learner to recall basic
integral formulae, explain
the double integration and
apply it to obtain the area
in Cartesian form.
Learner to recall basic
integral formulae, explain
double integral and apply it
to the given Cartesian form
function.
CO 4
Learner to recall basic
integral formulae, explain
the double integration apply
it to convert Cartesian form
as Polar form.
CO 4
Apply
Learner to recall basic
integral formulae, explain
triple integration and apply
it to obtain the volume in
Cartesian form.
CO 4
Learner to recall basic
integral formulae, explain
and apply the double
integration to obtain the
area in Polar form.
Learner to recall basic
integral formulae, explain
the double integration and
apply it to obtain the area
in Polar form.
CO 4
0
.
6
Evaluate by converting to polar
co-ordinates
Z a Z √a2 +x2 p
y x2 + y 2 dydx
0
7
CIE-II
Apply
0
.
Find the volume of tetrahedron
bounded by the co-ordinate planes
and the plane
x y z
+ + = 1.
a
b
c
8
Using double integral, find area of
the cardioid r = a(1 + cosθ) .
Apply
9
Evaluate the area of
ZZ
r3 dydx
Apply
over the region included between
the circles r=sinθ, r = 4cosθ
Page 15
CO 4
CO 4
10
If R is the region bounded by the
planes x=0, y=0, z=1 and the
cylinder
Apply
x2 + y 2 = 1
,evaluate
Learner to recall basic
integral formulae, explain
the triple integration and
apply the volume
integration to the Cartesian
form of bounded region.
CO 4
ZZZ
xyzdxdydz
R
1
2
3
4
5
6
PART-B LONG ANSWER QUESTIONS
Evaluate
Apply
Learner to recall basic
R 1 R 1−y the double integral
xydxdy.
integral formulae, explain
0 0
triple integration and apply
it to the given Cartesian
form function.
Evaluate the double integrall
Apply
Learner to recall basic
R π R a[1+cosθ] 2
integral formulae, explain
r cosθdrdθ.
0 0
triple integration and apply
it to the given Cartesian
form function.
Evaluate
Apply
Learner to recall basic
R √double
R 1the
x
integral 0 x [x2 + y 2 ]dxdy.
integral formulae, explain
triple integration and apply
it to the given Cartesian
form function.
Evaluate
Apply
Learner to recall basic
R x2double
R 5the
integral 0 0 x[x2 + y 2 ]dxdy.
integral formulae, explain
triple integration and apply
it to the given Cartesian
form function.
Evaluate the double integrall
Apply
Learner to recall basic
R 1 R π2
integral formulae, explain
0 0 rsinθdrdθ.
triple integration and apply
it to the given Cartesian
form function.
Evaluate
Apply
Learner to recall basic
R 1 R 2−x the double integral
xydxdy.
integral formulae, explain
0 x2
triple integration and apply
it to the given Cartesian
form function.
Page 16
CO 4
CO 4
CO 4
CO 4
CO 4
CO 4
7
Evaluate
the double integral
R a R √a2 −y2 2
x ydydx.
0 0
Apply
8
Evaluate
double integral
R log 2 R x the
[x+y]
dydx.
0
0 e
Apply
9
Evaluate
R a R √a2 −x2 p
a2 − x2 − y 2 dxdy.
0 0
Apply
10
RR
Find the value of
xydxdy taken
over the positive quadrant of the
2
2
ellipse xa2 + yb2 = 1.
Apply
CIE-II
Apply
11
Evaluate the double integral using
change
variables
R ∞ R ∞ of−(x
2 +y 2 )
e
dxdy
0
0
12
Find the volume of the tetrahedron
bounded by the plane
y
x
z
a + b + c = 1 and the coordinate
planes by triple integration.
Apply
13
By transforming into polar
RR x2 y2
coordinates Evaluate
dxdy
x2 +y 2
over the annular region between
the circles x2 + y 2 = a2 and
x2 + y 2 = b2 with b >a .
Apply
14
Find the area of the region
bounded by the parabola y 2 = 8ax
and x2 = 8ay.
Apply
Page 17
Learner to recall basic
integral formulae, explain
triple integration and apply
it to the given Cartesian
form function.
Learner to recall basic
integral formulae, explain
triple integration and apply
it to the given Cartesian
form function.
Learner to recall basic
integral formulae, explain
triple integration and apply
it to the given Cartesian
form function.
Learner to recall basic
integral formulae, explain
double integration and
apply it to obtain the area
in Cartesian form.
CO 4
Learner to recall basic
integral formulas, explain
double integration and
apply change of variable to
the given function.
Learner to recall basic
integral formulae, explain
and apply the triple
integration to value the
volume in Cartesian form.
Learner to recall basic
integral formulae, explain
double integration and
apply it to convert
Cartesian form as Polar
form.
Learner to recall basic
integral formulae, explain
double integration and
apply it to value the area in
Cartesian form.
CO 4
CO 4
CO 4
CO 4
CO 4
CO 4
CO 4
15
16
17
18
19
20
1
2
RR 3
Evaluate
r drdθ over the area
included between the circles
r = 2sinθ and r = 4sinθ
Learner to recall basic
integral formulae, explain
the double integration and
apply it to obtain the area
in Polar form.
Using triple integration find the
Apply
Learner to recall basic
volume of the
integral formulae, explain
spherex2 + y 2 + z 2 = a2 .
triple integration and apply
the triple integration to
value the volume in
Cartesian form.
Find the area of the
Apply
Learner to recall basic
cardioidr = a(1–cosθ).
integral formulae, explain
double integration and
apply it to value the area in
Polar form.
Find the area of the region
Apply
Learner to recall basic
3
bounded by the curves y = x and
integral formulae, explain
y = x.
double integration and
apply it to obtain the area
in Cartesian form.
RRR
Evaluate V dxdydz where v is
Apply
Learner to recall basic
the finite region of space formed by
integral formulae, explain
the planes x=0, y=0, z=0 and
double integration and
2x+3y+4z=12.
apply it to obtain the area
in Cartesian form.
Find the area bounded by curves
Apply
Learner to recall basic
xy = 2, 4y = x2 and the line y=4.
integral formulae, explain
double integration and
apply it to obtain the area
in Cartesian form.
PART-C SHORT ANSWER QUESTIONS
Evaluate the double integral
Apply
Learner to recall basic
integral formulae, explain
Z 2Z x
double integration and
ydxdy
apply it to the given
0
2
Cartesian form function.
CO 4
Evaluate the double integral
CO 4
Z
π
Z
Apply
Apply
acosθ
rdrdθ
0
2
..
Page 18
Learner to recall basic
integral formulae ,explain
double integration and
apply it to the given
function in polar form.
CO 4
CO 4
CO 4
CO 4
CO 4
CO 4
3
Evaluate the double integral
Z 3Z 1
xy(x + y)dxdy
0
4
.
State tFind the value of double
integral l
Z 2Z 3
xy 2 dxdy
.
Find the value of triple integral
Z 1Z 2Z 3
dxdydz
−2
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
CO 4
Apply
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
CO 4
Apply
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
CO 4
Apply
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
CO 4
Apply
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
CO 4
Apply
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
CO 4
Apply
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
CO 4
0
Evaluate the double integral
π
2
Z
1
Z
x2 y 2 dxdy
−1
0
8
Apply
−3
.
Evaluate the double integral
Z 2Z x
ydydx
0
7
CO 4
1
−1
6
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
0
1
5
Apply
.
Evaluate the double integral
π
Z
Z
asinθ
rdrdθ
0
9
2
.
Evaluate the double integral
∞Z
Z
0
10
π
2
2
e−r rdrdθ
0
Evaluate the double integral
Z
π
Z
a(1+cosθ)
rdrdθ
0
2
CIE-II
Page 19
11
12
13
14
15
16
State the formula to find area of
the region using double integration
in Cartesian form
Find the volume of the tetrahedron
bounded by the coordinate planes
and the plane x+y+z=1
Remember
State the formula to find volume of
the region using triple integration
in Cartesian form.
Find area of the ellipse using
2
2
double integration xa2 + yb2 = 1.
Remember
State the formula to find area of
the region using double integration
in polar form.
Find the area of the region
bounded by the parabolas y 2 = 4x
and x2 = 4y
Remember
Apply
Apply
Apply
17
Find the area of the curve
r = 2acosθ using double
integration in polar coordinates.
Apply
18
Find the area enclosed between the
parabola y = x2 and the line y = x
.
Apply
19
Find the area of the curve
r = 2asinθ .
Apply
20
Find area of the circle
.x2 + y 2 = a2 .
Apply
Page 20
CO 4
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
CO 4
CO 4
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
CO 4
CO 4
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
Learner to recall basic
integral formulae, explain
double integration and
apply it to the given
Cartesian form function.
CO 4
CO 4
CO 4
CO 4
CO 4
1
2
3
4
5
6
MODULE IV
VECTOR DIFFERENTIAL CALCULUS
PART A-PROBLEM SOLVING AND CRITICAL THINKING QUESTIONS
Verify Gauss divergence theorem
Apply
Learner to recall vector and
CO
for
scalar functions, explain
(x2 − yz)ī + (y 2 − zx)j̄ + (z 2 − xy)k̄
gradient and apply Gauss
taken over the cube bounded by
divergence theorem to
x=0, x=a, y=0, y=b, z=0, z=c.
obtain the transformation
between surface and volume
of a bounded region of cube.
Find the work done in moving a
Apply
Learner to recall vector and
CO
particle in the force field
scalar functions, explain
gradient and apply line
F̄ = 3x2 ī + (2zx − y)j̄ + z k̄ along
integral to obtain the work
the curve defined by
done by the force.
x2 = 4y, 3x3 = 8z from x=0 to x=2
.
Show that the force field given by
Apply
Learner to recall vector and
CO
3
2
3
2
2
F̄ = 2xyz ī + x z j̄ + 3x yz k̄ is
scalar functions, explain
conservative. Find the work done
gradient and apply line
in moving a particle from (1,-1,2)
integral to obtain the work
to (3,2,-1) in this force field.
done by the force.
Show that the vector
Apply
Learner to recall vector and
CO
(x2 − yz)ī + (y 2 − zx)j̄ + (z 2 − xy)k̄
scalar functions, explain curl
is irrotational and find its scalar
of the gradient, and apply it
potential function.
to obtain irrotational and
scalar potential function
Verify Gauss divergence theorem
Apply
Learner to recall vector and
CO
for F̄ = 3xī + (y + zx)j̄ + (xyz + z)k̄
scalar functions, explain
where V is the volume bounded by
gradient and apply Gauss
the plane 2x+3y+6z=12 in the first
divergence theorem to
octant .
obtain the transformation
between surface and volume
of a bounded region of
cylinder
Find the directional derivative of
Apply
Learner to recall vector and
CO
ϕ(x, y, z) = x2 yz + 4xz 2 at the
scalar functions, explain the
point (1, -2, -1) in the direction of
gradient, and apply it to
normal to the surface
obtain the direction
f (x, y, z) = x log z–y 2 at (-1,2,1).
derivative of the function.
Page 21
5
5
5
5
5
5
7
8
9
10
1
2
Using
Green’s theorem , evaluate
R
(2xy
− x2 )dx + (x2 + y 2 )dy where
c
C is the region bounded by y = x2
and y 2 = x .
Apply
Learner to recall vector and
scalar functions, explain
gradient and apply Green’s
theorem to obtain the
transformation between line
and double integral of a
bounded region of
parabolas.
Applying
Green’s
theorem
evaluate
Apply
Learner to recall vector and
R
2
2
scalar functions, explain
c (xy + y )dx + (x )dy where C is
the region bounded by y=x and
gradient and apply Green’s
2
y=x .
theorem to obtain the
transformation between line
and double integral of a
bounded region of
parabolas.
Verify Green’s Theorem in the
Apply
Learner to recall vector and
plane
for
scalar functions, explain
R
2
2
gradient and apply Green’s
c (3x − 8y )dx + (4y − 6xy)dy
theorem to obtain the
where C is the region bounded by
transformation between line
x=0, y=0 and x + y=1.
and double integral of a
bounded region of a plane.
Verify Stokes theorem for
Apply
Learner to recall vector and
F̄ = (y − z + 2)ī + (yz + 4)j̄ − xz k̄
scalar functions, explain
where S is the surface of the cube
gradient and apply Stoke’s
x=0, y=0, z=0 and x=2, y=2, z=2
theorem to obtain the
above the xy-plane..
transformation between line
and surface of a bounded
region of a plane.
PART-B LONG ANSWER QUESTIONS
Find the constants a,b,c the vector
Apply
Learner to recall vector and
f¯ = (x + 2y + az)ī + (bx − 3y −
scalar functions, explain
z)j̄ + (4x + cy + 2z)k̄ is irrotational
gradient and apply it to
and also find its scalar potential.
obtain solution of line
integral.
RR
Evaluate s F̄ .n̄ds if
Apply
Learner to recall vector and
2
2
F̄ = yz ī + 2y j̄ + xz k̄ and S is the
scalar functions, explain
Surface of the cylinder x2 + y 2 = 9
gradient and normal forces,
contained in the first octant
and apply it to value the
between the planes z=0 and z=2.
area of the cylinder .
Page 22
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
3
Find the work done in moving a
particle in the force field
F̄ = 3x2 ī + (2zx − y)j̄ + z k̄ along
the straight line from (0,0,0) to
(2,1,3) .
Find the circulation of
F̄ = (2x − y + 2z)ī + (x + y − z)j̄ +
(3x − 2y − 5z)k̄ along the circle
x2 + y 2 = 4 in the xy plane. .
Apply
5
Verify Gauss divergence theorem
for the vector point function
F̄ = (x3 − yz)ī − 2xy j̄ + 2z k̄ over
the cube bounded by x = y =z = 0
and x = y =z = a.
Apply
6
Verify Gauss divergence theorem
for 2x2 y ī − y 2 j̄ + 4xz 2 k̄ taken over
the region of first octant of the
cylinder y 2 + z 2 = 9 and x=2.
Apply
7
Verify
R Green’s theorem in the plane
for c (x2 − xy)dx + (y 2 − 2xy)dy
where C is a square with vertices
(0,0) ,(2,0) ,(2,2),(0,2).
Apply
8
Applying
Green’s theorem evaluate
R
(y
−
sinx)dx
+ cosxdy where C is
c
the plane triangle enclosed by y= 0
π
.y = 2x
π ,and x = 2 .
Apply
4
Apply
Page 23
Learner to recall vector and
scalar functions, explain
gradient and apply line
integral to value the work
done by the force.
Learner to recall vector and
scalar functions, explain
gradient and apply line
integral to value the work
done by the force.
Learner to recall vector and
scalar functions, explain
gradient and apply Gauss
divergence theorem to
obtain the transformation
between surface and volume
of a bounded region of cube
.
Learner to recall vector and
scalar functions, explain
gradient and apply Gauss
divergence theorem to
obtain the transformation
between surface and volume
of a bounded region of
cylinder.
Learner to recall vector and
scalar functions, explain
gradient and apply Green’s
theorem to obtain the
transformation between line
and double integral of a
square bounded region .
Learner to recall vector and
scalar functions, explain
gradient and apply Green’s
theorem to obtain the
transformation between line
and double integral of a
triangle bounded region
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
9
Apply Green’s Theorem in the
plane
for
R
2
2
2
2
c (2x − y )dx + (x + y )dy where
C is a is the boundary of the area
enclosed by the x-axis and upper
half of the circle x2 + y 2 = 16 .
Apply
10
Verify Stokes theorem for
f¯ = (2x − y)ī − yz 2 j̄ − y 2 z k̄ where
S is the upper half surface of the
sphere x2 + y 2 + z 2 = 1 bounded by
the projection of the xy plane.
Apply
11
Verify Stokes theorem for
f¯ = −y 3 ī + x3 j̄ where S is the
circular disc x2 + y 2 ≤1, z=0.
Apply
12
Find the directional derivative of
the function ϕ = xy 2 + yz 3 at the
point P (1,-2,-1) in the direction of
normal to the surface
x log z − y 2 = −4 at (-1,2,1)
2
If F̄ = 4xz
R ī − y j̄ + yz k̄
evaluate s F̄ .n̄ds where S is the
surface of the cube x= 0 ,x=a,
y=0,y=a.z=0,z=a
Apply
14
2
If f¯ = (5xy
R − 6x )ī + (2y − 4x)j̄
evaluate c f¯.dr̄ where C is curve
y = x3 plane from (1,1) to (2,8)
Apply
15
Evaluate
the line integral r
R 2
(x
+
xy)dx
+ (x2 + y 2 )dy where
c
C is the square formed by lines
x = ±1, y = ±1 .
Apply
13
Apply
Page 24
Learner to recall vector and
scalar functions, explain
gradient and apply Green’s
theorem to the
transformation between line
and double integral of a
bounded region of upper
half of the circle
Learner to recall vector and
scalar functions, explain
gradient and apply Stoke’s
theorem to the
transformation between line
and surface of a bounded
region of sphere.
Learner to recall vector and
scalar functions, explain
gradient and apply Stoke’s
theorem to the
transformation between line
and surface of a bounded
region of sphere.
Learner to recall vector and
scalar functions, explain the
gradient, and apply it to
obtain the direction
derivative of the function.
Learner to recall vector and
scalar functions, explain
gradient and apply Gauss
divergence theorem to
obtain the transformation
between surface and volume
of a bounded region of cube.
Learner to recall vector and
scalar functions, explain
gradient and apply line
integral to obtain the work
done by the force.
Learner to recall vector and
scalar functions, explain
gradient and apply line
integral to obtain the work
done by the force
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
16
17
18
19
20
1
2
3
4
5
If r̄ = xī + y j̄ + z k̄ show that
▽rn = nrn−2 r̄.
Apply
Learner to recall vector and
scalar functions, explain
gradient and apply it to
obtain the required solution
of normal forces
Evaluate
by Stokes theorem
Apply
Learner to recall vector and
R x
scalar functions, explain
c (e dx + 2ydy − dz)where c is the
curve x2 + y 2 = 4 and z=2 .
gradient and apply Stoke’s
theorem to obtain the
transformation between line
and surface of a bounded
region of a plane
Verify Stokes theorem for the
Apply
Learner to recall vector and
2
function x ī + xy j̄ integrated round
scalar functions, explain
gradient and apply Stoke’s
the square in the plane z=0 whose
theorem to obtain the
sides are along the line x=0,y=0
transformation between line
,x=a,y=a .
and surface of a square
bounded region
Verify Stokes theorem for
Apply
Learner to recall vector and
F̄ = (x2 + y 2 )ī − 2xy j̄ , taken
scalar functions, explain
around the rectangular bounded by
gradient and apply Stoke’s
the lines x=-a,x=a,y=0,y=a.
theorem to obtain the
transformation between line
and surface of a triangle
bounded region
Verify
Apply
Learner to recall vector and
R Green’s theorem in the plane
for c (3x2 − 8y 2 )dx + (4y − 6xy)dy
scalar functions, explain
where C is a region bounded by
gradient and apply Green’s
√
2
y = x and y = x .
theorem to obtain the
transformation between line
and double integral of a
bounded region of parabola
PART-C SHORT ANSWER QUESTIONS
Define gradient of scalar point
Remember —
function.
Define divergence of vector point
Remember —
function.
Define curl of vector point function. Remember —
State Laplacian operator.
Remember —
¯
Find curlf where
Remember —
f¯ = grad(x3 + y 3 + z 3 − 3xyz)
Page 25
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
6
Find the angle between the normal
to the surface xy = z 2 at the points
( 4,1,2) and (3,3,-3) .
Apply
7
Find a unit normal vector to the
given surface x2 y + 2xz = 4 at the
point (2,-2,3).
Apply
8
If is a vector then prove that
⃗
∇(ā,
r̄) = ā
Apply
9
Define irrotational vector and
solenoidal vector of .
Show that Curl(rn .r̄) = 0
Remember
10
Apply
Learner to recall vector and
scalar functions, explain
gradient and apply it to
obtain the angle between
the normal surfaces
Learner to recall vector and
scalar functions, explain
gradient and apply it to
obtain the unit vector of
normal surfaces.
Learner to recall vector and
scalar functions, explain
gradient and apply it to
obtain the required solution
of normal surfaces
—
CO 5
CO 5
CO 5
11
Prove that f = yz ī + zxj̄ + xy k̄ is
irrotational vector.
Apply
12
Show that
(x + 3y)ī + (y − 2z)j̄ + (x − 2z)k̄ is
solenoidal.
Apply
13
⃗ × (∇(ϕ))
⃗
Show that ∇
= 0 where
ϕ is scalar point function.
Apply
14
15
State Stokes theorem .
⃗ × f¯] = 0 where .
Prove that div[∇
Remember
Remember
Learner to recall vector and
scalar functions, explain
gradient and apply it to
obtain the required solution
of normal surfaces
Learner to recall vector and
scalar functions, explain
gradient and apply it to
obtain the irrotational
vector
Learner to recall vector and
scalar functions, explain
gradient and apply it to
obtain conservation of mass
Learner to recall vector and
scalar functions, explain
gradient and apply it to
obtain solution of scalar
point function
—
—
Remember
—
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
CO 5
CO 8
f = f1 i + f2 j + f3 k
16
Define line integral on vector point
function.
Page 26
17
18
19
20
1
2
3
4
5
Define surface integral of vector
point function F̄ .
Define volume integral on closed
surface S of volume V.
State Green’s theorem in plane
State Gauss divergence theorem .
Remember
—
CO 5
Remember
—
CO 5
Remember —
CO
Remember —
CO
MODULE V
PARTIAL DIFFERENTIAL EQUATIONS
PART A-PROBLEM SOLVING AND CRITICAL THINKING QUESTIONS
Form the partial differential
Apply
Recall dependent and
CO
equation by eliminating arbitrary
independent variables,
function lx + my + nz =
explain partial derivatives,
2
2
2
ϕ(x + y + z )
and applyit to form PDE by
.
eliminating arbitrary
function.
Form the partial differential
Apply
Recall dependent and
CO
equation by eliminating arbitrary
independent variables,
function xy + yz + zx = f (z/x + y).
explain partial derivatives,
and applyit to form PDE by
eliminating arbitrary
function.
2
2
Solve z(x − y) = px − qy
Apply
Recall dependent and
CO
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
2
2
Solve (z −2yz −y )p+(xy+xz)q =
Apply
Recall dependent and
CO
xy − zx
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
2
Solve z = pqxy
Apply
Recall dependent and
CO
independent variables,
explain partial derivatives,
and apply Char pit method
to solve nonlinear partial
differential equations.
Page 27
5
5
6
6
6
6
6
6
7
8
9
10
1
2
Find the integral surface of
x(y 2 + z)p − y(x2 + z)q =
(x2 − y 2 ).which contains the
straight line x+y=0,z=1.
Apply
Recall dependent and
independent variables,
explain partial derivatives,
and apply standard forms to
solve nonlinear partial
differential equations.
Solve (1+y)p+(1+x)q=z
Apply
Recall dependent and
independent variables,
explain partial derivatives,
and apply standard forms to
solve nonlinear partial
differential equations.
Solve xp2 + yq2 = z
Apply
Recall dependent and
independent variables,
explain partial derivatives,
and apply standard forms to
solve nonlinear partial
differential equations.
Solve xp + yq = 1.
Apply
Recall dependent and
independent variables,
explain partial derivatives,
and apply standard forms to
solve nonlinear partial
differential equations.
Solve (x-a)p+(y-b)q=z-c.
Apply
Recall dependent and
independent
variables,explain partial
derivatives, and apply Char
pit method to solve
nonlinear partial differential
equations.
PART-B LONG ANSWER QUESTIONS
Form the partial differential
Apply
Recall dependent and
equation by eliminating arbitray
independent variables,
function from
explain partial derivatives,
f (x2 + y 2 + z 2 , z 2 − 2xy) = 0
and applyit to form PDE by
eliminating arbitrary
function.
Form the partial differential
Apply
Recall dependent and
equation by eliminating a, b, c
independent variables,
y2
x2
z2
explain partial derivatives,
from a2 + b2 + c2 = 1
and apply it to form PDE
by eliminating arbitrary
constants.
Page 28
CO 6
CO 6
CO 6
CO 6
CO 6
CO 6
CO 6
3
Solve
(x2 − yz)p + (y 2 − zx)q = z 2 − xy
Apply
4
Solve px+qy=pq
Apply
5
Solve the partial differential
equation,
(mz −ny)p+(nx−lz)q = (ly −mx).
Apply
6
Find the differential equation of all
spheres whose centres lie on z-axis
with a given radius r.
Apply
7
Solve
(x2 − y 2 − yz)p + (x2 − y 2 − zx)q =
z(x − y)
Apply
8
Solve (x2 − y 2 − z 2 )p + 2xyq = 2xz
Apply
Page 29
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear .
CO 6
CO 6
CO 6
CO 6
CO 6
CO 6
9
Solve
x2 (y − z)p + y 2 (z − x)q = z 2 (x − y).
Apply
10
Solve px − qy = y 2 − x2 .
Apply
11
Solve px2 + qy 2 = z(x + y).
Apply
12
Solve p − x2 = y 2 + q.
Apply
13
Solve y 2 zp + x2 zq = xy 2 .
Apply
14
Solve ptanx + qtany = tanz.
Apply
Page 30
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first orderr.
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
CO 6
CO 6
CO 6
CO 6
CO 6
CO 6
15
Solve
(x − a)p + (y − b)q + (c − z) = 0.
16
Solve
17
Solve
18
Solve
19
Solve
20
Solve
Apply
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
pxy + pq + qy = yz.
Apply
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
2
2
z = p x + q y.
Apply
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
2
2
x p + y q = z.
Apply
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
z(x − y) = px2 − qy 2 .
Apply
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
(x − y)p + (y − x − z)q = z.
Apply
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order.
PART-C SHORT ANSWER QUESTIONS
Page 31
CO 6
CO 6
CO 6
CO 6
CO 6
CO 6
1
2
Define order and degree with
reference to partial differential
equation.
Form the partial differential
equation by eliminate the arbitrary
constants from z = ax3 + by 3 .
Remember
Apply
3
Form the partial differential
equation by eliminating arbitrary
function z = f (xz + y 2 )
Apply
4
Solve the partial differential
√
√
√
equation p x + q y = z
Apply
5
Form the partial differential
equation by eliminating a and b
from log(az − 1) = x + ay + b.
Apply
6
Form the partial differential
equation by eliminating the
constants from
(x − a)2 + (y − b)2 = z 2 cot2 α where
α is a parameter.
Apply
7
Eliminate the arbitrary constants
from z = (x2 + a)(y 2 + b)
Apply
Page 32
—
CO 6
Recall dependent and
independent variables,
explain partial derivatives,
and apply it to form PDE
by eliminating arbitrary
constants
Recall dependent and
independent variables,
explain partial derivatives
and apply it to form PDE
by eliminating arbitrary
function
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
Recall dependent and
independent variables,
explain partial derivatives,
and applyit to form PDE by
eliminating arbitrary
constants
Recall dependent and
independent variables,
explain partial derivatives,
and apply to form PDE by
eliminating arbitrary
constants
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
CO 6
CO 6
CO 6
CO 6
CO 6
CO 6
8
Solve the partial differential
equation
x(y − z)p + y(z − x)q = z(x − y).
Apply
9
Solve p + q = z
Apply
10
Solve zp + yq = x
Apply
11
12
Define Charpit’s method.
Solve xp + yq = 3z.
13
Solve px + qy = z
Apply
14
Solve p + 3q = 5z + tan(y − 3x)
Apply
Remember
Apply
Page 33
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
—
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first orders
CO 6
CO 6
CO 6
CO 6
CO 6
CO 6
CO 6
15
Solve 2p + 3q = 1
Apply
16
Solve z = p + q
Apply
17
Solve yq − px = z
Apply
18
Solve y 2 p − xyq = x(z − 2y)
Apply
19
Explain nonlinear partial
differential equation.
Types of first order nonlinear
differential equation.
20
CO 6
Remember
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
Recall dependent and
independent variables,
explain Lagrange’s Linear
equation, apply suitable
method to solve linear
partial differential equations
of first order
—
Remember
—
CO 6
Course Coordinator:
Mr.Satyanarayana.G, Assistant Professor
Page 34
HOD AE
CO 6
CO 6
CO 6
CO 6