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LINEAR ALGEBRA AND CALCULUS, Exercises of Engineering Mathematics

Jawaharlal Nehru Technological University, KakinadaEngineering Mathematics

ENGINEERING MATHEMATICS-1 FIRST YEAR 2024 QUESTION PAPER

Typology: Exercises

2023/2024

Available from 08/26/2025

veerabrahmam-yadaval
veerabrahmam-yadaval 🇮🇳

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|'''|'|'|''||'''||||
Code No: R231105
I B. Tech I Semester Regular Examinations, January-2024
LINEAR ALGEBRA AND CALCULUS
(Common to all Branches)
Time: 3 hours Max. Marks: 70
Note: 1. Question paper consists of two parts (Part-A and Part-B)
2. All the questions in Part-A is Compulsory
3. Answer ONE Question from Each Unit in Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART –A (20 Marks)
1. a) Define linear system of equations. [2M]
b) What is the normal form? [2M]
c) Find the matrix corresponding to quadratic form
+
4

+
2
. [2M]
d) Find the sum of the Eigen values of matrix
1
2
2
4
[2M]
e) State Cauchy’s mean value theorem. [2M]
f) Write the geometrical interpretation for Lagrange’s mean value theorem. [2M]
g) Find
,

for
,
=

+
+
2
[2M]
h) Find

if
=
+
,
−
[2M]
i) Let f(x, y) be a continuous function in R
2
where R =
{(x, y)/ a<x<b; c<y<d}
then
∬
,
 

 

=
?
[2M]
j) Evaluate
 
 

 

 

[2M]
PART – B (50 MARKS)
UNIT-I
2. a)
Find the rank of the matrix using echelon form
1
2
3
2
−
2
1
3
0
4
−
2
3
1
"
[5M]
b)
Solve the system of equations using Gauss elimination method
10
+
+
#
=
12
,
2
+
10
+
#
=
13
,
+
+
5
#
=
7
.
[5M]
(OR)
3. a)
Find the inverse using Gauss-Jordan method
2
−
1
3
1
1
1
1
−
1
1
"
[5M]
b)
Find the values of ‘a’ and ‘b’ for which the system of equations
+
+
#
=
3
,
+
2
+
2
#
=
6
,
+
'
+
3
#
=
(
has a unique solution.
[5M]
UNIT-II
4. a)
Determine the eigen values of
')*
where
*
=
−
3
−
7
−
5
2
4
3
1
2
2
"
[5M]
b)
Verify Cayley-Hamilton theorem for
*
=
2
1
1
0
1
0
1
1
2
"
[5M]
1 of 2
SET
-
1
R23
pf3
pf4
pf5
pf8

Partial preview of the text

Download LINEAR ALGEBRA AND CALCULUS and more Exercises Engineering Mathematics in PDF only on Docsity!

Code No: R

I B. Tech I Semester Regular Examinations, January-

LINEAR ALGEBRA AND CALCULUS

(Common to all Branches)

Time: 3 hours Max. Marks: 70

Note: 1. Question paper consists of two parts ( Part-A and Part-B)

2. All the questions in Part-A _is Compulsory

  1. Answer_ ONE Question from Each Unit in Part-B

PART –A (20 Marks)

  1. a) Define linear system of equations. [2M]

b) What is the normal form? [2M]

c) Find the matrix corresponding to quadratic form ᡶ

⡰

⡰

. [2M]

d)

Find the sum of the Eigen values of matrix 䙴

[2M]

e) State Cauchy’s mean value theorem. [2M]

f) Write the geometrical interpretation for Lagrange’s mean value theorem. [2M]

g)

Find

ㄅ〳

ㄅけ

ㄅ〳

ㄅげ

for ᡘ

⡰

[2M]

h)

Find

ㄅえ

ㄅけ

if ᡳ = ᡘ䙦ᡶ + ᡷ, ᡶ − ᡷ䙧

[2M]

i) Let f ( x , y ) be a continuous function in R

2

where R = {( x , y )/ a < x < b ; c < y < d } then

〙

[2M]

j)

Evaluate ᔖ ᔖ

⡰

⡨

⡩

⡨

[2M]

PART – B (50 MARKS)

UNIT-I

  1. a)

Find the rank of the matrix using echelon form㐩

[5M]

b) Solve the system of equations using Gauss elimination method

[5M]

(OR)

  1. a)

Find the inverse using Gauss-Jordan method 㐩

[5M]

b) Find the values of ‘a’ and ‘b’ for which the system of equations

ᡶ + ᡷ + ᡸ = 3 , ᡶ + 2 ᡷ + 2 ᡸ = 6 , ᡶ + ᡓᡷ + 3 ᡸ = ᡔ has a unique solution.

[5M]

UNIT-II

  1. a)

Determine the eigen values of ᡓᡖᡢᠧ where ᠧ = 㐩

[5M]

b)

Verify Cayley-Hamilton theorem for ᠧ = 㐩

[5M]

1 of 2

R

Code No: R

(OR)

Diagonalize the matrix ᠧ = 㐩−

㐳 and find ᠧ

⡲

using model matrix ‘P’

[10M]

UNIT-III

  1. a)

Verify Rolle’s mean value theorem ᡘ

う〶ぁけ

〲

㊙

in

[5M]

b) Write the Taylor’s series expansion for ᡘ

= log

about ᡶ = 0

[5M]

(OR)

Prove that

ゕ

⡴

⡩

⡳ √

⡱

⡹⡩

⡱

⡳

ゕ

⡴

⡩

⡶

[10M]

UNIT-IV

  1. a)

If ᡶ = ᡰ ᡕᡧᡱ ‖ , ᡷ = ᡰ ᡱᡡᡦ ‖ then prove that

‴

2

‖

‴ᡶ

2

‴

2

‖

‴ᡷ

2

[5M]

b) Determine whether the following functions are functionally dependent if so find the

relation between ᡡᡘ ᡳ = ᡶ㒓 1 − ᡷ

⡰

⡰

⡹⡩

⡹⡩

[5M]

(OR)

  1. a)

Find ᡶ

ㄅえ

ㄅけ

ㄅえ

ㄅげ

if ᡳ = log 㐶

ᡶ

2

+ᡷ

2

ᡶᡷ

[5M]

b) Find extreme valuesᡘ䙦ᡶ, ᡷ䙧 = 1 − ᡶ

⡰

⡰

[5M]

UNIT-V

  1. By change of Integration Evaluate

け

け

ㄘ

⡩

あ

[10M]

(OR)

Find the volume of the sphere x

2

+ y

2

+ z

2

= a

2

using triple integration.

[10M]

2 of 2

R

Code No: R

(OR)

Diagonalize the matrix ᠧ = 㐩

㐳 and find ᠧ

⡲

using model matrix ‘P’

[10M]

UNIT-III

  1. a) Write the Taylor’s series expansion for ᡘ

= log 䙦 1 − ᡶ䙧about ᡶ = 0

[5M]

b) Verify Rolle’s mean value theorem ᡘ䙦ᡶ䙧 = |ᡶ| in 䙰− 1 , 1 䙱 [5M]

(OR)

If ᡓ < ᡔ prove that

〩⡹〨

⡩⡸〩

ㄘ

⡹⡩

⡹⡩

〩⡹〨

⡩⡸〨

ㄘ

[10M]

UNIT-IV

  1. a) Expand f(x,y) = xy

2

+cos(xy) in powers of 䙦ᡶ − 1 䙧 ᡓᡦᡖ 䙦ᡷ −

ゕ

⡰

䙧 using Taylor’s

series.

[5M]

b)

If ᡳ =

ᡷ

ᡸ

ᡸ

ᡶ

then find ᡶ

ㄅえ

ㄅけ

ㄅえ

ㄅげ

ㄅえ

ㄅこ

[5M]

(OR)

  1. a)

Find ᡶ

ㄅえ

ㄅけ

ㄅえ

ㄅげ

if ᡳ = ᡱᡡᡦ

⡹⡩

け

げ

䙳 + tan

⡹⡩

げ

け

[5M]

b)

Show that , ,

x y z

u v w

y z z x x y

are functionally dependent.

[5M]

UNIT-V

  1. Evaluate by change of order of Integration

⡱〨 ⡹げ

げ

ㄘ

/⡲〨

⡰〨

⡨

[10M]

(OR)

  1. Evaluate ∭

⡰

⡰

〙

where R is the Region bounded by the

cylinder. x

2

  • y

2

= 1 and the planes z = 2 and z = 3 by changing it to cylindrical

coordinates.

[10M]

2 of 2

R

Code No: R

I B. Tech I Semester Regular Examinations, January-

LINEAR ALGEBRA AND CALCULUS

(Common to all Branches)

Time: 3 hours Max. Marks: 70

Note: 1. Question paper consists of two parts ( Part-A and Part-B)

2. All the questions in Part-A _is Compulsory

  1. Answer_ ONE Question from Each Unit in Part-B

PART –A (20 Marks)

  1. a) Find the rank of the singular matrix of order 4 × 4 [2M]

b) What type of the solutions exists for 2 ᡶ + 3 ᡷ = 5 , 4 ᡶ + 6 ᡷ = 10 system? [2M]

c) If 5 is an Eigen value of A the find the Eigen value of 4 ᠧ + 5 ᠵ

[2M]

d)

Write the quadratic form associated with 㐩

[2M]

e) Find the value of ‘c’ using Rolle’s ’s mean value theorem for ᡘ䙦ᡶ䙧 = ᡶ

⡰

ᡡᡦ 䙰− 1 , 1 䙱 [2M]

f) State Lagrange’s mean value theorem. [2M]

g)

Find

ㄅ〳

ㄅけ

ㄅ〳

ㄅげ

for ᡘ

⡰

⡰

[2M]

h)

Find

ㄅえ

ㄅけ

if ᡳ = ᡘ䙦ᡶ + 2 ᡷ, ᡶ − 2 ᡷ䙧

[2M]

i) If f ( x , y ) be a continuous defined over a Region R , were

R ={( x , y )/ x 1

< x < x 2

and c < y < d }then∬ ᡘ䙦ᡶ, ᡷ䙧 ᡖᡶ ᡖᡷ

〙

[2M]

j)

Evaluate ᔖ ᔖ

⡩

⡨

⡰

⡨

[2M]

PART – B (50 MARKS)

UNIT-I

  1. a)

Find the rank of the matrix using echelon form㐩

[5M]

b) Solve the system of equations

[5M]

(OR)

3 a) Test the consistency, if so, solve the system of equations

[5M]

b) Solve the system of equations using Gauss Seidel iteration method

[5M]

UNIT-II

  1. Reduce the quadratic form 2 ᡶ

⡰

⡰

⡰

− 2 ᡷᡸ − 2 ᡸᡶ − 2 ᡶᡷ to the canonical

form byorthogonal reduction. Hence find nature, rank, index, and signature.

[10M]

(OR)

1 of 2

R

Code No: R

I B. Tech I Semester Regular Examinations, January-

LINEAR ALGEBRA AND CALCULUS

(Common to all Branches)

Time: 3 hours Max. Marks: 70

Note: 1. Question paper consists of two parts ( Part-A and Part-B)

2. All the questions in Part-A _is Compulsory

  1. Answer_ ONE Question from Each Unit in Part-B

PART –A (20 Marks)

  1. a) The rank of 2 × 2 matrix with all elements are 3. [2M]

b) Write the condition for the homogeneous system of equations possess trivial

solutions.

[2M]

c)

Find the nature of the quadratic form 㐩

[2M]

d) Find the Eigen values of A

T

If 1 and 2 are the Eigen values of A. [2M]

e) Find the value of ‘c’ using Lagrange’s mean value theorem for ᡘ

= 2 ᡶ ᡡᡦ 䙰 0 , 1 䙱 [2M]

f) Write the Maclaurin’s series.

[2M]

g)

Find

ㄅ〳

ㄅけ

ㄅ〳

ㄅげ

for ᡘ

け

[2M]

h)

Find

ㄅえ

ㄅげ

if ᡳ = ᡘ䙦 2 ᡶ + ᡷ, ᡶ − 2 ᡷ䙧

[2M]

i) If the region ‘R’ is divided into two sub regions,ᡄ

⡩

⡰

then ∬

〙

[2M]

j)

Evaluate ᔖ ᔖ ᡖᡶ ᡖᡷ

⡩

⡨

⡩

⡨

[2M]

PART – B (50 MARKS)

UNIT-I

  1. a) Solve the system of equations

[5M]

b) Solve the system of equations using Gauss Jacobi iteration method

[5M]

(OR)

3 a)

Find the rank of the matrix using Normal form㐩

[5M]

b) Test the consistency, if so, solve the system of equations

[5M]

1 of 2

R

Code No: R

UNIT-II

Reduce the quadratic form 3 ᡶ

⡰

⡰

⡰

− 2 ᡷᡸ + 2 ᡸᡶ − 2 ᡶᡷ to the canonical

form by orthogonal reduction. Hence find nature, rank, index, and signature.

[10M]

(OR)

  1. a)

Find the Eigen values ᠧ

⡱

if ᠧ = 㐩

[5M]

b)

Verify Cayley-Hamilton theorem for ᠧ = 㐩

[5M]

UNIT-III

  1. Show that for any ᡶ > 0 , 1 + ᡶ < ᡗ

け

け

[10M]

(OR)

  1. a) Verify Cauchy’s mean value theorem ᡘ

⡰

⡱

in

䙱 [5M]

b)

Write the Taylor’s series expansion for ᡘ

= ᡱᡡᡦᡶ about ᡶ =

ゕ

⡲

[5M]

UNIT-IV

  1. a)

Prove that

2 2 2

2 2 2

u u u

x y z

if ᡳ =

⡰

⡰

⡰

⡹

ㄗ

ㄘ

[5M]

b) Find the maximum and minimum distance of the point 䙦 3 , 4 , 12 䙧 from the sphere

⡰

⡰

⡰

= 1 using Lagrange’s multiplier method

[5M]

(OR)

  1. a)

Find ᡶ

ㄅえ

ㄅけ

ㄅえ

ㄅげ

if ᡳ = ᡤᡧᡙ 䙲

け

ㄘ

⡸げ

ㄘ

けげ

[5M]

b) Find extreme values of the following functionsᡘ䙦ᡶ, ᡷ䙧 = ᡶᡷ䙦ᡓ − ᡶ − ᡷ䙧 [5M]

UNIT-V

Evaluate ∬

⡰

〙

where R is a triangle with vertices (0,0), (1, 0),

[10M]

(OR)

  1. Find the volume under the parabolic x

2

  • y

2

  • z = 16 over rectangle x = ±a, y = ±b [10M]

2 of 2

R