NARULA INSTITUTE OF TECHNOLOGY

An Autonomous Institute under MAKAUT

B.Tech /ALL /ODD/SEM_1/M 101/2021-2022

PAPER TYPE: REGULAR

YEAR: 2022
MATHEMATICS I
M 101
TIME ALLOTTED: 3 HOURS FULL MARKS: 70
The figures in the margin indicate full marks.
Candidates are required to give their answers in their own words as far as practicable

GROUP – A
(Multiple Choice Type Questions)
1. Answer any ten from the following, choosing the correct alternative of each question: 10×1=10

SL. NO. Question Marks CO

(i) If r  xi  y j  zk , then curl r is 1 CO1

a) i b) j c) k d) 0 .

(ii) The pair of the functions which satisfy the conditions 1 CO3
of Cauchy’s M.V.T. in [-1,1] is
a) ,
b) ,
c) |x|, x
d) ,
(iii) x  y3
3 1 CO3
sin 1 is a
x y
a) homogeneous function of degree 2
b) homogeneous function of degree 1
1
c) homogeneous function of degree
2
d) non-homogeneous function.
(iv) The value of a for which 1 CO2

   
2 x i  2 x  y j  3az  2 x k is a solenoidal is
2 3

a) -1
b)0
c) 1
d) 2.
(v) If 3  62  9  4 is the characteristic equation of a 1 CO1
square matrix A, then A-1 is
a) A  6 A  9 I
2

1 3 9
b) A 2  A  I
4 2 4
1 2 3 9
c) A  A
4 2 4
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An Autonomous Institute under MAKAUT

d) A  6 A  9
2

(vi) If Rank(A)=, then the matrix A3x3 is 1 CO2

a) singular
b) non singular
c) orthogonal
d) None

(vii) Identify the correct statements 1 CO1

a) A bounded monotonic sequence is divergent
b) A bounded monotonic sequence is convergent
c) A bounded monotonic sequence is oscillatory
d) A bounded monotonic sequence may not have finite
limit

(viii) Which of the following theorem can be applied on 1 CO2

f(x)=IxI in the interval [-1, 1]
a)Rolle’s theorem
b) Lagrange Mean value theorem
c) Cauchy Mean value theorem
d) None of these

(ix) 1 CO4

a) 1/2
b)1/6
c)2/3
d)4/3
(x) The series is 1 CO2

a) absolutely convergent
b) conditionally convergent
c) oscillatory
d) divergent
(xi)
 5 2 3 1 CO2
 
If A=  1 0 2  , then the sum of the eigen values of A
0 4 0
 
is
a) 3
b) 5
c) 6
 NARULA INSTITUTE OF TECHNOLOGY
An Autonomous Institute under MAKAUT

d) 10
(xii) 2 2 2 1 CO1
The value of    xyz dxdydz is
0 0 0

a) 1/2
b) 1/8
c) 8
d) 16
GROUP – B
(Short Answer Type Questions)
Answer any three from the following: 3×5=15

SL. NO. Marks CO

2. Use mean value theorem to prove that 5 CO3
x 
 tan 1 x  x when 0  x 
1 x 2
2
3. 5 CO3
Reduce the matrix A= to an equivalent

Echelon matrix and hence find its rank.

4.   n2 5 CO2
 1
Test the convergence of the series  1  
n 1  n
5 Find the eigen values and the corresponding eigen 5 CO1
vectors of the matrix

A=

6. 5 CO2
What is the value of   ydxdy over the region bounded

by y 2  x and y  x 2 .

GROUP – C
(Long Answer Type Questions)
Answer any three from the following: 3×15=45

SL. NO. Marks CO

No.
7. (a) Verify Cayley-Hamilton theorem for the matrix 5 CO2
 NARULA INSTITUTE OF TECHNOLOGY
An Autonomous Institute under MAKAUT

1 1 2 
3 1 1 . Hence, evaluate A-1.
 
2 3 1
(b) 1 1 1 5 CO3
Show that 1     ..... is convergent series.
22 32 42
(c) Solve by matrix method the system of equations 5 CO1

x+z=0
3x+4y+5z=2
2x+3y+4z=1.

8. (a) Use Cayley-Hamilton theorem to find A-1, where 5 CO1

A=
(b)  yx zx 4 CO3
If u  u  ,  show that
 xy zx 
u u u
x2  y2  z2 0
x y z
(c) State Lagrange’s Mean Value Theorem. Write Taylor’s 6 CO4
formula for the function
f ( x)  log(1  x),  1  x   about x  2 with
Lagrange’s form of remainder after 3 terms.
9. (a)
 
Show that A  6 xy  z 3 i  (3 x 2  z ) j  (3xz 2  y )k is 6 CO4

irrotational. Find the scalar  such that A   .

(b) Show that curl grad f  0 , where f  x2 y  2 xy  z 2 . 4 CO2

(c) Find the directional derivative of f ( x, y, z )  x 2 yz  4 xz 2 5 CO2

at the point (1,  2,  1) in the direction of the

vector 2i  j  2k
10. (a) Verify the divergence theorem for F(x, y, z) = xi + yj + 0k and 8 CO3
the surface x2 + y2 + z2 = 9.

 (3xydx  y dy)
(b) 2 3 CO2
Find the value of
C

Where C is the arc of the parabola

y  4 x2 from (0,0) to (1, 4)
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An Autonomous Institute under MAKAUT

  ydx  zdy  xdz  ,

(c) 4 CO3
Apply Stoke’s theorem to evaluate
C
where C is the curve of intersection of x2+y2+z2=9 and
x+z=3.
11. (a)  13  y 13  6 CO3
1  x 
If u  sin  , then prove that
 x y
 
 2u  2u 2  u
2
tan u  13 tan 2 u 
x2
 2 xy y    
x 2 xy y 2 12  12 12 
(b) Show that is irrotational. 3 CO2

(c) 6 CO4
Evaluate where V is the volume of
the cylinder =1 intercepted by the planes z=2 and z=3.

Course Outcomes (COs):

On successful completion of the learning sessions of the course, the learner will be able to
CO1: Recall the properties and formula related to matrix algebra, differential calculus,
multivariable calculus, vector calculus and infinite series.
CO2: Determine the solutions of the problems related to matrix algebra, differential calculus,
multivariable calculus, vector calculus and infinite series.
CO3: Apply the appropriate mathematical tools of matrix algebra, differential calculus,
multivariable calculus, vector calculus and infinite series for the solutions of the problems.
CO4: Analyze different engineering problems linked with matrix algebra, differential
calculus, multivariable calculus, vector calculus.