No.

of Printed Pages : 8 MTE–01

BACHELOR’S DEGREE PROGRAMME

(BDP)
Term-End Examination
December, 2023
(Elective Course : Mathematics)
MTE-01 : CALCULUS

Time : 2 Hours Maximum Marks : 50

Weightate : 70%
Note : (i) Question No. 1 is compulsory.

(ii) Attempt any four questions from

Question Nos. 2 to 7.
(iii) Use of calculator is not allowed.

1. Which of the following statements are true or

false. Give a short proof or a counter-example of
your answers : 2 each
x
(i) If f : R  R be defined by f  x   ,
1  x2
x
then f o  f o f  x   .
2
1  3x

P. T. O.
 [2] MTE–01

d  sin x
(ii)  cos t2 dt   cos x.

dx  x 

 
cos sin2 x  cos x2 .

(iii) If :
 x3  x2  16 x  20
 , x2
f  x    x  2 2

 k , x2

then the value of k if f is continuous at

x = 2 is 7.
(iv) If f   x  0  x ] a, b[ then f is
monotonically increasing on ] a, b [.
(v) The range of the function f defined by
1
f  x 
x   x

is ] 1,  [.

2. (a) Differentiate : 5
 1  x2  1 
1   w. r. t. tan1 x .
tan
 x 
 
(b) Integrate : 5
 x 
x
 e   dx
  1  x 2 
 
 [3] MTE–01

3. (a) Find the area enclosed between the

parabolas : 5
y2  4 a  x  a  and y2  4 a  x  a

(b) By dividing the interval [2,10] into four

equal parts, find the approximate value of

10 dx
2 2
x 4
using Simpsopn’s rule. 5

4. (a) Find the length of the arc of the curve

x  2y intercepted between y0 and

y  ln 2 . 5

(b) Differentiate :

 log x  x  xlog x
with respect be x. 5

2
5. Trace the curve y2   x  1 x  2  . Clearly

stating all the properties used for tracing it. 10

6. (a) If :

sin3 t cos3 t
x and y 
cos 2t cos 2t

dy 
then find at t  . 5
dx 6

P. T. O.
 [4] MTE–01

(b) If In   tan n x dx, prove that :

tan n 1 x
In  In 2 
 n  1

4 tan5
Using this find  0
xdx . 5

7. (a) Evaluate the integral : 5

2 5 x2
1 x2  4 x  3
dx

(b) Verify Lagrange’s mean value theorem for

the function f defined by : 5
f  x    x  1 x  2 x  3 , x  0, 4
 [5] MTE–01

MTE–01

2023

-01

%
(i)
(ii)

(iii)

x
(i) f :RR, f  x 
1  x2
x
] f o  f o f  x  
1  3 x2

P. T. O.
 [6] MTE–01

d  sin x
(ii)  cos t2 dt   cos x.

dx  x 

 
cos sin2 x  cos x2

(iii)

 x3  x2  16 x  20
 , x2
f  x    x  2 2

 k , x2

] x2 f k

(iv) f   x   0  x ] a, b[ ] f , ] a, b[

1
(v) f, f  x 
x   x
] ]1,  [

 1  x2  1 
1 
tan  tan 1 x
 x 
 

 x 
x
 e   dx
  1  x 2 
 
 [7] MTE–01

y2  4 a  x  a 
y2  4 a  x  a 

2,10
]
10 dx
2 2
x 4

x  ey y0 y  ln 2

 log x  x  xlog x x

2
y2   x  1 x  2 

sin3 t cos3 t
x y
cos 2t cos 2t

 dy
] t
6 dx

P. T. O.
 [8] MTE–01

In   tan n x dx, ]
tan n 1 x
In  In  2 
 n  1

4 tan5
 0
xdx

2 5 x2
1 x2  4 x  3
dx

f  x    x  1 x  2 x  3 , x  0, 4

f

MTE–01