sharda university noida and mte and their explanation about java and there is marks of all pages and importance of this project sharda university is located in noida and one of the best college for students

1. Page 1 of 9
Name: ………………………………………….
Student Sys Id.: ………….…………….…
Printed Pages:01
School of Engineering and Technology
Department of Computer Science and Engineering
Mid Term Examination (MTE), Sept-2020, Session :2020-21
[Programme] [Semester:] [Batch:]
Course Title: ………………………………………
Course Code: ………………………………………
Max Marks: 40
Instructions: 1. All questions are compulsory in Section A and Section B
Assume missing data suitably if any
CO1
CO2
Section A (1X20=20Marks)
All Questions are Compulsory
Sl. No Questions Marks CO
1.
The first derivative of 𝑦 = (1 − 𝑥/7)−7
at x=7 is
a.) 1
b.) 0
c.) ∞
d.) None of these
Correct option: (c)
1
CO1
2.
lim
𝑥→0
𝑒𝑡𝑎𝑛 𝑥
− 𝑒𝑥
𝑡𝑎𝑛 𝑥 − 𝑥
a.) 0
b.) 1
c.) ∞
d.) None of these
Correct option: (b)
1
3.
If 𝑥 = 𝑡 − 𝑠𝑖𝑛 𝑡 , 𝑦 = 1 − 𝑐𝑜𝑠 𝑡, value of 𝑑𝑦/𝑑𝑥 at 𝑡 = 𝜋/2 will be:
a) 0
b) 1
c) 𝜋
d) ∞
Correct option: (b)
1

2. Page 2 of 9
4.
If 𝑦 = 𝑙𝑜𝑔 𝑥 /𝑥, then
𝑑2𝑦
𝑑𝑥2 is given by
a)
2 𝑙𝑜𝑔 𝑥−3
𝑥3
b)
2(𝑙𝑜𝑔 𝑥−3)
𝑥3
c)
𝑙𝑜𝑔 𝑥−6
𝑥2
d) None of these
Correct answer: (a)
1
5.
If 𝑐𝑜𝑠 𝑥 = 𝑎0 + 𝑎1𝑥 + 𝑎2𝑥2
+ 𝑎3𝑥3
+ ⋯ , 𝑡ℎ𝑒𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 a3 =
a) 0
b) −1
c) 1
d)
1
3!
Correct answer: (a)
1
6.
𝑓(𝑥) = 𝑙𝑜𝑔 𝑥 can be expanded in powers of 𝑥 − 1 by using
a) Maclaurin’s theorem
b) Taylor’s theorem
c) Leibnitz theorem
d) Gregory’s theorem
Correct answer: (b)
1
7.
The maximum value of 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 is
a) 2
b) √2
c) 1
d) 1 + √2
Correct answer: (b)
1
8.
The function 𝑓(𝑥) = 𝑥3
− 6𝑥2
+ 24𝑥 + 4 has:
a) A maximum value at 𝑥 = 2
b) A minimum value at 𝑥 = 2
c) A maximum value at 𝑥 = 4 and a minimum at 𝑥 = 6
d) Neither maximum nor minimum at any point
Correct answer: (d)
1
9.
Maximum value of
𝑙𝑜𝑔 𝑥
𝑥
is
a) 𝑒
b)
1
𝑒
c) 0
d) 1
Correct answer: (b)
1

3. Page 3 of 9
10.
lim
𝑥→∞
𝑥 𝑡𝑎𝑛(1/𝑥) is:
a) 0
b) ∞
c) 1
d) −1
Correct answer: (c)
1
11.
If 𝑓(𝑥, 𝑦) = 𝑐, then
𝜕𝑦
𝜕𝑥
is:
a)
𝜕𝑓
𝜕𝑥
b)
𝜕𝑓
𝜕𝑦
c) −
𝜕𝑓
𝜕𝑥
𝜕𝑓
𝜕𝑦
d) −
𝜕𝑓
𝜕𝑦
𝜕𝑓
𝜕𝑥
Correct answer: (c)
1
12
If 𝑧 = 𝑐𝑜𝑠(𝑥𝑦3), then 𝜕2
𝑧/𝜕𝑥𝜕𝑦 =
a) −3𝑦2
𝑠𝑖𝑛(𝑥𝑦3) − 3𝑥𝑦5
𝑐𝑜𝑠(𝑥𝑦3)
b) 6𝑥𝑦 𝑠𝑖𝑛(𝑥𝑦3) − 9𝑥2
𝑦4
𝑐𝑜𝑠(𝑥𝑦3)
c) −6𝑥𝑦 𝑠𝑖𝑛(𝑥𝑦3) + 9𝑥2
𝑦4
𝑐𝑜𝑠(𝑥𝑦3)
d) 6𝑥𝑦 𝑠𝑖𝑛(𝑥𝑦3) + 9𝑥2
𝑦4
𝑐𝑜𝑠(𝑥𝑦3)
Correct answer: (a)
1
13
If 𝑧 = 𝑥𝑦𝑓(𝑥/𝑦), then 𝑥
𝜕𝑧
𝜕𝑥
+ 𝑦
𝜕𝑧
𝜕𝑦
=
a) 𝑧
b) 0
c) 1/𝑧
d) 2𝑧
Correct answer: (d)
1
14
If 𝑓(𝑥, 𝑦, 𝑧) = (𝑥2
+ 𝑦2
+ 𝑧2)−1/2
, then 𝑓𝑥𝑥 + 𝑓𝑦𝑦 + 𝑓𝑧𝑧 =
a) 0
b) 1
c) −1
d) None of these
Correct option: (b)
1
15
If 𝑢 = 𝑥2
− 𝑦2
, 𝑥 = 2𝑟 − 3𝑠 + 4, 𝑦 = −𝑟 + 8𝑠 − 5, then, 𝜕𝑢/𝜕𝑟 =
a) 4𝑥 + 2𝑦
b) 4𝑥 − 2𝑦
c) 8𝑥 − 6𝑦
d) 2𝑥 − 4𝑦
1

4. Page 4 of 9
Correct option: (a)
16
A function 𝑓(𝑥) has maximum value at 𝑥 = 𝑐 if:
a) 𝑓′(𝑐) = 0 and 𝑓′′(𝑐) > 0
b) 𝑓′(𝑐) = 0 and 𝑓′′(𝑐) < 0
c) 𝑓′(𝑐) = 0 and 𝑓′′(𝑐) ≠ 0
d) 𝑓′(𝑐) ≠ 0 and 𝑓′′(𝑐) < 0
Correct option: (b)
1
17
The critical point of 𝑓(𝑥, 𝑦) = 𝑥2
+ 𝑦2
+ 6𝑥 + 12 is
a) (−2,0)
b) (0, −3)
c) (3,0)
d) (−3,0)
Correct option: (d)
1
18
Conditions for 𝑓(𝑥, 𝑦) to the maximum are
a) 𝑓𝑥 = 0 = 𝑓𝑦; 𝑓𝑥𝑥𝑓
𝑦𝑦 < 𝑓𝑦𝑦
2
, 𝑓𝑥𝑥 < 0
b) 𝑓𝑥 = 0 = 𝑓𝑦; 𝑓𝑥𝑥𝑓
𝑦𝑦 > 𝑓𝑦𝑦
2
, 𝑓𝑥𝑥 < 0
c) 𝑓𝑥 = 0 = 𝑓𝑦; 𝑓𝑥𝑥𝑓
𝑦𝑦 > 𝑓𝑦𝑦
2
, 𝑓𝑥𝑥 > 0
d) 𝑓𝑥 = 0 = 𝑓𝑦; 𝑓𝑥𝑥𝑓
𝑦𝑦 = 𝑓𝑦𝑦
2
, 𝑓𝑥𝑥 < 0
Correct option: (b)
1
19
If 𝑓(𝑥) = 𝑐𝑜𝑠(𝑙𝑜𝑔 𝑥), then 𝑓(𝑥)𝑓(𝑦) −
1
2
[𝑓 (
𝑥
𝑦
) + 𝑓(𝑥𝑦)] has the value
a) −1
b)
1
2
c) −2
d) 0
Correct option: (d)
1
20
The 2𝑛𝑑 derivative of 𝑒𝑥
𝑐𝑜𝑠(𝑥) is
a) 2𝑒𝑥
𝑐𝑜𝑠 (𝑥 +
𝜋
2
)
b) 4𝑒𝑥
𝑐𝑜𝑠 (𝑥 +
𝜋
4
)
c) 𝑒𝑥
𝑐𝑜𝑠 (𝑥 +
𝑛𝜋
4
)
d) None of these
Correct option: (a)
1
Section B (1X20=20 Marks)
All Questions are Compulsory
21 1

5. Page 5 of 9
If ∫ 𝑔(𝑥)𝑑𝑥 = 𝑓(𝑥), then ∫ 𝑓(𝑥)𝑔(𝑥)𝑑𝑥 is equal to
a) 𝑙𝑜𝑔|𝑓(𝑥)|
b)
1
2
[𝑔(𝑥)]2
c)
1
2
[𝑓(𝑥)]2
d) None of these
Correct option: (c)
CO2
22
∫ 3𝑥√1 − 2𝑥2 𝑑𝑥 =
a) −
1
2
(1 − 2𝑥2)3/2
+ 𝐶
b) −(1 − 2𝑥2)3/2
c) (1 − 2𝑥2)3/2
+ 𝐶
d) None of these
Correct option: (a)
1
23
∫ 𝑠𝑒𝑐 𝑥 𝑑𝑥 =
a) 𝑙𝑛|𝑠𝑒𝑐 𝑥| + 𝐶
b) 𝑙𝑛|𝑠𝑒𝑐 𝑥 + 𝑡𝑎𝑛 𝑥| + 𝐶
c) 𝑙𝑛|𝑡𝑎𝑛 𝑥| + 𝐶
d) 𝑁𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒
Correct option: (b)
1
24
∫
1
𝑥2 − 4
𝑑𝑥 =
a)
1
4
𝑙𝑛 |
𝑥−2
𝑥+2
| + 𝐶
b)
1
4
𝑙𝑛 |
𝑥+2
𝑥−2
| + 𝐶
c) 𝑙𝑛 |
𝑥−2
𝑥+2
| + 𝐶
d) None of these
Correct option: (a)
1
25
∫ 𝑒𝑥
𝑠𝑖𝑛 𝑥 𝑑𝑥
a) 𝑒𝑥(𝑠𝑖𝑛 𝑥 − 𝑐𝑜𝑠 𝑥) + 𝐶
b)
𝑒𝑥
2
(𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥) + 𝐶
c) 𝑒𝑥(𝑠𝑖𝑛 𝑥 − 𝑐𝑜𝑠 𝑥)
d)
𝑒𝑥
2
(𝑠𝑖𝑛 𝑥 − 𝑐𝑜𝑠 𝑥) + 𝐶
Correct option: (d)
1
26 1

6. Page 6 of 9
∫
1
√1 − (𝑥 + 1)2
𝑑𝑥 =
a)
1
2
sin−1
𝑥
b) sin−1
𝑥
c) 2 sin−1
𝑥
d) None of these
27
The integration of ∫ 𝑓(𝑥)𝑔(𝑥)𝑑𝑥 using by parts is given by
a) 𝑓(𝑥) ∫ 𝑔(𝑥)𝑑𝑥 − ∫
𝑑𝑓(𝑥)
𝑑𝑥
(∫ 𝑔(𝑥)𝑑𝑥)𝑑𝑥
b) 𝑓(𝑥) ∫ 𝑔(𝑥)𝑑𝑥 + ∫
𝑑𝑓(𝑥)
𝑑𝑥
(∫ 𝑔(𝑥)𝑑𝑥)𝑑𝑥
c) 𝑓(𝑥) ∫ 𝑔(𝑥)𝑑𝑥 − ∫
𝑑𝑔(𝑥)
𝑑𝑥
(∫ 𝑓(𝑥)𝑑𝑥)𝑑𝑥
d) 𝑓(𝑥) ∫ 𝑔(𝑥)𝑑𝑥 + ∫ (
𝑑𝑔(𝑥)
𝑑𝑥
∫ 𝑓(𝑥)𝑑𝑥) 𝑑𝑥
Correct option: (a)
1
28 1
29
∫ √(𝑥2 + 𝑎2) 𝑑𝑥 =
a)
1
2
𝑥√𝑥2 + 𝑎2 +
𝑎2
2
𝑙𝑜𝑔|𝑥 + √𝑥2 + 𝑎2| + 𝐶
b) −
𝑎2
2
𝑙𝑜𝑔|𝑥 + √𝑥2 + 𝑎2| + 𝐶
c)
1
2
𝑥√𝑥2 + 𝑎2 + 𝐶
d) None of these
Correct option: (a)
1
30 1
31
Which of the following is false?
a) ∫ 𝑓(𝑥)𝑑𝑥
𝑏
𝑎
= ∫ 𝑓(𝑡)𝑑𝑡
𝑏
𝑎
b) ∫ 𝑓(𝑥)𝑑𝑥
𝑎
𝑏
= ∫ 𝑓(𝑡)𝑑𝑡
𝑏
𝑎
c) ∫ 𝑓(𝑥)𝑑𝑥
𝑏
𝑎
+ ∫ 𝑓(𝑥)𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥
𝑐
𝑎
𝑐
𝑏
d) None of these
Correct option: (b)
1

7. Page 7 of 9
32 1
33
The value of the double integral ∫ ∫ 𝑑𝑦𝑑𝑥
𝑒𝑥
1
2
0
is:
a) 𝑒2
+ 3
b) 𝑒2
− 3
c) 𝑒2
+ 12
d) 𝑒2
− 12
Correct option: (b)
1
34
After changing the order of integration, the integral ∫ ∫ 𝑑𝑦𝑑𝑥
𝑒𝑥
1
2
0
,
reduces to which of the following form:
(1) ∫ ∫ 𝑑𝑥𝑑𝑦
𝑒𝑥
1
2
0
(2) ∫ ∫ 𝑑𝑥𝑑𝑦
2
log 𝑦
𝑒2
1
(3) ∫ ∫ 𝑑𝑥𝑑𝑦
log 𝑥
1
𝑒
1
(4) ∫ ∫ 𝑑𝑥𝑑𝑦
𝑒𝑥
2
log𝑦
0
Correct answer: (b)
1
35
Maclaurin’s series of 𝑐𝑜𝑠 𝑥 is given as
a)𝑥 −
𝑥3
3
+
𝑥5
5
− − − − −
b) 1 −
𝑥2
2
+
𝑥4
4
− − − − −
c) 𝑥 −
𝑥2
2!
+
𝑥4
4!
− − − − −
d) 1 −
𝑥2
2!
+
𝑥4
4!
− − − − −
Answer: d
1
36
Minimum value of 2 sin𝑥 is given by
a) -1
b 1
c -2
d) 0
Answer : c
1

8. Page 8 of 9
Instructions:
i. There are Two section consist of 40 Questions in MTE Question paper.
ii. All questions are compulsory.
iii. The two tables (given below) must be filled by the paper setter.
iv. Paper setter must see the COs of the courses described in syllabus to frame the questions in
alignment with expected COs and Blooms taxonomy
Table1: Distribution of question & marks among the Units of syllabus
Unit Questions Numbers Total Marks
1 1to 20 20
2 21 to 40 20
37
If R is the region bounded by 𝑥𝑦 = 16, 𝑥 = 0, 𝑦 = 0 and 𝑥 = 8, then the
value of the double integral ∬ 𝑥2
𝑑𝑥𝑑𝑦
.
𝑅
is:
(1) 256 (2) 512 (3) 324 (4) 128
Correct option: (b)
1
38
If 𝑢 =
𝑥3+𝑦3
√𝑥+√𝑦
then 𝑢 is a
a) Homogeneous function of degree 5/2
b) Homogeneous function of degree 3/2
c) Non-Homogeneous function of Degree 5/2
d) None of these
Answer : a
1
39
The stationary point of the function 𝑓(𝑥, 𝑦) = 𝑥3
𝑦2
(1 − 𝑥 − 𝑦) for
extreme values is:
a) (1/2, 1/3)
b) (1/3, 1/2)
c) (2/3, 3/2)
d) None of these
Correct answer: (a)
1
40
The expansion of 𝑐𝑜𝑠(𝑥) in ascending powers of 𝑥 −
𝜋
4
is obtained by using
a) Maclaurin’s theorem
b) Taylor’s Theorem
c) Euler’ Theorem
d) Leibnitz theorem
Correct option: (b)
1

9. Page 9 of 9
Table 2: Mapping between COs and questions
(Number of COs may vary from course to course)
COs
Knowledge level
(Blooms taxonomy, K1,
K2. ..)
Questions Numbers Total Marks
CO1 K1,K2,K3 1to 20 20
CO2 K1,K2,K3 20 to 40 20
…. ….
(Name of Question paper setter)
Contact No:
* You may type the questions into this template and RENAME THE FILE with the subject code
Example: ECE223.
* Unit/chapter number can be considered for the syllabus if the same are not in modular form.
* Font to be used for ETE is “Times New Roman” and the text size is 11.