Ajay Kumar Garg Engineering College Gzb

COURSE: B. TECH. SEMESTER: IInd

SUBJECT: ENGINEERING MATHEMATICS II SUBJECT CODE:
KAS-203T

Module:1

Ordinary Differential Equation of Higher Order

MULTIPLE CHOICE QUESTIONS
4
𝑑𝑦 𝑑3 𝑦
1. What are the order and degree of the differential equations 𝑦 = 𝑥 𝑑𝑥 + (𝑑𝑥 3 )
(a) order = 4 and Deg = 2
(b) order = 3 and Deg = 4
(c) order = 2 and Deg = 1
(d) order = 4 and Deg = 3
(Ans: b)
2. What is the complementary function (𝐷3 − 3𝐷2 + 4)𝑦 = 0,
(a) 𝑐1 𝑒 −𝑥 + (𝑐2 + 𝑐3 𝑥) 𝑒 2𝑥

(b) 𝑐1 𝑒 −𝑥 + (𝑐2 + 𝑐3 𝑥)𝑒 𝑥

(c) 𝑒 −𝑥 + (𝑐2 + 𝑐3 𝑥)𝑒 3𝑥

(d) 𝑐1 𝑒 −𝑥 + 𝑐1 𝑒 2𝑥

(Ans:a)

3. What is Particular Integral of (4𝐷2 + 4𝐷 − 3)𝑦 = 𝑒 2𝑥

1
(a) 21 𝑒 2𝑥

1 2𝑥
(b) 𝑒
2

1
(c) 𝑒𝑥
21

1
(d) 21 𝑒 3𝑥

(Ans: a)

4. Suppose 𝑦𝑝 (𝑥) = 𝑥 𝑐𝑜𝑠 2 𝑥 is a particular solution of 𝑦 ″ + 𝛼𝑦 = −4 𝑠𝑖𝑛 2 𝑥. Then the

constant 𝛼 equals.
(a) 1

(b) -2

(c) 2

(d) 4

(Ans d)
𝑑2 𝑦
5. Solve the equation 𝑑𝑥 2 + 9𝑦 = 𝑠𝑖𝑛 4 𝑥
1
(a) 𝑦 = (𝑐1 𝑐𝑜𝑠 3𝑥 + 𝑐2 𝑠𝑖𝑛 3𝑥) − 7 𝑠𝑖𝑛 4 𝑥

1
(b) 𝑦 = (𝑐1 𝑐𝑜𝑠 𝑥 + 𝑐2 𝑠𝑖𝑛 𝑥) − 7 𝑠𝑖𝑛 4 𝑥

1
(c) 𝑦 = (𝑐1 𝑐𝑜𝑠 3𝑥 + 𝑐2 𝑠𝑖𝑛 𝑥) − 7 𝑠𝑖𝑛 𝑥

1
(d) 𝑦 = (𝑐1 𝑐𝑜𝑠 2𝑥 + 𝑐2 𝑠𝑖𝑛 2𝑥) + 𝑠𝑖𝑛 4 𝑥
7

(Ans : a)
𝑑2 𝑦 𝑑𝑦
6. The solution 𝑦(𝑥) of the differential equations𝑑𝑥 2 + 4 𝑑𝑥 + 4𝑦 = 0 Satisfying the
𝑑𝑦
condition 𝑦(0) = 4, 𝑑𝑥 (0) = 8 is
(a) 4𝑒 2𝑥
(b) (16𝑥 + 4)𝑒 −2𝑥
(c) 4𝑒 −2𝑥
(d) 4𝑒 −2𝑥 + 16𝑥𝑒 2𝑥
(Ans: b)
𝑑4 𝑦
(7) Solve 𝑑𝑥 4 − 𝑚4 𝑦 = 𝑐𝑜𝑠 𝑚 𝑥

𝑥
(a) 𝑦 = 𝑐1 𝑒 𝑚𝑥 + 𝑐2 𝑒 −𝑚𝑥 − 4𝑚3 𝑠𝑖𝑛 𝑚 𝑥

𝑥
(b) 𝑦 = 𝑐1 𝑐𝑜𝑠 𝑚 𝑥 + 𝑐2 𝑠𝑖𝑛 𝑚 𝑥 − 4𝑚3 𝑠𝑖𝑛 𝑚 𝑥

𝑥
(c) 𝑦 = 𝑐1 𝑒 𝑚𝑥 + 𝑐2 𝑒 −𝑚𝑥 + 𝑐3 𝑐𝑜𝑠 𝑚 𝑥 + 𝑐4 𝑠𝑖𝑛 𝑚 𝑥 − 4𝑚3 𝑠𝑖𝑛 𝑚 𝑥

(d) 𝑦 = 𝑐1 𝑒 𝑚𝑥 + 𝑐2 𝑒 −𝑚𝑥 + 𝑐3 𝑐𝑜𝑠 𝑚 𝑥 + 𝑐4 𝑠𝑖𝑛 𝑚 𝑥 − 𝑠𝑖𝑛 𝑚 𝑥

(Ans: c)
1
8. Find the Value of𝜑(𝐷)(𝐷−𝑎)𝑛 𝑒 𝑎𝑥 is equal to –
𝑥 𝑛 𝑒 𝑎𝑥
(a) 𝜑(𝑎)
𝑥 𝑛 𝑒 𝑎𝑥
(b) 𝑛 ! 𝜑(𝑎)
𝑥 𝑛 𝑒 𝑎𝑥
(c) ,  𝜑(𝑎) ≠ 0
𝜑(𝑎)
𝑥 𝑛 𝑒 𝑎𝑥
(d) , 𝜑(𝑎) ≠ 0
𝑛 ! 𝜑(𝑎)
(𝐴𝑛𝑠: 𝑑)
𝑑𝑥 𝑑𝑦
9. The points x & y lie on where x & y are solution for = −𝜔𝑦,  = 𝜔𝑥.
𝑑𝑡 𝑑𝑡

(a) parabola
(b) straight line
(c) circle
(d) ellipse
(Ans: c)
𝑑𝑥 𝑑𝑦
10. Solve + 4𝑥 + 3𝑦 = 0, 𝑑𝑡 + 2𝑥 + 5𝑦 = 0
𝑑𝑡

(a) 𝑥 = 𝑐1 𝑒 2𝑡 + 𝑐2 𝑒 5𝑡 , 𝑦 = 𝑐3 𝑒 7𝑡 + 𝑐4 𝑒 2𝑡

(b) 𝑥 = 𝑐1 𝑒 −2𝑡 + 𝑐2 𝑒 −7𝑡 ,𝑦 = 𝑐3 𝑒 −2𝑡 + 𝑐4 𝑒 −7𝑡

(c) 𝑥 = 𝑐1 𝑒 3𝑡 + 𝑐2 𝑒 4𝑡 , 𝑦 = 𝑐3 𝑒 2𝑡 + 𝑐4 𝑒 5𝑡

(d) 𝑥 = 𝑐1 𝑒 2𝑡 + 𝑐2 𝑒 𝑡 ,𝑦 = 𝑐3 𝑒 2𝑡 + 𝑐4 𝑒 −7𝑡

(Ans: b)

𝑑2 𝑦
11. The solution of 𝑑𝑥 2 − 𝑦 = 𝑘 (here k is a non-zero constant), which vanishes when x=0
and which tends to finite limit as x tends to infinity is
(𝑎) 𝑦 = 𝑘(1 + 𝑒 −𝑥 )
(𝑏) 𝑦 = 𝑘(𝑒 −𝑥 − 1)
(𝑐)𝑦 = 𝑘(1 + 𝑒 −𝑥 + 𝑒 𝑥 )
(d) 𝑦 = 𝑘(1 + 2𝑒 −𝑥 )
(𝐴𝑛𝑠: 𝑏)
𝑑2 𝑦 𝑑𝑦
12. Solve the Linear Diff Eq. 𝑑𝑥 2 − 𝑐𝑜𝑡 𝑥 𝑑𝑥 − (1 − 𝑐𝑜𝑡 𝑥)𝑦 = 𝑒 𝑥 𝑠𝑖𝑛 𝑥
𝑒 𝑥 𝑐𝑜𝑠 𝑥
(a) 𝑦 = 𝑐2 𝑒 𝑥 − − 𝑒 −𝑥 (2 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥)
2

𝑒 𝑥 𝑐𝑜𝑠 𝑥 𝑐1
(b) 𝑦 = 𝑐2 𝑒 𝑥 − 2
− 5
𝑒 −𝑥 (2 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥)
 𝑒 𝑥 𝑐𝑜𝑠 𝑥 𝑐1
(c) 𝑦 = 𝑐2 − − 𝑒 −𝑥 (2 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥)
2 5

𝑒 𝑥 𝑐𝑜𝑠 𝑥 𝑐1
(d) 𝑦 = 𝑐2 𝑒 𝑥 − − (𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥)
2 5

(Ans: b)
𝑑2 𝑦 𝑑𝑦
13. If y = x is a solution of of 𝑥 2 +𝑥 − 𝑦 = 0 , then the second linearly independent
𝑑𝑥 2 𝑑𝑥
solution of this equation is
1
(𝑎) 𝑥
1
(𝑏) 𝑥 2
(𝑐) 𝑥 2
(d) Constant
(Ans: a)

𝑑2 𝑦 𝑑𝑦 2
14. Solve by normal form − 4𝑥 𝑑𝑥 + (4𝑥 2 − 1)𝑦 = −3𝑒 𝑥 𝑠𝑖𝑛 2 𝑥
𝑑𝑥 2

2
(a) 𝑦 = 𝑒 𝑥 (𝑐1 𝑐𝑜𝑠 𝑥 + 𝑐2 𝑠𝑖𝑛 𝑥 + 𝑠𝑖𝑛 2 𝑥)
2
(b) 𝑦 = 𝑒 𝑥 (𝑐1 𝑐𝑜𝑠 2𝑥 + 𝑐2 𝑠𝑖𝑛 2𝑥 + 𝑠𝑖𝑛 2 𝑥)
2
(c) 𝑦 = 𝑒 𝑥 (𝑐1 𝑐𝑜𝑠 2𝑥 + 𝑐2 𝑠𝑖𝑛 𝑥 + 𝑠𝑖𝑛 𝑥)

(d) 𝑦 = 𝑒 𝑥 (𝑐1 𝑐𝑜𝑠 𝑥 + 𝑐2 𝑠𝑖𝑛 𝑥 + 𝑠𝑖𝑛 2 𝑥)

(Ans: a)

15. A part of C.F for 𝑦 ″ − 𝑐𝑜𝑡 𝑥 𝑦 ′ − (1 − 𝑐𝑜𝑡 𝑥)𝑦 = 𝑒 𝑥 𝑠𝑖𝑛 𝑥 is

(a) cot x (b) sin x (c) 𝑒 𝑥 (d) 𝑒 −𝑥

(Ans: c)
𝑑2 𝑦 𝑑𝑦
16. Solve by changing the independent variable: (1 + 𝑥 2 )2 𝑑𝑥 2 + 2𝑥(1 + 𝑥 2 ) 𝑑𝑥 + 4𝑦 =
0
(a) 𝑦 = 𝑐1 𝑐𝑜𝑠(𝑡𝑎𝑛−1 𝑥) + 𝑐2 𝑠𝑖𝑛(2 𝑡𝑎𝑛−1 𝑥)

(b) 𝑦 = 𝑐1 𝑐𝑜𝑠(2 𝑡𝑎𝑛−1 𝑥) + 𝑐2 𝑠𝑖𝑛(2 𝑡𝑎𝑛−1 𝑥)

(c) 𝑦 = 𝑐1 𝑐𝑜𝑠(2 𝑡𝑎𝑛−1 𝑥) + 𝑐2 𝑠𝑖𝑛(𝑡𝑎𝑛−1 𝑥)

(d) 𝑦 = 𝑐1 𝑐𝑜𝑠(𝑡𝑎𝑛−1 𝑥) + 𝑐2 𝑠𝑖𝑛(𝑡𝑎𝑛−1 𝑥)

(Ans: b)

𝒅𝟐 𝒗 𝒅𝟐 𝒚 𝒅𝒚 𝟏 𝟏 𝟔
17. If 𝒅𝒙𝟐 + 𝑰𝒗 = 𝑺is the normal form of 𝒅𝒙𝟐 + 𝒙−𝟏/𝟑 𝒅𝒙 + (𝟒𝒙𝟐/𝟑 − 𝟔𝒙𝟒/𝟑 − 𝒙𝟐 ) 𝒚 =
𝟎 obtained by solving change of dependent variable, then the value of I is
 (a) 1
(b) 0
(c) 6x −2
(d)- 6x −2
(Ans: d)

𝑑2 𝑦 2
18.Solve by method of variation of parameters 𝑑𝑥 2 − 𝑦 = 1+𝑒 𝑥
(a) 𝑦 = 𝑒 𝑥 𝑙𝑜𝑔(𝑒 −𝑥 + 1) + 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥 − 𝑒 −𝑥 𝑙𝑜𝑔(𝑒 𝑥 + 1)
(b) 𝑦 = 𝑒 𝑥 𝑙𝑜𝑔(𝑒 −𝑥 + 1) + 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥 − 𝑒 −𝑥 𝑙𝑜𝑔(𝑒 𝑥 ) − 1

(c) 𝑦 = 𝑒 𝑥 𝑙𝑜𝑔(𝑒 −𝑥 + 1) + 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥 − 𝑙𝑜𝑔(𝑒 𝑥 + 1) − 1

(d) 𝑦 = 𝑒 𝑥 𝑙𝑜𝑔(𝑒 −𝑥 ) + 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥 − 𝑒 −𝑥 𝑙𝑜𝑔(𝑒 𝑥 + 1)

(Ans: c)
𝑑2 𝑦 𝑑𝑦 4
19. A particular solution of 4 𝑥 2 + 8𝑥 +𝑦 = is
𝑑𝑥 2 𝑑𝑥 √𝑥

𝑙𝑜𝑔𝑥 𝑙𝑜𝑔 (𝑙𝑜𝑔𝑥)2

(𝑎) (𝑏) (𝑐) 𝑥 2 𝑙𝑜𝑔𝑥 (d)
𝑥 𝑥2 2√𝑥

(Ans: d)
𝑑2 𝑦 𝑑𝑦
20. Solve by method of variation of parameters 𝑑𝑥2 − 2 𝑑𝑥 = 𝑒 𝑥 𝑠𝑖𝑛 𝑥
(a) 𝑦 = 𝑐1 + 𝑐2 𝑒 2𝑥 − 𝑒 𝑥 𝑠𝑖𝑛 𝑥
1
(b) 𝑦 = 𝑐1 + 𝑐2 𝑒 2𝑥 − 2 𝑒 𝑥 𝑠𝑖𝑛 𝑥

1
(c) 𝑦 = 𝑐1 + 𝑐2 𝑒 2𝑥 − 3 𝑒 𝑥 𝑠𝑖𝑛 𝑥

1
(d) 𝑦 = 𝑐1 + 𝑐2 𝑒 𝑥 − 2 𝑒 𝑥 𝑠𝑖𝑛 𝑥

(Ans: b)

21. The differential equation 𝑆𝑖𝑛2 𝑥 𝑦 ′′ + 𝑆𝑖𝑛𝑥 𝐶𝑜𝑠𝑥 𝑦 ′ + 4 𝑦 = 0 is solved by changing

the Independent
variable 𝑥 into independent variable 𝑧 then
x x x
(a) z = 2logtan 2 (b) z = 2logcot 2 (c) z = 2logcos 2 (d) z =
2cosx
(Ans: a)

22. Find particular integral of (𝐷 − 2)2 𝑦 = 8𝑥 2

(a) 2𝑥 2 − 4𝑥 − 3
(b) 2𝑥 2 + 4𝑥 − 3

(c) 2𝑥 2 − 4𝑥 + 3

(d) 2𝑥 2 + 4𝑥 + 3
(Ans:d)
𝑑
23. The particular integral of (𝐷3 + 𝑎2 𝐷)𝑦 = 𝑠𝑖𝑛𝑎𝑥 , 𝐷 ≡ 𝑑𝑥 is

−𝑥 −𝑥 −𝑥 −𝑥
(𝑎) cos 𝑎𝑥 (𝑏) cos 𝑎𝑥 (𝑐) 2𝑎2 𝑠𝑖𝑛𝑎𝑥 (𝑑) 2𝑎2 cos 𝑎𝑥 𝑠𝑖𝑛𝑎𝑥
2𝑎 2𝑎2

(Ans: c)

24. Find the P.I. of (𝐷3 − 𝐷)𝑦 = 2𝑐𝑜𝑠ℎ𝑥 , is

1
(𝑎) (𝑒 𝑥 + 𝑒 −𝑥 )
2
1
(𝑏) 𝑥 2 (𝑒 𝑥 + 𝑒 −𝑥 )
1
(𝑐) 𝑥 2 (𝑒 𝑥 + 𝑒 −𝑥 )
2
1
(𝑑) (𝑒 𝑥 − 𝑒 −𝑥 )
2
(Ans: b)

𝑑2 𝑦 𝑑𝑦
25. 𝑑𝑥 2 + 𝑐𝑜𝑡𝑥 𝑑𝑥
+ 4 𝑐𝑜𝑠𝑒𝑐 2 𝑥. 𝑦 = 0, reduce form by changing the independent
variable:

𝑑2 𝑦
(a) −𝑦 =0
𝑑𝑧 2
𝑑2 𝑦
(b) +𝑦 =0
𝑑𝑧 2

𝑑2 𝑦
(c) + 2𝑦 = 0
𝑑𝑧 2

𝑑2 𝑦
(d) − 2𝑦 = 0
𝑑𝑧 2

(Ans: b)

𝑑2 𝑦 𝑑𝑦
26. If 𝑑𝑥 2 + 𝑐𝑜𝑡 𝑥 𝑑𝑥 − (1 − 𝑐𝑜𝑡 𝑥)𝑦 = 𝑒 𝑥 𝑠𝑖𝑛 𝑥 , Find one part of solution i.e. value of u
(a) 𝑢 = 𝑒 −𝑥
(b) 𝑢 = 𝑒 𝑥

(c) 𝑢 = 𝑒 2𝑥

(d) 𝑢 = 𝑒 3𝑥

(Ans: a)
𝑑𝑦 𝑑𝑦
27. Find order and degree of (𝑥 + y)(𝑥 + 𝑦)2 (𝑥 𝑑𝑥 + y) = 𝑥y (1+ 𝑑𝑥
)
(a) (1,1)
(b) (1,2)
(c) (2,1)
 (d) None of these
(e) (Ans: a)
28. When three roots of a linear differential equation are equal then its C.F is
(a) (𝑐1 + 𝑥𝑐2 + 𝑥 2 𝑐3 )𝑒 𝑚𝑥
(b) (𝑐1 + 𝑥𝑐2 + 𝑥 2 𝑐3 )𝑒 −𝑚𝑥
(c) (𝑐1 − 𝑥𝑐2 − 𝑥 2 𝑐3 )𝑒 𝑚𝑥
(d) None of these
(Ans: a)
29. Find P.I. of (𝐷 2 − 4𝐷 + 3)𝑦 = 𝑒 3𝑥
𝑥
(a) P.I.= 𝑒 −2𝑥 𝑒 𝑒
𝑥
(b) P.I.=2 𝑒 3𝑥
𝑥
(c) P.I.= 𝑒 −2𝑥 𝑒 𝑒
(d) None of these
(Ans: b)
𝑑3 𝑦 𝑑2 𝑦 𝑑𝑦
30. P.I of 𝑑𝑥 3
--2 𝑑𝑥2 + 4 𝑑𝑥 -8y = 8
(a) 𝑦=1
(b) 𝑦 = −1
(c) 𝑦=2
(d) None of these
(Ans: b)
31. Determine the differential equation whose general solution is𝑦 = (𝑐1 + 𝑐2 𝑥 + 𝑐3 𝑥 2 )𝑒 𝑥 .
(a) 𝑦"′ − 3𝑦′′ + 3𝑦′ − 𝑦 = 0
(b) 𝑦 ′′′ + 3𝑦 ′′ + 2𝑦 ′ = 0
(c) 𝑦 ′′′ − 3𝑦 ′′ + 2𝑦 ′ = 0
(d) None of these
(Ans: a)
𝑑4 𝑦
32. If --𝑚4 𝑦 = 0 then its C.F will be
𝑑𝑥 4
(a) (𝑐1 𝑒 𝑚𝑥 + 𝑐2 𝑒 −𝑚𝑥 + 𝑐3 𝑐𝑜𝑠𝑚𝑥 + 𝑐4 𝑠𝑖𝑛𝑚𝑥)
(b) (𝑐1 𝑒 −𝑚𝑥 + 𝑐2 𝑒 −𝑚𝑥 + 𝑐3 𝑐𝑜𝑠𝑚𝑥 + 𝑐4 𝑠𝑖𝑛𝑚𝑥)
(c) (𝑐1 𝑒 𝑚𝑥 + 𝑐2 𝑒 𝑚𝑥 + 𝑐3 𝑐𝑜𝑠𝑚𝑥 + 𝑐4 𝑠𝑖𝑛𝑚𝑥)
(d) None of these
(Ans: a)
𝑑2 𝑦 𝑑𝑦
33. If +2 + 10y + 37𝑠𝑖𝑛3𝑥 = 0 , It P.I will be
𝑑𝑥 2 𝑑𝑥
(a) 6𝑐𝑜𝑠3𝑥 − 𝑠𝑖𝑛3𝑥
(b) 𝑐𝑜𝑠3𝑥 − 𝑠𝑖𝑛3𝑥
(c) 𝑐𝑜𝑠3𝑥 − 6𝑠𝑖𝑛3𝑥
(d) None of these
(Ans: a)
𝑑2 𝑦
34. If – y = 1, which vanishes when x = 0 and tends to a finite limit as x tending to infinity
𝑑𝑥 2
the y will be
(a) 𝑒 2𝑥 − 1
(b) 𝑒 𝑥 − 1
(c) 0
(d) None of these
(Ans: b)
𝑑2 𝑦 𝑑𝑦
36. If −2 + y = 𝑒 𝑥 𝑥 𝑠𝑖𝑛𝑥, It P.I will be
𝑑𝑥 2 𝑑𝑥
𝑥
(a) 𝑒 (−𝑥𝑐𝑜𝑠𝑥 + 2𝑠𝑖𝑛𝑥)
(b) 𝑒 𝑥 (𝑥𝑐𝑜𝑠𝑥 + 2𝑠𝑖𝑛𝑥)
(c) 𝑒 𝑥 (𝑥𝑐𝑜𝑠𝑥 − 2𝑠𝑖𝑛𝑥)
(d) None of these
 (Ans:a)
37. The process of formation of the differential equation is given in the wrong order, select the
correct option from below given options.
(1) Eliminate the arbitrary constant
𝑑𝑦
(2) Differential equation which involves x,𝑥, 𝑦, 𝑑𝑥
(3) Differentiating the given equation w.r.t x as many times as the number of arbitrary
constants.
(a) 1,2,3
(b) 3,1,2
(c) 2,1,3
(d) None of these
(Ans:c)
𝑑2 𝑦 𝑑𝑦
38. The differential equation 𝑑𝑥2 + 2 𝑑𝑥 + y = 0
(a) Second order linear
(b) Nonlinear
(c) Linear with fixed constants
(d) Undeterminable to be linear or nonlinear
(Ans: a)
39. A differential equation is considered to be ordinary if it has
(a) One dependent variable
(b) More than one dependent variable
(c) One independent variable
(d) More than one independent variable
(Ans: a)
40. If the root of A.E. are (-1000, -1000) then C.F.
(a) (𝑐1 + x𝑐2 )𝑒 −1000𝑥
(b) (𝑐1 − x𝑐2 )𝑒 −1000𝑥
(c) (𝑐1 + x𝑐2 )𝑒 1000𝑥
(d) None of these
(Ans: a)
41. The general solution of (x 2 D2 – x D) y = 0 is
(a) y = (c1 + c2 ex )
(b) y = (c1 + c2 x)
(c) y = (c1 + c2 x 2 )
(d) None of these
(Ans: c)
42. For what value of ω does the system described by y′′ + 9y = 4 cos(ωt) exhibit resonance?
(a) 0
(b) 3
(c) 4
(d) None of these
(Ans:b)
𝑑4 𝑦
43. P.I. of 𝑑𝑥 4 − 𝑦 = 𝑐𝑜𝑠 𝑥 𝑐𝑜𝑠ℎ 𝑥
(a) 𝑐𝑜𝑠ℎ𝑥 𝑐𝑜𝑠𝑥
(b) 𝑐𝑜𝑠ℎ𝑥 𝑐𝑜𝑠ℎ2𝑥
1
(c) − 5 𝑐𝑜𝑠 𝑥 𝑐𝑜𝑠ℎ 𝑥
(d) None of these
(Ans: c)
𝑑2 𝑦 𝑑𝑦
44. Solution of 𝑑𝑥 2
+ 2𝑝 𝑑𝑥 + ( 𝑝2 + 𝑞 2 )𝑦 = 𝑒 2𝑥
𝑒 2𝑥
(a) 𝑦 = 𝑒 −𝑝𝑥 (𝑐1 𝑐𝑜𝑠𝑞𝑥 + 𝑐2 𝑠𝑖𝑛𝑞𝑥) + (2+𝑝)2 +𝑞2
 𝑒 2𝑥
(b) 𝑦 = 𝑒 𝑝𝑥 (𝑐1 𝑐𝑜𝑠𝑞𝑥 + 𝑐2 𝑠𝑖𝑛𝑞𝑥) + (2+𝑝)2 +𝑞2
(c) 𝑦 = 𝑒 𝑝𝑥 (𝑐1 𝑐𝑜𝑠𝑞𝑥 + 𝑐2 𝑠𝑖𝑛𝑞𝑥)
(d) None of these
(Ans:a)
𝑑2 𝑦 𝑑𝑦 𝜋
45. The complete solution of 𝑑𝑥 2
+ 2 𝑑𝑥 + 10𝑦 + 37 𝑠𝑖𝑛3𝑥 = 0, given that when 𝑥 = 2
being
𝑑𝑦
given that 𝑦 = 3 𝑎𝑛𝑑 𝑑𝑥
= 0 when 𝑥 = 0
(a) 𝑦 = 0
(b) 𝑦 = 1
(c) 𝑦 = 𝑠𝑖𝑛𝑥
(d) None of these
(Ans: b)
𝑑4 𝑦 𝑑2 𝑦
46. Solution of 𝑑𝑥 4
+ 2𝑛2 𝑑𝑥 2 + 𝑛4 𝑦 = 𝑐𝑜𝑠𝑚𝑥 when m ≠ 𝑛
𝑐𝑜𝑠𝑚𝑥
(a) 𝑦 = (𝑐1 + 𝑥𝑐2 )(𝑐3 𝑐𝑜𝑠𝑛𝑥 + 𝑐4 𝑠𝑖𝑛𝑛𝑥) + (𝑛2 −𝑚2 )2
𝑠𝑖𝑛𝑚𝑥
(b) 𝑦 = (𝑐1 + 𝑥𝑐2 )(𝑐3 𝑐𝑜𝑠𝑛𝑥 + 𝑐4 𝑠𝑖𝑛𝑛𝑥) + (𝑛2 −𝑚2 )2
(c) 𝑦 = (𝑐1 + 𝑥𝑐2 )(𝑐3 𝑐𝑜𝑠𝑛𝑥 + 𝑐4 𝑠𝑖𝑛𝑛𝑥)
(d) None of these
(Ans:a)
47. Solution of 𝑦 ′′ -2𝑦 ′ +2y= 𝑥 + 𝑒 𝑥 𝑐𝑜𝑠 𝑥 is
1 1
(a) 𝑦 = 𝑒 𝑥 (𝑐1 𝑐𝑜𝑠 𝑥 + 𝑐2 𝑠𝑖𝑛 𝑥) + 2 (𝑥 + 1) + 2 𝑥𝑒 𝑥 𝑠𝑖𝑛 𝑥
1 1
(b) 𝑦 = 𝑒 𝑥 (𝑐1 𝑐𝑜𝑠 𝑥 + 𝑐2 𝑠𝑖𝑛 𝑥) + 2 (𝑥 + 1) + 2 𝑥𝑒 𝑥 𝑐𝑜𝑠𝑥
1 1
(c) 𝑦 = 𝑒 𝑥 (𝑐1 𝑐𝑜𝑠 𝑥 + 𝑐2 𝑠𝑖𝑛 𝑥) + 2 (𝑥 + 1) − 2 𝑥𝑒 𝑥 𝑠𝑖𝑛 𝑥
(d) None of these
(Ans:a)
48. Solution of (𝐷 2 − 4𝐷 + 4)𝑦 = 8𝑥 2 𝑒 2𝑥 𝑠𝑖𝑛 2 𝑥 is
(a) 𝑦 = 𝑒 2𝑥 [𝑐1 + 𝑐2 𝑥 + (3 − 2𝑥 2 ) 𝑠𝑖𝑛 2 𝑥 + 4𝑥 𝑐𝑜𝑠 2 𝑥]
(b) 𝑦 = 𝑒 2𝑥 [𝑐1 + 𝑐2 𝑥 + (3 + 2𝑥 2 ) 𝑠𝑖𝑛 2 𝑥 − 4𝑥 𝑐𝑜𝑠 2 𝑥]
(c) 𝑦 = 𝑒 2𝑥 [𝑐1 + 𝑐2 𝑥 + (3 − 2𝑥 2 ) 𝑠𝑖𝑛 2 𝑥 − 4𝑥 𝑐𝑜𝑠 2 𝑥]
(d) None of these
(Ans: b)
𝑑3 𝑦 𝑑2 𝑦 𝑑𝑦
49. The solution of the homogeneous linear differential equation is 𝑥 3 𝑑𝑥 3 + 3𝑥 2 𝑑𝑥 2 + 𝑥 𝑑𝑥 =
24𝑥 2
(a) 𝑦 = 𝑐1 + 𝑐2 𝑙𝑜𝑔 𝑥 + 𝑐3 (𝑙𝑜𝑔 𝑥)2 − 3𝑥 2
(b) 𝑦 = 𝑐1 + 𝑐2 𝑙𝑜𝑔 𝑥 + 𝑐3 (𝑙𝑜𝑔 𝑥)2 + 3𝑥 2
(c) 𝑦 = 𝑐1 + 𝑐2 𝑙𝑜𝑔 𝑥 − 𝑐3 (𝑙𝑜𝑔 𝑥)2 − 3𝑥 2
(d) None of these
(Ans: b)
d
50. The solution of the differential equation (D 2 + 1)y = 2 is D 
dx
(a) y = C1 cos x + C2 sin x + 2
(b) y = C1 cos x + C2 sin x − 2
(c) y = C1 + C2 x + 2
(d) None of these
(Ans: a)
𝑑
51. The complete solution of (𝐷2 − 3𝐷 + 4)𝑦 = 0, 𝐷 ≡ is
𝑑𝑥
(a) y = c1e − x + c 2 e 4 x
 (b) y = c1x + c 2 + x
(c) y = (c1 + c 3 x )e x
(d) None of these
(Ans:d)
52. The complementary function of (𝐷2 + 2𝐷 + 1)𝑦 = 𝑥 − 1, is
(a) (c1 + c 2 x )e x
(b) (c1 + c 2 x )e − x
(c) y = c1e x + c 2 e − x
(d) None of these
(Ans:b)
1 1
53. To find 𝑓(𝐷) 𝑒 𝑎𝑥 𝑋,if we have brought e ax to the left from right of 𝑓(𝐷),then D
must be replaced by
(a) D-a
(b) a
(c) D+a
(d) None of these
(Ans:c)
54. Particular integral of (𝑥 2 𝐷2 + 5𝑥𝐷 + 4)𝑦 = 𝑥 𝑙𝑜𝑔 𝑥is given by
1
(a) x log x
9
2
(b) x
27
1 2
(c) x log x − x
9 27
(d) None of these
(e) (Ans:c)
𝑑2 𝑦 𝑑𝑦
55. The complete solution of 𝑥 2 𝑑𝑥 2 − 𝑥 𝑑𝑥 + 2𝑦 = 𝑥 𝑙𝑜𝑔 𝑥is
(a) xc1 cos(log x) + c 2 sin (log x)
(b) x log x
(c) (a)+(b)
(d) None of these
(Ans:c)
𝑑2 𝑦 𝑑𝑦
56. If P+Qx=0, then the part of complementary function of 𝑑𝑥 2 + 𝑃 𝑑𝑥 + 𝑄𝑦 = 𝑅is
(a) y=x
(b) y = x 2
(c) y = x 3
(d) y = x 4
(Ans:a)
𝑑2 𝑦 𝑑𝑦
57. y = x 2 is a part of complementary function of 𝑑𝑥 2 + 𝑃 𝑑𝑥 + 𝑄𝑦 = 𝑅if
(a) 1+P+Q=0
(b) 2 + 2Px + Qx 2 = 0
(c) 1-P+Q=0
 (d) P+Qx=0
(Ans:b)
58. If y = e mx is a solution of linear differential equation of 2nd order
d2y dy
+P + Qy = R ,then
dx 2
dx
(a) m 2 + Pm + Q = 0
(b) m 2 + Pm + Q = 1 .
(c) m 2 + Pm + Q  0
(d) None of these
(Ans:d)
2
𝑑4 𝑦 𝑑3 𝑦 𝑑2 𝑦 𝑑𝑦
59. The order of the differential equation − 3( ) +4 −5 + 6𝑦 = 0is
𝑑𝑥 4 𝑑𝑥 3 𝑑𝑥 2 𝑑𝑥
(a) 3
(b) 6
(c) 4
(d) 2
(Ans: c)
60. Particular integral of the differential equation (𝐷2 + 𝐷 + 1)𝑦 = 𝑒 𝑥 is
1
(a) e x
3
(b) 3e x
(c) e x
(d) None of these
(Ans:a)
61. The solution of differential equation (𝐷2 + 1)𝑦 = 0is
(a) y = c1e − x + c 2 e x
(b) y = c1 cos x + c 2 sin x
(c) (c1 + c 2 x)cos x + (c 3 + c 4 x)sin x
(d) None of these
(Ans:b)
62. For the differential equation F(D )y = e ax ,if F(a) = 0 ,then
1 ax
e is given by
F(D )
1 ax
(a) e
F(a )
1
(b) e ax
F(− a )
1
(c) e ax
F(D + a )
(d) None of these
(e) (Ans: d)
63. If F(− a 2 ) = 0 ,then which of the following is correct:
1 1
(a) sin ax = sin ax
F(D )2
F(− a 2 )
 1 x
(b) sin ax = − cos ax
D +a
2 2
2a
1 x
(c) 2 sin ax = cos ax
D + a2 2a
(d) None of these
(Ans:b)
𝑑2 𝑦 𝑑𝑦
64. The solution of the differential equation 𝑑𝑥 2 − 3 𝑑𝑥 + 2𝑦 = 𝑒 𝑥 is
(a) y = c1e x + c 2 e 3x + x
(b) (c1 + c 2 )e x − xe x
(c) y = c1e x + c 2 e 2 x − xe x
(d) y = c1 x + c 2 e x − e 2 x
(Ans: c)
𝑑2 𝑦 𝑑𝑦
65. Complementary function of the differential equation 𝑥 2 𝑑𝑥 2 + 4𝑥 𝑑𝑥 + 2𝑦 =
𝑒 𝑥 is
(a) c1 + c 2 x
(b) c1x −2 + c 2 x −1
(c) c1e − x + c 2 e x
(d) None of these
(Ans: b)
𝑑3 𝑦 𝑑2 𝑦
66. Complementary function of the differential equation 𝑥 4 𝑑𝑥 3 + 2𝑥 3 𝑑𝑥 2 −
𝑑𝑦
𝑥 2 𝑑𝑥 + 𝑥𝑦 = 1is
(a) (c1 + c 2 x )e x + c 3 e − x
(b) c1e x + c 2 e − x + c 3 e 2 x
c
(c) (c1 + c 2 log x )x + 3 +
1
log x
x 4x
(d) None of these
(Ans:d)
𝑑2 𝑦 𝑑𝑦
67. Complementary function of the equation (𝑥 + 𝑎)2 𝑑𝑥 2 − 4(𝑥 + 𝑎) 𝑑𝑥 + 6𝑦 =
𝑥is
(a) 𝑐1 𝑥 2 + 𝑐2 𝑥 3
(b) 𝑐1 (𝑥 + 𝑎)2 + 𝑐2 (𝑥 + 𝑎)3
(c) 𝑐1 𝑒 3𝑥 + 𝑐2 𝑒 2𝑥
(d) None of these
(Ans: b)

68. The differential Equation formed by eliminating the arbitrary constant from
𝑦 2 = (x-c)2
(a) y’2 = -1
(b) y’2 = 1
(c) y2 = -1
(d) y2 = -1
(Ans:b)
 𝑑4 𝑦 𝑑𝑦
69. The degree of the differential equation (𝑑𝑥 4 )3/2 = 1 - 2 𝑑𝑥 is
(a) 4
(b) 3
(c) 2
(d) 1
(Ans:b)
70. Solution for (D2 + 1)2 (D – 2) y = 0 is
(a) y = (c1 + c2x ) e-x + (c3 + c4 x ) ex + c5e2x
(b) y = (c1 + c2x ) cos x + (c3 + c4 x ) sin x + c5e2x
(c) y = (c1 + c2x ) cos x + (c3 + c4 x ) sin x + c5ex
(d) None of these
(Ans:b)
71. P.I. for y” + 6y’ + 9y = 7e2x
𝑒 2𝑥
(a) 7 25
𝑒 2𝑥
(b) 7
17
𝑒 2𝑥
(c) 25
𝑒𝑥
(d) 7 25
(Ans:a)
72. Solution of the differential equation y’’ +2 y’+ y = 0 , y (0) = 1, y’(0) = -1 is
(a) xe-x
(b) -xe-x
(c) -e-x
(d) e-x
(Ans: d)
 Ajay Kumar Garg Engineering College Gzb .
Course: B. Tech. Semester: IInd
Subject: Engg. Mathematics-I Subject Code: KAS-203T

Module:2

MULTIVARIABLE CALCULUS II
MULTIPLE CHOICE QUESTIONS

1. Value of 𝜞(𝒏 + 𝟏)

(a) 𝛤(𝑛 + 1)
(b) n
(c) 𝛤(𝑛)
(d) n𝛤(𝑛)

Ans: (d)
∞
2. Evaluate ∫𝟎 √𝒙𝒆−𝒙 𝒅𝒙

(a) 12 √𝜋
(b) √𝜋
(c) 32 √𝜋
(d) 12 𝜋
Ans: (a)
𝝅 𝝅
𝒅𝜽
3. Value of ∫𝟎 𝟐 𝒙 ∫𝟎 √𝒔𝒊𝒏 𝜽 𝒅𝜽
𝟐
√𝒔𝒊𝒏 𝜽

(a) 3 𝜋
(b) 2𝜋
(c) 𝜋
(d) √𝜋
Ans: (c )

𝟏
4. Value of ∫𝟎 𝒙𝒎−𝟏 (𝟏 − 𝒙𝟐 )𝒏−𝟏 𝒅𝒙

1 𝑚
(a) 2 𝐵 ( 2 , 𝑛)
Page 1 of 11
 (b) 13 𝐵 (𝑚3 , 𝑛)
1 𝑚 𝑛
(c) 𝐵( , )
2 2 2
(d) 12 𝐵 (𝑚, 𝑛
2
)
(Ans: a)

∞ 𝒙𝒎−𝟏
5. Value of ∫𝟎 𝒅𝒙, where m, n, a and b are positive.
(𝒂𝒙+𝒃)𝒎+𝒏

(a) 𝐵(𝑚,𝑛)
𝑎𝑚𝑏𝑛
(b) 𝑎𝑚𝑏𝑚
𝐵(𝑚,𝑚)

(c) 𝑎𝐵(𝑛,𝑛)
𝑚 −𝑏 𝑛

(d) 𝑎𝐵(𝑚,𝑛)
𝑚 +𝑏 𝑛

(Ans:a)

𝟐
6. Value of ∫𝟎 𝒙𝟒 (𝟖 − 𝒙𝟑 )−𝟏/𝟑 𝒅𝒙

(a) 163 𝐵 (53 , 23)

(b) 13 𝐵 (53 , 23)
(c) 163 𝐵 (53 , 34)
(d) 163 𝐵 (35 , 23)
(Ans:a)

7. Assuming 𝜞(𝒏)𝜞(𝟏 − 𝒏), 𝟎 < 𝒏 < 𝟏,

(a) 𝑠𝑖𝑛𝜋𝑛𝜋
(b) 𝑠𝑖𝑛2 𝜋𝑝𝜋
(c) 𝑠𝑖𝑛𝜋 𝜋
(d) 𝑠𝑖𝑛 𝜋2 𝑝𝜋
(Ans:a)

𝟏
8. Value of 𝜞(𝒎)𝜞 (𝒎 + )
𝟐

Page 2 of 11
 (a) 2√2𝑚
𝜋
𝛤(2𝑚)
(b) 22𝑚+1
√𝜋
𝛤(2𝑚)
(c) 22𝑚−1
√𝜋
𝛤(2𝑚)
(d) 22𝑚+1√𝜋
𝛤(2𝑚 + 1)
(Ans:c)

9. Apply Dirichlet’s integral to evaluate ∬𝑽 𝒙𝟐l-𝟏 𝒚𝟐𝒎−𝟏 𝒅𝒙𝒅𝒚 for all positive values of
x and y such that 𝒙𝟐 + 𝒚𝟐 ≤ 𝒂𝟐

𝟐𝒍+𝟐𝒎
(a) 𝒂 𝟒 𝜞(𝒍+𝒎+𝟏)
𝜞(𝒍)𝜞(𝒎)

𝒂𝟐𝒍+𝟐𝒎 𝜞(𝒍)𝜞(𝒎)
(b) 𝟒 𝜞(𝒍+𝒎)
𝒂𝟐𝒍+𝟐𝒎 𝜞(𝒍)𝜞(𝒎)
(c) 𝟒 𝜞(𝒍+𝒎−𝟏)
𝒂𝒍+ 𝒎 𝜞(𝒍)𝜞(𝒎)
(d) 𝟒 𝜞(𝒍+𝒎+𝟏)
(Ans:a)

10.Using Dirichlet’s integral, find the area in the first quadrant bounded by the
𝒙 𝜶 𝒚 𝜷
curve (𝒂) + (𝒃) = 𝟏
𝟏 𝟏
𝒂𝒃 𝜞(𝜶)𝜞(𝜷)
(a) 𝜶𝜷 𝟏 𝟏
𝜞( + )
𝜶 𝜷
𝟏 𝟏
𝒂𝒃 𝜞(𝜶)𝜞(𝜷)
(b) 𝜶𝜷 𝜞(𝟏 +𝟏 +𝟏)
𝜶 𝜷
𝟏 𝟏
𝜞( )𝜞( )
(c) 𝒂𝒃 𝜶
𝜶𝜷 𝜞(𝟏 +𝟏 −𝟏)
𝜷

𝜶 𝜷
𝟏 𝟏
𝜞( )𝜞( )
(d) 𝜶
𝟏 𝟏
𝜞( + +𝟏)
𝜷

𝜶 𝜷

(Ans:b)

𝒙 𝒚 𝒛
11.The plane + + =1 meets the axes in A, B and C. Apply Dirichlet’s integral to
𝒂 𝒃 𝒄
find the volume of the tetrahedron OABC. Also find its mass if the density at
any point is kxyz.
2 2 2 2 2 2
(a) V = 𝑎 𝑏6 𝑐 , M = 𝑘𝑎720
𝑏 𝑐

(b) V = 𝑎𝑏𝑐
6
,M=
𝑘𝑎𝑏𝑐
720

Page 3 of 11
 2 2 2
(c) V = 𝑎𝑏𝑐
6
,M=
𝑘𝑎 𝑏 𝑐
720
𝑎2 𝑏2 𝑐 2
(d) V = 6 , M = 702
𝑎𝑏𝑐

(Ans:a)

12.Apply Dirichlet’s integral to find the mass of a sphere 𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐 = 𝒂𝟐 ,the

density at any point being 𝝆 = 𝒌𝒙𝟐 𝒚𝟐 𝒛𝟐
9
(a) 4𝑘𝑎
945
𝜋

𝑘𝑎9 𝜋
(b) 945
9
(c) 4𝑘𝑎
945
4𝑎9 𝜋
(d) 945
(Ans:a)

13.If l, m, n are all positive∭ 𝒙𝒍−𝟏 𝒚𝒎−𝟏 𝒛𝒏−𝟏 𝒅𝒙𝒅𝒚𝒅𝒛, where the integral is taken
𝒙𝟐 𝒚𝟐 𝒛𝟐
throughout the part of ellipsoid 𝟐
+ 𝟐
+ = 𝟏 which lies in positive octant.
𝒂 𝒃 𝒄𝟐

𝑙 𝑚 𝑛
𝑎𝑙 𝑏𝑚 𝑐 𝑛 𝛤( )𝛤( )𝛤( )
(a) 2
𝑙 𝑚 𝑛
2 2
𝛤( + + +1)
2 2 2
𝑙 𝑚 𝑛
𝑎 𝑏 𝑐 𝑛 𝛤( )𝛤( )𝛤( )
𝑙 𝑚
(b) 2
𝑙 𝑚 𝑛
2 2
8𝛤( + + )
2 2 2
𝑙 𝑚 𝑛
𝑎 𝑏𝑚+1 𝑐 𝑛 𝛤( )𝛤( )𝛤( )
𝑙+1
(c) 2
𝑙 𝑚 𝑛
2 2
8𝛤( + + +1)
2 2 2
𝑙 𝑚 𝑛
𝛤( )𝛤( )𝛤( )
(d) 2 2
𝑙 𝑚 𝑛
2
8𝛤( + + +1)
2 2 2

(Ans:a)
𝒙𝟐 𝒚𝟐
14.Apply Dirichlet’s integral to find the mass of an octant of the ellipsoid 𝟐
+ +
𝒂 𝒃𝟐
𝒛𝟐
= 𝟏,the density at any point being 𝝆 = 𝒌𝒙𝒚𝒛
𝒄𝟐

2 2 2
(a) 𝑎 48
𝑏 𝑐

2 2
(b) 𝑘𝑎48𝑐
2 2
(c) 𝑘𝑎48𝑏
2 2 2
(d) 𝑘𝑎 48𝑏 𝑐
(Ans:d)
Page 4 of 11
15.Find the value of ∬𝑫 𝒙𝒍−𝟏 𝒚𝒎−𝟏 𝒅𝒙𝒅𝒚, where D is the domain 𝒙 ≥ 𝟎, 𝒚 ≥ 𝟎 and
x+y≤1.

(a) 𝛤(𝑙)𝛤(𝑚)
𝛤(𝑙+𝑚)
(b) 𝛤(𝑙+𝑚+1)
𝛤(𝑙)𝛤(𝑚)

(c) 𝛤(𝑙)𝛤(𝑚)
𝛤(𝑙+1)
(d) 𝛤(𝑙+𝑚+1)
𝛤(𝑙+1)𝛤(𝑚)

(Ans:b)

16.Evaluate ∭𝒗 𝒆(𝒙+𝒚+𝒛) 𝒅𝒙𝒅𝒚𝒅𝒛taken over the positive octant such that 𝒙 + 𝒚 + 𝒛 ≤

𝒂

(a) 12 [𝑒 𝑎 (𝑎2 − 2𝑎) − 2]

(b) 12 [𝑒 𝑎 (𝑎2 − 2𝑎 + 2)]
(c) 12 [𝑒 𝑎 (𝑎2 − 2𝑎 + 2) − 2]
(d) 12 [(𝑎2 − 2𝑎 + 2) − 2]
(Ans:c)

17. Evaluate ∭ 𝒍𝒐𝒈( 𝒙 + 𝒚 + 𝒛)𝒅𝒙𝒅𝒚𝒅𝒛,the integral extending over all positive and
zero values of x, y and z subject to 𝟎 < 𝒙 + 𝒚 + 𝒛 ≤ 𝟏.

(a) 18
(b) − 181
(c) 19
(d) 32
(Ans:b)
𝒅𝒙𝒅𝒚𝒅𝒛
18. Evaluate ∭𝑽 taken throughout the volume of the sphere 𝒙𝟐 + 𝒚𝟐 +
√𝒂𝟐 −𝒙𝟐 −𝒚𝟐 −𝒛𝟐
𝒛𝟐 = 𝒂𝟐 lying in positive octant.
2 2
(a) 𝜋 8𝑎
2
(b) 𝜋 8𝑎
2
(c) 𝜋8𝑎

Page 5 of 11
 (d) 𝜋8𝑎
(Ans:a)

∞ 𝒅𝒙
19.Find the value of ∫𝟎 ,𝒂 > 𝟎
𝒂𝟐 +𝒙𝟐
(a) Convergent, 2𝑎
𝜋

(b) Divergent, 2𝑎
𝜋

(c) Convergent, 𝜋𝑎
(d) Divergent, 𝜋𝑎
(Ans: a)

∞ 𝒅𝒙
20.Find the value of ∫𝒆
𝒙(𝒍𝒐𝒈𝒆 𝒙)𝟑

(a) Convergent, 13
(b) Divergent, 12
(c) Convergent, 12
(d) Divergent, 13
(Ans:a)

∞
21.Find the value of ∫𝟏 𝒙𝒆−𝒙 𝒅𝒙

(a) Convergent, 1𝑒
(b) Divergent, 2𝑒
(c) Convergent, 2𝑒
(d) Divergent, 1𝑒
(Ans:c)

22. The curved surface of the solid generated by revolution about the x-axis, of the
area bounded by the curve 𝒚 = 𝒇(𝒙), x-axis and ordinates 𝐱 = 𝒂, 𝐱 = 𝒃 is

(a) ∫𝒂𝒃 𝟐𝝅𝒚𝒅𝒔

(b) ∫𝒂𝒃 𝝅𝒚𝒅𝒔
(c) ∫𝒂𝒃 𝟐𝒚𝒅𝒔
(d) ∫𝒂𝒃 𝒚𝒅𝒔
(Ans:a)

Page 6 of 11
23.The area of the parabola 𝒚𝟐 = 𝟒𝒂𝒙 lying between the vertex and the latus rectum
is revolved about the x-axis. Then volume generated.
(a) 𝝅𝒂𝟑
(b) 𝝅𝒂𝒃𝟐
(c) 𝟐𝝅𝒂𝟐
(d) 𝟐𝝅𝒂𝟑
(Ans:d)
𝒙𝟐 𝒚𝟐
24.Find the volume of the solid generated by revolving the ellipse 𝟐 + 𝟐 = 𝟏 about
𝒂 𝒃
the x-axis.

(a) 𝟒𝟑 𝒂𝒃𝟐
(b) 𝝅𝒂𝒃𝟐
(c) 𝟒𝟑 𝝅𝒂𝒃𝟐
(d) 𝟒𝟑 𝝅𝒂𝒃
(Ans:c)
∞ 𝟏
25.The value of integral ∫𝟏 𝒅𝒙 is
𝒙𝟐

(a) 0
(b) 1
(c) ∞
(d) None of these.
(Ans:b)

𝟎
26.The value of ∫−∞ 𝒙𝒔𝒊𝒏𝒙 𝒅𝒙

(a) 0
(b) ∞
(c) −∞
(d) 1
(Ans:c)

∞
27.The value of ∫𝟎 √𝒙𝒆−𝒙 𝒅𝒙 is

(a) √𝜋
1
(b) √𝜋
2
(c) 0
(d) ∞
Page 7 of 11
 (Ans:b)

𝟎
28.Value of ∫−∞ 𝒆−|𝒙| 𝒅𝒙 is
(a) 0
(b) 1
(c) ∞
(d) −∞
(Ans:b)

29. 𝜷(𝟑, 𝟐) is equal to

(a) 3/2
1
(b)
12
5
(c)
8
(d) ¾
(Ans:b)

𝟑 𝒅𝒙
30.The integral ∫𝟎 converges to
√𝟗−𝒙𝟐
(a) 1
(b) 𝜋
(c) 𝜋/2
(d) 𝜋/4
(Ans:c)

31. 𝜷(𝟐, 𝟏) + 𝜷(𝟏, 𝟐) is equal to

(a) 1
(b) 0
(c) 2
(d) −1
(Ans: a)

32.Value of ┌(𝟑. 𝟓) is
(a) √𝜋
(b) 15/8
15
(c) √𝜋
8
(d) None of these
(Ans: c)

Page 8 of 11
 𝝅
33. ∫𝟎𝟐 𝒔𝒊𝒏𝟕 𝜽√𝒄𝒐𝒔𝜽𝒅𝜽 is equal to
3 3
(a) 𝛽 ( , )
4 4
3
(b) 𝛽 (4, )
4
1 3
(c) 𝛽 (4, )
2 4
(d) 𝛽(4,4)
(Ans:c)

𝒙𝟐 𝒚𝟐
34.What is the volume generated when the ellipse 𝟐 + 𝟐 = 𝟏 is revolved around its
𝒂 𝒃
Minor axis?
(a) 4𝑎𝑏 Cubic units
4
(b) 𝑎2 𝑏 Cubic units
3
4
(c) 𝑎𝑏 Cubic units
3
(d) 4 cubic units
(Ans:b)

35. Volume of sphere of radius a is given by

4
(a) 𝜋𝑎
3
(b) 4𝜋𝑎
4
(c) 𝜋𝑎2
3
4
(d) 𝜋𝑎3
3
(e) (Ans:d)

36. The surface area of the solid obtained by rotating = √𝟗 − 𝒙𝟐 , −𝟐 ≤ 𝒙 ≤ 𝟐 about x-

axis is equal to
(a) 12𝜋
(b) 24𝜋
(c) 6𝜋
(d) None of these.
(Ans:b)

∞ 𝐥𝐧 𝒙
37.The integral ∫𝟏 𝒅𝒙 is
𝒙𝟐
(a) Convergent
(b) Divergent
(c) Oscillatory
(d) None of these
(Ans:a)

Page 9 of 11
 𝟏 𝒅𝒙
38.The value of ∫𝟎 is
√−𝒍𝒐𝒈𝒙
(a) √𝜋
√𝜋
(b)
2
(c) 2√𝜋
(d) None of these
(Ans:a)
∞ 𝟒
39.The value of ∫𝟎 𝟒𝒙𝟒 𝒆−𝒙 𝒅𝒙 is
𝟏
(a) ┌( )
𝟒
1 𝟏
(b) ┌ ( )
4 𝟒
𝟑
(c) ┌ (− )
𝟒
(d) 0
(Ans:b)

∞
40.The integral ∫𝟎 𝒆−𝟒𝒙 𝒄𝒐𝒔𝟓𝒙𝒅𝒙 is
(a) Convergent, value = −4/41
(b) Convergent, value = 4/41
(c) Divergent, value =−∞
(d) Oscillatory
(Ans:b)
∞
41.The integral ∫−∞ 𝒆−𝒙 𝒅𝒙 is
(a) Divergent
(b) Oscillating finitely
(c) Convergent
(d) Oscillating infinitely
(Ans:a)
∞ 𝒅𝒙
42.Value of ∫√𝟐 is
𝒙√𝒙𝟐 −𝟏
𝜋
(a)
2
𝜋
(b) −
4
𝜋
(c)
4
(d) 0
(Ans:c)
∞ 𝒅𝒙
43.The value of ∫𝟎 , 𝒂 > 𝟎 is
𝒂𝟐 +𝒙𝟐
𝜋
(a)
2
𝜋
(b)
2𝑎
(c) −∞

Page 10 of 11
 (d) ∞
(Ans:b)

𝟎
44.The integral ∫−∞ 𝒙𝒔𝒊𝒏𝒙 𝒅𝒙
(a) Oscillates finitely
(b) Convergent
(c) Diverges to −∞
(d) Diverges to ∞
(Ans:c)

Page 11 of 11
 Ajay Kumar Garg Engineering College, Ghaziabad

COURSE: B. TECH. SEMESTER: II

SUBJECT: ENGINEERING MATHEMATICS II SUBJECT CODE: KAS-203T

MODULE-3
SEQUENCE AND SERIES
MULTIPLE CHOICE QUESTIONS

1. Period of cos 3x is
a) 
2
b)
3
c) 2 
d) None of these
(Ans:b)
2. The periodic function of period 2  represented by the following graph is

 2 3 4

A

A if 0  x  
a) f x   
 A if   x  2
 A if 0  x  
b) f x   
A if   x  2
c) f x   A
d) None of these
(Ans: a)
a0   nx nx 
3. If f x      a n cos  b n sin  then value of a n is
2 n 1  l l 

f x  cos nx dx
1 2
a)
  0

1 c 2l nx
b)  f x  cos dx
l c l

1
 1 c 2l nx
c)  f x  cos dx
c c c
d) None of these
(Ans:b)

4. If f x   x 4 in (-1,1),then the value of b n is

1
a)
n4

b)
 1
n

n4
c) 0
d) None of these
(Ans:c)
 1  1  t  0
5. If f t    ,then f(t) is
1 0  t 1
a)
Even function
b)
Odd function
c)
Periodic function
d)
Constant function
(Ans:b)
6. Fourier Series expansion of an even function in   ,  has
a) Only sine terms
b) Only cosine terms
c) Both sine & cosine terms
d) None of these
(Ans:b)
 x,   x  0
7. If f x    ,then f is
x, 0 x
a) Even function
b) Odd function
c) Periodic function
d) None of these
(Ans: a)
8. If f x   x 2 in -2<x<2, f(x+4) = f(x) ; then a n is
a) 0
f x  cos nx dx
1 2
2 0
b)
nx
c) 2 f x  cos
1 2
dx
2 2
d) None of these
(Ans: c)

2
9. If f (x)  x 2 in   ,  ,then b n is
1
a)
n2

b)
 1n
n2
c) 0
d) None of these
(Ans:c)
 2x
1   ,   x  0
10.If f x    is expanded in Fourier Series ,then
1  2x , 0 x
 
a) a n  0
b) b n  0
c) Both a) and b)
d) None of these
(Ans:b)
11.If f (x)  e x is expanded in Fourier Series in 0,2  ,then a n is
a)
n
1 n2

1  e  2 

b) 2 1   1n
n
2


c)  2 1   1n
n
2

d) None of these
(Ans: d)
12.The Fourier Series expansion of x sinh x in   ,   contains
a) Only sine terms
b) Only cosine terms
c) Both sine & cosine terms
d) None of these
(Ans: b)
13.At x   ,the Fourier Series of f x   x  x 2 in   ,  converges to
a)     2
b)    2
c) 0
d)  2
(Ans: d)
14.The series 1 + + + +………..is

(a) Divergent (b) Convergent (c) Oscillates finitely (d) Oscillates infinetly
(Ans: b)

3
 √( )
15.For which real number m does the infinite series ∑ converges

(a) m> 1/3 (b) m > 1/2 (c) m > 1 (d) m > 3/2
(Ans: (d)
16.The series ∑ ( ) for | |>1 is
(a) Divergent (b) Convergent (c) oscillatory (d) None of these
(Ans:a)

17.The series + + +………..

(a) Diverges for x >1
(b) Converges for x >1
(c) Diverges for x 1
(d) None of these
(Ans:a)

18.For an even function f(x) which of the following vanishes?

(a) a0 (b) an (c) bn (d) None of these
(Ans: c)

( )
19.For f(x) = √ ,0<x<2 a0 is given by
(a) (b) (c) (d)
(Ans:d)

20.For f(x) = | | ; -2 x 2 ,anis given by

(a){ } (b) { }

(c) { } (d){ }

(Ans:a)
21.For f(x) = x2 , -2 x 2 bn is given by
(a) (b) 0 (c) 1 (d) not defined
(Ans: b)

22.In the Fourier series expansion of f(x) = 2x + sinx, -1 < x < 1, the coefficient that
vanishes is
4
 (a) a0 (b) bn (c) an (d) both a0 & an
(Ans: d)
23.In the half range cosine series for f(x) = x ( – x) ; 0 < x < , a0 is given by
(a) (b) (c) (d)
(Ans:c)
24.The Sequence is
(a) Convergent
(b) Divergent
(c) Oscillatory
(d) None of these
(Ans: a)
25.Test the Convergence of
(a) Convergent
(b) Divergent
(c) Oscillatory
(d) None
(Ans: a)
26.The Series ∑
(a) 1
(b) less than 1
(c) greater than 1
(d) None
(Ans:c)
27. The sequence is
(a) Diverge to 5
(b) Converge to 0
(c) Converge to 5
(d) None
(Ans:c)
28.The Sequence is
(a) Divergent
(b) Convergent
(c) Not Divergent
(d) Not Convergent
(Ans:c)
29.The sequence ( ) is Converge to
(a) 1
(b) 2
(c) infinite
(d) e
5
 (Ans:d)

30.The Sequence
(a) 2
(b) 1
(c) 3
(d) None
(Ans:a)
31.The P-Series is convergent if
(a) P = 1
(b)
(c)
(d) None
(Ans:c)
32. The convergence of the series
(a) Divergent
(b) Convergent
(c) Not Divergent
(d) Not Convergent
(Ans:a)
33. Determine the value ∑
(a) Convergent to 0
(b) Convergent to 12
(c) Divergent
(d) Not Convergent
(Ans: b)
34.The series ∑
(a) Divergent
(b) Not Convergent
(c) Not Divergent
(d) Convergent
(Ans: a)
35. The series ∑
(a) p- series
(b) G.P
(c) Alternating
(d) None
(Ans:a)
36.Test the convergence of ∑ is
(a) Divergent

6
 (b) Not Convergent
(c) Not Divergent
(d) Convergent
(Ans:a)
37. Test the convergence of ∑ is
(a) Divergent
(b) Convergent
(c) Not Divergent
(d) Not Convergent
(Ans: b)
38.Which test is use to check the convergence of ∑
(a) Comparison Test
(b) Cauchy Test
(c) Raabie’s Test
(d) Ratio Test
(Ans:d)
39.
( )
(a)
( )
(b)
( )
(c)

(d) None of these

(Ans: a)
40.
(a) Comparison Test
(b) P-series
(c) Raabie’s Test
(d) Ratio Test
(Ans: b)
41.Test for Convergence of the series
(a) Cgt if
(b) Cgt if
(c) Cgt if
(d) Cgt if
(Ans:b)
42. Which are the Fourier coefficient of f(x)
(a)
(b)
(c)
(d)
7
 (Ans:d)
43. Period of ( )
(a)
(b)
(c)
(d)
(Ans:c)
44.The Fourier coefficient of ( ) in are
(a)
(b)
(c)
(d) a & c
(Ans:d)
45.The Fourier coefficient of ( ) | | in are
(a)
(b)
(c)
(d) b & c
(Ans: d)
46. ( ) | | in as Fourier series. At which point this series
. Obtained
(a) x=0
(b)
(c)
(d) a & c
(Ans: a)
47.The Fourier coefficient of for ( ) , . are
(a)
(b)
(c)
(d) None of these
(Ans: a)
48.The Fourier coefficient of ( ) | | in the interval ( ) is
(a)
(b)
(c)
8
 (d) None of these
(Ans: a)
49. The Fourier coefficient of ( ) in the interval ( ) is
(a)
(b)
(c)
( )
(d)
(Ans: d)
50. Test for Convergence of the series
(a) Cgt if
(b) Cgt if
(c) Cgt if
(d) Cgt if
(Ans: d)
51.The Fourier coefficient of ( ) in the interval ( ) is
(a)
(b)
(c)
(d) None
(Ans: b)
52.The Fourier coefficient of ( ) in the interval ( ) is
(a) 0
(b) 2
(c) 4
(d) None
(Ans: a)
53.The sufficient condition for the uniform convergence of a Fourier series are known as
(a) Dirichlet’s condition
(b) Euler’s Formula
(c) Fourier coefficient
(d) All
(Ans:a)
54.Essential condition for obtaining Fourier series is
(a) Continuity
(b) differentiability
(c) Periodic
(d) None
(Ans: c)
55.Fourier series of discontinuous functions is also known as

(a) Mean value

9
 (b) Geometric value

(c) sum value

(d) none

(Ans: a)

56. Behaviour of Fourier series at a point of discontinuity is called

(a) Eulers Formula

(b) Gibbs Phenomenon

(c) Dirichlet’s condition

(d) none
(Ans: b)

57. Which Test is use to check the convergence of the series

(a) Ratio Test

(b) Raabe’s Test
(c) both a & b
(d) none
(Ans: b)

58. Which Test is use to check the convergence of Alternative series

(a) Leibnitz’s Test

(b) Leibnitz’s theorem

(c) Leibnitz’s Rule

(d) none

(Ans: a)

59. Necessary condition for Convergence a series ∑ is

(a)

(b)

(c)

10
 (d) None

(Ans: a)

60. Test the convergence of the series

(a) convergent

(b) divergent
(c) oscillate

(d) none

(Ans: a)
61. A sequence * + is convergent if is ……
a) finite b) not finite c) oscillating d) None of these.
(Ans: a)
62. Every convergent sequence is ……
a) unbounded b) bounded c) bounded above d) None of these.
(Ans: b)
63. An infinite geometric series is convergent if ……
a) | | b) c) d) .
(Ans: a)
64. The series is ……

a) divergent b) c) oscillating infinite d) None of these.

(Ans: b)
65. The series is ……

a) divergent b) c) oscillating finite d) None of these.

(Ans:b)
66. The infinite series ∑ is convergent if

a) b) c) d) p .
(Ans: b)

11
67. The series √ is ……

a) not convergent b) c) coverges to 1 d) None of these

(Ans: a)
68. If √( ) then
a) ∑ converges but ∑ diverges b) ∑ diverges but ∑
converges
c) ∑ and∑ both converges d) ∑ and ∑ both diverges
(Ans: b)

69. If , then the infinite series ∑ is ……

a) divergent b) c) not convergent d) None of these.

(Ans: b)

70. The series + …. is convergent if

a) b) c) d) for all x.

(Ans: a)

71. The series ∑ . /is

a) convergent b) may be convergent c) divergent d) conditionally convergent.

(Ans: c)

72. The series is divergent if ……

a) b) c) d) .
(Ans: a)
73. The series ∑[ ]is ……
a) divergent b) c) oscillating infinite d) None of these.
(Ans: b)
n2
74. If , then the infinite series ∑ is ……
2n

a) not convergent b) c) divergent d) None of these.

12
(Ans: b)
75. .By D’ Alembert’s ratio test, an infinite series ∑ is convergent if is ……

a) b) c) d) .
(Ans: a)
76. The series∑ is
a) convergent b) may be convergent c) divergent d) None.
(Ans: a)
77. If ( ) is discontinuous at x then the Fourier series converges to _________ where
( ), ( ) are respectively right hand and left hand limits of ( )

( ) ( ) ( ) ( )
a) b)
( ) ( ) ( ) ( )
c) d)
(Ans: a)
78. The period of the function ( ) is
a) b) c) 2 d)

(Ans: c)
79. If f  x   x sin x in ( ,  ) then the value of bn is

a) b) c) d)

(Ans: d)
80. If ( ) in ( ) and ( ) ( ), then the value of is

a) ∫ . / b) ∫ . /

c) ∫ . / d) Both (a) & (b)

(Ans: d)
81. In Fourier series expansion of function ( ) , the value of is

13
 a) b) c) d)

(Ans: d)
82. In Fourier series expansion of function ( ) , the value of is

( ) ( ) ( )
a) b) c) d) None of

these.
(Ans: c)
83. Half range Fourier sine series for the function ( ) is given by

, ( ) - , ( ) -
a) ∑ b) ∑
, ( ) -
c) ∑ d) None of these
(Ans: b)

14
 Ajay Kumar Garg Engineering College, Ghaziabad

COURSE: B.TECH. SEMESTER: II

SUBJECT: ENGINEERING MATHEMATICS II SUBJECT CODE: KAS-203T

MODULE-4
COMPLEX VARIABLE – DIFFERENTIATION
MULTIPLE CHOICE QUESTIONS

1. The conjugate harmonic function of is …………….

(a) + c (b) + c (c) + c (d) None of these

(Ans: b)

2. The harmonic conjugate of is …………….

(a) +c (b) +c (c) (d) None of these

(Ans: d)

3. If f(z) = u + iv is an analytic function then u and v both satisfy Laplace‟s

equation.
(a) Statement is correct (b) Statement is false (c) None
(Ans: a)

4. A function f(z) = is……..

(a) Analytic everywhere (b) Analytic nowhere (c) only differentiable (d) None
(Ans: a)
5. If f(z) = u + iv is an analytic fn. in the z-plane, then the C-R equations are satisfied by
it‟s real and imaginary parts i.e …….

(a) , (b) ,

(c) , (d) ,
(Ans: a)

6. Milne- Thomson method is used to construct …….

a) analytic function b) Continuous function

c) differentiable function d) None of these.

(Ans: a)

7. Write the Milne-Thomson‟s method to construct an analytic function F(z) = u + iv when

the real part u is given:….

a) ∫* ( ) ( )+ b) ∫* ( ) ( )+
 c) ∫* ( ) ( )+ d) None of these.
(Ans: b)
8. Let ( ) be a complex valued function. Where then is analytic for
any value of
a) is analytic for suitable value of
b) is analytic only when constant
c) can‟t be analytic for any value of
d) .
(Ans: d)
9. The value of „s‟ such that ( ) ( ) is an analytic function.
a) 1 b) 2 c) ⁄ d)
(Ans: a)
10. If a function ( ) is continuous at , then
a) ( ) is differentiable at
b) ( ) is not necessarily differentiable at
c) ( ) is analytic at
d) None of the above.
(Ans: b)
11. The only function among the following that‟s analytic, is
) ( ) ( ) b) ( ) ( ) c) ( ) ̄ d) ( ) .
(Ans: d)
12. An analytic function is
a) Infinitely differentiable
b) not necessarily differentiable
c) finitely differentiable.
d) None of these.
(Ans: a)
13. Let ( ) ( ) for all real x and y then the imaginary part of u, such that
( ) ( ) ( ) is analytic, is
a) ( )
b) ( )
c) ( )
d) ( )
(Ans: a)
14. Which of the following can not be the real part of an analytic function.
a) b) c) d) ( )
(Ans: a)
15. The harmonic conjugate of ( ) ( ) is

a) ( )
b) ( )
c) ( )
d) ( )
 (Ans: a)
16. The invariant points of the transformation are

a) b) c) d)
(Ans: b)
17. Under the mapping the image of line y  0 is,
a) ( ) b) ( ) c) ( ) d) ( )
(Ans: b)
18. The mapping ( ) ̄ is

a) Conformal b) isogonal c) neither conformal nor isogonal d) analytic

(Ans: b)
19. The bilinear transformation w which maps the point in the plane onto the
points- in plane is
a) b) c) d)

(Ans: c)

20. A function f(z) may be differentiable in a domain except for a finite number of points,
these points are called…..

a) Isolated points b) Invariant points c) Singular points d) Conjugate points

(Ans: c)

21. The points which coincide with their transformations are called………and can be
obtained by the condition…..

a) Singular points, f(z)=w b) Isolated points, f‟(z)=z

c) Conjugate points, f(z)=0 d) Invariant points, f(z)=z

(Ans: d)

22. A transformation w= is known as……

a) Mobius transformation b) Linear transformation

c) Inverse transformation d) Mapping

(Ans: a)

23. The image of the circle | | in complex plane under the transformation wz=1
is….

a) v= b) v= c) u= d) u

(Ans: c)
24. The transformation represents….

a) Magnification b) Rotation c) Translation d) Inversion

(Ans: b)

25. Angle of rotation and coefficient of magnification are given by….

a) | ( )| | ( )| b) | ( )| | ( )|
c) | ( )| | ( )| d) | ( )| | ( )|

(Ans: d)

26. “If function f(z) & g(z) are analytic within & on a closed curve C and | ( )| | ( )| on
C then f(z) & f(z)+g(z) have same number of zeros inside C” this statement is known as…

a) Liouville‟s theorem b) Cauchy‟s theorem

c) fundamental theorem of algebra d) Rouche‟s theorem

(Ans: d)

27. A transform is called Bilinear Transformation, if

(a)
(b)
(c)
(d)
(Ans: b)

28. Under the transformation , the image of the line in z-plane is

(a)
(b)
(c)
(d) None
(Ans:b)

29. Under the transformation the image of the line y=0 in the z-plane is

(a) v=-1
(b) v=1
(c) u=1
(d) u = -1
(Ans: a)

30. A mobius transformation maps circle into

(a) Straight line

(b) Circle
(c) Parabola
 (d) none
(Ans: b)

31. The Bilinear Transformation that maps the points ( ) ( )

(a)
(b)
(c)
(d)
(Ans: d)

32. The condition of a conformal mapping in terms of Jacobian is given by

(a) ( )
(b) ( )
(c) ( )
(d) None
(Ans: a)

33. The Bilinear Transformation which carries 0, i,-i into 1,-1,0 respectively is given by

(a)
(b)
(c)
(d)
(Ans: b)

34. The transformation transform the unit circle in the w-plane into straight line in
z-plane if

(a)
(b) | | | |
(c) | | | |
(d) None
(Ans: b)
35. The transformation is said to be normalized if ad-bc is equal to

(a) 0
(b) 1
(c)
(d) None
(Ans: b)

under the transformation | | transform into

( )
 (a)

(b)

(c)

(d) None

(Ans: b)

37. The Bilinear transformation having only one fixed point then it is called

(a) Parabolic

(b) Hyperbolic

(c) circle

(d) none

(Ans: a)

38. A Bilinear transformation having two fixed point as p and q then which is true

(a)

(b)

(c) w= pqz

(d) None

(Ans: a)

39. A Bilinear transformation is called Elliptic if

(a) | |

(b) | |

(c) | |

(d) None

(Ans: a)

40. A Bilinear transformation is called Hyperbolic if

(a) | |
 (b) | |

(c) | |

(d) k is Real

(Ans: d)

41. A Bilinear transformation is neither hyperbolic nor Elliptic or parabolic is called

(a) Orthogonal

(b) Normalized

(c) loxodromic

(d) none

(Ans: c)

42. Which is not meaning to an Analytic function.

(a) Regular Function

(b) Harmonic Function

(d) Holomorphic Function

(d) Monogenic Function

(Ans: b)

43. which are the correct C-R equations

(a)

(b)

(c)

(d) None

(Ans: a)

44. C-R equation are the condition of an analytic function

(a) Sufficient

(b) Necessary

(c) a&b
 (d) None

(Ans: b)

45. if is a part of an analytic function then f(z) will be

(a) logz +c

(b) log | | +c

(c) sinz +c

(d) None

(Ans: a)

46. At which point for ( ) √| | is satisfied C-R equation

(a) (0,0)

(b) (1,1)

(c) (0,1)

(d) (1,0)

(Ans: a)

47. for what value of a and b ( ) ( ) is an analytic function

(a)

(b)

(c)

(d) None

(Ans: a)

48. Find image of | | under the mapping wz=1

(a) 2v+1=0

(b) 2v-1=0

(c) u=v

(d) None

(Ans: a)
49. Which of the following is a Bilinear transformation

(a) w = z

(b)

( )
(c) ( )

(d) None

(Ans: a)
50. If f(z) is analytic then f'(z) is:
a) u  i v
x x
u v
b) i
y y
u v
c) i
x x
u v
d) i .
y y
(Ans: a)
51. CR equations for w = u+ i v are:
a) ux  vx , uy  vy .
b) ux  vy , u y  vx
c) ux  vy , u y  vx .
d) ux  vy , u y  vx
(Ans: b)
52. Which is correct for w=f(z):
dw w
a) 
dz x
dw w
b) 
dz x
dw w
c)  .
dz y
dw w
d) 
dz y
(Ans: a)
53.A function u(x,y) is harmonic if:
a)  u2   v2  0
2 2

x x
 2u  2 v
b)  0
x 2 y 2
 2u  2u
c)  0
x 2 y 2
d) None
(Ans: c)
54. A derivative of f(z) = log z is:
 a) z
b) 1
z
c) 0
d) none.
(Ans: b)
55.f(z)= u+iv is analytic iff u and v are:
a) harmonic.
b) continuous.
c) differentiable.
d) satisfies CR equation.
(Ans: d)
56.If f is analytic in domain D, then in that domain
a) f is continuous only.
b) f is differentiable only.
c) both a and b.
d) None
(Ans: c)
57.If w = f(z) is conformal mapping in D, then f(z) is:
a) analytic in D.
b) not always analytic
c) never analytic.
d) none
(Ans: a)
58.The bilinear transformation which maps z  1, z  0, z  1 onto w  i , w  0, w  i is
a) w  iz
b) w  z
c) w  i(z  1) .
d) None
(Ans: a)
59.Which is analytic function?
a) sin z .
b) z .
c) Im .( z ) .
d) Re(iz) .
(Ans: a)
60.A function f(z) is analytic function if
a) Real part of f(z) is analytic
b) Imaginary part of f(z) is analytic
c) Both real and imaginary part of f(z) is analytic
d) none of the above
(Ans: c)
61.If f (z) = x + y + i ( x + y) is analytic then , , equals to
a) = 1 and = -
b) = 1 and = -
c) = 1 and = -
d) = = =1
 (Ans: a)

62.Harmonic conjugate of u(x,y) = eycosx is

ex cos y + C
a)
ex sin y + C
b)
ey sin x + C
c)
–ey sin x + C
d)
(Ans: d)
63.If the real part of an analytic function f (z) = x2 – y2 – y, then the imaginary part is
a) 2xy b) x2+2xy c) 2xy–y d) 2xy +x
(Ans: d)
64.There exist no analytic function f such that
a) Re f (z) = y - 2x b) Re f (z) = y2 - 2x c) Re f (z) = y2 – x2 d)Re f (z) = y – x
(Ans: b)
65.If ( ) ( and f(0) = 0 then f(z) is
)

a) Continuous but not differentiable at z = 0 b) differentiable at z = 0

c) Analytic everywhere except at z = 0 d) not differentiable at z = 0
(Ans: d)
66.The function ex (cosy- i siny) is
(a) Analytic (b) not analytic (c) analytic when z = 0 (d) analytic when z = i
(Ans: b)
67.Harmonic conjugate of u = log √ is
a) ( b) ( c) tan-1( ) d) tan-1( )
) )

(Ans: d)

68.f (z) =| ̅| is differentiable

a) nowhere b) only at z = 0 c) everywhere d) only at z = 1
(Ans: b)
69.w = is a bilinear transformation when
a) ad – bc = 0 b)ad – bc 0 c) ab – cd 0 d) None of these
(Ans: b)
70.A mapping that preserves angles between oriented curves both in magnitude and in sense
is called
a) informal b) isogonal c) conformal d) formal
(Ans: c)
71.f(z) = ez is analytic
a) only at z = 0 b) only at z = i c) nowhere d) everywhere
(Ans: d)
72.The points that coincide with their transformation are known as
a) fixed points b) critical points c) singular points d) None of these
(Ans: a)
73.The fixed points of the transformation w = z2 are
a) 0,-1 b) 0,1 c) -1,1 d) – i , i
(Ans: b)
74.The invariant points of the mapping w = are
a) 1, -1 b) 0,-1 c) 0,1 d) – 1 , -1
(Ans: c)
 Ajay Kumar Garg Engineering College, Ghaziabad

COURSE: B. TECH. SEMESTER: II

SUBJECT: ENGINEERING MATHEMATICS II SUBJECT
CODE: KAS-203T

MODULE-5
COMPLEX VARIABLE – INTEGRATION

MULTIPLE CHOICE QUESTIONS

1. The value of the integral of 1 / z along a semicircular arc from -1 to 1 in the clockwise
direction will be ……………….

a) zero (b) –πi (c) πi (d) None of these

(Ans:b)
2. Evaluate ∫𝐶 |𝑧|𝑑𝑧 , where C is the contour i.e the left half of unit circle |𝑧| = 1 from 𝑧 =
−𝑖 to 𝑧 = 𝑖.
a) 0 b)2i c) i d) 1
(Ans: b)
3. If f(z) is an analytic function and f’(z) is continuous at each point within and on a simple
closed curve C then,
a) ∮𝑐 𝑓(𝑧)𝑑𝑧 = 0 b)∮𝑐 𝑓(𝑧)𝑑𝑧 = 1
c) ∮𝑐 𝑓(𝑧)𝑑𝑧 = 2𝜋𝑖 d) doesn’t exist
(Ans:a)
4. If f(z) is an analytic function in region R between two simple closed curve c1 and c2, then

a) ∮𝑐1 𝑓(𝑧)𝑑𝑧 = ∮𝑐2 𝑓(𝑧)𝑑𝑧 b) ∮𝑐1 𝑓(𝑧)𝑑𝑧 = 1-∮𝑐2 𝑓(𝑧)𝑑𝑧

c) ∮𝑐2 𝑓(𝑧)𝑑𝑧 + ∮𝑐1 𝑓(𝑧)𝑑𝑧 = 0 d) no relation exist.
(Ans:a)
5. If f(z) is an analytic function within and on a simple closed curve C and ‘a’ is any point
within C, then
𝑓(𝑧) 𝑓(𝑧)
a)∮𝑐 𝑑𝑧 = 2𝜋𝑖𝑓(𝑎) b) ∮𝑐 𝑑𝑧 = −2𝜋𝑖𝑓(𝑎)
(𝑧−𝑎) (𝑧−𝑎)
𝑓(𝑧)
c) ∮𝑐 𝑑𝑧 = 0 d) does not exist
(𝑧−𝑎)
(Ans:a)
6. Evaluate  (z − a) n dz , where n is integer and C is the circle, with center a and radius r.
C

a) 0 b) i c) i-1 d) 1+i
(Ans:a)
𝑒𝑧
7. The Poles of function f(z)=𝑧 2+𝑎2 are……

a) ±2𝑖 b) 0,1 c) ±𝑎𝑖 d) None of these.

(Ans:c)
 𝑒𝑧
8. Sum of the residues at poles of f(z) = 𝑧 2+𝑎2 is…….

1 1 3
a) 𝑎 sina b)− 2 c) 2 d) None of these.
(Ans: a)
1+𝑧
9. The residue of f(z)= 𝑧 2−2𝑧 4 at z = 0 is……….

a) -1 b) 0 c) 1 d ) None of these
(Ans: c)

𝑧 2 −4
10. Simple poles of f(z) =𝑧 2+5𝑧+4 are………..

a) 1,4 b)-1,4 c) -1,-4 d) None of these.

(Ans:d)
𝑧 2 +3
11. For the function f(z)= 𝑧 2(𝑧 2+4) , the pole z=0 has the order……..

a) 1 b)2 c)0 d) None of these.

(Ans:b)
1
12. Singular points of function f(z)=𝑧(𝑧−1)2 are………

(a) 0, 1, -1 ( b) 0 , 1, 1 (c) 1, -1, -1 (d) None of these

(Ans:b)
1
13. The expansion of 𝑧−2 is valid for

(a) |𝑧| < 1 (b) |𝑧| > 3 (c) |𝑧| < 2 (d)None
(Ans: c)
1
14. The region of validity of 𝑧+1 for its Taylor’s series expansion about z = 0 is…..

(a)|𝑧| < 1 (b) |𝑧| > 1 (c) |𝑧| = 1 (d) None

(Ans: a)

15. If F(z) is an analytic function at z = a, then it has a power series expansion about z = a.

(a)Statement is true ( b) Statement is false (c) None of these

(Ans: a)

 (z − z )dz , where C is the upper half of the circle z−2 = 3.

2
16. Evaluate the integral
C
a) 0 b) 10 c) -20 d) 30

z dz , where C is the boundary of the triangle with vertices 0 ,1 + i ,−1 + i .

2
17. Evaluate
C

a) 1+2i b) -1-i c) 0 d) 3i
(Ans: c)
 cos z
18. Evaluate c z −  dz , where C is the circle z − 1 = 3 .
a)2πi b) -2 i c) -πi d) 1
(Ans:b)
e 3iz
19. Evaluate  dz , where C is the circle |𝑧 − 𝜋| = 16/5
c
(z +  )3

a) 𝑒 3𝑖 b) 𝑒 2𝑖 c) 𝑒 𝑖 d) 𝑒 0 − 1
(Ans: d)
𝑓(𝑧)
20. The relation ∫𝑐 𝑑𝑧 = 2𝜋𝑖 𝑓 ′ (𝑎) 𝑖𝑠 known as:
(𝑧−𝑎)2
a) Cauchy integral theorem b) Cauchy residue theorem
c) Cauchy integral formula d) Cauchy integral formula for derivative
(Ans: d)
𝑐𝑜𝑠 𝜋𝑧
21. Find the zeros of (𝑧) = (𝑧−1)2 .
𝜋 𝜋
a) 𝑧 = (2𝑛 + 1) 2 , n is integer b) 𝑧 = 2
𝜋
c) 𝑧 = 𝑛 2 , n is integer d) 𝑧 = 0
(Ans: a)
1 
22. Find the singularities of the following function and discuss its nature: f (z ) = cos 
z−2
a) Isolated essential singularity at z=2 b) Removable singularity at z=2
c) Isolated singularity at z=2 d) Essential singularity at z=2
(Ans:d)
2z + 1
23. Find the poles of 2 and the residue at each pole.
z −z−2
1 5 1 5
a) Poles are z=1, -2 Residues are , b) Poles are z=-1,-2 Residues are ,
3 3 3 3
1 5 1 5
c) Poles are z=-1,2 Residues are , d) Poles are z=1,2 Residues are ,
3 3 3 3
(Ans: c)
24. Residue of f(z) at 𝑧 = ∞ is….

a) 0 b) lim 𝑧𝑓(𝑧) c) lim (− 𝑧𝑓(𝑧)) d) ∞

𝑧→∞ 𝑧→∞
(Ans:c)
25. The region in which function f(z) has no singularity is known as:

a) Radius of curvature b) Region of convergence

b) Region of divergence d) None of the above
(Ans: b)

26. The limit point of zeros of a function f(z) is an…..

a) essential singularity b) removable singularity
c)isolated singularity d) isolated essential singularity
(Ans: d)
 𝑧2
27. Integration of the complex function f(z) = 𝑧 2−1 in the counter clockwise direction,
around |z−1| =1 is
(a) 𝜋𝑖
(b) −𝜋𝑖
(c) 0
(d) None of these
(Ans: a)
28. If 𝑓(𝑧) is an analytic function and all its derivatives are continuous at each point in the
cluse curve c, then which of the following is true
(a) ∫𝑐 𝑓(𝑧)𝑑𝑧 = 0
(b) ∫𝑐 𝑓(𝑧)𝑑𝑧 = 2 𝜋𝑖
(c) ∫𝑐 𝑓(𝑧)𝑑𝑧 = −2 𝜋𝑖
(d) None of these
(Ans:a)
𝑒 −𝑧
29. Value of the integral ∫𝑐 𝑧+1 𝑑𝑧 where c is the circle |𝑧| = 1
(a) 𝜋𝑖
(b) 0
(c) −𝑖
(d) None of these
(Ans:b)
3𝑧 2 +7𝑧+1
30. Value of the integral ∫𝑐 𝑧+1 𝑑𝑧 where c is the circle |𝑧 + 𝑖| = 1
(a) 0
(b) 2
(c) i
(d) None of these
(Ans: a)
𝑒 −𝑧
31. Value of the integral ∫𝑐 𝑧+1 𝑑𝑧 where c is the circle |𝑧| = 2
(a) 𝜋𝑖
(b) 2𝜋𝑖𝑒
(c) 2𝜋𝑖
(d) None of these
(Ans: b)
3𝑧 2 +7𝑧+1
32. Value of the integral ∫𝑐 𝑧+1 𝑑𝑧 where c is the circle |𝑧| = 1.5
(a) 0
(b) 2
(c) −6𝜋𝑖
(d) None of these
(Ans: c)
𝑑𝑧
33. Value of the integral ∫𝑐 𝑧 2+9 where c is the circle |𝑧 − 3𝑖| = 4
𝜋
(a) 3
2𝜋
(b) 3
(c) 0
(d) None of these
(Ans:c)
34. At a point where 𝑓(𝑧) = 0 is called
 (a)Zeros of 𝑓(𝑧)
(b)Derivative of 𝑓(𝑧)
(c)Continuity of 𝑓(𝑧)
(d)None of these
(Ans:a)
1
35. Singularity of 𝑓(𝑧) = 𝑠𝑖𝑛 𝑧−1 at z = 1
(a) Non isolated
(b) Isolated essential
(c) Essential
(d) None of these
(Ans:b)
𝑒𝑧
36. Singularity of 𝑓(𝑧) = 𝑧 2 at z = 0
(a) Non isolated
(b) Isolated essential
(c) Essential
(d) None of these
(Ans: b)
𝑧2
37. The poles of the function 𝑓(𝑧) = (𝑧−1)(𝑧−2)2 are
(a)z = 1 is a simple pole and z = 2 is a pole of order 2
(b)z = 2 is a simple pole and z = 1 is a pole of order 2
(c)z = 1 is a simple pole and z = 2 is a simple pole
(d)None of these
(Ans:a)
38. When z = a is a simple pole then its residue can be evaluated by
(a) lim(𝑧 − 𝑎)𝑓(𝑧)
𝑧→𝑎
(b) lim(𝑧 + 𝑎)𝑓(𝑧)
𝑧→𝑎
(c) lim(𝑧 − 𝑎)𝑓(𝑧)
𝑧→0
(d) None of these
(Ans:a)
(𝑧−1)
39. Value of residue at z = 2 of 𝑓(𝑧) = (𝑧−2)(𝑧+1)2
1
(a) 9
2
(b) 9
−1
(c) 9
(d) None of these
(Ans:a)
(12𝑧−7)
40. Value of the integral ∮𝑐 (2𝑧+3)(𝑧−1)2 by Cauchy’s residue theorem for |𝑧| = 2 and
|𝑧 + 𝑖| > √3
(a) 𝜋𝑖
(b) - 𝜋𝑖
(c) 0
(d) None of these
(Ans:c)
𝑐𝑜𝑡𝜋𝑧
41. The nature of the singularity of 𝑓(𝑧) = (𝑧−𝑎)2 at 𝑧 = ∞
(a) Non isolated essential
 (b) Non isolated
(c) Isolated
(d) None of these
(Ans: d)
𝑒 𝑧 −1
42. The principal part of Laurent series of 𝑓(𝑧) = 𝑧2
(a) 𝑧
1
(b) 𝑧
1
(c) 𝑧 2
(d) None of these
(Ans:b)
43. Which of the following is related to Cauchy residue theorem?
(a) ∫𝑐 𝑓(𝑧)𝑑𝑧 = 0
(b) ∫𝑐 𝑓(𝑧)𝑑𝑧 = 2 𝜋𝑖
(c) ∫𝑐 𝑓(𝑧)𝑑𝑧 = 2 𝜋𝑖(𝑆𝑢𝑚 𝑜𝑓 𝑅𝑒𝑠𝑖𝑑𝑢𝑒𝑠)
(d) None of these
(Ans:c)
1+k𝑖
44. For the integral ∫𝑜 𝑧̅ 𝑑𝑧 along the curve given by 𝑦 = 2𝑥 the value of k is
(a) k = 2
(b) k =1
(c) k = 0
(d) None of these
(Ans:a)
1−𝑐𝑜𝑠𝑧
45. If 𝑓(𝑧) = expanded by Laurent’s series about the point z = 0 then the coefficient
𝑧3
of 1/z is
1
(a) 2
1
(b) 3
1
(c) 4
(d) None of these
(Ans:a)
𝑒 2𝑧
46. Value of the integral ∮𝑐 (𝑧+1)4 by Cauchy’s integral formula for |𝑧| = 3
8𝑖𝜋
(a) 3𝑒 2
𝑖𝜋
(b) 3𝑒 2
8𝑖𝜋
(c) − 3𝑒 2
(d) None of these
(Ans:a)
2
47. Value of the integral ∮𝑐𝑒 𝑠𝑖𝑛𝑧 𝑑𝑧 where c is the region bounded by |𝑧| = 1
(a) 0
(b) 1
(c) 2
(d) None of these
(Ans: a)
 (2𝑧 2 +5)
48. Value of the integral ∮𝑐 (𝑧 2+4)(𝑧+2)3 𝑑𝑧 where c is the square with the vertices at 1 +
𝑖, 2 + 𝑖, 2 + 2𝑖, 1 + 2𝑖
(a) 0
(b) 2i
(c) 1
(d) None of these
(Ans:a)
49. The limit point of the zeros of a function𝑓(𝑧) is an isolated essential singularity
(a) Yes
(b) No
(c) May be
(d) None of these
(Ans:a)
50. Residue at infinity of a function 𝑓(𝑧) is equal to
1
(a) − 2𝜋𝑖 ∮𝑐 𝑓(𝑧)𝑑𝑧
1
(b) ∮ 𝑓(𝑧)𝑑𝑧
2𝜋𝑖 𝑐
1
(c) − 2 ∮𝑐 𝑓(𝑧)𝑑𝑧
(d) None of these
(Ans:a)
1
51. The kind of singularity of the function 𝑓(𝑧) = 1−𝑒 𝑧 at 𝑧 = 2𝜋𝑖
(a) Non isolated essential
(b) Non isolated
(c) Pole of order 1
(d) None of these
(Ans:c)
52. The process of integration along a closed curve is called
(a) Contour integration
(b) Line integration
(c) Surface Integration
(d) None of these
(Ans:a)

53. When all pole of the complex function 𝑓(𝑧) lies outside the closed curve c then its
integration is
(a) Always 0
(b) Always 1
(c) Always negative
(d) None of these
(Ans:a)
54. If a function f(z) is analytic inside and on a simple closed curve C, then  f(z)dz = 0 .
C

This result is known as

a) Cauchy’s integral formula
b) Cauchy’s integral theorem
c) Cauchy’s theorem for multi-connected region
d) Cauchy’s residue theorem
(Ans:b)
55. A function f(z) is not analytic at a point z = a. Then z = a is called
e) Residue
f) Singularity
g) Critical point
h) Cardinality
(Ans:f)
z
56. The poles of the function f (z ) = are
(z + 2 )(z − 3)
i) 2 and 3
j) -2 and 3
k) 2 and -3
l) -2 and -3
(Ans: j)
1
57. The Taylor’s expansion of the function f (z ) = for z  3 is
z+3
n
1  z
m)  (− 1)n  
3 n =0 3
n
1  z
n)  
3 n =0  3 
n
1  3
o)  (− 1)n  
z n =0 z
n
1  3
p)  
z n =0  z 
(Ans:o)

e
z2
58. The value of dz for the triangle having vertices as z = 0, z = 2+i and z = 1-i is
C

q) 2i
2i
r)
e2
s) 0
t) None of these
(Ans:s)

sin z
59. The residue of  z(z + 1) dz at z = 0 is
C

u) 2i
v) sin 1
w) 0
x) None of these
(Ans: w)
 z2 + 1
60. The value of  dz where C is given by|z|= 2 is
C z+1

y) 2i
z) 4i
aa) i
bb) None of these
(Ans:z)
e iz
61. The value of  3 dz where C is given by|z|= 2  is
C
z
cc) 2i
dd) 4i
ee) − i
ff) None of these
(Ans: ee)
e 2z
62. The value of  dz where C is given by|z|= 20 is
c
(z + 1 )5

4i
gg)
3e 2
2
hh) 2
3e
ii) 0
jj) None of these
(Ans: gg)
sin z
63. The value of  dz where C is given by|z|= 30 is
C
(z + )(z + 2)
kk) 2i
ll) − 2i
mm) 0
nn) None of these
(Ans: mm)

cos z x2 y2
64. The value of  z(z + 4)
C
dz where C is the ellipse given by
4
+
9
= 1 is

i
oo)
2
1
pp)
4
qq) 0
rr) None of these
(Ans: oo)
65. A region which is not simply connected is called
 ss) Multiple curve
tt) Jordan Curve
uu) Connected curve
vv) Multiply connected
(Ans: vv)
66. If f(z) is analytic and f ’(z) is continuous at all points in the region bounded by simple
closed curves C1 and C2, then
ww)  f (z )dz =  f (z )dz
C1 C2

xx)  f (z )dz   f (z )dz

C1 C2

yy)  f ' (z )dz   f ' (z )dz

C1 C2

zz) None of these

(Ans: ww)
67. If f(z) is analytic and f ’(z) is continuous at all points in the region bounded by simple
closed curves C1 and C2, then
aaa)  f (z )dz =  f (z )dz
C1 C2

bbb)  f (z )dz   f (z )dz

C1 C2

ccc)  f ' (z )dz   f ' (z )dz

C1 C2

ddd) None of these

(Ans: aaa)
68. If the principle part contains an infinite number of non zero terms of z-a, then z = a is
called
eee) Pole
fff) Essential singularity
ggg) zero of the function
hhh) Removable singularity
(Ans: fff)
69. If f(a) = 0 and f ’(a) ≠ 0 , then z = a is called

(a) Simple curve

(b) Zero of order n
(c) Simple pole
(d) None of these
(Ans: c)

2𝜋
70. Integral of the type ∫0 𝐹(𝑐𝑜𝑠𝜃 , 𝑠𝑖𝑛𝜃)𝑑𝜃 can be written as

𝑧 + 𝑧 −1 𝑧− 𝑧 −1 𝑑𝑧
(a) ∮𝐶 𝐹( , ) ; C:|𝑧| = 1
2 2𝑖 𝑧
 𝑧 + 𝑧 −1 𝑧− 𝑧 −1 𝑑𝑧
(b) ∮𝐶 𝐹( , ) ; C:|𝑧| = 1
2 2𝑖 𝑖𝑧
𝑧− 𝑧 −1 𝑧+ 𝑧 −1 𝑑𝑧
(c) ∮𝐶 𝐹( , ) 𝑖𝑧 ; C:|𝑧| = 1
2 2𝑖
(d) None of these
(Ans: b)
2𝜋 𝑐𝑜𝑠3𝜃
71. Value of ∫0 𝑑𝜃 is
5−4𝑐𝑜𝑠𝜃

(a) 0
𝜋
(b) 3
𝜋
(c) 12
−𝜋
(d) 12
(Ans: c)
𝑧 2 −1
72. Expansion of the function f(z) = ( 𝑧+2 )( 𝑧+3 ) in the region 2 <|𝑧|< 3 is

(a) Taylor series

(b) Fourier series
(c) Laurent series
(d) Can’t be found
(Ans: c)
𝑠𝑖𝑛 𝑧
73. Expansion of about z = 𝜋 is
𝑧−𝜋

(𝑧−𝜋)2 (𝑧−𝜋)4
(a) 1+ + + …….
3! 5!
(𝑧−𝜋)2 (𝑧−𝜋)4
(b) 1+ - + …….
3! 5!
(𝑧−𝜋)2 (𝑧−𝜋)4
(c) -1+ + + …….
3! 5!
(𝑧−𝜋)2 (𝑧−𝜋)4
(d) -1+ - + …….
3! 5!
(Ans: d)
𝑑𝑧
74. The value of ∮𝐶 where C is the circle |z-a|=r , is
𝑧−𝑎

(a) 0
(b) 𝜋𝑖
(c) 2𝜋𝑖
(d) 3𝜋𝑖
(Ans: c)

75. The value of ∮𝐶 (𝑧 − 𝑎)𝑛 𝑑𝑧 , where n is not equal to -1 and C is the circle |z-a|=r , is

(a) 0
(b) 1
(c) 2
 (d) 3
(Ans: a)

76. The value of ∮𝐶 (𝑥 2 − 𝑦 2 + 2𝑖𝑥𝑦)𝑑𝑧 , where C is the circle |z |=1 , is

(a) 0
(b) 4
(c) −2
(d) 3
(Ans: a)