Tiêu đề MP DPP Sheet 11.pdf ✅

1 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 IIT JAM PHYSICS 2025(Online Batch) SECTION: MATHEMATICAL PHYSICS Daily Practice Problem (DPP) Sheet 11: Complex Analysis (Basic Preview of Complex Numbers - I) ; PART - A: MULTIPLE CHOICE QUESTIONS (MCQ) Q.1. If x + iy =   100 i 3 1 , then the co-ordinate of point P(x, y) will be (a)   99 99 2 ,2 3 (b)   99 99 2 , 2 3  (c)   99 99 2 ,2 3 (d) none of these Q.2. Let 1 z and 2 z are two non-zero complex numbers such that 1 2 1 2 z z z z    then the phase difference between z 1 and z 2 is (a) 300 (b) 450 (c) 600 (d) 900 Q.3. The area of triangle formed by the complex numbers z iz , , z iz  is equal to (a) 1 2 2 z (b) 1 2 2 z (c) 2 z (d) 2 z Q.4. If complex numbers 1 2 z z, and origin form an equilateral triangle, then which of following relation is TRUE? (a) 2 2 1 2 1 2 z z z z    0 (b) 2 2 1 2 1 2 z z z z    0 (c) 2 2 1 2 z z   0 (d) 2 2 1 2 1 2 z z z z    2 0 Q.5. If z  0 is a complex number such that arg  4 z   , then (a)     2 2 Re Im z z  (b)   2 Re 0 z  (c)   2 Im 0 z  (d) None of these Q.6. Two complex numbers z z 1 2 & such that 1 2 z z  . Which of the following is always CORRECT? (a) 1 2 z z  (b) 1 2 i z e z   (c) 1 2 1 z z  (d) 2 2 1 2 z z  1 Q.7. Let 1 z and 2 z are two non-zero complex numbers such that 1 2 z i z   0 and argz z1 2    then arg z is equal to (a)  / 4 (b)  / 4 (c) 3 / 4  (d) 3 / 4 
2 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 Q.8. Given that i  1 , then i i will be (a) Purely Real (b) Purely Imaginary (c) Complex Number (d) Not defined Q.9. Multiplying a complex number z by 1 3  i rotates the radius vector of z by an angle of (a) 60o clockwise (b) 30o anticlockwise (c) 30o clockwise (d) 60o anticlockwise Q.10. The locus of the point z defined by the equation arg 4   4 z    , is (a) circle (b) parabola (c) ellipse (d) straight line Q.11. If z ia complex number of unit modulus and argument  , then 1 arg 1 z z         is equal to (a)  (b) 2   (c)   (d)  Q.12. The complex numbers ‘z’ for which arg 1   6 z    and arg 1 2   3 z    , is (a) 3 2 2 i   (b) 3 2 2 i   (c) 1 3 2 2  i (d) 1 3 2 2  i Q.13. The value of the following series:   3 5 i i i n    .... upto 2 1 terms will be (a) i (b) – i (c) 1 (d) – 1 Q.14. If z is complex number such that 1 z 2cos z    , then   2 2 cos n n z z n   is equal to (a) 1 (b) 0 (c) -1 (d) -n Q.15. If   2 z i       2 3 7 ( is a real parameter and i  1 ), then the locus of z will be (a) circle (b) ellipse (c) parabola (d) hyperbola PART - B: Numerical Answer Type (NAT) Questions Q.16. The phase of the complex number 1i i in the polar representation is _____________ (in degree) [Your answer should be AN INTEGER] Q.17. The smallest positive integer n for which 1 1         n i i is a positive real quantity, will be _________________ [Your answer should be AN INTEGER]
3 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 Q.18. If 101 1 k k a ib i     , then the values of a b   will be ________________________________________ [Your answer should be AN INTEGER] Q.19. Let 1 z and 2 z are two non-zero complex numbers such that 1 2 z z 1 and arg arg / 2 z z 1 2       The value of 1 2 z z is equal to ________________________________________________________ [Your answer should be AN INTEGER] Q.20. The value of  for which 2 3 sin 1 2 sin i i     is purely imaginary, is equal to ________________________ [Your answer should be upto ONE DECIMAL PLACES] Q.21. If the phrase of the complex number 99 99 2 2 i i        is given by  , then the value of  is _________ [Your answer should be upto ONE DECIMAL PLACES] PART - C: Multiple Select Questions (MSQ) Q.22. For which of the following numbers, z 1? (a) 999 1 2    i     (b) 111 1 1 i i         (c) 555 1 2 2 3 3 i        (d) 777 1 2 1 3 i i         Q.23. Let 1 z and 2 z are two non-zero complex numbers such that 1 2 1 2 z z z z    Which of the following statements is/are CORRECT? (a) Re 0 z z1 2   (b) Re 0 z z1 2   (c) Im 0 z z1 2   (d) Im 0 z z1 2   Q.24. If   2 for 1 1 z z z   is a purely real number, then the point z lies on (a) real axis (b) imaginary axis (c) circle passing through origin (d) circle with center at the origin Q.25. If sum & products of two complex numbers are real, then which of the following statements are CORRECT? (a) both complex numbers are real (b) both complex numbers are purely imaginary (c) one of the complex numbers are real (d) both are complex conjugate of each other
4 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 Answer Key PART - A: MULTIPLE CHOICE QUESTIONS (MCQ) 1. (c) 2. (d) 3. (b) 4. (b) 5. (b) 6. (b) 7. (d) 8. (a) 9. (a) 10. (d) 11. (d) 12. (d) 13. (a) 14. (c) 15. (b) PART - B: NUMERICAL ANSWER TYPE (NAT) QUESTIONS 16. (45) 17. (4) 18. (1) 19. (-1) 20. (35.2 to 35.4) 21. (-0.5) PART - C: MULTIPLE SELECT QUESTIONS (MSQ) 22. (a, b, c) 23. (a, c) 24. (a, c) 25. (a, d)