Title MP DPP Sheet 10.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 2024 (Online Batch) SECTION: MATHEMATICAL PHYSICS Daily Practice Problem (DPP) Sheet 10: FOURIER SERIES (Complex Form of Fourier Series, Dirichlet’s Condition, Convergence Theorem) PART - A: MULTIPLE CHOICE QUESTIONS (MCQ) Q.1. Consider the following periodic function:    1 1 such that 2      x f x e x f x f x        The complex form of Fourier series expansion of f x  can be expressed as (a)     1 cosh1 1 n in x n e in        (b)   1 cosh1 1 in x n e in       (c)     1 sinh1 1 n in x n e in        (d)   1 sinh1 1 in x n e in       Q.2. Consider the following periodic function:       8 for 0 5 such that 20 0 for 5 20 t f t f t f t t           In the complex Fourier Series expansion of the given function, the value of 2 C1 will be (a) 2 16 /  (b) 2 32 /  (c) 2 64 /  (d) 2 8 /  Q.3. Consider the following periodic functions: f t t f t t t 1 2        6cos 6sin for / /            In the complex Fourier series expansion of the given functions, 1 Cn and 2 Cn represents complex Fourier co-efficients for f t 1   and f t 2   respectively. The value     1 2 1 1 C C/ is (a) 0 (b) 6 (c) 1 (d) 36 Q.4. Consider the following periodic function:       1/ 2 for 2 0 such that 4 1/ 2 for 0 2 x f x f x f x x            In the complex form of the Fourier Series Expansion, the value of 2 1 3 C C/ is (a) 3 (b) 1 (c) 6 (d) 9
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.5. Consider the following periodic function:       0 0 0 0 0 for 2 2 for such that 2 2 0 for 2 2 T t f t A t f t T f t T t                          In the complex Fourier series expansion of the given funnction, the value of Cn will be (a) 0 0 sin A n n T   (b) 0 0 cos A n n T   (c) 0 0 sin 2 A n n T   (d) 0 0 cos 2 A n n T   Q.6. Consider a periodic function f t  having period 1 unit and having the following Fourier co-efficients 1 for 0 3 0 for 0 n n n C n             The periodic function f t  will be (a)   2 3 3 i t f t e     (b)   2 3 3 i t f t e    (c)   2 3 3 i t f t e     (d)   2 3 3 i t f t e    Q.7. Consider the following periodic function:         5 3 f x x x x x f x f x         4 3 for 1 1 such that 2 If the Fourier Series Expansion of the function f x  can be written as       0 1 1 cos sin n n n n f x a a n x b n x            The sum of the series 0 n n a    will be (a) 0 (b) 1 (c) -1 (d) 2 Q.8. The constant term in the Fourier Series Expansion of the function     1 cos 0 2 2 x f x x      is (a) 0 (b) 4 / (c) 4 /  (d) 2 / Q.9. Which one of the following periodic functions have only sine harmonic terms in their Fourier Series Expan sion in the interval      x ? (a) 2 x (b) 3 sin x x  (c) 2 cos nx x  (d) 2 2x x  Q.10. In the Fourier series expansion of the periodic function f x x x           having period 2 , the ratio of the Fourier co-efficients for the second harmonic and fifth harmonic will be (a) 2 (b) -2 (c) -5/2 (d) 5/2
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.11. Consider the following function:       for 0 such that 2 for 0 x f x f x f x x x                At x   / 2, the sum of the Fourier Series will be (a)  / 2 (b)  / 4 (c) 3 / 4  (d) 3 / 2  Q.12. The avearge value of the function:   0 2 such that 2      x f x e x f x f x         is (a) 2 1 e     (b) 2 1 e    (c) 2 1 2 e     (d) 2 1 2 e    Q.13. From the Complex form of Fourier series expansion of the function     x f x e x       , indicate the correct relation from the following: (a) 1 1 1 1 2 .......... sinh 2 5 10              (b) 1 1 1 1 2 .......... sinh 2 5 10              (c) 1 1 1 1 2 .......... sinh 2 5 10              (d) 1 1 1 1 2 .......... sinh 2 5 10              COMMON DATA FOR Q.14 & Q.15 If   2 f x x x   (     x ) is expanded into the following Fourier series:       2 2 4 2 1 cos 1 sin 3 n n f x nx nx n n         Q.14. The sum of the series 2 2 2 2 1 1 1 1 ...... 1 2 3 4     is equal to (a) 2 12  (b) 2 12   (c) 2 4  (d) 2 4   Q.15. The sum of the series 2 2 2 2 1 1 1 1 ...... 1 2 3 4     is equal to (a) 2 2 (b) 2  (c) 2 3  (d) 2 6  Q.16. The function   2 f x x x    , 0 2 has a Fourier series representation 2 2 2 1 4 4 4 cos sin 3 n x nx nx n n               At x = 0 and 2 , the series converges to 2 2 . Then the value of the series 2 1 1 is: n n    (a) 2 2 (b) 2  (c) 2 3  (d) 2 6 
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 Q.17. If the Fourier series expansion of f x  in the interval  ,  is given as following     exp n n f x C inx      For which one of the following functions the Fourier Co-efficients Cn will be a real quantity? (a) 2 x x  4 (b) 3 sin 5 nx x  (c) 2 cos 3 nx x  (d) 3 2 4 3 x x x   Q.18. Consider the following periodic function:         2 f x x x f x f x          such that 2 If the Fourier series expansion of f x  can be written in the form     exp n n f x C inx      the co-efficient Cn will be (a) real number (b) complex number with non-zero real part (c) purely imaginary number (d) cannot say anything from the given information Q.19. The value of f 0 in the Fourier Series of the periodic function       2 1 for 0 such that 2 2 1 for 0 x x f x f x f x x x                     (a) -1 (b) +1 (c) -2 (d) -2 Q.20. For which of the following periodic functions, Fourier series expansion is NOT POSSIBLE? (a)   for 0 for 0 x x f x x x            (b)   for 0 for 0 k x f x k x            (c)   for 0 for 0 k x f x x            (d)   2 2 for 0 for 0 x x f x x x            PART - B: Numerical Answer Type (NAT) Questions Q.21. The constant term in the Fourier Series Expansion of the function f x x x        2 2 will be ________________________________________________ [Your answer should be AN INTEGER] Q.22. The co-efficient of the Fundamental Cosine Harmonic term in the Fourier Series Expansion     5 f x x x x        will be __________________ [Your answer should be AN INTEGER]