Title Special Function DPP Sheet 01.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 CSIR-NET/JRF DEC 2024 (Online Batch) SECTION: MATHEMATICAL PHYSICS Daily Practice Problem (DPP) Sheet S1: SPECIAL FUNCTIONS (Legendre & Hermite Polynomials) Q.1. The solution of the differential equation   2 1 12 0 d dy x y dx dx          will be (a)   1 2 3 1 2 x  (b)   1 2 3 1 2 x  (c)   1 3 5 3 2 x x  (d)   1 3 5 3 2 x x  Q.2. The roots of the Legendre polynomials P x 5   are (a) imaginary (b) real & equal (c) real & positive (d) real and lies between -1 & +1 Q.3. Let P x n   be the Legendre polynomial of order n. Then   ' P x n  is equal to (a)     1 ' 1 n P x n   (b)     ' 1 n  P x n (c)     1 n  P x n (d)     '' 1 n  P x n Q.4. The expression 2 ' (0) P n can be written as (a) 0 (b) 1 (c) 1 2n (d) ( 1) . ! 2 n n  n Q.5. The value of P P 99 100    1 1    is (a) 1 (b) 2 (c) -2 (d) 0 Q.6. Express the following polynomials into legendre polynomials: (i) 3 2 4 6 7 2 x x x    [Ans.         3 2 1 0 8 47 4 4 5 5 P x P x P x P x    ] (ii) 3 2 4 2 3 8 x x x    [Ans.         3 2 1 0 8 4 3 22 5 3 5 3 P x P x P x P x    ] (iii) 3 x x  1 [Ans. 3 1 0       2 8 5 5 P x P x P x   Q.7. The function 2 f x x ( )  can be written in terms of Legendre Polynomials as (a) 2 0 2 ( ) ( ) 3 P x P x  (b) 1 2 2 ( ) ( ) 3 P x P x  (c) 2 1 ( ) ( ) 3 P x P x  (d) 2 0 ( ) ( ) 3 P x P x 
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. The function 2 f x x x ( )   can be written in terms of Legendre Polynomials as (a) 2 1 0 3 ( ) ( ) ( ) 2 P x P x P x   (b) 2 1 0 2 ( ) 3 ( ) ( ) 3 P x P x P x   (c) 2 1 0 2 ( ) ( ) ( ) 3 P x P x P x   (d) 2 1 0 2 ( ) ( ) ( ) 3 P x P x P x   Q.9. The value of the integral   1 2 3 1 P x dx ( )  is equal to (a) 2/3 (b) 2/5 (c) 2/7 (d) 1/5 Q.10. The value of the integral   1 1 ( ) 0 P x dx n n    is equal to (a) 0 (b) 1 (c) 1 2 1 n  (d) 2 2 n 1 Q.11. The value of the integral   1 1 2 2 1 ( ) 1 2 P x xz z dx n      is equal to (a) 2 2 1 n  (b) 2 2 1 n z n  (c) 2 1 n z n  (d) 0 Q.12. Let P x n   be the Legendre polynomial of order n >1. The value of the integral     1 1 1 n x P x dx     is (a) 2 / 2 1  n   (b) 2/3 (c) 2/5 (d) 0 Q.13. The value of the integral       1 1 1 99 100 0 1 1 1 P x dx P x dx P x dx         is (a) 0 (b) 1 (c) 2 (d) 199 Q.14. Given the recurrence relation for the Legendre polynomials:           1 1 2 1 1 n n n n xP x n P x nP x       The value of the integral 1 2 1 1 1 ( ) ( ) n n x P x P x dx     is (a) 2 ( 1) (2 1)(2 1)(2 3) n n n n n     (b) 2 ( 1) (2 1)(2 1)(2 3) n n n n n     (c) ( 1) (2 1)(2 1)(2 3) n n n n n     (d) ( 1) (2 1)(2 1)(2 3) n n n n n     Q.15. Given the recurrence relation for the Legendre polynomials:           1 1 2 1 1 n n n n xP x n P x nP x       The value of the integral 1 1 1 ( ) ( ) n n xP x P x dx   is (a) 2 2 4 1 n n  (b) 2 2 1 n n  (c) 2 2 1 n n  (d) 2 2 4 1 n n 

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.25. Given that   2 2 0 ! n t tx n n t H x e n       , the value of H9 0 is (a) 0 (b) 3024 (c) -3024 (d) -1512 Q.26. The value of the integral   2 4 3 2 4 64 12 6 8 x e x x x H x dx            will be (a) 384  (b) 768  (c) 1536  (d) 0 QUESTIONS ON NEW PATTERN OF GATE EXAMINATION PART I: Numerical Answer Type (NAT) Questions Q.27. The value of the integral   2 2 2 0 4 2 x e x dx     is __________________________________________ [Your answer should be upto ONE DECIMAL PLACES] Q.28. If the Hermite polynomial H x 3   becomes zero at    , , , then       is _________________ [Your answer should be AN INTEGER] PART II: Multiple Select Questions (MSQ) Q.29. Which of the following statement is CORRECT about the Legendre polynomials P x 5   ? (a) P x P x 5 5       0.5 0.5    (b) P x P x 5 5      0.5 0.5    (c) Graph of P x 5  intersects x-axis 5 times. (d) Graph of P x 5  intersects x-axis 10 times. Q.30. Which of the following statements is CORRECT about the solution of the differential equation? 2 2 2 8 0 d y dy x y dx dx    (a) y x  has even parity. (b) y x  has odd parity. (b) y x  has 8 nodes. (d) y x  has 4 nodes Answer Key 1. (c) 2. (d) 3. (a) 4. (a) 5. (d) 7. (a) 8. (b) 9. (c) 10. (a) 11. (b) 12. (d) 13. (c) 14. (a) 15. (a) 16. (d) 17. (b) 18. (a) 19. (c) 20. (d) 22. (b) 23. (c) 24. (c) 25. (a) 26. (c) 27. (7.0 to 7.2) 28. (0) 29. (a, c) 30. (a, d)