Tiêu đề MP DPP Sheet 17.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 17: Miscellaneous Topics (Gamma, Beta Integral and Jacobian) 1. The value of the integral 4 1 0 2 e x x dx    (a) 24e (b) 48e (c) 12e (d) 36e 2. The value of the integral 0 e x x dx    (a) 4 (b) 6 (c) 8 (d) 12 3. The value of the integral  3 2 1 ln x dx x   (a) 2 (b) 4 (c) 8 (d) 6 4. The value of the integral 2 0 2 x xe dx    (a) 1/ 4 (b) 3/ 4 (c) 5 / 4 (d) 7 / 4 5. The value of the integral 2 6 4 0 x x e dx    (a) 15 512  (b) 15 1024  (c) 15 2048  (d) 15 256  6. The value of the integral   1 1 0 1 ln 1,0, 1, 2,.... a dx a x                  (a) a / 2 (b) 3 / 2 a (c) a (d) 2 a
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 7. The value of the integral     1 0 ln 1,2,3,.... n x dx n   (a) n 1 ! (b)   1 1 ! n n   (c) n! (d)   1 . ! n  n 8. The value of the integral     1 0 .ln 1,2,3,.... n x x dx n   (a)     1 1 ! 1 n n n n    (b)   1 ! n n n n  (c) ! n n n (d)   1 ! 1 n n n   9. The value of the integral     0 1 k ax n e dx n  k x     (a) 1 1 ak n e a n   (b) 1 1 ak n e a n    (c) 1 1 ak n e a n    (d) 1 1 ak n e a n   10. The value of the integral   1 4 5 0 7 1 x x dx   (a) 1/45 (b) 1/90 (c) 1/120 (d) 1/180 11. The value of the integral /2 5 4 0 sin . cos d      (a) 8/315 (b) 4/315 (c) 8/225 (d) 4/225 12. The value of the integral 1 4 2 0 x x dx . 1  (a) 8  (b) 4  (c) 32  (d) 16  13. The value of the integral 4 3 0 4 x dx  x  (a) 2048/35 (b) 4096/35 (c) 1024/35 (d) 512/35 14. The value of the integral /4 2 4 0 sin 2 . cos 2 d      (a) 1/35 (b) 2/35 (c) 1/7 (d) 2/7
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 15. The value of the integral   1 2 5 3 0 x x dx 1  (a) 1/12 (b) 1/24 (c) 1/36 (d) 1/48 16. Consider the following transformation: 2 3 2 2 u x y z v x yz w z xy       3 , 4 , 2 The Jacobian of u v w , ,  w.r.t.  x y z , ,  at 1, 1,0   is (a) 10 (b) 20 (c) -20 (d) -10 17. Consider the following transformation: 2 4 x u v y uv    , The Jacobian of  x y,  w.r.t u v,  will be (a) 2 2 2 2 u v  (b) 2 2 2 4 u v  (c) 2 2 u v  4 (d) 2 2 u v  2 18. Consider the following transformation: 2 2 2 2 u x y v x y x r y r       2 , 2 where cos , sin   The Jacobian of u v,  w.r.t r,  will be (a) 2 r sin 2 (b) 3 6 sin 2 r  (c) 2 6 sin 2 r  (d) 6 sin 2 r  19. Consider the following transformation: p x y z q y z r z       , , The Jacobian of  p q r , ,  w.r.t.  x y z , ,  is (a) -1 (b) 0 (c) 1 (d) 2 20. Consider the following transformation: 1 1 , tan tan 1 x y u v x y xy        The Jacobian of u v,  w.r.t  x y,  will be (a)  2 1 x y  (b) sin x y   (c) tan  x y   (d) 0 21. Consider the following transformation: u a x y v a x y   cosh cos , sinh sin The Jacobian of u v,  w.r.t  x y,  will be (a)   1 2 cosh 2 cos 2 2 a x y  (b)   1 2 sinh 2 sin 2 2 a x y  (c)   1 2 sinh 2 sin 2 2 a x y  (d)   1 2 cosh 2 cos 2 2 a x y 
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 22. Consider the transformation from  x y,  coordinates to u v,  coordinates, such that cos , sin x x u v e y u v e y     The Jacobian of u v,  w.r.t.  x y,  is (a) 1 2 2 xe (b) 1 2 2  xe (c) 0 (d) 2 xe Answer Key 1. (b) 2. (a) 3. (d) 4. (b) 5. (c) 6. (c) 7. (d) 8. (a) 9. (a) 10. (d) 11. (a) 12. (c) 13. (b) 14. (a) 15. (c) 16. (b) 17. (b) 18. (b) 19. (c) 20. (d) 21. ( ) 22. (b)