Tiêu đề MP DPP Sheet 18.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 18: Miscellaneous Topics (Taylor Series Expansion & Stationary Points) 1. The Taylor series expansion of   6x f x e  about x  4 is (a)     24 0 6 4 ! n n n e x n      (b)     24 0 6 4 ! n n n e x n      (c)     24 0 6 4 ! n n n e x n       (d)     24 0 6 4 ! n n n e x n      2. The Taylor series expansion of f x x     ln 3 4   about x  0 is (a)   1 1 1 4 ln 3 ! 3 n n n n x n             (b)   1 0 1 4 ln 3 ! 3 n n n n x n             (c)   1 1 1 4 ln 3 3 n n n n x n             (d)   1 0 1 4 ln 3 3 n n n n x n             3. The Taylor Series expansion of f x x    cos 4 about x  0 is (a)     2 0 4 2 ! n n n x n     (b)     2 0 16 2 ! n n n x n     (c)     2 0 16 4 ! n n n x n      (d)     4 0 4 4 ! n n n x n       4. The taylor series expansion of f x x    cos about 3 x   will be (a)   2 1 3 1 ........... 2 2 3 4 3 f x x x                     (b)   2 1 3 1 ........... 2 2 3 4 3 f x x x                     (c)   2 3 1 3 ........... 2 2 3 4 3 f x x x                     (d)   2 3 1 3 ........... 2 2 3 4 3 f x x x                     5. Suppose a real-valued function f x  such that       ' " f f f 1 1, 1 1, 1 1    and all other higher derivatives of f x  at x = 1, are zero. The value of f x  at x = 1/3 will be (a) 1/3 (b) 4/9 (c) 5/9 (d) 7/9
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 6. The taylor series expansion of the function   2 4 3x f x x e  about x = 0 will be (a)   2 4 0 3 ! n n n x n      (b)   2 4 1 3 ! n n n x n      (c)     2 4 1 3 2 ! n n n x n      (d)     2 4 0 3 2 ! n n n x n      7. The taylor series expansion of the function   3 2 f x x x    10 6 about x = 3 will be (a)       2 3 57 33 3 3 3       x x x (b)       2 3        57 33 3 3 3 x x x (c)       2 3        57 33 3 3 3 x x x (d)       2 3        57 33 3 3 3 x x x 8. The coefficient of  4 x 1 of Taylor series expansion of   2 1 f x x  about x = 1, will be (a) -5 (b) 5 (c) -4 (d) 4 9. The Taylor series expansion of the function f x x x     cos . ln 1  about x = 0, will be (a) 2 3 ..... 2 6 x x     x (b) 2 3 ..... 2 6 x x x    (c) 2 3 ..... 2 6 x x x    (d) 2 3 ..... 2 6 x x     x 10. The coefficient of  3 x 1 of Taylor series expansion    1 x f x x e   about x = 1, will be (a) e / 6 (b) e / 2 (c) e / 2 (d) e / 6 11. Consider the following function:   3 2 f x x x x     12 36 17 The function has a local minima at (a) x = 6 (b) x = -6 (c) x = 2 (d) x = -2 12. The function   1 f x x ln x  has a local maxima at (a) x = 1 (b) x = e (c) x = 2 (d) x = 1/e 13. The minimum value of the function   2 2 4 9 x x f x e e   will be (a) 10 (b) 11 (c) 12 (d) 14 14. The function     x f x x x R    attains a maximum value at (a) x = 2 (b) x = 3 (c) x = e (d) x = 1/e 15. If the function     3 2 2 f x x ax a x a      2 9 12 1 0 , attains a maximu & minimum value at p and q respectively such that 2 p q  , then a is equal to (a) 0 (b) 1 (c) 2 (d) -1
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 16. A positive real number x when added to it’s inverse, gives the minimum value of the sum at (a) x = 2 (b) x = -2 (c) x = 1 (d) x = -1 17. If P1,1, Q3,2 and R is a point between them on x-axis, then value of PR RQ  will be minimum at (a) 5 / 3,0 (b) 4 / 3,0 (c) 1/ 3,0 (d) 8 / 3,0 18. Suppose a cubic polynomial   3 x px q p q    , 0 has three distinct real roots. Which of the following statements is CORRECT? (a) Polynomial has a maxima at p / 3 and minima at  p / 3 . (b) Polynomial has maxima at p / 3 and  p / 3 . (c) Polynomial has minima at p / 3 and  p / 3 . (d) Polynomial has a minima at p / 3 and maxima at  p / 3 . 19. Consider a function:   0 5 sin for 0, 2 x f x t t dt x          Which of the following statements is CORRECT? (a) Function has a maxima at  and minima at 2 . (b) Function has maxima at  and 2 . (c) Function has minima at  and 2 . (d) Function has a minima at  and maxima at 2 . 20. Let     2 2 f x b x bx     1 2 1 and m b  be the minimum value of f x . As b varies, the maximum possible value of m b  will be (a) 0 (b) 1 (c) -1 (d) 2 Answer Key 1. (a) 2. (c) 3. (b) 4. (b) 5. (c) 6. (a) 7. (c) 8. (b) 9. (a) 10. (b) 11. (a) 12. (b) 13. (b) 14. (d) 15. (b) 16. (c) 17. (a) 18. (d) 19. (a) 20. (b)