Title MP DPP Sheet 09.pdf ✅

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.6. Suppose f x  is defined as following:       0 for - 0 such that 2 2 for 0 x f x f x f x x               In the fourier series expansion of f x , the coefficient of cos2x is (a) 0 (b) 1/2 (c) 1 (d) 1/  Q.7. Consider the following periodic function:       3 f t t t a t a f t a f t        for such that 2 The Fourier Co-efficient n b will be proportional to (Symbols have their usual meanings) (a) 2 1/ n (b) 3 1/ n (c) 2 n (d) 3 n Q.8. Suppose f x  is defined as following:       0 for - 0 such that 2 for 0 4 x f x f x f x x x                 then the co-efficient an will be (a)   2 1 4 n n  (b)   2 1 1 4 n n   (c)   1 1 4 n n   (d)   1 4 n n  Q.9. An alternating current after passing through the rectifier has the following form:       4sin for 0 such that 2 0 for 2 t t f t f t f t x               The Fourier Coefficient 1 b will be (Symbols have their usual meanings) (a) 2 (b) 4 (c) 1 (d) 0 Q.10. Let f x( ) be a function of period 2 such that 1 0 ( ) 0 0 x f x x           Then Fourier series of f x( ) in the interval     x is (a) 1 2 1 1 sin sin 3 sin 5 .... 2 3 5 x x x           (b) 1 2 1 1 sin sin 3 sin 5 .... 2 3 5 x x x           (c) 1 2 1 1 cos cos 3 cos5 .... 2 3 5 x x x           (d) 1 2 1 1 cos cos3 cos 5 .... 2 3 5 x x x          
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 periodic function:       1 2 for 1 0 such that 2 1 2 for 0 1 t t f t f t f t t t              The ratio 1 3 a a/ will be (Symbols have their usual meanings) (a) 1/3 (b) 1/9 (c) 3 (d) 9 Q.12. Consider the following function:       2 f t t f t f t     1 sin such that 2 Which of the following is CORRECT? (Symbols have their usual meanings) (a) 0 1 2 3 1 , 0, 2 2 a a a     (b) 0 1 2 3 1 , 0, 2 2 a a a    (c) 0 1 2 1 1, 0, 2 a a a    (d) 0 1 2 1 2, 0, 2 a a a     Q.13. The Fourier series for an arbitrary periodic function with period 2L is given by   0 1 1 cos sin 2 n n n n a n x n x f x a b L L            . For the particular periodic function, the value of a0 is f(x) 1 1⁄2 –2 –2 –1 0 1 x (a) 1/2 (b) 1 (c) 0 (d) 2 Q.14. In the fourier expansion for f x x x x       cos     . Which of the following statement is CORRECT? (a) 0 for all n n b  (b) 2 2 for all even n 1 n n b n   (c) 2 2 for all even n 1 n n b n    (d) 2 2 1 n n b n    Q.15. In Fourier series expansion of f x x    sin      x , which of following co-efficients are NON-ZERO? (a) an for odd n (b) an for even n (c) bn for odd n (d) bn for even n. Q.16. Fourier series expansion of the function 2 2 2 2 1 ( ) 0 1 x f x x x                     (a) 1 2 2sin 2sin 2 sin 3 3 x x x            (b) 1 2 2sin 2sin 2 sin 3 3 x x x             (c) 1 2 2cos 2cos 2 cos3 3 x x x            (d) 1 2 2cos 2cos 2 cos3 3 x x x            
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 f x  be an even, period function of x which is continuous. Then, it can be represented by (a) Series containing cosine terms (b) Series containing sine terms (c) Series containing both sine & cosine terms (d) Not possible to comment from the given information Q.18.   2 1 f x x  sin and   2 2 f x x  cos are expanded in their corresponding fourier series in the interval      x . Which of the following is CORRECT? (a) 1 2    2 2 a b    1/ 2, 1/ 2 (b) 1 2    2 2 b a    1/ 2, 1/ 2 (c) 1 2    2 2 a a    1/ 2, 1/ 2 (d) 1 2    2 2 b b    1/ 2, 1/ 2 Q.19. Consider the Fourier series corresponding to the function   3, 0 5 3, 5 0 x f x x          Then the fourier series of the given function contains (a) Cosine terms only (b) Both cosine and sine terms (c) Sine terms only (d) Sine terms and non-zero constant terms. Q.20. For which of the following periodic functions, the Fourier Series does NOT contain any sine terms? (a)   cos for 0 0 for 0 t t f t t            (b)   cos for 0 cos for 0 t t f t t t            (c)   sin for 0 sin for 0 t t f t t t            (d)   sin for 0 0 for 0 t t f t t            PART - B: Numerical Answer Type (NAT) Questions Q.21. In the Fourier series expansion of the finction     4 f x x   in 2,2 , the Fourier co-efficient 4 b will be equal to ____________________________________________ [Your answer should be AN INTEGER] Q.22. The Fourier coefficient 3 a in the Fourier series expansion of f x x x f x f x          2 2 2 ; 4       is _____________________________ [Your answer should be upto TWO DECIMAL PLACES] Q.23. The constant term in the Fourier series expansion of f t t     sin in ,     , will be _______________________________ [Your answer should be upto TWO DECIMAL PLACES] Q.24. The Fourier coefficients an in the Fourier series expansion for   1, 0 1, 0 x f x x            ______________________________________________ [Your answer should be AN INTEGER]