B.Sc. Part 1 Mathematics
TRIGONOMETRY
Ques1: Prove that the sin z and cos z both are periodic functions of period 2π. [Kanpur B.Sc. 2005; Rohil. 2005]
Ques2: Prove that tan z is a periodic function of period π. [Kanpur B.Sc. 2004]
Ques3: Prove that sinh z = -i sin iz. [Meerut1995]
Ques4: Show that cos (ix) = cosh x. [Lucknow 2007, 08]
Ques5: Prove that
1+tanh x/1-tanh x = sinh 2x + cosh 2x. [Kanpur B.Sc. 2002; Gorakh. 2005]
Ques6: Prove that:
cosh 2x + cosh 5x + cosh 8x + cosh 11x = 4 cosh 13/2 x cosh 3/2 x cosh 3x. [Purvanchal 2012]
Ques7: Separate into real and imaginary parts:
exp {sinh (x + iy)}. [Kumaun 2008]
Ques8: If tanh (u + iv) = sin (x + iy), prove that
tanh u = sin x/cosh y and tan v = sinh y/cos x. [Agra 1994]
Ques9: Prove that Log (i) = iπ(2n+ 1/2). [Kanpur B.Sc. 2004; Purvanchal 2012]
Ques10: Prove that :
Log (-i) = iπ(2n - 1/2). [Agra 2007]
Ques11: Find the principal and general values of
log (-1+i). [Lucknow 2009]
Ques12: Show that logarithm of any real negative quantity is many-valued. Also, find the general value of log (-3). [Lucknow 2006; Rohil. 2006, 08; Avadh 1991; Bundel. 2006]
Ques13: Prove the following :
(i) Log (-4) = log 4 + i(2n + 1)π. [Avadh 1993]
(ii) Log (1 + i) = 1/2 log 2 + i(2n + 1/4) π. [Avadh 2014; Bundel. 2004; Lucknow 2007]
Ques14: Find the general and principal values of log √i, and compare them with the corresponding values of 1/2 log i. [Kanpur B.Sc. 1999, B.A. 1998; Lucknow 2008; Rohil. 2007]
Ques15: Prove that
1 - 2 ( 1/3.5 + 1/7.9 + 1/11.13 + ... ) = π/4. [Kanpur B.Sc. 1995]
Ques16: Prove that :
1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 - ... = π/2√2. [Kanpur B.Sc. 2005, 08, B.A. 1997; Rohil. 2012; Agra 1991, 95, 98]
Ques17: Prove that :
π/8 = 1/1.3 + 1/5.7 + 1/9.11 + ... . [Agra 2007, 08; Lucknow 2007; Meerut 2007, 09; Kanpur B.Sc. 1998, 2012]
Ques18: Prove that :
cos 2π/2n+1 + cos 4π/2n+1 + cos 2nπ/2n+1 = - 1/2. [Kanpur B.Sc. 2005]
Ques19: Sum the series :
cos x - cos3x/3! + cos5x/5! - ... ... . [Kanpur B .Sc. 2013, 14]
Ans: sin ( cos x ) cosh ( sin x ).
Ques20: Sum the series :
cos π/3 + 1/2 cos 2π/3 + 1/3 cos 3π/3 + ... ... . [Kanpur B.Sc. 2003, 08]
Ques21: Sum the series :
cos π/3 + 1/3 cos 2π/3 + 1/5 cos 3π/3 + 1/7 cos 4π/3 + ... . [Kanpur B.A. 1997; Rohil. 1993; Agra 1991, 95, 97]
Ques4: Show that cos (ix) = cosh x. [Lucknow 2007, 08]
Ques5: Prove that
1+tanh x/1-tanh x = sinh 2x + cosh 2x. [Kanpur B.Sc. 2002; Gorakh. 2005]
Ques6: Prove that:
cosh 2x + cosh 5x + cosh 8x + cosh 11x = 4 cosh 13/2 x cosh 3/2 x cosh 3x. [Purvanchal 2012]
Ques7: Separate into real and imaginary parts:
exp {sinh (x + iy)}. [Kumaun 2008]
Ques8: If tanh (u + iv) = sin (x + iy), prove that
tanh u = sin x/cosh y and tan v = sinh y/cos x. [Agra 1994]
Ques9: Prove that Log (i) = iπ(2n+ 1/2). [Kanpur B.Sc. 2004; Purvanchal 2012]
Ques10: Prove that :
Log (-i) = iπ(2n - 1/2). [Agra 2007]
Ques11: Find the principal and general values of
log (-1+i). [Lucknow 2009]
Ques12: Show that logarithm of any real negative quantity is many-valued. Also, find the general value of log (-3). [Lucknow 2006; Rohil. 2006, 08; Avadh 1991; Bundel. 2006]
Ques13: Prove the following :
(i) Log (-4) = log 4 + i(2n + 1)π. [Avadh 1993]
(ii) Log (1 + i) = 1/2 log 2 + i(2n + 1/4) π. [Avadh 2014; Bundel. 2004; Lucknow 2007]
Ques14: Find the general and principal values of log √i, and compare them with the corresponding values of 1/2 log i. [Kanpur B.Sc. 1999, B.A. 1998; Lucknow 2008; Rohil. 2007]
Ques15: Prove that
1 - 2 ( 1/3.5 + 1/7.9 + 1/11.13 + ... ) = π/4. [Kanpur B.Sc. 1995]
Ques16: Prove that :
1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 - ... = π/2√2. [Kanpur B.Sc. 2005, 08, B.A. 1997; Rohil. 2012; Agra 1991, 95, 98]
Ques17: Prove that :
π/8 = 1/1.3 + 1/5.7 + 1/9.11 + ... . [Agra 2007, 08; Lucknow 2007; Meerut 2007, 09; Kanpur B.Sc. 1998, 2012]
Ques18: Prove that :
cos 2π/2n+1 + cos 4π/2n+1 + cos 2nπ/2n+1 = - 1/2. [Kanpur B.Sc. 2005]
Ques19: Sum the series :
cos x - cos3x/3! + cos5x/5! - ... ... . [Kanpur B .Sc. 2013, 14]
Ans: sin ( cos x ) cosh ( sin x ).
Ques20: Sum the series :
cos π/3 + 1/2 cos 2π/3 + 1/3 cos 3π/3 + ... ... . [Kanpur B.Sc. 2003, 08]
Ques21: Sum the series :
cos π/3 + 1/3 cos 2π/3 + 1/5 cos 3π/3 + 1/7 cos 4π/3 + ... . [Kanpur B.A. 1997; Rohil. 1993; Agra 1991, 95, 97]
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