B.Sc. Part 1 Mathematics
INTEGRAL CALCULUS
Ques1: Find the area of the quadrant of an ellipse. [Bundel. 2010; Agra 2000]
Ques2: Find the area bounded by the following curves : y = | x-1 |, y = 0, | x | = 2. [Avadh 2006]
Ques3: Show that the length of an arc of the cycloid x = a ( t - sin t ), y = a ( 1 - cos t ) is 8a. [Kanpur B.Sc. 1999; Avadh 2012; Purvanchal 2014; Meerut 2006; Kashi 2011; Agra 2002]
Ques4: Show that the length of an arc of the curve x sin t + y cos t = f ' (t), x cos t - y sin t = f '' (t) is given by s = f(t) + f '' (t) + c, where c is the constant of integration. [Agra 2003]
Ques5: Find the length of an arc of the curve
x = a (t + sin t), y = a (1 - cos t). [Avadh 2002; Agra 2009; Garhwal 2000]
Ques6: Find the intrinsic equation of the curve
y = a log sec (x/a). [Avadh 2011]
Ques7: Find the volume of a sphere of radius 'a' by the method of integration. [Kanpur B.Sc. 1998; Agra 2010; Avadh 2010]
Ques8: Find the volume of the solid generated by the revolution of an ellipse around its minor axis. [Kanpur B.Sc. 1995, 2012]
Ques9: Find the volume of the solid generated by the revolution of an arc of the catenary y = c cosh (x/c) about the x-axis. [Meerut 2009]
Ques10: If m > 0 and n > 0, prove that
B(m,n) = B(m + 1,n) + B(m, n + 1). [Kanpur B.Sc. 2001, 05, 11; Avadh 1994, 96, 2006, 11, 14; Bundel. 2012; Gorakh. 2005]
Ques11: Prove the following relation :
B(m + 1, n)/B(m, n) = m/(m + n). [Purvanchal 2013]
Ques12: Show that the length of the curve y = log sec x between the points where x = 0 and x = π / 3 is log (2 + √3). [Kanpur B.Sc. 2003, 05; Rohil. 2014; Bundel. 2013; Avadh 2003]
Ques13: Find the curved surface of a hemisphere of radius r. [Kanpur B.Sc. 2014]
Ques14: Show that the surface of the spherical zone contained between two parallel planes is 2πah, where a is the radius of the sphere and h the distance between the planes. [Kanpur B.Sc. 2009]
Ques15: Find the surface generated by the revolution about the x-axis of an arc of the curve y = c cosh (x/c) from the vertex to any point (x,y). [Meerut 2000, 04, 07, 10; Avadh 1991, 96]
Ques16: For a catenary y = c cosh (x/c), prove that cS = 2V = πc (cx + sy), where s is the length of the arc from the vertex, S and V are respectively the area of the curved surface and volume of the solid generated by the revolution of the arc about the x-axis. [Avadh 2000]
Ques17: The plane x/a + y/b + z/c = 1 meets the axis in A, B, C. Find the volume of the tetrahedron OABC. [Avadh 2010]
Ques18: The plane x/a + y/b + z/c = 1 meets the co-ordinate axes in the points A, B, C. Use Dirichlet's integral to evaluate the mass of the tetrahedron OABC, the density at any point (x, y, z) being kxyz. [Garhwal 2003]
Ques10: If m > 0 and n > 0, prove that
B(m,n) = B(m + 1,n) + B(m, n + 1). [Kanpur B.Sc. 2001, 05, 11; Avadh 1994, 96, 2006, 11, 14; Bundel. 2012; Gorakh. 2005]
Ques11: Prove the following relation :
B(m + 1, n)/B(m, n) = m/(m + n). [Purvanchal 2013]
Ques12: Show that the length of the curve y = log sec x between the points where x = 0 and x = π / 3 is log (2 + √3). [Kanpur B.Sc. 2003, 05; Rohil. 2014; Bundel. 2013; Avadh 2003]
Ques13: Find the curved surface of a hemisphere of radius r. [Kanpur B.Sc. 2014]
Ques14: Show that the surface of the spherical zone contained between two parallel planes is 2πah, where a is the radius of the sphere and h the distance between the planes. [Kanpur B.Sc. 2009]
Ques15: Find the surface generated by the revolution about the x-axis of an arc of the curve y = c cosh (x/c) from the vertex to any point (x,y). [Meerut 2000, 04, 07, 10; Avadh 1991, 96]
Ques16: For a catenary y = c cosh (x/c), prove that cS = 2V = πc (cx + sy), where s is the length of the arc from the vertex, S and V are respectively the area of the curved surface and volume of the solid generated by the revolution of the arc about the x-axis. [Avadh 2000]
Ques17: The plane x/a + y/b + z/c = 1 meets the axis in A, B, C. Find the volume of the tetrahedron OABC. [Avadh 2010]
Ques18: The plane x/a + y/b + z/c = 1 meets the co-ordinate axes in the points A, B, C. Use Dirichlet's integral to evaluate the mass of the tetrahedron OABC, the density at any point (x, y, z) being kxyz. [Garhwal 2003]
No comments:
Post a Comment