2006 Bangalore University B.Sc Mathematics MATHEMATICS Question paper
I Sem B.SC Examination, Mar/Apr 2006 (Y2K7 scheme)(200708 and onwards) MATHEMATICS Time:3 hrs Max marks:80 Instruction:Answer all the Sections SECTIONA Answer any ten questions(10*2=20) 1)Define symmetric and skew symmetric matrix and give an example for each 2) Find the chararcteristic roots of the matrix 1 4 3 2 3)State the necessary and sufficient condition for a non empty subset of group G To be a subgroup of G 4)Find the cross product of vector a=2i+3j2k and vector b=12i4j+8k are Orthogonal 5)Find the cross product of vector a=2i+3j+4k and vector b=ik 6)The centroid of a triangle is(4,1,1).If two of the vctorices are(4,2,7) and (5,4,1) find the third vertex. 7)Find the direction cosine of a line equally inclined with the coordinate axes 8)Find the centre and the radius of the sphere 4x2+4y2+4z216x24y+16=0 9)Find the second derivative of log(sin x) w.r.t. x 10)Find the nth derivative of y=sin3x 11 if z=xy find delta z/delt x,delta z/delta y 12)Evaluate Integration cos tita/root of 4sin2 tita dtita With lower limit 0 and higher limit pie/2 13)Evaluate Integration of 1/2x2+x3 dx 12)Solve dy/dx=ex+y 15)Solve dy/dx +(cot x)y=1
SECTIONB
II Answer any two questions(2*5=10) 1)solve the following system of equation by matrix method: 2x+3y=8,2yz=1,2x+3z=11 2)Verify Cayley Hemilton theorem for the Matrix a= [3 2 an hence find A1 1 4] 3)Find a matrix P such that P1AP is a diagonal matrix where A=[3 4 2 3] III Answe any two questions(2*5=10) 1)Write the multiplication modulo 6 table for the set G={1,2,3,4,5} is(G,X6) a Group?Give reason 2)Show that the set of all fourth root of unity form a group under usual Multiplication 3)Find the area of the parallelogram whose diagonals are 3i2jk and 4i+2j+k 4)Prove that[vectora*b;vectorb*c;vectorc*a]=[vectora,vectorb,vectorc]2 IV Answer any two questions(2*5=10) 1)Find the angle between the diagonals of a cube 2)Find the equation of the plane passing through the point (2,4,0)(3,2,9) And parallel to the line whose direction ratio are 3,2,3 3)Derive the equation of a sphere wi9th a centre (x1,y1,z1) and radius R 4)Find the equation of the right circular cone whose vertex is(2,3,4) axis Makes equal angles with the coordinatues axes and semivertical angle is 300
V Answer any 4 (4*5=20) 1)Find the nth drivative of cos4x 2) If y=sin1x ,show that (1x2)yn+2(2n+1)xyn+1n2yn=0 3) If u=log(x4+y4/xy), show that x *delta u/delta x +y*delta u/delta y=3 4)Evaluate the integration of(x1)/(x2)(x3)dx 5) Evaluate the integration of 1/(3cos x) dx 6)Evaluate the integration of x *sin3x*cos x dx 7) Evaluate the integration sec4tita d tita with upper limits pie/4 and lower limits 0
VI Answe any two (2*5=10 1)Solve y dy/dx=ex+y 2)Solve dy/dx=x2+y2/xy 3)Verify the exactness and solve the equation (5x4+3x2y22xy3)dx+(2x3y3x2y25y4)dy=0 4)Solve cos2x dy/dx +y=tan x
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