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B0-R4

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B0-R4
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    B0-R4 Page 1 of 2 January, 2012B0-R4: BASIC MATHEMATICSNOTE:Time: 3 Hours Total Marks: 1001. a) Express the following in the form of A + iB    ( ) ( ) ibaibaibaiba +−−−+ 22 .b) Find the area enclosed by the curve 4and 1,0,4 2 ==== y y x x y .c) Find the characteristic roots of the matrix  200320321 .   d) Solve the differential equation ax x ydxdy ++++==== .e) Evaluate  xeelim  x x x −−−−→→→→ −−−− 0 .f) Test the convergence of the series ...! x! x x ++++++++++++ 332 2  g) If t sina y ,t cosa x ========   , find dxdy . (7x4)   2. a) Show that x =  2 is one of the root of 02 2 3 3 3 2 1 6  =+−−−−=∆  x x x x x .b) Find the inverse of matrix by Gauss Elimination method A =  2 1 1 3 2 3 1 2 1 . (9+9)3. a) Find all the asymptotes of the curve 0axy3yx 33 ====−−−−++++ .b) The function ( )  x f  is defined as ((   )) 4x5xf  −−−−==== when 10 ≤< x  =  x x 34 3 − when 21 << x  Discuss its continuity at x  = 1.c) Find the domain and range of the function 1x1)x(f  2 −−−−==== . (8+6+4)     1. Answer question 1 and any FOUR from questions 2 to 7.2. Parts of the same question should be answered together and in the samesequence.    B0-R4 Page 2 of 2 January, 20124. a) Evaluate dx xb xa  x ∫ + π   02222  sincos .b) If k  jia  2 +−= → and k  jib  43 −+= → , find →→ ⋅ ba , →→ × ba .c) Find the lengths of major and minor axes of the ellipse x 2 +4y 2 =16. (8+6+4)   5. a) Find the point of local maxima and minima and corresponding maximum and minimum valuesof the function ( )( ) 2 21 +− x x .b) Discuss the convergence of series ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅++++++++⋅⋅⋅⋅++++++++⋅⋅⋅⋅++++++++⋅⋅⋅⋅ bababa 222 332211 .c) Expand ( )  x e + 1log using Maclaurin series upto four terms. (6+6+6)   6. a) Find the eccentricity, centre, foci, directrices and latus rectum of the hyperbola. 0163272169 22 =−−+− y x y x  b) Solve the second order homogenous differential equation 2222 243 x ydxdy xdx yd  x =+− . (9+9)   7. a) Find the equation of the plane passing through the intersection of the planes 6 =++ z y x  and 05432 =+++ z y x and the points ( ) 1 ,1 ,1 .b) Find the coordinates of the point on the curve 43 2 ++= x x y the tangent at which passesthrough the srcin.c) If a + b + c  = 0, find the characteristic roots of the matrix A =  cababcbca   . (6+6+6)
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