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NMA A = DEC 2014, , =, , SN - 249, , | Semester B.A./B.Sc. Examination, Nov./Dec. 2014, (2014-15 and Onwards) (Semester Scheme) (CBCS) (Fresh), , MATHEMATICS - I, , Time : 3 Hours, , Instruction: Answer all questions., , PART-A, 1. Answer any five questions :, a) Define:, i) Equivalent matrices, , ii) Row reduced Echelon form of a matrix., , Max. Marks : 70, , (5x2=10), , b) Find the value of 4, for which the system 3x + y — Az=0;4x-2y—3z=0;, , 22x + 4y + AZ =0 has a non-trivial solution., , c) Find the nth derivative of sin?x., , If 1 th Sa ee, d) If z = cos”'(xy) prove that Bay ~ ayax, , e) Using reduction formula, evaluate fin? xdx., , r, , f) Evaluate 5 x cos4xdx., 0, , g) Show that the planes x + 2y - 3z + 4 = 0 and 4x + 7y + 6z +2 = 0 are, , perpendicular., , h) Write the condition for co-planarity of two lines., , P.T.O., , PRAVEEN NV.
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sn-249 2 DEC 2014 2 8, , PART -B, Answer any one full question. (ieTS= 13), , 2. a) Find the rank of the matrix, , 2-1-3 -1, 1 2 Sof, 1014 1, Oo 1 4-1, , b) Solve completely the system equations x + 3y — 2z = 0; 2x —y + 4z = 0;, x-Tly+ 14z=0., , c) Verify Cayley-Hamilton’s theorem and hence find inverse of the matrix, , 10 8, 214-1, 1-147, 123 4, , 3. a) Reduce the matrix E 4 4 4) into normal form and hence find the rank., , OR, , b) Show that the following system of equations is consistent and solve them », X+2y4+2z=1; 2x+y+zZ=2;3x+2y+2z=3; y+z=0., c) Find the eigen values and the corresponding eigen vectors of the matrix, , 1-11, 100 |, -1 4-1], , , , PART -—CG, Answerany two full questions. (2x15=30), 4, a) Find then! detivative of ———-——, . a) Find the n" derivative o' (x4 12(x—1)°, , b) State and prove Leibnitz’s theorem., c) If y = sin-'x show that (1 - x?) y,,5— (2n +1), OR, , *Yng1 — ny, = Q,, , PRAVEEN NV.
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nt 0 a 5 DEC 2014 SN, -3- hai} i 4 —249, , 5. a) Ifz=sin(ax + ) + Cos Pz 2 02, m ax— — = —, y ( y) prove tha ae By?, , b) State and Prove Euler's theorem for homogeneous functions., ¢) Find the total derivative of u wrt 't' where u = e* siny where x = logt, y = t2, , 6. a) Ifu=f(r) where r2 = x2 4 y2 4 22 7 oY #0 _ e424, y prove that = a +e = We ST., , (u,v, w), , A(x, y, Z) *, , c) Obtain the reduction formula for fin’ xdx where n is a positive integer., OR, , b) lfu=z—x,v=y-zandw=x+y+4z find, , 7. a) Obtain the reduction formula for jtan®xdx., , b) Evaluate :, , 25, i) J x2V2—x dx, 0, , ii) [ xsin® x.cos* xdx0, x? -1, , dx logx, , , , 1, c) Evaluate J, 0, , Where ais a parameter., PART-D, , Answerany one full question. (1x15=15), , 8. a) Find the equation of a plane passing through the line of intersection of the, planes x + 2y + 3z = 4, 2x + y—z +5 = 0 and perpendicular to the plane, 5x + 3y + 62+ 8=0., , b) Obtain the condition for co-planarity of two lines in both vector and Cartesian, forms., , PRAVEEN NV.
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SN-249 DEC 2014 4 {ARITA, , c) Find the equation of the right circular cone through the point (2, 1, 3) with, vertex at the point (1, 1, 2) and axis parallel to the line, , 2 -4° 3, OR, . . . x-1 y-2 z-3, 9. a) Find the shortest distance between the skew lines a = 3) = 7. and, , x-2_y-6_ z-5, 3 5°, , b) Show that the plane 2x + 3y + 4z = 58 touches the sphere, x2 + y? +2? — 4x — 6y — 8z = 0 and find the point of contact., c) Find the equation of the right circular cylinder whose axis is the line, , X-a -B zxe =P =aot and whose radius is ‘r’., | m n, , PRAVEEN NV.
