Engineering Mathematics-I Past Paper PDF May-Jun 2024

Total No. of Questions : 9] SEAT No. : 8 23 PB3586 -1...

Total No. of Questions : 9] SEAT No. : 8 23 PB3586 -1 [Total No. of Pages : 5 ic- F.E. tat 0s ENGINEERING MATHEMATICS-I 9:0 (2019 Credit Pattern) (Semester -I/II) (107001) 02 91 9:5 Time :2½Hours] [Max. Marks :70 0 40 Instructions to the candidates: 5/0 13 1) Q.1 is Compulsory. 0 2) Answer Q.2 or Q.3, Q.4 or Q.5, Q.6 or Q.7, Q.8 or Q.9. 5/2.23 GP 3) Figures to the right indicate full marks. 4) Assume suitable data, if necessary. E 5) Neat diagrams must be drawn wherever necessary. 81 8 C 23 6) Use of electronic pocket calculator is allowed. ic- Q1) Write the correct option for the following MCQs. 16 tat 8.2 0s  2u a) If u  x  y then 3 3  ?.24 9:0 xy 91 49 9:5 i) 3 ii) 3 30 40 iii) 2 iv) 0 01 02 5/2 u  ( x, y ) GP b) If x  uv , y  the   ?  (u, v) 5/0 v CE 81 8 2u 23.23 i) ii) uv v ic- 16 tat v v 8.2 0s iii) iv) 2u 2u.24 9:0 91 49 9:5 1 1 1  30   40 c) Rank of matrixA= 0 1 0  is....? 01 02 1 1 1  5/2 GP 5/0 i) 0 ii) 1 CE iii) 2 iv) 3 81.23 16 8.2.24 -1 1 P.T.O. 49  1 4  8 d) Using Cayley Hamilton theorem A for the matrix A=   is 23 -1 2 3 ic- given by; tat 0s 1 1 i) (A+4I) ii) (A+5I) 9:0 5 4 02 91 9:5 1 1 0 iii) (A  5I) iv) (A  4I) 40 4 5/0 13 5 0 e) If A1 =A ' then matrix A is....? 5/2.23 GP i) Orthogonal ii) Singular iii) Non-Singular iv) None of above E 81 8 C 23 u If u  x 3  4 y  3 x , ic- f) =....? x 16 tat 8.2 i) 4 ii) 3 x 2  3 0s.24 9:0 iii) 3 x 2  4 y iv) 3 x 2  1 91 49 9:5 30 40  2u 01 Q2) a) If u  x  y ,find y x 02 xy 5/2 GP  x3  y 3  2  u 2  2u 2  u 2 5/0 b) If u  log  2 2  , find the value of x  2 xy y x y  x 2 xy y 2 CE 81 8 23 u u u.23   0 c) If u  f ( y  z , z  x, x  y ) , Prove that x y z ic- 16 tat 8.2 0s OR.24 9:0  u   x  1 91 49 If x  au  bv and y  au  bv , prove that  x   u   2 9:5 Q3) a) 2 2   y  v 30 40 01 u u 02 1  y  If u  sin    x  y , find the value of x  y 2 2 b) 5/2 x x y GP 5/0 cos sin CE x ,y z  f ( x, y ) ,then 81 c) If and show that u u.23 z z z z u   ( y  x)  ( y  x ) 16 u  x y 8.2.24 -1 2 49  uv  (u, v) 8 If x  uv and y  23 Q4) a) , find u v  ( x, y ) ic- tat 0s b) Examine for functional dependence: 9:0 02 91 x y 9:5 u , v  tan 1 x  tan 1 y. If dependent find the relation between 1  xy 0 40 them. 5/0 13 0 5/2.23 GP c) Discuss maxima and minima of f ( x, y )  x 3  y 3  3axy a  0. E 81 8 C 23 OR ic- 16 tat 8.2 0s Q5) a) Prove that JJ '  1 for the transformation x  u cos v, y  u sin v.24 9:0 b) Find the percentage error in computing the parallel resistance r of two 91 49 9:5 1 1 1 30 resistances r1 and r2 from the formula r  r  r where r1 and r2 are 40 01 1 2 02 both in error by +2% each. 5/2 GP Find maximum value of u  x 2 y 3 z 4 such that 2 x  3 y  4 z  a by 5/0 c) CE 81 langrange’s method. 8 23.23 ic- 16 tat Q6) a) Find for what values of k, the set of equations 8.2 2x  3y+6z  5t=3 0s y  4z+t=1.24 9:0 91 4x  5y+8z  9t=k 49 9:5 has i) No solution 30 40 ii) An intinite number of solutions. 01 02 b) Examine for linear dependence of vectors 5/2 GP (1,  1,1) , (2,1,1) and (3,0,2) 5/0 CE 1 2 2  81 1  Show that A= 3  2 1 2  is orthogonal..23 c)  2 2 1  16 8.2 OR.24 -1 3 49 Q7) a) Examine for consistency the following set of equations and obtain the 8 solution if consistent. 23 2x  y  z=2 ic- x+2y+z=2 tat 4x  7y  5z=2 0s b) Examine for linear dependence of vectors 9:0 02 91 (1,2,4),(2,  1,3),(0,1,2). 9:5 0 40 c) 5/0 13 Determine the currents in the network given in figure below. 0 5/2.23 GP E 81 8 C 23 ic- 16 tat 8.2 0s.24 9:0 Q8) a) Find the eigen values and eigen vectors of the following matrix. 91 49 9:5 30 1 1 1 40   A= 0 2 1. 01 02 0 0 3 5/2 GP 5/0 CE 81 8  1 2 2  23.23   b) Verify Cayley - Hamilton theorem for A=  1 3 0  and use it to ic- 16 tat  0 2 1  8.2 0s.24 Find A-1 9:0 91 c) Find the modal matrix P which transform the matrix 49 9:5 30 40  1 1 2    01 02 A=  1 2 1  to the diagonal form. 5/2  0 1 1 GP 5/0 CE 81.23 OR 16 8.2.24 -1 4 49 Q9) a) Find the eigen values and eigen vectors of the following matrix 8 23  1 2  ic- A=  . tat  5 4  0s 9:0 0 1 0  02 91   9:5 b) Verify cayley Hamilton theorem for A= 0 0 1 . Hence find A-1. 0 40 5/0 13 1 3 3  0 5/2.23 GP c) Reduce the following quadratic form to the Sum of the squares form. E 81 8 3 x 2  3 y 2  3 z 2  2 xy  2 xz  2 yz. C 23 ic- 16 tat 8.2 0s.24 9:0 91 49 9:5 30 40 01 02 5/2 GP 5/0  CE 81 8 23.23 ic- 16 tat 8.2 0s.24 9:0 91 49 9:5 30 40 01 02 5/2 GP 5/0 CE 81.23 16 8.2.24 -1 5 49