Download DBATU B-Tech 1st Year 2019 Mid Semester BTBS101 Engineering Mathematics I Question Paper

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Course: Common to All Branches Div: A/B/C/D/E Sem: I

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SVKM INSTITUTE OF TECHNOLOGY, DHULE
Mid Semester Exam 2019-20

Max Marks: 20 Date:-3/10/2019 Duration:- 1 Hr.

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Instructions to the Students:

1. All Questions are Compulsory
2. Use of Non-Programmable calculator allowed

Q.1 Write a correct option of following questions

1. The Product of Eigen values of Matrix A equal to
   

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   (a) |A| (b) 0 (c) 1 (d) None
2. Eigen values of triangular matrix are
   (a) Non Principle diagonal (b) Principle Diagonal (c) Zero (d) None
3. The Eigen values of A and A’ are always
   (a) Different (b) Same (c) Cannot be decided (d) None

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Q.2

1. If z = e^xy then ?z/?x =
   (a) e^xy (b) e^xyy (c) e^xy x (d) e^xy xy
2. If u = x^y then ?u/?x =
   (a) x^ylogx (b) x^ylogy (c) yx^y-1 (d) 0

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3. If u = x²+2xy+y², then x ?u/?x + y ?u/?y =
   (a) u (b) 0 (c) 3u (d) 2u

Q.3 Solve Any Two of the following.

1. Reduce the Matrix A to Normal form and find its Rank A =

   1	2	3
   1	4	2
   2	6	5

2. If u = f(x-y, y-z, z-x) then show that ?u/?x + ?u/?y + ?u/?z = 0

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3. Prove that ?(f, f)/?(y, z) = ?(f, f)/?(x, y) if f(x,y) = 0 and f(y,z) = 0

Solve Any One of the following.

1. Verify Cayley-Hamilton theorem to A =

   1	1
   3	-2

   and hence find A^-1 also deduce that A⁴ = 625I
2. If u = cosec^-1 (x²+y²)/(x³+y³) prove that
   x² ?²u/?x² + 2xy ?²u/?x?y + y² ?²u/?y² = tanu[12 tan²u + 12 tan²u]

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