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# Himachal Pradesh University (HPU) 2010 Model applied maths-I - Question Paper

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Applied Maths-I

B

B.Tech. 1st Semester

Examination

Applied Mathematics-I

Time Allowed: 3 Hours Maximum Marks: 100

Attempt Five questions in all, selecting one question from each Section A, B, C and D. Section E is compulsory. Each Section carry 40 marks.

SECTION-A

1. (a) If ; show that (7)

(b)   A rectangular box, which is open at the top, has a capacity of 256 cubic feet. Determine the dimensions of the box such that the least material is required for the construction of the box. Use Lagrange,s method of multipliers to obtain the solution. (8)

1. (a) To prove that . (7)

(b) Evaluate over the tetrahedron bounded by the plane x = 0, y = 0, z = 0 and x + y

+ z = 1. (8)

SECTION-B

3        (a) Solve by method of variation of parameters

(7)

(c) Reduce the quadratic form equation into sum of squares. (8)

4        (a) Find the characteristic roots and characteristic vectors of

. (7)

(b) Reduce the quadratic form equation into sum of squares.

(8)

SECTION-C

5        (a) Solve the differential equation . (7)

(b) Solve the differential equation . (8)

6 (a) Solve the differential equation . (7)

(b) Solve the differential equation

. (8)

SECTION-D

6        (a) If x, y and z are unequal and

=0

Prove that . (7)

(b) Show that the equations x + 2y z = 3, 3x y + 2z

= 1, 2x 2y + 3z = 2, x y + z = -1 are consistent and solve them. (8)

7. (a) Given, x = u +v + w, , ;show that . (7)

(b) By definition of continuity, prove that function , is continuous at (0,0). (8)

Section E

8(a) If ; prove that . (5)

(b) If ,prove that . (5)

(c) Define orthogonal matrix and prove that . (5)

(d) Prove that the modulus of each characteristic root of unitary matrix is unitary. (5)

(e) Use method of Variation of parameters to solve

. (5)

(f) Solve the differential equation

. (5)

(g) State and prove Eulers theorem for function of three variables. (5)

(h) Explain Taylors series. (5)