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

Applied Maths-I

** **

** **

** B.Tech. 1 ^{st}
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**

** **

**(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)**

**(a)**To prove that

**(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 *+ 2*y * *z
*= 3, 3*x * *y *+ 2*z *

= 1, 2*x * 2*y *+ 3*z
*= 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)

Earning: Approval pending. |