Mathematics-1

Uttarakhand Technical University

B.tech (first year)

2011-12

 Mathematics-1

time : 3hr

Total marks :100

SECTION A

Q1:- Attempt any four of the following :

4x5=20

1. Reduce the matrix
   to normal form and hence find its rank.
2. Show that the vectors :
   X₂  = (1,2,4), X₂ = (2,-1,3), X₃ = (0,1,2), X₄ =(-3,7,2) are linearly dependent. Find the relation between them.
3. Show that the matrix
   is Hermitian and iA is skew Hermitian.
4. Verify Cayley-Hamilton theorem for
   and hence find A^-1.
5. Prove that the eigenvalues of a unitary matrix are of unit modulus.
6. Test the consistency for the following system of equation and if system is consistent solve them:
   x + y + z =6,
   x + y + 3z= 14,
   x + 4y + 7z =30.

SECTION B

Q2:- Attempt any four of the following: 

4x5=20

1. If  y = e ^asin-1x   , show that
   (1-x²)y_n+2 – (2n+1)xy_n+1 – ( n²+a²)y_n =0  
2. If  y = sin(msin^-1x), find y_n at x =0.
3. If u = x²tan^-1(y/x) – y² tan^-1 (x/y), show that
   
4. If 
   
5. If u = f(r,s,t) and r =x/y, s = y/z , t = z/x,
   show that
   
6. Expand f (x,y) = tan^-1y/x  in the neighborhood of (1,1) up to third degree term.

SECTION C

Q3:- Attempt any two of the following:

10X2=20

1. If u= xyz, v=x² + y² + z², w =x + y + z
   Find the Jacobian
   
2. The power P required to propel a steamer of length l at a speed u is given by p =λu³l³, where λ is constant .If u is increased by 3% and l is decreased by 1%. Find the corresponding increase in  p.
3. Using the Lagrange method of undetermined multiplier find the point upon the plane                 ax + by + cz=p qt which the function f = x² + y² + z²,has a minimum value and find this minimum f.

SECTION D

sub (Mathematics-1) b tech 1st year  utu

Q4:- Attempt any two of the following:

10X2=20

1. Evaluate the integral
   by changing the order of integration.
2. Show that
   , where symbols have their usual meaning.
3. Evaluate
   ,over the area between y= x²  and y=x

SECTION E

Q5:- Attempt any two of the following:

10X2=20

1. Find the angles between the normal surface xy=  z²  at the point (4,1,2) and (3,3,-3).
2. Prove that div (grad rⁿ) =n(n+1) r^n-2  ,where r²= x² + y² + z²   hence show that 
   
3. Explain the Stoke's theorem .using stoke theorem evaluate the integral  where F=y²i^{^}+x²j^{^}-(x+z)k^{^},and c is the boundary of the triangle with vertices (0,0,0), (1,0,0) and (1,1,0).