MATHEMATICS-1
PAPER NO. 1          
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PUT ON: 6th Jan, 2K3
                                               AMA-101
                                          MATHEMATICS-1
                                  (B.Tech 1st & 2nd Semester,2122)
Time : 3 Hours                                                                                        Maximum Marks : 60
NOTE:- Attempt Five questions in all. Question No. 1 is compulsory. Attempt two questions
                from each section.

1(a) If z = u2 + v2 and u = at2, v = 2at, find dz/dt.
(b) Expand sin x in powers of x.
(c) Prove that:
            
    where S is the length of the arc measured from fixed point and p is the radius of curvature.
(d) Change the order of integration in
            
(e) Find the centre of pressure of a triangular area immersed in a homogeneous liquid with one side in the free surface.
(f) Show that :
            
(g) Find the value of Log (i)i.
(h) Find the general value equation of cone which touches the three co-ordinate planes.
(i) The order of convergence of Newton-Raphson method is :
        (i) 1.168
        (ii)1
        (iii) 2
        (iv) 3
(j) Apply Gauss-Jordan method to solve:
    x + y + z = 9
    2x - 3y + 4z = 13
    3x +4y + 5z = 40.

                                              Section-A

2. (a) The period of a simple pendulum with small oscillation is T = 2π(l/g)1/2. If T is computed using l = 8 ft,
         and g = 32 ft./sec2, find the % error in T if the true values are l = 8.05 ft. and g = 32.01 ft/sec2. What is
         the approximate error in T ?
    (b) If u = tan-1(y2/x)., prove that :
            
3. (a) Find the maximum and minimum distances of the point (3, 4, 12) from the sphere x2 + y2 + z2 = 1.
    (b) For the cardiode show that p2/r is constant.
4. (a) If the density at any point of the solid octant of the ellipsoid
            
    varies as x,y,z find the co-ordinates of C.G. of the solid.
    (b) By using the transformation x + y = u, y = uv, show that :
            
5. (a) Using double integral, find the M.I. about x-axis of the area enclosed by the lines: x = 0, y = 0, (x/a) + (y/b) = 1.
    (b) Transform the following to Cartesian form and hence evaluate :
            

                                               Section-B


6. (a) What is necessary condition for the convergence of a positive term series? Test the convergence of
            
    (b) State and prove Cauchy's root test. Test the convergence of
            
7. (a) Sum the series :
            
    (b) Use De Moivre's theorm to solve :
            x4 - x3 + x2 - x + 1 = 0.
8. (a) A plane passes through a fixed point (a, b, c), show that the locus of the foot of the perpendicular from the
         origin on the plane is a sphere.
    (b) The radius of a normal section of a right circular cylinder is 2 unit, the axis lies along the straight line:
            

    Find the equation of cylinder.
9. (a) Using Regula-Falsi method, compute a root of xex = sin x. Correct to three decimal places.
    (b) Solve by Guass elimination method:
         2x + 2y + z = 7
         x - 2y - u = 2
         3x - y - 2z - u = 3.
         x - 2u = 0.
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