B.SC. Part One 3rd Year
Paper:Mathematics (Paper“A”)                                                                               Time Allowed:3 hours Max.Marks:30
Note:     Attempt any five questions,selecting ONE question from section-A and Two from               section -B and Two from section- C.All
                                                                 SECTION-A
Q.1        (a) State and prove Lagrange's mean value theorem.
              (b) Using lagrange's mean theorem to show that: 
              |                         |∠|b-a|if x ≠ 0
Q.2        (a) Find Lim          Sin3 x     
                                          x-0sin1x
              (b) State Roll's theorem and check the validity of its hypothesis and conclusion in case of a                     function.
                    F(x) =2x with 1a=1,b=1
Q.3        (a) State and prove the parallelogram law of additio0n of vectors.
              (b) By using vectors prove the parallelogram law of addition of vectores.
Q.4        (a) Prove that the derivative of a vectore “a”of constant magnitude is orthogonal tc “a”.
              (b) Find a vector v which satisfies
                    d3v - 2 d2v -3 dv =0 such that v-i dv =j and d2v = k
                    dt3        dt2      dt                          dt             dt2
                                                                 SECTION-B
Q.5        (a) Prove that:
Q.5        (a) Find the centroid of the arc of the cycloid x=a (0 + sin0) y=(1- Cos0) which lies in the first                    quadrant.
              (b) Find the least force which will set into motion a particle at rest ona rough horizontal plane.
Q.6        (a) Prove that the force field.
                    F-(y2 -2xyz3)i + (3 + 2xy - x2 z3) j + (6z3 - 3x2 yz2) k
                    is conservation and determine its potential.
              (b) Find 0∫(3Sinti + Costj) dt
Q.7        (a) Find the pedal equation of x2 + y2 =1
                                                            9       4
              (b) Evaluate ∫1 xdx
                                    √ x
Q.8        (a) Prove that the effect of a couple upon a regid body is unaltered if it is replaced by any other                     couple.
              (b) If a rigid body is equilibrium under the action of three coplanner forces prove that the liner of                     action of them forces must be either concurrent or parallel.
Q.9        (a) Define continuous function. Prove that the function f(x) = sin2xis continuous for all x ε lR
              (b) Find the equation of a tangent to the hyperbola  x2 - y2  =1 in the form x  Cos h0
                                                                                           a2    b2                        a
Q.10      (a) Show that o∫[    ]  dx = π /n2  
              (b) Evaluate: ∫√ a2 + x2 dx
Q.11      (a) Find the asymptotes of curve r = asec0+b tan 0
              (b) Locate all relative maxima, relative minima and saddle points for the function f (x,y)
                    = 2y2x-yx2 + 4xy
Q.12    . (a) Prove that in elliptic orbit the velocity at any point of the orbitis given by V2 = μ(2/r - 1/a)
              (b) Find d2y when x3 + y3 + 3axy
                          dx2     
                                                                          SECTION-C   
              (a) F(x) =1/4 x4 -x25
              (b) If u = In (x2+y2+x2) prove that x r2u  = yr2u = z r2u
                                                                     ryrz    rzyx    rxry         
Q.13      (a) Evaluate the following.
              (i)  ∫ex  1 - Sinx                                     (ii)  ∫ x2 + 2x + 3           
                          1 - Cosx                                           (x2 + 2)(x2 + 1) 1/2    
              (b) Find the inverse of A=[    ]
Q.14      (a) Find the fourier series representing the identity function f(x) = x, 0<x<2π and sketch its graph                     from x = -4π to x = 4π.
              (b) Construct the fourier series to represent the function.
                     ƒ (x) = {                }