Roll No. ...................... B.Tech. Total No. of Questions : 09] [Total No. of Pages : 02 st (Sem. - 1 ) ENGINEERING MATHEMATICS - I SUBJECT CODE: BTAM – 101 Paper ID : [A1101] (2011 Batch) Time : 03 Hours Maximum Marks : 60 Instruction to Candidates: 1) Section - A is Compulsory. 2) Attempt any Five questions from Section B & C. 3) Select atleast Two questions from Section B & C. Section - A Q1) (2 Marks each) a) b) c) d) 2 Identify the symmetries of the curve r = cos . Find the Cartesian co-ordinates of the point ( 5, tan-1(4/3) ) given in polar co-ordinates. u u u 0 If u= F( x-y , y-z , z-x ) , then show that x y z If u is a differentiable vector function of t of constant magnitude, then du show that u . 0 dt 1 e) 2x - x Change the Cartesian integral 2 f ( x, y)dxdy into an i) 0 x equivalent polar integral. For what values of a,b,c the vector function f = (x + 2y + az ) i ( bx 3y z ) j + ( 4x + cy + 2z ) k is irrotational. Give the physical interpretation of divergence of a vector point function. y2 z 2 x2 1? What surface is represented by 2 3 2 If x = r cos and y = r sin , then find the value of . j) Given that F( x,y,z)=0, then prove that ( f) g) h) Q2) a) Section – B (8 Marks each) Show that radius of curvature at any point (x, y) of the hypocycloid 2 x3 b) J - 1379 2 y3 2 a3 is three times the perpendicular distance from the origin to the tangent at (x, y) Trace the curve r = 1+ cos by giving all salient features in detail. 1 P.T.O.
Q3) a) b) Q4) a) b) Q5) a) b) Find the area included between the curve xy2= 4a2 (2a x) and its asymptote. The curve y2 (a + x) = x2 (3a x) is revolved about the axis of x. Find the volume generated by the loop. r2 If = t n e 4t then find the value of n that will make 1 (r 2 ) 2 r t r r State Eulers theorem and use it to prove that u u 1 x3 y3 x y sin2u, where u tan 1 x y 2 x y The temperature T at any point (x,y,z) in the space is T= 400 x y z 2. Use lagranges multiplier method to find the highest temperature on the surface of the unit sphere x2 + y2 + z2 =1. Expand x2y + 3y-2 in ascending powers of x-1 and y + 2 by using Taylors theorem. Section – C (8 Marks each) 12 x Q6) a) Evaluate: xy dx dy , by changing the order of integration. 0 x2 b) Q7) a) b) Find the volume bounded by the cylinder x2+y2 = 4 and the planes y + z = 4 and z = 0. Prove that: grad div F = curl curl F + 2 F . Usethe stokes theorem to evaluate [( x + 2y)dx + ( x - z)dy + (y - z)dz] c Q8) a) b) Q9) a) Where C is the boundary of the triangle with vertices (2,0,0), (0,3,0), and (0,0,6) oriented in the anti-clock wise direction. Find the directional derivative of f(x,y,z) = x y2 + yz3 at (2,-1,1) in the direction of i + 2 j + 2K . Find the area lying inside the cardiode r = 2(1+cos ) and outside the circle r = 2. State greens theorem in plane and use it to evaluate (y sinx)dx cosx dy , where C is the triangle enclosed by y=0, c x = , and y = (2/ )x. b) J - 1379 State Divergence theorem use it to evaluate F . nd s s where F = (4x3 i - x2y j + x2z k and S is the surface of the cylinder x2+y2 =a2 bounded by the planes z = 0 and z = b. 2