Roll No. Total No. of Pages : 03 Total No. of Questions : 09 B.Tech. (Sem.–1st) ENGINEERING MATHEMATICS-I Subject Code : AM-101 (2005-2010 Batches) Paper ID : [A0111] Time : 3 Hrs. Max. Marks : 60 INSTRUCTION TO CANDIDATES : 1. 2. 3. 4. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each. SECTION-B & C have FOUR questions each. Attem p t any FIVE questions from SECTI ON B & C ca rrying EIGHT marks each. Select atleast TWO questions from SECTION - B & C. SECTION-A l. Write briefly : u x 1 (a) If x2 = au + bv, y2 = au – bv, then prove that . x y y v 2 (b) If u = log (x2 + xy + y2), then use Euler’s theorem to show that x u u y 2 x y (c) Express the integral 4 e x dx in terms of gamma function. 0 (d) Test the convergence/divergence of the series n 1 [N-1- 1055 ] n2 3n .
2 a 3a x (e) Change the order of integration of 0 (f) Test whether the series f ( x, y ) dx dy . x 2 /4 a (1) n 1 is absolutely convergent or not. 2n 1 n 1 Explain how? (g) Use De-Moivre’s theorem to express sin 5 cos2 in a series of sines of multiples of . (h) Find the general value ii. (i) Verify that fxy = fyx, when f (x, y) = sin –1 (y/x). (j) Obtain the real part of tan (A + iB). SECTION-B 2. (a) Trace the curve y2 (a + x) = x2 (3a – x) be giving all salient features in detail. (b) Find the radius of curvature at the point (r, ) of the curve r = a(1 – cos ) and show that 2 varies as r2. 3. (a) Find the area included between the curves r = a(1 – cos ) and r = a(l + cos (b) Find the surface of the solid generated by the revolution of the curve x = a cos3t, y = b sin 3 t about y-axis. 4. (a) If xx yy zz 2z = c, then show that at x = y = z, = – ( x log e x)–1. xy (b) Find the points on the surface z2 = xy + 1 nearest to the origin 5. (a) Find the centre of gravity of the arc of the curve x2/3 + y2/3 = a2/3 in the first quadrant. (b) Expand f (x, y) = sin x y in ascending powers of (x – 1) and (y – /2) by using Taylor’s theroem up to second degree terms. [N-1- 1055 ]
SECTION-C 2a 2 a x x2 ( x2 y 2 ) dx dy , by changing into polar co-ordinates. 6. (a) Evaluate 0 0 (b) Find the volume bounded by the co-ordinate planes x =0, y = 0, z = 0 and the plane x y z = 1. a b c 7. (a) Find the centre and radius of the circle x2 + y 2 + z2 – 2x – 4y – 6z – 2 = 0, x + 2y + 2z – 20 = 0. (b) Find the equation of a cone whose semi vertical angle is /4, has its vertex at origin and axis along the line x = –2y = z. 8. (a) If cos + cos + cos = 0 = sin + sin + sin , then prove that cos2 + cos2 + cos2 = 3 = sin 2 + sin 2 + sin 2 2 (b) If sin –1 (u + iv) = + i then prove that sin 2 and cosh 2 are the roots of the equation x2 – (1 + u 2 + v 2)x + u2 = 0. 9. (a) Test for what values of x does the series x x 2 x3 x4 .... 1.2 3.4 5.6 7.8 converges/diverges. (b) Examine the convergence of the following series : (i) n 1 en 1 e2 n (ii) n 1 [N-1- 1055 1 1 22 32 .... n 2 ]