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- Advanced Engineering Mathematics - AEM
- 2015
- PYQ
**I K G Punjab Technical University - IKGPTU**- Automobile Engineering
- B.Tech
**3 Views**- Uploaded 3 months ago

Roll No. Total No. of Pages : 02 Total No. of Questions : 09 B.Tech. (2011 onwards) (Sem.–1) ENGINEERING MATHEMATICS-I Subject Code : BTAM-101 Paper ID : [A1101] Time : 3 Hrs. Max. Marks : 60 INSTRUCTION TO CANDIDATES : 1. 2. 3. 4. 5. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each. SECTION - B & C. have FOUR questions each. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each. Select atleast TWO questions from SECTION - B & C. Use of non programmable calculator is allowed. SECTION-A l. Write briefly : (a) Find the radius of curvature at the origin for the curve : y 4 + x 3 + a (x 2 + y 2 ) – a 2 y = 0. (b) Find the area of the cardioad r = a(1– cos ) (c) Find the volume of a sphere of radius ‘a’. (d) If u = x log xy, where x 3 + y 3 + 3xy = 1, find du . dx (e) F in d t h e e q u a t i o n s o f t h e ta n g e n t a n d n o r ma l to th e s u r f a c e x 3 + y 3 + 3xyz =3 at (l,2,–l). (f) Show that the area between the parabolas y 2 = 4ax and x 2 = 4ay is 16 2 a . 3 (g) Calculate the volume of the solid bounded by the planes x = 0, y = 0, x + y + z = 1 and z = 0. (h) A particle moves along the curve x = t3 +1, y = t2, z = 2t + 3, where t is a time. Find the components of its velocity and acceleration at t = 1 in the direction iˆ ˆj 3kˆ . [MCode - 54091 ] (S-1) 05

(i) Find div F and curl F , where F = grad(x 3 + y 3 + z 3 – 3xyz). (j) State Stoke’s theorem. SECTION-B 2. Trace the folium of Descartes : x3 + y 3 =3 axy stating the salient points. 3. Find the moment of inertia of one loop of the lemniscates: r2 = a2 cos 2 about the initial line. 4. If u = f(r) and x = r cos , y = r sin , then prove that : 2u x 2 2u y 2 f (r ) 1 f (r ) r 5. Find the maximum and minimum distances of the point A(3,4,12) from the sphere x 2 + y 2 + z 2 = 1. SECTION-C x 6. Evaluate (i) – xe x2 y dy dx by change of order of integration, 0 0 c b a (ii) x 2 y 2 z 2 dx dy dz. –c –b –a 7. (i) Find the directional derivative of (x,y,z) = xy 2 + yz 3 at the point (2,–l, l). (ii) A vector field is given by F = (sin y) iˆ + x(l + cos y) ĵ . Evaluate the line integral over a circular path given by x 2 + y 2 = a 2 , z = 0 . 8. Apply Green’s theorem to evaluate : y 1 – sin x dx cos x dy , where C is the C plane triangle enclosed by the lines y = 0, x = 2 and y = x. 2 9. Verify divergence theorem for F x 2iˆ zjˆ yzkˆ taken over the curve bounded by x = 0, x = 1, y = 0, y = 1, z = 0 and z = 1. [MCode - 54091 ] (S-1) 05

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