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- Advanced Engineering Mathematics - AEM
- 2014
- PYQ
**I K G Punjab Technical University - IKGPTU**- Automobile Engineering
- B.Tech
**3 Views**- Uploaded 3 months ago

Roll No. Total no of pages :3 Total No. of Qustions :09 B.Tech (Sem.1st) Engineering Mathematics-I Subject Code :BTAM-101 Paper ID : [ A1101 ] Time: 3 Hrs. Max. Marks :60 Note:- (1) Q uestion-I is compulsory to attempt, Consisting of ten short answer type question carrying two marks each. (2) Attempt five questions (carrying eight marks each) by electing at least two questions each form Section-A and Section B Q1. (a) Graph the set of points whose polar co-ordinates satisfy the conditions. r ≤ 0,θ=π/4 (b) Obtain the local extreme values of the function ƒ(x,y)=x2+2xy (c) (d) ππ Evaluate ∫∫ 0x sin y dydx. y ∞ ∂ƒ ∂ƒ If ƒ (x,y)= ∑ (xy)n, given |xy|<1, then find ∂x , ∂y . n=0 → → → (e) F→and G are irrotational vector point functions, then show that F xG is a solenoidal function. (f) State Stoke’s Theorem. (g) If F→ =grad (x3+y3+z3-3xyz),then find curl F→ . ( ( (h) Find the work done by the force field F→ =xy +(y-x)j over the straight line from i (1,1) to (2,3). (i) Rectify the curve x=acos3 t,y=asin3t. (j) Find the possible percentage error in computing the resistance r from the formula (2x10=20) 1 = 1 + 1 , if r and r in error by 2%. r r 1 r2 1 2 M-54091 1 P.T.O

Section-A Q2. (a) Trace the curve x3+y3=3axy by giving all salient features in detail. (b) Find the volume of the solid generated by the revolution of the curve r = a(1+cosθ) about initial line. Q3. (a) If ρ be the radius of curvature at any point P of the curve y2=4ax and S is its focus, then show that ρ varies as (SP) 2 3 (5,3) (5,3) (b) Find the area between the curve y2(2a-x)=x3 and its asymptote. Q4. (a) Find the points on the surface z2= xy +1 nearest to the origin. (b) Expand ƒ(x,y)=tan-1 xy in ascending powers of (x-1) and (y-1) up to second degree terms. ∂2u (4,4) ∂2u +y ∂u Q5. (a) If u=tan-1xx-y , then prove that x2 ∂x2 +2xy ∂x∂y +y2 ∂y2 =2 sin u cos3u 3 3 2 (4,4) (b) If u=log (x3+y3+z3-3xyz), then prove that ( ∂x∂ + ∂y∂ + ∂z∂ )2 u= -9(x+y+z) -2 Section-B Q6. (a) Using triple integral find the volume of the tetrahedron bounded by the co-ordinate planes and the plane x + y + z = 1 a b c (b) Evaluate : 12-x xy dxdy by changing the order of integration. ∫∫ 0x (4,4) 2 → → → ∫∫S F•N Fds, where S is the portion of the sur→ ( ( ( (b) If F =yzi -xzj +xyk , then evaluate 2 ∆ → ( Q7. (a) Prove that Curl (Curl F ) =grad (divF ) - (3,5) face of the sphere x +y +z =1,in the first octant. 2 2 2 Q8. (a) Apply Stoke’s theorem to evaluate the line integral ∫c[ydx+zdy+xdz], Where C is the curve of intersection of the sphere x2+y2+z2=a2 and the plane x+z=a. (b) Using Green’s theorem evaluate the line integral ∫c[(y-sinx)dx+cosxdy] Where C is the plane triangle bounded by the lines y=0, x= π /2, y=(2 / π) x M-54091 2 P.T.O

( ( ( Q9. (a) Verify Divergence theorem for F→=(x2-yz)i +(y2-xz)j +(z2-xy)k Taken over the rectangular parallelepiped 0 ≤ x ≤ a,0 ≤ y ≤ b,0 ≤ z ≤ c. (b) Evaluate ∫∫e x2+y2 dxdy, where R is semicircular region bounded by the x-axis and R the curve y=√1-x2 , by changing to polar co-ordinates. ***** M-54091 3 (5,3)

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