Calculus — Winter 2020

Objective Questions (MCQS) lim The value of x (log x) is 1. x0

1 ( b) 0 ( c ) -1 ( d )  2. lim x  y Find (x,y)(0,0) x  y

1 ( b) -1 ( c ) e ( d ) Limit does not exist 3. x2  y2  u u If u  sin 1   then x  y  ?  x  y  x y

sinu ( b) tanu ( c )  sinu ( d ) tanu 4. The curve a2x2  y3 (2a y) is symmetric about

xaxis ( b) yaxis ( c ) origin ( d ) x y line1 (x,y) If x  rcos and y rsin then  ? 5. (r,)

rcos (b) rsin (c) r (d)  r 6. 12345.....n  ? n(n1) n(n1)(2n1) n (n1)

(b) (c) (d) n(n1)262  n2 7. Test the convergence of  n n 0

Convergent (b) Divergent (c) Finitely Oscillating (d) Infinitely Oscillating

Find the equations of the tangent plane and normal line to the surface x22y23z2 12 at (1, 2,1).  x y z u u u

If u  f  , ,  , Prove that x  y  z  0.  y z x x y z  x  y 

Expand x2 y 3y 2 in terms of (x 1) and (y  2)

Find the Maximum and Minimum values of x3 3xy2 3x2 3y2 4

1 1 1 Show that the series 1   ...... is Convergent.234

 2n 1 Test the convergence of the series  3n 1 n  1

lim log (sin x) Evaluate . x0 (cot x)

 1 Define the third type of Improper Integral and Evaluate  dx x20

Change the order of the integration of   f(x,y) d y dx 0 x 1 1  x x  y

Evaluate    dz d y dx000

2 4 8 16 Prove that 1    .......... converges and Find its sum.

Find the volume of the tetrahedron bounded by the plane xyz  2 and the planes x 0, y 0, z 0