Calculus — Winter 2019

Objective Question (MCQ) Mark

07 1. 1 1 1 The sum of the series 1   ...248 (A) 1 (B) 2 (C) 3 (D) Infinity 2. The series  n  11 n is (A) convergent (B) divergent (C) Oscillating (D) none 3.  sinn The series is n2 n1 (A) convergent (B) divergent (C) Oscillating (D) none The curve 4. 9 y 2  x  x  1  2 is symmetric about (A) x-axis (B) y-axis (C) Line y=x (D) origin 5. Apoint  a , b  is said to be a saddle point if at  a , b  (A) rts2 0 (B) rts2 0 (C) rts2 0 (D) rts2 0 The volume of solid generated by revolving a circle 6. x 2  y 2  9 about x- axis (A) 43  (B) 3 6 (C) 3 63  (D) 3 6  7. The value of l i m x    s i n x x  (A) 1 (B) 0 (C) 2 (D) Infinity

07 1. Which of the following is homogeneous function of degree one? (A) x2 (B) x2 (C) y y2 x  y 2 (D) y x y  x x y 2. x y The value of lim x,y1,1 x2  y2 (A) 2 (B) 0 (C) Infinity (D) 3. If x r c o s , y r s i n     x then the value of r (A) c o s  (B) s e c  (C) cosec (D) s i n  4. The value of 21 x0 l n x x d x d y (A) 2ln2-2 (B) 2ln2-1 (C) ln2 (D) 0 5. The value of l i m x  0 x x (A) 1 (B) e (C) x (D) 0 6. 1 x y The value of dxdydz000 (A) 1 (B) (C) 13 (D) 16 7. The value of2 sin0 r 3 d r d      (A)32  (B)2  (C) 364  (D) None Define Jacobian and show that J∗Jˈ = 1.

. Find the equations of tangent plane and normal line to

x 2  y 2  z 2  8 1 at the point   1 ,  4 , 8 04

Arectangular box open the top is to have a volume of 108 c.c. find the dimension of the box requiring least material for its construction.

Show that  2 u   v    2  v  u03 where  yxy  x2y2 ;x,y0,0  x4 4y4

Discuss the continuity of f x,y 1 ;x,y0,0  5

State and prove Euler’s Theorem for Homogeneous functions.    x2  y2  Also, if u sin1 then show that  3 3     x2  y2  i. x u x  y u y 1 t a n u ii. x x u x x  2 x y u x y  y y u y y 1  t a n 3 u  t a n u 07

Evaluate A r s i n d r d     over the area of the curve r  1 c2 o s     above the initial line. Evaluate the integral by the changing the order of integration,

80 23 y x 4  1 d x d y04

i. Use triple integral to find the volume of the cylinder x2  y2 1 between the planes z  1 and z  203 . ii. Evaluate 0 0 e   x 2  y 2 d x d y04 by changing to polar coordinates

Test the convergence of the series  n  1 n  n1  1 03 , if convergent then find its value.

Test the convergence of the series135 + + +⋯, 1∗2∗3 2∗3∗4 3∗4∗5

For which value of x does the series x2  x3  x4  x5  ... is absolute or conditionally convergent or divergent? What is the radius of convergent of x2  x3  x4  x5  . . .07 ?

Determine the convergent of  n  1 t a1 n   1 n n2

Find the volume of the solid generated by revolving the region bounded by x  y 2 and the lines x  0 , x  204 about the x-axis.

Trace the curve r a1cos;a0.

  Expand sin x in powers of x by using the Taylor’s series. Also,    4 find the value of s i n 4 603 .

1 ex e2x e3x 3 Find lim  x0 3 

Discuss the convergence of the following integrals: 1 1 

 dx (ii) ex2 dx x2 1 0