Calculus — Summer 2023

Objective Question (MCQ) Mark

07 1. The ln i→ m 2 n −4 + n1 = _ _ _ _ _

1 (b) 2 (c) -3 (d) -1 2. The sum of the series 1 +2 +4 +2 + i s . . . . .

3 (b) 5 (c) 11 (d) 3. If the equation of the curve contains only even powers of x and y then the curve is symmetric about…..

X-axis (b) Y – axis (c) line Y=X (d) both X-axis and Y-axis 4. lim(x −1)(x−1) = _______ The x→1

3 (b) 1 (c) 2 (d) 5. The value of  11 x d x is ……

-2 (b) 5 (c) 1 (d) 6. When the area bounded by the curve y=f(x), the ordinates x=a, x=b and the x-axis is revolved about x-axis then the volume of the solid generated is given by …….

b a y d x   b 2 ydx

(c) a b a y 2 d x

b a y 2 d x   7. 2xy lim The value of is……. (x,y)→(5,1) x + y

(b) (c) (d)

07 1. If f ( x , y , z ) = 2 x s i n ( y + 5 z ) t h e n   f z i s . .

1 0 x c o s ( y + 5 z )

x c o s ( y + 5 z ) 10cos(y +5z) 5xcos(y +5z)

(d) 2. If Uis a homogeneous function of degree n in the variables x and y then x   u x + y   u y = _ _ _ _ _

(n-1)u (b) (n+1) u (c) nu (d) (2n+1)u 3. 22 1 6 x y 2 d y d x = _ _ _ _

35 (b) 84 (c) 20 (d) 4. Maclaurin’s series of e − x i s . . . . .

n  = 0 ( − 1 n ) n ! x n  xn 

(c) n! n=0 n  = 0 ( − ) n n ! x n

n  = 0 ( ( − n 1 ) + n1 x ) n ! 5. The degree of the homogeneous function f ( x , y ) = x x + + y y i s . . . .

1

1

1

1 6. What does the polar equation r = a , a > 0 represent ?

Line (b) circle (c) rectangle (d) Parabola 7. l i m x → 0  1 x − c o s e c 2 x  is of the form _________0   − 1

(b) (c) (d)0

Test the convergence of n  = 1 t a n − 1 n − t a n − 1 ( n + 1 )

Find the values of a and b such that l i m x → 0 x ( 1 + a c o s x x3 ) − b s i n x = 104

x2 + y2 u = (1) If then show that x + y x   u x + y   u y =3 u (2) If u = l o g ( x 3 + y 3 − x 2 y − x y 2 ) then show that x  2 x u2 + 2 x y   x2  u y + y  2 y u2 = − 303

Find the local extreme values of the function x 3 + 3 x y 2 − 1 5 x 2 − 1 5 y 2 + 7 2 x03

If y = f ( x + c t ) + g ( x − c t ) then prove that  2 t y2 = c  2 x y2

If u f ( x , y ) w h e r e x r c o s , y r s i n   = = = then prove that u x2 u y2 u r2 r1 u2      +     =     +    07

(2x − y2)dxdy Evaluate where Ris the triangular region R R enclosed between the lines y = − x + 1 , y = x + 1 a n d y = 3 .03

Evaluate a 0 a 2 − 0 x ( x 2 + y 2 ) d y d x04 by changing to polar coordinates where a > 0.

Sketch the region and by changing the order of integration evaluate1 01 x s i n y 2 d y d x07

Test the convergence of the series n  = 1 n ( n2 + n 1 ) − ( n1 + 2 )03

Test the convergence of  3n   1 1    + (1) (2)   5n+1  3n 4n  n=1 n=1

Examine for which values of x the series n  = 1 ( − 1 ) n n + +1 x n + 1

If u = f ( e y − z , e z − x , e x − y ) then show that u u u + + = 0 x y z

 dx  Prove that converges when xp1 p 1 a n d d i v e r g e s w h e n p  104

Expand t a n − 1 ( x + h ) in powers of h and hence find the value of t a n − 1 ( 1 . 0 0 3 )07

Find the volume of solid of revolution obtained by rotating the area 2x+3y = bounded below the lines in the first quadrant about the x-axis.

Using slicing method find the volume of a solidball of radius a.

Trace the curve y 2 ( 2 a − x ) = x 3