Engineering Mathematics Workbook_Final

GATE/ESE 2021




























ENGINEERING




MATHEMATICS






WORKBOOK






















Recommended for








CE/ME/EE/ECE/

CS & IT/CH/PI











AUTHOR: K. UMAMAHESWARA RAO

Table of contents





S. No Chapter Pages



1 Linear Algebra 1



2 Calculus 10



3 Vector Calculus 52



4 Differential Equations & Partial Differential Equations 82



5 Complex Variables 134



6 Probability & Statistics 162



7 Numerical Methods 208



8 Laplace Transforms 228



9 Fourier Series 238























1

1


















Linear




















Algebra

Linear Algebra


1. Let A be the n n matrix with   12 24 5  
−
1
integer entries and assume that A also 4. Let A =   x 6 2  . The value of
has only integer entries, then     − 1 − 2 3    

(a) det A = n x for which the matrix A is not
invertible is ________
(b) det A =  1
(a) 6 (b) 12
=
2
(c) det A n (c) 3 (d) 2
(d) det A will depend on the entries of A [IISC 2007, IISC 2008]
−
1
and A
  p 
[IISC 2007] 5. Let A =     where p, q, r are
  q r  

2. Let A be a 3 3 matrix with Eigen rational numbers. A = the value of
0
−
values 1, 1, and 3. Then __________
2
2
+ ______
2
(a) A + A is non-singular
(a) 2
2
(b) A − A is non-singular (b) 1
2
(c) A + 3A is non-singular (c) 0

2
(d) A − 3A is non-singular (d) cannot be determine using given data
f
[JAM 2005] 6. For n > 1, let ( ) n be the number of

 cos  − sin   n n real matrices A such that
3. The Eigen values of     A + 2 I = 0. Then
 sin   cos   
are (a) f = 0


(a) cos and sin (b) ( ) 0f n = its n is even

(b) tan and cot
(c) ( ) 0f n = its n is odd
i
(c) e and e − i
(d) f = 
(d) 1 and 2
[IISC 2007]
[IISC 2007]
 a b 
7. Let A =     be a 2 2 real
  c d 

matrix with det A = 1. If A has no real
Eigen values then



2

Linear Algebra

+
(a) (a d )  2 4    1 4 8   

11. The rank of the matrix 2 10 22 

+
(b) (a d ) = 2 4    0 4 12   
is __________
+
(c) (a d )  2 4
(a) 3 (b) 2

+
(d) (a d ) = 2 16 [IISC 2008] (c) 1 (d) 0
[JAM 2007]
8. For a square matrix A, Let Tr(A) denote
the sum of its diagonal entries. Let I be 12. The least positive integer n such that

the identity matrix. If A and B are 2 2     n
matrices with real entries such that   cos sin  
A = B = 0 and ( ) 0 then limit   4 4   is identity matrix
tr B 
of ( + ) as t → 0 is ______  − sin  cos   
( + )   4 4  
(a) 0 (b)  of order 2_____


tr A (a) 4 (b) 8
)
( +
(c) (d) det A B
tr B
(c) 12 (d) 16
[IISC 2008] [JAM 2008]

9. If A and B are 3 3 real matrices such 13. Let A be 10 10 matrix with each row
that rank (AB) = 1 then rank(BA) has exactly one entry equal to 1, the
cannot be _______ remaining nine entries of the row being

(a) 0 (b) 1 0. Which of the following is not a
passible value for det A.
(c) 2 (d) 3
(a) 0 (b) 1−
[JAM 2006]
(c) 10 (d) 1
10. Let A be a matrix of order 2 with real
entries such that AB = BA for an [IISC 2009]
matrices of order 2 then,
14. Consider the system of equations
(a) A is always the zero matrix
a x b y c z = d ,
+
+
1
1
1
1
=
R
I
(b) A  for some   a x b y c z = d ,
+
+
2
2
2
2
(c) A is always invertible a x b y c z = d where a , b , c ,
+
+
3
i
i
3
i
3
3

(d) A is never invertible [IISC 2007] d are real for 1 i  3. If
i

3

Linear Algebra

b 1 c 1 d 1 17. Let P be a 4 4matrix whose det =

b 2 c 2 d  0 then the above 10.the determinant of 3P− is
2
−
−
b 3 c 3 d 3 (a) 810 (b) 30
system has ________ (c) 30 (d) 810

(a) atmost one solution [JAM 2012]

(b) always exactly one solution 18. Let A be a 5 5 matrix all of whose
eigen values are zero. Then which of the
(c) more than one but finitely many following statements is always true?
solution
T
(a) A = − A (b) A = − A
(d) infinitely many solutions
5
T
0
(c) A = A (d) A =
[IISC 2009]
[IISC 2010]

15. Let A be a 3 3 matrix with trace A = 3 19. Let A be n n matrix with real entries

and det A = 2. If 1 is an eigen value of such that A + 2 I =
A, then the Eigen values of the matrix 0. Then ______
A − 2I are (a) n is an odd integer
2

(i
(a) 1, 2 − ) 1 , − ( 2 i + ) 1 (b) n is an even integer
(c) n has to be 2
(i
(b) 1, 2 + ) 1 , − ( 2 i + ) 1
(d) n could be any positive integer
−
(i
(i
(c) 1, 2 − ) 1 , 2 + ) 1 [IISC 2011]
 1 2 3 4 5 1 
−
(i
(d) 1, 2 − ) 1 , − ( 2 i + ) 1    2 3 4 8 6 3  
[JAM 2011] 20. Let P  =   
 2 4 6 7 10 3 
 a b     4 7 10 14 16 7  
16. Consider the matrix A =     with
  c d  then the rank of the matrix P is ______
real entries suppose it has repeated eigen (a) 1 (b) 2
values. Pick the correct statement.
(c) 3 (d) 4
0
(a) bc =
[MS 2005]
(b) A is always a diagonal matrix
21. Consider the following equation
+
0
(c) det A x + y z = 3, x az b+ = ,
y + 2z = 3. This system has infinitely
(d) det A can take any real value
many number of solutions. If _____
[IISC 2009]




4

Linear Algebra

(a) a = − 1 b = 0 24. Let A be a 4 4 non-singular matrix
and B be the matrix obtained from A by
(b) a = 1 b = 2 adding to its third row twice the first
)
−
1
det
(c) a = 0 b = 1 row. Then (2A B = __________
(d) a = − 1 b = 1 [MS 2005] (a) 2 (b) 4

 1 0 1 x 1 x  (c) 8 (d) 16
+
+
 
  0 1 1 1   [MS 2007]
22. Let P  =    .
+
+
 1 1 x 0 1 x  25. Let A, B be 2 2 matrices with real
  − =
  1 1 x 1 x 0   entries and assume that AB BA CI
+
+
for some constants C, where I is identity
Then the determinant of the matrix P is matrix. Then C is _______
______
(a) 0 (b) 1
3
(a) ( 3 x + ) 1 (c) 2 (d) 4

2
(b) ( 3 x + ) 1 [IISC 2007]
 a  
(c) ( 3 x + ) 1    
26. Let A =    1   where a is a real.
(d) ( x + 1 )(2x + ) 3 [MS 2005]   49 
Then A is invertible _____
23. Let AX = B be a non-homogeneous
2
system of linear equations. The (a) for all a  22
augmented matrix
 1 1 − 2 1 M 1  (b) for all a  180  2 49
 
( AB  ): −  1 2 3 − 1 M 0   . (c) for all a  22 (or) a  180  2
2
  49
   0 3 1 0 M − 1   
Which of the following statement is (d) for all rational values of a
true? [IISC 2007]


(a) Rank of A is 3 27. If A an n n matrix with real (or)
complex entries and A = 0 then ____
3
(b) The system has no solution.
3
(a)( + ) =0
(c) The system has unique solution.
(b) I + A is invertible
(d) The system has infinite number of
solution (c) I + A is not invertible

(d) necessarily A = 0
[MS 2006]
[IISC 2007]


5

Linear Algebra

28. If A is a 3 3 non zero matrix such that   6  

A = 0, then the number of non-zero (a) 0 
2

eigen values of A is _________     0    

(a) 0 (b) 1
 1 
(c) 2 (d) 3   6  
(b)   0  
[MS 2008]  
  0  

a
29. Let A = ( ) be an orthogonal matrix  
ij
1  2 + 1 
of order n such that a = ,   4  
ij
n  
(c)  0 
1 n n  
i = 1,2,........n. If a = 2   a then  0 
n i= 1 j= 1 ij    

n
  n ( ij a ) 2 = ____ (d) not determined uniquely
a −
i= 1 j= 1
[IISC 2006]
n + 1 n − 1
(a) (b) 31. The determinant
n n
+
+
a b c d e 1
n + 1 n − 1 b c d + a f 1
2
2
+
(c) (d)
n n c d a b g 1 =_________
+
+
+
[MS 2009] d + a b c h 1

30. Let M denote a 3 3 real matrix such (a) 0
 1 
     4   4  (b) 1
1
1
         

 
that M      5 =  1 
2 = 
2 , M 

+
+
) e
       5 (c) (a b )(c d + + f + g + h
       0      0   
3
3
 
 
 
 
+ + +
 6    (d) (a b c d )(e + f + g + ) h
 
then M   0  is [IISC 2006]
 
   0    32. Let P =   be a 50 50 matrix,

p
ij
where
p = ij min (  ,i j ) i j = , 1,2,......50 
then the rank of P is _________


6

Linear Algebra

(a) 1 (b) 2 x p q 1

(c) 25 (d) 50 a x r 1
36. If = 0 then x = ______
[MS 2010] a b x 1
a b c 1
 1 8 
33. An Eigen vector of the matrix    
  0 1  [IISC 2000]

is ___________ 37. The characteristic polynomial of the
 0 0 0 
T
T
1,2
(a) ( ) (b) (5,0 )    

matrix 1 0 1  is ____________

T
T
1,1
(c) (0,2 ) (d) ( )    0 1 0   
[MS 2011] 2 2
(a) ( x x + ) 1 (b) ( x x − ) 1
34. An Eigen vector of the matrix
 2 1 0  2
  (c) ( x x + ) 1 (d) ( x x − ) 1
2
 
 1 2 1
 
   0 0 2    [IISC 2002]

 1   0  38. Let A be a 3 3 real matrix. Suppose
        A = 0. Then A has ______
4


(a) 0 (b) 1
   
   0       0    (a) exactly two distinct real eigen values
 0   2  (b) exactly one non-zero real Eigen
        value


(c) 0 (d) 2
    (c) exactly 3 distinct real Eigen values
   1       2   
(d) No non-zero real eigen value
[MS 2012]
[IISC 2002]

35. Let A be a 3 3 real matrix with Eigen
values 1, 2, 3 and det B = A + − 1 A . 39. The determinant
2
Then the trace of the matrix B equal to x 0 x 1 x 2 x 3 x 4
________ x 0 x x 2 x 3 x 4
x x x x x = _________
91 95 0 1 3 4
(a) (b) x x x x x
6 6 0 1 2 4

97 101 x 0 x 1 x 2 x 3 x
(c) (d)
6 6


[MS 2013]



7

Linear Algebra

(a)  a a a 
  1 2 3 


   x 0 (x x− 1 )(x x− 2 )(x x− 3 )(x x− 4  )   4 42. Let A be the matrix b 1 b 2 b  c 3 3   

c
c


1
)
−
−
−
−
x
(b) ( x x 1 )( x x 2 )( x x 3 )( x x where a i a j a k , 2 
+
+
0
4
3
1
2
+
+
+
+
(c) b i b j b k , c i c j c k are
2
1
2
3
3
1
−
−
−
−
x 0    (x x 1 )(x x 2 )(x x 3 )(x x 4  )   4 three mutually orthogonal unit vector
matrix . Then ________
(d) x x x x x x
4
3
2
1
0
−
2
1
(a) A = A (b) A = A
[IISC 2005]

−1

(c) = (d) =
40. The determinant [IISC 2004]
+
1 x 1 1 1
−
1 1 x 1 1 43. For real numbers a, b, c the following
linear system of equations
+
1 1 1 y 1 x + + = 1, ax by cz = 1,
+
+
y z
−
1 1 1 1 y
a x b y c z = 1 has a unique
2
+
2
+
2
(a) xy solution if and only if _________

=
(a) b a and b c
2
xy
(b) ( )

=
(b) a b and a c
−
−
(c) (1 x 2 )(1 y 2 ) (c) a c and a b
=




2
2
(d) x + y [IISC 2005] (d) a b b c and a c
41. Let a, b, c are arbitrary real numbers. Let [IISC 2003]
 1 a b 


  44. Let P be a 3 2 matrix, Q be a 2 2

A be the matrix A =   0 1 c  . Let I matrix and R be a 2 3 matrix such
 
   0 0 1    that PQR is equal to the identity matrix.

be the 3 3 identity matrix. Then Then ____
(a) rank of P is equal to 2
1
2
(a) A − 3A + 3I = A
(b) Q is non-singular
1
2
(b) A + 3A + 3I = A (c) Both A and B are true
1
+
2
(c) A + A I = A (d) There are no such matrices P, Q and

[IISC 2012]

R
(d) A is not invertible [IISC 2004]

8

Linear Algebra

+
1 a 1 a 2 .................a n
+
a 1 a .............a
45. 1 2 n
............ ...............................
+
a a 1 a
1 2 n
[NBHM 2009]

  1 
46. Given that the matrix     has an
  2 3 

Eigen value 1, compute trace and its
determinant.

[NBHM 2009]

47. Let ∈ ( ) such that trace
2
2
cos A= and det A = 3. The
characteristic Polynomial of A =
−
1
_________
[NBHM 2005]














































9

2


























Calculus

Calculus

1  1 z  − 2    y 
 
  

)

5
f
1. The integral     dx dy dz = 4. Let ( , x y = x y 2 tan − 1       . Then
0  0   0     x 
___________ f  f 
x + y = _______
1  1 y  − 2    x  y
  
 


(a)     dx dz dy
0  0   0    (a) 2f (b) 3f
1  1 y  − 1   (c) 5f (d) 7f
  
 


(b)     dx dz dy
0  0   0    [JAM 2006]
2  2  1 z   cos x t − 2
−
  
 


f
(c)     dx dz dy 5. Let ( ) x =  e dt . Then
0  0   0    sin x
1
f  )
( / 4 equals _______
2  2  1 y  
−
 

  

(d)     dx dz dy
0  0   0    (a) 1 e (b) − 2 e
[JAM 2005]
(c) 2 e (d) − 1 e
2 n + 1 + 3 n + 1
2. Lt
n
n→  2 + 3 n [JAM 2006]
(a) 3 (b) 2  1 cx 1 x 
+
6. If Lt       = 4 then
−
(c) 0 (d) 1 x→ 0  1 cx 
+
[JAM 2006]  1 2cx 1 x 
Lt   −     is _________
17
f x =
3. Let ( ) (x − 2 ) (x + ) 5 24 then x→ 0  1 2cx 
_______ (a) 2 (b) 4
(a) f does not have a critical point at (c) 16 (d) 64
[IISC 2008]
(b) f has maxima at 2
2
y
(c) f has minima at 2 7. Evaluate   xe dx dy where R is
R
(d) f has neither minima nor maxima at 2 the region bounded by the lines x = 0, y
2
[JAM 2006] = 1 and the parabola y = x .

[JAM 2006]








11

Calculus

8. Find the volume of the solid bounded ln ( x + y 2 )
2
above the surface z = − 2 y and 12. The value of   x + y 2 dx dy
1 x −
2
2
0
below by the plane z = .
where G = ( { , x y  ) R ,
2
[JAM 2006]

1 x + y  e 2 } is _____
2
2
9. Using change of variables evaluate
  xy dx dy where the region R is (a)  (b) 2
bounded by the curves xy = 1, xy = 3, (c) 3 (d) 4
y = 3x and y = 5x in the first
[JAM 2010]
quadrant. [JAM 2006]
n
10. Let A(t) denote the area bounded by the 13. The value of Lt  2 1 is
n→ 
=
curve y e − x , the x-axis and the k 1 = n + kn
straight lines x = − t and x t = . Then ________
)
Lt A ( ) t is _______ (a) 2 ( 2 1 (b) 2 2 1
−
−
t→ 
(a) 2 (b) 1 1
)
−
(c) 2 − 2 (d) ( 2 1
1 2
(c) (d) 0
2 [JAM 2015]

[JAM 2007] 14. Let V be the region bounded by the
planes x = 0, x = 2, y = 0, z = 0 and y + z
11. The function f defined by
 e 1/x x  0 = 1. Then the value of

f ( ) x =     y dx dy dz is __________
 0 x  0 v

1 4
(a) is differential for all real values of x (a) (b)
2 3
(b) is not differential at x = 0
1
(c) is not differential for x < 0 (c) 1 (d)
3
(d) is not differentiable for x > 0
[JAM 2011]
[IISC 2008] 1 z y

2 3
15. The value of    xy z dx dy dz
z= 0 y= 0 x= 0
is ___________


1 1
(a) (b)
90 50


12

Calculus

(c) f is discontinuous only at one point
(0,1)
1 1
(c) (d) (d) f is continuous only at one point
45 10
(0, 1)
[JAM 2012]
[JAM 2013]

16. The value of the integral
2  2
  1 1 y 1 x dx dy = 3 _______ 19. The value of    x   dx dies between
+
0 y 0
______

−
2 2 1
(a) 2 2 (b) (a) 1 and 1.5 (b) 1.5 and 2
2
(c) 2 and 2.5 (d) 2.5 and 3
−
−
2 2 1 2 2 1
(c) (d) [IISC 2010]
8 9
2
2
20. The value of   x + y dx dy ,
[IISC 2009] D
2
2
2
,
17. Let [x] denote the greatest integer D = ( { , x y  ) R x  x + y  2 } x
function of x. The value of  for which is _______
the function
  2 7
 sin − x   , x  0 (a) 0 (b) 9




f ( ) x =      − x 2   is
 14 28
 ,  x = 0 (c) (d)

continuous at x = 0 is _____ 9 9
[JAM 2013]
(a) 0 (b) sin ( ) 1−
)
21. The area of the region ( { , x y ,
(c) sin 1 (d) 1
3
0 x y  , 1,  +  3 } is _____

x
y
[JAM 2013] 4 2
18. Let the function f(x) is defined by 9 7
 e x , x is rational (a) (b)

f ( ) x =  16 16
 e 1 x , x is irrational
−

for x  ( ) . Then ______ (c) 13 (d) 19
0,1
32 32
(a) f is continuous at every point in (0,1) [MS 2005]

(b) f is discontinuous at every point in
(0,1)


13

Calculus

− + − + − +
( 2n
f
x
22. Let ( ) x = x x − x − 1 , −   1 2 3 4 5 6 .......+ − )
Lt =
2
which of the following statements is n→  n + 1 + n − 1
2
true.
1
(a) f is not differentiable at x = 0 and x = (a)  (b) 2
1
1
(b) f is differentiable at x = 0 but not (c) 0 (d) −
differentiable at x = 1 2
[MS 2007]
(c) f is not differentiable at x = 0 but
differentiable at x = 1 26. By changing order of integration
)
(d) f is differentiable at x = 0 and x = 1.   1 ex f ( ,x y dy dx can be expressed
0 1
[MS 2006] __________

23. Let 1 ln y
)
f x = 1 )( x − 2 )(x − 3 )(x − 4 )(x − ) 5 (a)   f ( , x y dx dy
( ) ( x −
0 1
x
, −  . The number of distinct

real roots of the equation (b) ∫ ∫ ( , )
d 1 1
dx ( f ( )) 0x = is exactly ______ 1 ln y
)
(c)   f ( , x y dx dy
0 0
(a) 2 (b) 3
1
(c) 4 (d) 5 (d) ∫ ∫ ( , )
1
[MS 2006] [MS 2007]

 2 1 27. Let ( )   x and
f x =


f
24. Let ( ) x =  x sin x x  0 . Then 
  x 0  x  1
 0 x = 0 




g ( ) x =  x − 1 1 x  2 for
1
x
f
(a) ( ) x is continuous at x = 0  x − 2 2   3

 0 x = 3


(b) ( ) 0f 1 exists
x    . Then ( ) x + g ( ) x is
0,3
f
(c) f 11 ( ) x is continuous at x= 0 _______
(d) f 11 ( ) 0 exists [MS 2007] (a) discontinuous at points 1 and 2
(b) continuous on [0, 3] but not derivable
25. on (0,3)


14

Calculus

(c) differentiable once but not twice on    2    3 3 2
(0,3) (a)   +   (b)  +
 
 
 
 
 2   2  4
(d) twice differentiable on (0, 3)
3 2
[MS 2008] (c)  − (d) 0
4
28. The area of the region enclosed by the
2
curve y = x and x + y = 2 is [MS 2008]
_______

27
(a) 3 (b)
2 1 1
31. The value of   3 x + 3 1 dx dy =
9 0 y
(c) (d) 9
2 2 5/2 2

(a) (b) (2 3/2 − ) 1
[MS 2008] 3 3

2
)
)

−
−
29. Define ( , x y = f ) x + y 2 cos 2 (t + ) x dt . (c) 2 ( 2 1 (d) 2 ( 2 1
0 3
df
)
Then (0, y =__________ [IISC 2006]
dx
32. Let [x] denote the greatest integer
(a) 0 1   1  
( )
cos y 2 function. The value of 1  2   x  2    dx =
(b)
2 ____________

( )
cos y 2 (a) 1 + 1 − 1 (b) 1 + 1
2
( )
(c) + cos y 2 3 2 2 3 2
2
1 1 1 1 1
2
y +
( )
(d) cos 2 ( ) cos y 2 (c) + + (d) −
3 2 2 3 2
[IISC 2006] [IISC 2006]


3
30. If  x f ( ) t dt = x 2 sin x + x , then 33. The area of the region bounded by
3
y
2 0 and y = is
0 y = x , x + − = 0
   ________
f       is _________
 2  (a) 0.25 (b) 0.50
(c) 0.75 (d) 1.0




15

Calculus

[MS 2009] 38. Let the function  : 0, ) → R is given
f
2 −
3 − 2 x by ( ) x = x e . Then the maximum
x
x
f
34. Lt
x
x→ 0 4 − 3 x value of f is___
3 2 − 1 − 2
(a) (b) (a) e (b) 4e
4 3 − 3 − 4
3 ( ) (c) 9e (d) 16e
3
2
(c) loge (d) log
4 2 [MS 2011]
3
[IISC 2006] 39. Let D be the triangle bounded by y axis,
=
the line 2y  and the line y = x .
1
35. Lt (n + 5 4n 3 ) − 5 n equals ________ cos y
n→  Then the value of   dx dy =
D y
4
(a) 4 (b) _______
5
1 3
5 (a) (b)
(c) 0 (d) 2 2
4
(c) 1 (d) 2
[IISC 2006]
[MS 2011]
36. The value of
1    1 e − 1 2 ( x + 2 y 2 ) dx dy = ________ 40. The volume of the solid revolution

2 − − generated by revolving the area bounded
by the curve y = x and the straight
 1
(a) (b) lines x = 4, y = about the x-axis, is
0
4 2
___________
1 1
(c) (d) (a) 2 (b) 4
4 2

[MS 2010] (c) 8 (d) 12

+
37. The value of   e − x y dx dy , whose [MS 2012]
S 41. The value of∫ ∫ 2
1
2


S = ( { , x y ), 0   1, y  0,1 x +  2} 0 2
x
y
equals _____________
(a) e + 1 (b) 1
(a) 1 (b) 2
2
−
−
2
2
1
e
(c) e − e (d) e − (c) e − 1 (d) e
[IISC 2005]
[MS 2010]

16

Calculus

+
x  (a) 8 (b) 7
g
42. Let ( ) x =  sin y 2
( ) dy . Then
x 
−
(c) 10 (d) 10 − 14
g 1 ( ) x = _________
[IISC 2001]
2
(a) sin x 46. Let f be a real function defined by
 ax b if x  − 1
+

2
sin (x  + ) + sin (x  − ) 2   2

1
(b) f ( ) x =  x + 1 if − 1 x 

2  − ax b if x  1
+


2
2
(c) sin x  ( + ) − sin (x  − ) where a and b are real numbers. If f is
continuous on the real line then the
product ab is ________
2
2
(d) cos x  ( + ) − cos (x  − )
−
(a) 2 (b) 4
−
[IISC 2005] (c) 2 (d) 0
2 −
x
f
43. At x = 2, ( ) x = x e has [IISC 2001]
)
__________ 47. f ( , x y = x + 100x y + 200xy + 10y
6
7
7
5
2
2
2
(a) local minimum, but not global then x fxx + 2xy fxy + y fyy =
minimum _______
(b) local maximum, but not global (a) 42x + 4200x y + 8400xy + 420y
7
7
2
6
5
maximum
6
7
5
2
7
(b) 42x + 500x y + 200xy + 10y
(c) global minimum
5
2
6
7
7
(c) 42x + 1000x y + 1200xy + 420y
(d) global maximum
2
7
7
5
6
(d) 7x + 700x y + 1400xy + 70y
[IISC 2005]
[IISC 2002]
44. The value of lim ( 1 + 1 + ⋯ ⋯ +
→∞ +1 +2
1 ) is ___________ 48. For a real number y, let [y] denote the
+ largest integer smaller than or equal to y.
2  2
(a) 0 (b) ln2 The value of    x   dx = _________
0
2
(c) e (d) e
(a) 1 (b) 5 − 2 − 3
[IISC 2005]
8
45. The maximum value of (c) 3 − 2 (d)


10 − 3cos − 4sin + 9 for 3
 
0   2 [IISC 2004]

17

Calculus
(
f
49. Let : 0, ) → R be the function 52. Find ‘C’ of Rolle’s Theorem for
f ( ) x = e x (sin x − cos ) x in
e x
defined by ( ): . Then   
f x
x x  / 4,5 / 4
Lt f ( ) x = ________  
x→  (a) / 2 (b) 3 / 4
(c)  (d) does not exist
(a) does not exist (b) 0
53. The value of  of
(c) 1 (d) e
1
f ( ) b − f a = − ) ( )
( ) (b a f  for the
[IISC 2003] function ( ) x = Ax + Bx c in the
+
f
2
50. Let  ,  be two real numbers and interval [a, b] is ________
→
  0. The function : f R R
+
−
(a) b a (b) b a
defined by
−
+
b a b a
 0 if x  0 (c) (d)
 2 2

f ( ) x =   1 
 x  sin   if x  0
      54. If the Rolle’s Theorem holds for the
  x  function ( ) 2f x = x + ax + bx in the
3
2
is differentiable at 0 iff __________ 1

−
(a)   = (b)    interval  1,1 for the point c = 2 then
(c)    (d)   1 value of a & b are
1
2
[IISC 2003] (a) a = , b = −
2
5 − 3 x
x
51. The value of the limit Lt is 1
x
x→ 0 3 − 2 x (b) a = − 2 , b = 2
___________


10 ( ) (c) a = 1 , b = 2
5
(a) log e 9 (b) log 3 3 2
2
1
log 5 (d) a = − , b = −
2
(c) 2 (d) log 5 2
log 3 2
2
1 1
f
g
[IISC 2004] 55. If ( ) x = , ( ) x = in [1, 2] then
x x 2
Mean Value Theorems the mean value C of Cauchy’s mean
value theorem is






18

Calculus

4 5 3
(a) (b) (c) (d) 0
3 4 4

5   x sin x 6
4
(c) (d) none of these 60. cos x dx =
3 0

56. Find the mean value ‘c’ of L.M.V.T for (a) 3 2 / 512 (b) 5 2 / 256
1/3
f x = (4 x ) in [1, 6] (c) 3 2 /128 (d) none of these
−
( ) 2 +
(a) 3 
61.  cos x dx = _____
−
(b) 4 0

(c) 8 62.  n   x dx = _____ , where [x] is a step
0
(d) cannot be applied function and ‘n’ is an integer.

8 ( n n + ) 1 ( n n − ) 1
f
1
57. If ( ) x = and f(0) = 1 (a) (b)
x + 2 3x + 4 2 2
then the lower and the upper bounds of f
(1) estimated by mean value theorem are (c) n (d) n + 1
2 2
5 1
(a) 2 and 3 (b) and 
6 10 2
)
63.  log (tan x dx =
3 4 0
(c) and (d) 7 and 5
4 3 
(a) 0 (b)
Definite Integrals 2

1  1 x  (c) (d) 
+
x
58.  2 cos log       dx = ____ 4
−
− 1  1 x 
2 d   e sin x
64. Let   F ( ) x   = , x  0. If
(a) 0 (b) 1 dx x
 sin x  2
 4   2e  dx =   F ( ) k − F ( ) 1 then k
(c) (d) none of these 1  x 
2  
= ________
4
59.   sin x dx =  2 
−    x   u  
65. If  = log     then x + y =
  y   x  y
(a)  (b)
2



19

Calculus

(a)  (b) 2 dy
69. If x + y y = x c then at (1, 1) is
dx
(c) 0 (d) 1
−
(a) 1 (b) 1
x y
2
66. If  = 5 5 then (c) 0 (d) 2
x + y 2
2
x  + 2 xy  + y  = ____ 70. If u = x e z where y = a − 2 x ,
2
2
2
y
xx xy yy
du
2
1 1 z = sin x then find at (0, 1, 1) is
(a)  (b) −  dx
4 4 ______
3 3 −
(c)  (d) −  (a) e (b) e
4 4
−
1
(c) e (d) 2e
  1 4 1 4  
z
y
67. If z = sin − 1   x + y 1   then 71. If u = f (2x − 3 , 3y − 4 , 4z − 2x )
 1 6    then 6u + 4u =
6
 x − y  x y
x z + 2xy z + y z = ______ (a) 2u− z (b) 8u
2
2
xy
yy
xx
z
−
1 (c) 3u (d) 3u
2
(a) tan z (tan z − 11 ) z z
144
Maxima & Minima
1
(b) tan z 72. The maxima & minima of the function
12 x
2
f ( ) x =  (t − 3t + ) 2 dt occurs
tan z 0
2
(c) (sec z − 11 ) respectively at
144
(a) x = − 2 and x = 1
(d) None of these
68. If (b) x = − 1 and x = 2
u ( , x y = ) x 2 tan 1 ( / y x − ) y 2 tan − 1 ( / x y ), (c) x = 2 and x = 1

x  0, y  0then
(d) x = 1 and x = 2
(
+
x 2 ( 2 / u  x 2 ) 2xy  2 / u  x ) y + y 2 ( 2 / u  y 2 ) =
73. The maximum value of the function
f ( ) x = x − 9x + 24x + in [1, 6] is
5
2
3
−
(a) u (b) u ______
(c) 2u (d) 3u 74. The values of ‘a’ and ‘b’ for which the
function ( ) x = x + ax + bx has
3
f
2

20

Calculus

local minima at x = 4 and point of   4 x 2 / y x dy dx =
inflection at x = 1 are 79. e _______
0 0
(a) a = − 3, b = − 24 3 4
8
7
(a) 4e − (b) 3e −
(b) a = − 3, b = 24
4
9
4
(c) 3e + 7 (d) 3e −
(c) a = 3, b = − 24
80. The value of   xy dx dy where ‘R’
(d) a = 3, b = 24 R
is the region bounded by x – axis,
2
75. The function ordinate x = 2a and the curve x = 4ay ,
)
2
f ( ,x y = x − 3x + 4y − 10 at (2, is
3
2
0) has
a 3 a 4
(a) (b)
(a) a maximum (b) a minimum 4 3
(c) a saddle point (d) both (a) & (b) a 4 a 4
(c) (d)
76. The function 6 8
)
x
2
f ( , x y = x y − 3xy + 2y + has − y
81. The value of     e dy dx is
(a) No local extremum 0 x y

(b) One local minimum but no local 1 (b) − 1
maximum (a) 2 2

−
(c) One local maximum but no local (c) 1 (d) 1
minimum
82. By changing the order of integration, the
(d) One local minimum and one local 4 2 ax )
a
maximum double integral   f ( , x y dy dx
0 x 2
4a
77. The distance between origin and a point can be expressed as
+
nearest to it on the surface z = 2 1 xy   q s f ( , x y dx dy then q r =
)
is p r

2
(a) 3 (b) 2 (a) y (b) y

(c) 1 (d) None (c) 0 (d) y


78.  2  3 xy dx dy = _____ 1 1 x 2 1 x − − 2 y 2 dz dy dx
−
y= 0 x= 0 83.    = ____
−
2
2
0 0 0 1 x − y − z 2
(a) 9 (b) 18
(c) 27 (d) 6


21

Calculus

 2  sinh x − sin x
(a) (b) 88. Lt is
2
2 8 x→ 0 x sin x

  2 1 1
(c) (d) (a) (b)
12 8 2 4


1 1 x
1
84. The value of    − x dz dx dy is (c) 1 (d) 1
0 y 0 3 6
1 1 1
(a) (b) 89. Lt e x (cos x ) sin x =
2
12 16 x→ 0

1 (a) 1 (b) e − 1/2
(c) 12 (d)
21
(c) e 1/2 (d) e


a
85. The value of    x y x y z dz dy dx is  
0 0 0  n n n 
90. Lt  + + ...+ 
n→    n 2 n + 1 2 n + 2 (n − ) 1 2  
2
a 4 a 4  
(a) (b)
16 12  
(a) (b)
a 6 a 4 4 3
(c) (d)
48 4 1
(c) 0 (d)
 1   x 1  4


86. Lt e − 1    5x + sin     =  
5x
x→       5 x  Lt  1 + 1 + ........+ 1 
91.
+
+
n→     1 n 2 n n n +  
(a) -1 (b) 1
(a) log2 (b) 2
(c) 0 (d) does not exist
1 


2 8cos x   8    8       (c) (d)

87. Lt   sin   + x −   sin       is 2 4
x→  8x   6   6  
n
1 92. Lt  1 =
+
(a) 1 (b) n i → k= 13n k
2
  4     3  
3 (a) log     (b) log    
(c) (d) 3  3   4 
2





22

Calculus

 3   5  (c) a = 3, b = 2 (d) a = 2, b = 3
(c) log       (d) log      
 2   4  98. Which of the following function is
continuous at x = 3.
1
 ! n n 
93. Lt   n     

n→   n   2, x = 3

 
(a) ( ) x =  x − 1, x  3
f
(a) 0 (b) e 

 x + 3 , x  3
 
1  3
(c) 1 (d)
e  4, x = 3

f
(b) ( ) x = 
+
sin2x a sin x  −  8 x x  , 3
94. If Lt = b where ‘b’ is
n→ 0 x 3  x + 3, x  3

f
finite then a = ____, b = ________ (c) ( ) x = 
 x − 4, x  3

(a) -2, -1 (b) 2, 1
1
(d) ( ) x = f , x  3
(c) 2, -1 (d) -2, 1 x − 27
3
95. The values of a and b such that  x + 3x a , x  1
+
2

f
+
2
a sin x b log (cos x ) 1 99. If ( ) x =  is
Lt =  bx + 2, x  1

n→ 0 x 4 2
differentiable for ‘x’ then a, b =
(a) -1, -2 (b) 1, 2
(a) a = 3, b = 5 (b) a = 1, b = 2
(c) -1, 2 (d) 1, -2
(c) a = 1, b = 3 (d) a = 3, b = 1
x
96. If y = + x + x + ....... then 100. If 4x − 7 = 5, then the value of
y ( ) 2 = 2 x − − x is
(a) 4 or 1 (b) 4 only 1 1
(a) 2, (b) ,3
(c) 1 only (d) undefined 3 2

97. The values of a and b for which the 3 2
function (c) ,9 (d) ,9
3
2
 2x + 1 if x  1

 
f

2
f ( ) x =  ax + b if 1 x  is 101. If ( ) x = x − 1 + x − 2 is not
3

 5x + 2a if x  3 derivable at x =


continuous every where (a) x = 0 (b) x = 1, 2
(a) a = 2, b = 1 (b) a = 1, b = 2 (c) x = 3 (d) none

23

Calculus

102. A real function tan x tan x
  x + 2  , x for x  0 (a) (b)

f ( ) x =  2y − 1 2y − 1
  x +  x + 5sin , x x  0
2
3

2
f
. If ( ) x is twice differentiable then (c) sec x (d) sec x
2y − 1 2y − 1
(a)  1, =  = 0 (b)  1, =  = 5 dy
+
(c)  5, = = − 10 (d)  5, = = 5 106. If x y = a b (x + ) y a b then dx =

 2x  x 1
103. The derivative of sin − 1   2     with (a) (b)
+
 1 x  y y
 2x 
respect to tan − 1   2     is equal to 1 y
−
 1 x  (c) (d)
x x
(a) 0 (b) 1
107. By applying, Rolle’s theorem for
2x sin x

(c) (d) 2 f ( ) x = in 0, , the value of
−
1 x 2 e x
)
c  (0, is
104. If x = ( a  sin − ) ,

2
d y  
y ( a  = cos − ) then = (a) (b)
dx 2 6 4

1  
(a) −  (c) (d)
a sin 2 2 3
2
108. Which of the following function satisfied
1  all the conditions of Rolle’s Theorem in
(b) − cosec 4
4a 2 the interval [0,1]

1 f x = 
(c) − sec  2 / 2 cosec   4 / 2 (a) ( ) tan x
4a
 1
1  x , 0  x 

(d) sec  2 / 2 cosec   4 / 2  2
f
4a (b) ( ) x = 

 − , 1  x  1
1 x
 2

105. If y = tan x + tan x + tan x + .....
dy 109. By applying Lagranges mean value for
then = the function
dx




24

Calculus
)
f x = + )log (1 x on [0,1] the 113. For the function ( ) x = x , the mean
+
( ) (1 x
f
value of c  ( ) is value theorem does not held in the
0,1
interval….
4 1
(a) (b)   1  

−
e e (a)  1,0 (b) 0, 2  



−
−
4 e 1 e
(c) (d)  1 


e e (c) 0, 2   (d)  1,1
−


110. By applying mean value theorem, for the  
=
=
function ( ) x = f x − 2 4 on [2,4], the 114. If x r cos , y r sin then the
)
value of c  (2,4 is value of  2  +  2  =
 x 2  y 2
(a) 6 (b) 8
(a) 0 (b) 1
(c) 2.5 (d) 3.14 x y
(c) (d)
111. By applying Rauchy’s mean value r r
2
theorem for ( ) x = x , ( ) x = x
3
g
f
115. The Taylor series expansion of sin x
over [1,2] the value of c  ( ) is 
1,2
about x = is
6
14 13
(a) (b)
9 7 1 3      1    2


(a) +   x −   −   x −   + ......
2 2  6  4  6 
2 3
(c) (d)
3 2 x 3 x 5
(b) x − 3! + 5! − .......
=
3
112. A curve ‘C’ is defined as x a cos  ,

y = a sin  in 0, / 2 . What will be 1

3
(c)
the point P on curve C where the tangent 2
to the curve is parallel to the chord 3
−
−
joining points (a,0) & (0, a) x  / 6 (x  / ) 6
(d) − + .....
 a a  1! 3!
(a) (a, a) (b)   ,    
 2 2  116. The Taylor series expansion of
sin x
 a a   a a  f ( ) x = x  at x = is
−
(c)   ,     (d)   ,    
 2 2   2 2 2 2 




25

Calculus

(x  ) 2 e 4 e 2
−
(a) 1+ + ..... (c) (d)
3! 4! 2!


(x − ) 2 x 2 y 2 z 2
(b) 1− + + .....
3! 119. If u = x y z then
1 1 1
−
(x  ) 2
(c) 1− + ..... u + u + u =
3! x y z

(a) 0 (b) 1
−
(x  ) 2
(d) 1− + .....
−
+
+
3! (c) x y z (d) ( 2 x + y + ) z
)
+
117. The Taylor series expansion of 120. If u = f ( , r s where r = x y ,
 1 x 
+
−
log       at x = 0 is s = x y then u + x u = y
−
 1 x 
(a) 2u (b) 2u
 x 3 x 5  r s




(a) 2 x + + + .... − −


 3 5  (c) 2u (d) 2u
s
r
)
−
=
 x 3 x 5  121. If z = f ( , x y where x e + u e ,
v


(b) 2 x − + + ... − u v




 3 5  y = e − e then z − u z = v
 x 2 x 4  (a) xz − yz (b) xz + yz

(c) 2    + + .... x y x y


 2 4  (c) xz + yz (d) xz − yz
y x y x
4
 x 2 x 4    x + y 4  

(d) 2    − + .... 122. If u = log   x + y   then


 2 4   
 2 u +  2 u + 2  2 u
2
x
118. In the taylor series expansion of e , x  x 2 2xy   y  y 2 =
x y
4
about x = 2, the coefficient of ( x − ) 2
(a) 0 (b) 3
is
(c) -3 (d) 1/3
1 e 2
(a) (b)
4! 4!

26

Calculus

 1 x + y 3  x
3
x
123. If u = tan −      then (c) sin y (d) u
 x − y 
 u  u 2 2
x + y = x y
 x  y 127. If u = x + y then


2
2
(a) sin 2u (b) cos 2u x u + 2xyu + y u =
xx xy yy
(c) tan 2u (d) cot 2u
(a) u (b) 2u
x + 2y + 3z
124. If u = sinu = then (c) 4u (d) 6u
x + 8 y + 8 z 8
2
 u  u  u x 2 ( x − y 2 ) 3
x + y + z = 128. If u = then
 x  y z  ( x + y 2 ) 2
2

1 xu + x yu = y
−
(a) tanu (b) 7tanu
7
(a) u (b) 3u
1 1
(c) secu (d) − tanu (c) 4u (d) 24u
7 7
=
129. By change of variables x r cos ,
  n   y r sin in  f ( , x y dx dy
 
)
=
x
x
n  
−
125. If u = x f 1  + y f 2   then
 
 
y
y
    changes to
( cos , sin 
 u  u  2 u  2 u  2 u   f r  r ) ( , r  )dr d
x + y + x 2 + 2xy + y 2 =

 x  y  x 2    y 2 then ( , r  ) =
x y
(a) 0 (b) ( n n + ) 1 u
1
(a) r (b)
2
(c) n u (d) ( n n − ) 1 u r
2
x + y 3 (c) r (d) 1
3
126. If u =
x − y By change of variables x uv , y = v
=
3
x + y 3   x   130. u
u = + x sin   then
)
x − y   y   in   f ( ,x y dx dy changes to
x u + 2xyu + y u =   v  
2
2
  
)
xx xy yy f uv , u      ( ,u v du dv then

 x + y 3   )
3
(a) 0 (b) 2       ( ,u v =
 x − y 

27

Calculus

 u (c) 41 (d) 46
(a) uv (b)
v 135. The maximum value of
2
3
5
 v x − 9x + 24x + is
(c) (d) 1
u (a) 21 (b) 25
250 (c) 41 (d) 46
f
131. The function ( ) x = x + 2 at x =
x 136. The maximum value of the function
5, attains   

5Cos + 3Cos    +     + 3 is
(a) maxima  3 
(b) minima (a) 5 (b) 10


(c) neither maxima nor minima (c) 11 (d) 9

(d) none 1/x
f
137. The function ( ) x = x has maxima
sint at x = _______
f
132. The function ( ) t = at t = 0
t
attains (a) 1 (b) e
2
(a) maxima (c) e (d) 2

(b) minima 138. Maxima slope of the curve
2
3
− x + 6x + 2x + 1 is
(c) neither maxima nor minima
(a) 14 (b) 16
(d) none
(c) 19 (d) -13
133. The right circular cone of largest volume
that can be enclosed by a sphere of 139. The function
2
2
radius 1m has a height of f ( ,x y ) 2x= 4 + y − x − 2y has a
1 2 relative _____
(a) m (b) m
3 3  1 

(a) maxima at   ,1
 

2 2 4  2 
(c) m (d) m
3 3  1 

(b) minima at   ,1

 
134. The maximum value of  2 
2
3
f ( ) x = x − 9x + 24x + is
5
(c) maxima at (0, 1)
(a) 21 (b) 25 (d) minima at (0, 1)


28

Calculus

140. The function 145. 0   /2 log (1 tan x dx+ )
) 4x +
2
8
f ( , x y = 2 6y − 8x − 4y +
)
the optimal value of ( ,f x y is (a) 0 (b) 1
1
(a) a minimum value equal to 10/3 (c) (d) 2
2
(b) a maximum value equal to 8/3 
146. 0  /2 log + )
(1 tan x dx
(c) a maximum value equal to 10/3

(d) a minimum value equal to 8/3 (a) 0 (b) log2
4
141. The distance between origin and the
point nearest to it on the surface (c)  log2 (d)  log2
+
z = 2 1 xy is 8 2

2
+
3 147. 0   /2 (a 2 cos x b 2 sin 2 ) x dx =
(a) 1 (b)
2 
2
(a) 0 (b) (a + b 2 )
(c) 3 (d) 2 2



2
2
142. The value of 1  e x ln x dx = (c) (a + b 2 ) (d) (a + b 2 )
4 8
4 2 2 4
3
3
)
(a) e + (b) e − 148. 0   /2 log (sin x dx =
9 9 9 9
2 4 4 2 − 
3
3
(c) e + (d) e − (a) 0 (b) log2
9 9 9 9 2
143. − 1 1  x 2 4 sin x dx = (c) − log2 (d) log2
x +
1
5
3
149. 0   /2 sin x cos xdx =
(a) 0 (b) 
1 1
 (a) 16 (b) 24
(c) 2 (d)
2
1 1
 +  (c) (d)
 /2 10  1 sin x  48 96
144. −   /2 x log     dx =
−
 1 sin x  1
150. s 0   dx =
+
(a) 0 (b) 2 a 2 cos x b 2 sin x
2
2
(c) / 2 (d) 


29

Calculus

 (c) 16 (d) 14
(a) 0 (b)
ab  /4
f
156. If ( ) x = 0  tan x dx
 f ( ) 3 + f ( ) 1 =

(c) ab (d)
a + b 2
2
1
4
6
151. 0   x sin x cos x dx = (a) 1 (b) 2
3 2 5 2 3
(a) (b) (c) (d) 2
512 256 2


3 2 157. The value of 0  1 0  x 2 e dy dx = 0
/ y x
(c) (d) 0
256
1 1
  − x 2 /2 (a) 2 (b) 3
152. − e dx =


1 1
(a) (b) 2 (c) (d) 0
2 8


(c) 1 (d)  2 2x− 4 2y − 1
158. The value of 0  0  x + 1 dy dx =

e
153. 0   − y 3  y dy = 1/2
+
−
3
(a) 42 36log
e

(a)  (b) 3
+
−
3 (b) 36 42log
e
 (c) 42 36log+ 3
(c) (d) 0 e
2
−
3
(d) 42 36log
e
−
154. 0  1 x 6 1 x dx = 2 1
159. The value of 1  2 0  x dy dx =
2
5 5 x + y 2
(a) (b)
256 128  1
(a) log2 (b) log2
5 3 4 4
(c) (d)
512 512  

2 x ) (c) 2 log2 (d) 2 log4
155. − 2  0  (sin x y dy dy =

(a) 0 (b) 32 160. The value of 0   x sin x dx =
+
2
1 cos x

30

Calculus

  (a) 1/4 (b) 3/2
(a) (b)
4 2 (c) 4/3 (d) 3

 2  2 166. The value of
(c) (d)
4 8 0   x   0  2 sin y dx dy dx =
y
0  1 x 6
161. The value of dx =
−
1 x 2 (a) -2 (b) 2
(c) -4 (d) 4
 3
(a) (b)
2
32 32 167. The area bounded by y = 4x and
2
5 x = 4y is
(c) (d) 0
32
(a) 16/3 (b) 32/3

0  1 y  1 1 x dx dy = 3
+
162. The value of y (c) 8 (d) 16
2
−
2 2 1 168. The area bounded by 2y = x and
(a) 2 2 (b) x = y − 4 is
2
(a) 6 (b) 18
−
−
2 2 1 2 2 1
(c) (d)
8 9 (c) 16 (d) 

0   x   1 − y /2 169. The value of 0   /2 x   /2 cos y dy dx =
163. The value of e dy dx =
y y

(a) 1 (b) 2 (a) -1/2 (b) -1


(c) -2 (d) 0 (c) 1/2 (d) 1

2
x
164. Let 170. The value of 0  − 1 2 y  2 e dx dy =
)
E = ( { ,x y  R 2 ,0   1,0  y  } x
x
E  (a) e (b) e − 1
4
4
then ( x + ) y dy dx =
4
(a) -1 (b) 0 e − 1 e 4
(c) (d)
4 4
(c) 1/2 (d) 1
171. By change the order of integration in
165. Let 0  8 x  2 f )
E = (  ,x y ) R 2 ,0 x  y ,0     s q /4 ( ,x y dydx changes to

y
)
 E  ye − (x y+ ) r  p  f ( ,x y dx dy then q =
then dx dy =

31

Calculus

2
(a) 4y (b) 16y 1 1
(c) (d)
6 12
(c) x (d) 8
−
2 xy dx dx =
176. The value of 0  1 x  2 x
172. Let
0   1 y xy sin xy 0   a b xy sin xy dy dx 3 1
1
( ) dx dy =
1
(a) a= 0, b = x (b) a = 1, b = x (a) (b)
8 6
(c) a = 0, b = 1 (d) a = -1, b = x 5 1
(c) (d)

173. For n N , the value of 24 2
 n 1− ( / x n ) n dx = 177. − 2 1  1+ x dx =
−
0 n x
(a) 3.5 (b) 5.5
(a) 0 (c) 4 (d) none

1 1 1 1
(b) 1+ + + ...+ 178. The integral 0  1 dx converges
2 3 n 1 x 2
−
to
1 1
(c) 1+ + ........+
2 n + 1 (a)  (b) 0
(c) / 2 (d) 
1 1
(d) 1+ + ........+ 1
2 n + 2 179. The integral  − 1 x 2 dx converges to

+
174. The value of  xy dx dy taken over
the region bounded by two axis and the (a) 0 (b) 
straight line x + y = 1 2

(c) − (d) 
1 1 2
(a) (b)
20 24
2
180. The area bounded by y = x and the
1 1 lines x = 0
(c) (d) 4 and y = is
30 40
64
0
175. If R is the region bounded by x = , (a) 64 (b) 3
y = 0 and x + y = 1 then 128 128
 R  ( x + 2 y 2 ) dx dy = (c) 3 (d) 4



1 1
(a) (b)
3 5



32

Calculus

181. By change the order of integration (a) 0.27 (b) 0.67
0  2 x  2x ( , x y dy dx may be (c) 1 (d) 1.22
)
2 f
represented as 186. The function

)

2 f
(a) 0  2 x  2x ( ,x y dy dx f ( ) x =  x a sin 1 , x  0 is

x
)

(b) 0  2 y  y f ( ,x y dy dx  0 , x = 0

0
)
(c) 0  4 y  /2 y f ( , x y dy dx differentiable at x = for all al in the
interval
)
(d) x  2x 0  2 f ( , x y dy dx (a) (− ,1 ) (b) ( 1,− ) 
2
182. The volume generated by revolving the ) (d) ( ,1
area bounded by y = 8x and the line x (c) (1, )
2
= 2, about y-axis is [JAM CA 2006]

128 5 x
2
f
(a) (b) 187. Let ( ) x =  (t − 1 )(t − 5t + ) 6 dt ,
5 128 0
for all x R . Then
127 32
(c) (d)
5 5 (a) f is continuous but not differentiable
on R
183. The value of integral of the function
4
Q ( , x y = ) 4x + 3 10y along the (b) f’ is bounded on R
straight line segment from the point (0,0) (c) f’ has exactly three zeroes
to the point (1, 2) in the xy plane is
(d) f is continuous and bounded on R
(a) 33 (b) 34 [JAM CA 2011]
(c) 40 (d) 56 4
−
184. The value of 0   0   e − ( x + 2 y 2 ) dx dy = 188. For the function y = 1 x , the point
x =
0 is a point of
 (a) inflection
(a) (b) 
2 (b) minima

(c)  (d) (c) maxima
4

(d) absolute minima [JAM CA 2005]
2
185. The length of the arc y = x 3/2
3 189. The value of a and b for which the
3
2
f
between x = 0 and x = 1 is function ( ) x = x + ax + bx has


33

Calculus


local minima x = 4 and point of 0 x  5. Then F has local minimum at
inflection at x = 1 are the points

(a) 3, 24 (b) -3, -24 (a) {0,2,4} (b) {1,3,5}


(c) -3, 24 (d) 0, 0 (c) {0,3,4} (d) {3,4,5}

[JAM CA 2005] [JAM CA 2007]

190. The value of x and x with x  x 193. Consider the function
1 2 1 2 2
x

2
such that (12 − x − x 2 ) dx has the f ( ,x y ) (x= + ) y − (x + y ) 1+ .
x 1 The absolute maxima value and the
largest value are absolute minimum value of the function

(a) -3, 3 (b) -4, 1 on the unit square.

x
(c) -4,4 (d) -4,3 (  , x y ):0   1,0  y   1 ,

[JAM CA 2005] respectively are

f x =
+
191. For ( ) (1 sin x )cos x , where (a) 3 and 3 (b) 3 and 3
0  x  2p , where of the following 2 2 4
statements is true 3 3
(c) 3 and (d) 2 and
 4 4
f
(a) ( ) x has a local maxima at x =
6 [JAM CA 2007]

(b) ( ) x has a local minima at x = 194. Let ( ) x = x − x + 1, 0 x  1.

2
3
f
f
3
Then the absolute minima value of
f
(c) ( ) x has a local maxima at f ( ) x is
5
x = 14 5
3 (a) (b)
27 9
3
(d) ( ) x has a local minima at x = 23
f
4 (c) (d) 1
27
[JAM CA 2006] x
2
F
195. Let ( ) x =  (t − 3t + ) 2 dt . Then

0
192. Let F has
f ( ) x =  x (t − 1 )(t − 2 )(t − 3 )(t − ) 4 dt , (a) a local maximum at x = 1 and a local
0 minimum at x = 2



34

Calculus

(b) a local minimum at x = 1 and a local 199. For the function
maximum at x = 2  y  1   x 
z = x tan − 1       + y sin −      + 2 ,
(c) local maxima at x = 1 and x = 2  x   y 

(d) local minima at x = 1 and x = 2 the value of x z  + y z  at (1, 1) is
 x  y
[JAM CA 2008]
 − − 1
2
196. If ( ) x = f  x ( t t − ) 1 dt , then (a) 4 sin 1
a

−
+
1
(a) f has a local maximum at x = 0 and a (b) + sin 1 2
local minimum at x = 1 4
(b) f has local minima at x = 0 and x = 1 (c)  + sin 1 2
−
−
1
4
(c) f has a local maximum at x = 1 and a
local minimum at x = 0  −
1
(d) + sin 1 [JAM CA 2005]
(d) f has local maxima at x = 1 and x = 0 4
=
=
2
3
x
f
197. If ( ) x = ax + bx + + 1 has a 200. For x r cos , y r sin , which of
local maximum value 3 at x = − 2, then the following is correct?
r    − 1
3 5 3 5 (a)  = sec and  = sin
(a) a = ,b = (b) a = ,b = x x r
4 2 2 4 r   
(b) = sec and = cosec
3 5 3 5  x  x
(c) a = ,b = (d) a = ,b =
4 4 2 2 r    1
(c) = cos and =
 x  x r cos
17
f x =
198. Let ( ) (x − 2 ) (x + ) 5 24 . Then r    − sin
(d) = cos and =
 x  x r
(a) f does not have a critical point at 2
[JAM CA 2005]
(b) f has a minimum at 2
 x 3

(c) f has a maximum at 2   , ( ,x y ) (0,0 )
)
201. If ( ,f x y =  x + y 2
2

(d) f has neither a minimum nor a  0 , otherwise
 
maximum at 2 then at (0,0)
[JAM MA 2006] f  f 
(a) and exist and are equal
 x  y






35

Calculus

f  f   1 ln x 
+
(b) and exist but not equal (c) lncot       + c
 x  y  2 
 ln x 
f  f  (d) lnsin 1+       + c
(c) exists but does not  2 
 x  y
f  f  [JAM CA 2005]
(d) exists but does not
 y  x 1
205. The value of 1  e e (1 ln x+ ) dx is
[JAM CA 2006] x 2

 x   y  (a) 1 (b) 1/e

202. Suppose z = x sin       + y sin       , (c) e (d) 0
 y   x 
z  z  206. If a real valued function f is given by
xy  0 . Then x + y is equal to f ( ) t
 x  y  x dt = 2 x b x  where a
+
,
0
a t 2
(a) -z (b) 0 > 0 and b are areal constants, then f(4) is
(c) z (d) 2z equal to

=
=
203. If z = e xy 3 , x t cost , y t sint then (a) 4 (b) 6
(c) 8 (d) 10
dz at t =  is
dt 2 [JAM CA 2010]


2
t −
cos x
f
(a)  3 / 8 (b)  3 / 4 207. Let ( ) x =  sin x e dt , then
−
(c)  3 / 2 (d)  3 / 8 ( ' f  / ) 4 equals
[JAM CA 2009]
(a) 1/ e (b) − 2 / e
 dx (c) 2 / e (d) − 1/ e
204. The value of
+
−
x 1 cos 2 (1 ln x )
[JAM MA 2006]
is
208. Let :f R R→ be a continuous function.
 1 ln x 
+
(a) ln tan       + c 0  x f ( )dt = x sin ( ) x
2t
 2  If  for all
 1 ln x  
−
(b) ln tan       + c x R , then f(2) is equal to
 2 
(a) -1 (b) 0
(c) 1 (d) 2

[JAM MA 2007]




36

Calculus

3
2
) ( ) t dt ,
209. Let ( ) x = f 0  x ( x − 2 t g 213. If 0  x f ( ) t dt = x 2 sin x + x . Then
where g is real valued continuous f     
function on R. Then f’(x) is equal to   2     is

( )
3
(a) 0 (b) x g x    2    3 3 2




 x x (a)   2   +   2   (b)  + 4
(c) g ( ) t dt (d) 2x 0  g ( ) t dt    
0 3 2
(c)  − (d) 0
[JAM MA 2008] 4
210. Let a be non-zero real number. Then e 1 +
1 214. The value of 1  2 e (1 ln ) x dx is
lim 2  x sin t 2 x
( ) dt equals
2
→
x a x − a a
(a) 1 (b) 1/e
1 1 (c) e (d) 0
( )
( )
(a) sin a 2 (b) cos a 2
2a 2a [JAM CA 2005]
1 1
( )
( )
(c) − sin a 2 (d) − cos a 2
2a 2a  dx
215. The integral 1  2 x )
+
[JAM MA 2009] x (1 e
→
211. Let : f R R be defined as (a) converges and has value < 1
(b) converges and has value equal to 1
 tant


f ( ) t =  t , t  0 (c) converges and has value > 1
 (d) diverges
 1, t = 0

0
216. For   , the value of the integral
1 3 0   −  x 2
Then the value of lim 2  x 2 f ( ) t dt e dx equals
x→ 0 x x

1  
(a) is equal to (-1) (b) is equal to 0 (a) (b)
(c) is equal to 1 (d) does not exist 2  2
2 
[JAM MS 2006] (c) (d) 2
 
d sin x 2
t
212.  e dt is equal to [JAM CA 2007]
dx 0
)
x
2
2
(a) e sin x cos x (b) e sin x 217. The integral 0   /2 min (sin ,cos x dx
)
2
(c) (2sin x e sin x (d) e 2sin x equals



37

Calculus

2
(a) 2 − (b) 2 − 2 (a) 2 (b) 1
1
(c) 2 2 (d) 2 + 2 (c) (d) 0
2
218. The value of the integral [JAM MA 2007]
tan x
0   /2 tan x + cot x dx is 223. The value of the integral

−
(a) / 6 (b) / 2 −    /4 1 cos2x dx is

(c) 0 (d) / 4 2
−
1   log x (a) 1− 1 (b) 1− 1
e
219. The integral dx 2 2
x
1 1
(a) converges to e (c) 3− 2 (d) 2 − 2
1
(b) converges to
e [JAM GP 2008]

(c) converges to 1 224. The value of the integral
 is
(d) diverges 9 dy
0 y 1+ y
 1 dx
220. The value of is
0 x (1 x− ) ( )
−
(a) 4 (b) 4 10 1
 (c) 8 (d) 12
(a) 0 (b)
2 [JAM CA 2012]
(c)  (d) 2
2
x
225. Area enclosed by the curves y = and
2n+
f
221. Let ( ) x = n sin 1 x cos x . Then the 2
n
value of y = 2x − 1 lying in the first quadrant
lim 0   /2 f ( ) x dx − 0   /2 ( lim f ( )) is
x dx
n→  n n→  n
(a) 1/6 (b) 1/4
is
(c) 1/2 (d) 1/3
(a) 1/2 (b) 0 [JAM CA 2005]
(c) -1/2 (d) −

2 xy
222. Let A(t) denote the area bounded by the 226. The value of 0  1 y  1 x e dx dy
curve y = e − x , the x-axis and the e + 2 e − 2
=
straight lines x = − t and x t . Then (a) 2 (b) 2
lim A ( ) t is equal to
t→ 




38

Calculus

e − 1 e + 1 (d)  2/3 1 2 ( / 1 v− ) ( f u uv− ,uv )u du dv
(c) (d) 1/ (1 v− )
2 2
[JAM CA 2006]
[JAM CA 2005]
231. The area bounded by the curve
0  1 y  1 x y = (x + 2
227. The value of dx dy is ) 1 , its tangent at (1,4) and the
2
( x + y 2 ) x-axis is

  1 2
(a) (b) (a) (b)
4 2 3 3

  4
(c) (d) (c) 1 (d)
3 5 3


[JAM CA 2005] 232. If  denotes the region bounded by the
x-axis and the lines y = x and x = 1,
228. The value of the integral then the value of the integral

0   x   e − y dy dx cos 2x
( )
y   dx dy is
 x
(a) 0 (b) 1
sin2 cos2
(a) (b)
(c) 2 (d)  2 2

229. The entire area bounded by the curve (c) cos 2 (d) sin 2
r = a cos2 is
2
[JAM CA 2007]
(a) a (b) 2a
233. Let D be the region in the first quadrant


(c) a (d) 2 a lying between x + 2 y = 2 1 and
230. The double integral x + 2 y = 2 4. The value of the integral
1  2 x  2x f ( ,x y dy dx under the   sin ( x + y 2 ) dx dy is
)
2
transformation x u= (1 v− ), y uv is D
=
transformed into 
(a) (cos1 cos2− )

2
( / 1 v
−
(a)  1/2  2/3 1/ − − ) ) ( f u uv ,uv )du dv 4
(1 v
 − )
( / 1 v−
(b)  1/2  2/3 1/ 2 (1 v− ) ) ( f u uv− ,uv )u du dv (b) (cos1 cos4
4

( / 1 v−
(c)  1/2  2/3 1/ 2 (1 v− ) ) ( f u uv− ,uv )v du dv (c) (cos1 cos2− )
2


39

Calculus

 1   u v 2u v  
−
+
)
−
(d) (cos1 cos4 [JAM CA 2007] (d) 0  4 − u u  /2  f ,    dv du
2 3  3 3 
234. Consider the double integral 236. The area of the region bounded by the
0  1 x  2 x f ( ,x y dy dx . After curves x = 2y and y = 2x is
)
+
2
2
reversing the order of the integration, the
integral becomes (a) 1 (b) 2
3 3
(a)
0   0 y− 2 f ( , x y dx dy + 1   0 1 f ( , x y dx dy (c) 4 (d) 4
)
)
2
1
)
+ 2  3 y  1 f ( , x y dx dy 3
(b) [JAM CA 2008]
)
0   0 y f ( , x y dx dy + 1   0 1 f ( , x y dx dy237. The value of the integral
)
2
1
)
+ 2  3 y−  1 2 f ( , x y dx dy 0  3 0  3x dy dx is
x + y 2
2
(c)
0   0 y f ( , x y dx dy + 1   0 y f ( , x y dx dy (a) 3log 2 + 3
)
(
)
)
2
1
)
+ 2  3 y−  1 2 f ( , x y dx dy
(
)
(b) 3log 2 − 3
(d)
0   0 y− 2 f ( , x y dx dy + 1   0 y f ( , x y dx dy
)
)
2
1
)
+ 2  3 y  1 f ( , x y dx dy (c) 3log 2
)
(
3
(d) log 2 + 3 [JAM CA 2008]
[JAM CA 2008] 2
235. The double integral 238. Changing the order of integration of
0  2 x  4 x f ( , x y dydx under the −  1 1  1 x 2 f ( , x y dy dx gives
)
−
)
−
−
transformation u = x y , v = y − 2x − 1 x 2
+
)
+
1 y
is transformed into (a) 0  1  1 y f ( ,x y dy dx +
−
 u v 2u v 
−
+
(a) 0  4 u  u /2  f  ,     dv du −  0 1  1 y 2 f )
−
 3 3  − 1 y 2 ( , x y dy dx
−
+
−
)
+
(b) 3 0  4 u  u /2  f   u v , 2u v      dv du (b) 0  1  1 y f ( ,x y dy dx
−
 3 3  1 y
)
+
−
−
1 4 u   u v 2u v   − −  0 1  1 y 2 f ( ,x y dy dx
(c) 0  u  /2  f ,    dv du − 1 y 2
−
3  3 3 

40

Calculus

0  1 1 y ) 242. The area of the region bounded by the
+
(c) − − f ( , x y dy dx
1 y
curves r = 1 and r = cos2 ,
3
)

−
+ −  0 1  − 1 y 2 2 f ( , x y dy dx 0    , is
−
2
1 y
)
+
(d) 0  1 − 1 y f ( , x y dy dx (a)  (b) 
−
1 y
2 3
)
−
− −  0 1  − 1 y 2 2 f ( , x y dy dx  
−
1 y
(c)
4 (d) 8
 
239. The value of x +  y dx dy , where
D [JAM CA 2010]
[x + y] is the greatest integer less than or
equal to x + y is the region bounded by x 243. The area included between the curves
2
= 0, y = 0 and x + y = 2, is x + 2 y = 2 a and
b x + a y = a b (a  0,b  ) 0 , is
2
2 2
2 2
2
3 1
(a) (b)
2 2  a
−
(a) a b
1 2
(c) (d) 0
4
(b)  a − 2 3ab b
+
2
2
x
240. The area bounded by the curves y =
(c) a a b −
and x = y is
2
2
2
(a) 1/3 (b) 2/3 (d)  a − b [JAM CA 2011]
(c) 4/3 (d) 5/3 244. The area bounded by the curves
−
[JAM CA 2009] x = 2 4 2y and x = 2 y + 4 is
241. The value of the integral (a) 16 (b) 24
0   /2 x   /2 sin y dy dx is (c) 30 (d) 36

y
245. The value of the integral
1    x x e − x 2 / y dy dx is
(a) 0 (b) 0 y= 0
2
(a) 0 (b) 1/2
(c) 1 (d) 2
(c) 4 (d) 1


[JAM GP 2005]





41

Calculus

1
246. If (c) 1+ + .....+ 1
h
1 y+
5
4
  4 dx dy =   1 dy dx +   ( ) g dy dx 2 n + 1
=
y= 0 x= 0 x= 0 y= 0 x= 4 y g ( ) x 1 1
, then the functions g(x) and h(x) are, (d) 1+ + .....+ n + 2 [CSIR]
2
respectively

(a) (x – 4) and 1 (b) (x + 4) and 1
 /4
(c) 1 and (x – 4) (d) 1 and (x + 4) 250.  log + )
(1 tan x dx
0
[JAM GP 2009]

(a) 0 (b) log 2
247. The surface area obtained by revolving 4

y = 2x , for x  0,2 , about y-axis is  
(c) log 2 (d) log 2
8 2
(a) 2 5 (b) 4 5
 /2
+
2
(c) 2 5 (d) 4 5 251.  (a 2 cos x b 2 sin 2 ) x dx =
0
[JAM CA 2009] 
2
(a) 0 (b) (a + b 2 )

=
248. If the line y mx, 0 x  2 is rotated 2


about the line y = − 1, then the area of (c) (a + b 2 ) (d) (a + b 2 )
2
2
the generated surface is 4 8
[JNU]
+
+
(a) 4 (1 m ) 1 m
252. Let ‘ f ’ be a real valued function of a
real variable defined as
+
+
4
(b) (1 m 2 ) 1 m f ( ) x = x −  , where   x denote the
x
+
2
(c) 4 ( 1+ m ) 1 m largest integer less than or equal to x
1.25
. The value of  f ( ) x dx _________
0.25
+
+
2
(d) 4 (1 m ) 1 m is (up to 2 decimal places).
[GATE-2018 (EE)]
[JAM CA 2007] 253.  2 1 x dx = ________
−

249. For n N , the value of 0
(a) 0 (b) -1
n 1− ( / x n ) n
 dx = (c) 1 (d) 2
−
0 n x e 1
+
254. The value of  e (1 ln x ) dx is
(a) 0 1 x 2
1 1 1 (a) 1 (b) 1/e
(b) 1+ + + .........+
2 3 n
(c) e (d) 0


42

Calculus

)
x
255. The integral  0  / 2 min (sin , cos x dx (a) 2 (b) 2 10
10
equals
(c) 10  (d) 2
(a) 2 − (b) 2 − 2
2
[ESE PRELIMS-2017]
(c) 2 2 (d) 2 + 2

256. The value of the integral 261. The value of the integral

tan x
2
 0  / 2 tan x + cot x dx is  (x − 2 ) 1 sin x − 2 ( ) 1 dx is



(a) /6 (b) /2 0 (x − ) 1 + cos (x − ) 1

(c) 0 (d) /4 (a) 3 (b) 0
(c) -1 (d) -2

5 / 2
257. The value of  f ( ) x dx, where [ME, GATE-2014 : 1 MARK]
− 5 / 2

 / 2
)
3
f ( ) x = e  x 2 . sin x + 4cos , x 262.  log (sin x dx =
0
equals_____ − 
(a) 0 (b) log2
(a) 4 (b) 8 2
5 − 5 (c) 
−
(c) (d) log2 (d) log2
2 2 2
=
100 263. If  x sin x dx k , then the values
258. The value of  sin x dx . 0
0 of k is equal to_________.

(a) 100 (b) 100 [GATE-2014 (CS-SET 3)]
(c) 200 (d) 200  / 2
5
3
264.  sin x cos x dx =
 1 0
259. 2  dx =
2
+
2
a cos x b 2 sin x 1 1
0
(a) (b)
 16 24
(a) 0 (b)
ab (c) 1 (d) 1
 48 96
(c) ab (d)
2
a + b 2 
6
4
265.  x sin x cos x dx =
260. The value of the integral 0
 0 2    9 sin   + 3 2   d is



43

Calculus

3 2 5 2 4 4 16
(a) (b) (c) and 0 (d) and
512 256   

3 2 [GATE-17-CSIT]
(c)  ab (d) 0
256 270. The value of the integral

−
x
sin x
266. The value of  0  1 cos x dx =  −   / 4 1 cos2x dx is
+
2
2
  1 1
(a) (b) (a) 1− (b) 1−
−
4 2 2 2
 2  2
(c) (d) (c) 3− 1 (d) 2 − 1
4 8 2 2

267. Consider the following definite 271. The value of the integral
−
1
1 ( sin x ) 2
9
integral I =  dx .  dy is
−
0 1 x 2 0 y 1+ y
The value of the integral is
)
−
 3  3 (a) 4 (b) 4 ( 10 1
(a) (b)
24 12 (c) 8 (d) 12
 3  3
A
(c) (d) 272. Let ( ) t denote the area bounded by
48 64
x
the curve y = e , the x - axis and the
[GATE-17-CE] straight lines x = t − and x t , then
=
lim ( ) t is equal to
( )
2
t→
268. The value of  0  / 4 x cos x dx correct (a) 2 A (b) 1
to three decimal places (assuming (c) 1/2 (d) 0
that  = 3.14 ) is ____. 

1/ 2
273. e − y 3 .y dy =
[GATE-18-CSIT]
0
  x   
1
269. If ( ) x = R sin   + S , ' f   = 2 
f
 
  
 

2
 2    (a)  (b)
3
and  0 1 f ( ) x dx = 2R , then the constants 


(c)
2 (d) 0
R and S are respectively.
2 16 2 274. The value of  1 dt is ________
(a) and (b) and 0 0 − lnt
  

44

Calculus

 1 1
(a) (b)  (a) (b)
2 5015 15015

−  1 1
(c) -  (d) (c) (d)
2 5005 5001
 / 2



275. The value of e − 5x 4 x dx is ______ 280. The value of  cot  d is ______
0 0
 
3  2  (a) (b)
(a) (b) 2 3
2 3
 
  (c) (d)
(c) (d) 2 4
3 2
 x
a
276. The value of   x dxis _________ 281. The value of  1 x 6 dx is ________
+
0 a x 0
a a + 1  
(a) (b) a+ (a) (b)
(ln a ) a (ln a ) 1 3 3 3


a a + 1  
(c) a+ (d) (c) (d)
(ln a ) 1 (ln a ) a 2 2 2
 4 + 5 )
1 x ( 1 x
−
277.  x 6 1 x dx = 2 282. The value of  (1 x ) 15 dx is _____
+
0 0
    1 1
(a) (b) (a) (b)
256 128 5005 5001

  3 1 1
(c) (d) (c) (d)
512 512 1001 10001
1 x 6 283. The length of the arc y = x 3/ 2 , z = 0
278. The value of  − 2 dx =
0 1 x from (0, 0, 0) to (4, 8, 0) is ________
 3 8 3/ 2
(a) (b) (a) ( 10 + ) 1
32 32 27

5 8 3/ 2
(c) (d) 0 (b) ( 10 − ) 2
32 27

1 8 3/ 2 −

279. The value of x 4 (1 5x dx− ) 5 is _____ (c) ( 10 ) 1
27
0


45

Calculus

8 288. A parametric curve defined by
(d) ( 10 3/ 2 + ) 2
27   u    u 
x cos   , y = sin   in the range
284. Let f be increasing, differentiate  2   2 


function. If the curve y = f ( ) x passes 0 u  1 is rotated about the X-axis by
0
through (1, 1) and has length 360 . Area of the surface generated is

2 1 (a) (b) 
1 x 
L =  1+ 2 dx  2, then curve 2
1 4x (c) 2 (d) 4
is __________ [GATE-17-ME]

)
x −
(a) y = ln ( ) 1 289. Let W = f ( ,x y , where x and y are
functions of t. Then, according to the
−
(b) y = 1 ln x dw
)
(c) y = ln 1+ ( x chain rule, dt is equal to


+
x
(d) y = 1 ln ( ) (a) dw dx + dw dt
dx dt dy dt
285. The length of the arc  w x   w x 
)
x = ( a t − sint ), y = a (1 cost between (b)  x t  +  y t 
−
t = 0 to t = 2 is _____ (c)  w dx +  w dx

(a) 8a (b) 4a  x dt  y dt
dw x  dw x 
(c) 4 2a (d) 2 2a (d) +
dx t  dy t 
)
+
286. The length of the arc r = a (1 cos
290. The surface area obtained by
between = 0 to  is _____ revolving y = 2x for x 0,2 about y –

287. Consider a spatial curve in three-
axis is ____
dimensional space given in parametric
(a) 2 5 (b) 4 5
from by x ( ) t = cos , y ( ) sin ,t = t
t
(c) 2 5 (d) 4 5
2 

z ( ) t = t , 0 t 
 2 291. The surface area generated by
The length of the curve is ___________ rotations

3
3

[ME, GATE-2015 : 2 MARKS] x = a cos  , y = a sin  ,0   
about y- axis



46

Calculus

12 5 295. The volume generated by revolving
2
2
(a)  a (b)  a
5 12 the area bounded by y = 8x and the
2
6 5
−
2
2
2
(c)  a (d)  a line x = , about y axisis
5 6
128 5
292. The surface area of the solid (a) (b)
5 128
generated by revolving line segment
127 32
y = x + 2 for 0  x  1about the line (c) 5 (d) 5
y = 2 is __________

(a) 2 (b) 2

2
(c) 2 2 (d) 2 296. The area bounded by x = 2y and
=
2

293. If the line y mx, 0 x  2 is y = 2x is _______
rotated about the line y = − 1 then the 1 2
(a) (b)
area of the generated surface is 3 3
4
_________ (c) (d) 4
3
+
+
(a) 4 (1 m ) 1 m
297. The area between the parabolas
+
+
4
(b) ( 1 m 2 ) 1 m y = 4ax and x = 4ay is
2
2
2
+
2
(c) 4 ( 1 + m ) 1 m (a) a (b) 14 a
2
2
3 3
2
+
+
(d) 4 (1 m ) 1 m 16 17
2
2
(c) a (d) a
3 3
294. The volume generated by revolving
[EE, ESE-2019]
the area bounded by the parabola
298. The area of the region bounded by the
y = 2 8 , x y axis and the lines y = − 4 to
−
parabola y = x + 2 1and the straight
y = 4 about y-axis is
line x y+ = 3 is
32
(a) 32 (b) 59 9
5 (a) (b)
128 6 2
(c) (d) None of the above 10 7
5 (c) (d)
3 6
[EE-1994]


47

Calculus

2
299. Let I = c  xy dxdy , where R is the 2 x 1
R 302.   2 3/ 2 dy dxtransforms to
1 0 ( x + ) 2
region shown in the figure and
a c
−
4
c = 6 10 . The value of I equals ____.  1 2  dr d in polar coordinates then a,

(Give the answer up to two decimal 0 b r
b, c respective are ____________
places)


(a) ,sec  and 2sec
4


(b) , 2sec , 4sec
4


(c) , 2sec , 4sec
2

(d)  , sec and 2sec

300. The area bounded by the curves 2

−
x = 2 4 2y and x = 2 y + 4 is 303. The value of  xydx dy over the
D
(a) 16 (b) 24
region common to the circles
(c) 30 (d) 36
+
x + 2 y = 2 x and x y = y is _________
301. Let D be the region in the first
1 1
quadrant lying between x + 2 y = 2 4 . The (a) (b)
192 96
value of the integral 1 1
(c) (d)
 sin ( x + y 2 ) dx dy is 48 24
2
D
304. What is the area common to the circle

(a) (cos1 cos2− ) = a and r = 2a cos ?
4
2
2
(a) 0.524 a (b) 0.614 a

)
−
(b) (cos1 cos4 (c) 1.047 a (d) 1.228 a 2
2
4
[GATE-2006]

(c) (cos1 cos2− ) 305. A surface ( , x y = 2x +
)
3
2 S 5y − is
 integrated once over a path consisting of
(d) (cos1 cos4− )
2 the points that satisfy




48