MTH301 Calculus II Solved Final Term Papers For Final Term Exam Preparation

Size: px

Start display at page:

Download "MTH301 Calculus II Solved Final Term Papers For Final Term Exam Preparation"

Jasmine Knight
5 years ago
Views:

1 MTH301 Calculus II Solved Final Term Papers For Final Term Exam Preparation Question No: 1 Laplace transform of t is 1 s 1 s 2 e s s Question No: 2

2 Symmetric equation for the line through (1,3,5) and (2,-2,3) is x 2 y 2 z x 2 y 3 z x 1 y 3 z x 1 y3 z Question No: 3

3 The level curves of f(x, y) = y Cscx are parabolas. True. False. Question No: 4 The equation z r is written in Rectangular coordinates Cylindrical coordinates Spherical coordinates None of the above

4 Question No: 1 Intersection of two straight lines is Surface Curve Plane Point Question No: 2 Plane is a surface. One-dimensional Two-dimensional Three-dimensional Dimensionless

5 Question No: 3 Let w = f(x, y, z) and x = g(r, s), y = h(r, s), z = t(r, s) then by chain rule w r w x w y w z x r y r z r w x w y w z r r r r r r w x x w y y w z z x r s y r s z r s w r w r w r r x r y r z Question No: 4 What are the parametric equations that correspond to the following vector equation? ^ ^ r (t) sin 2 t i(1cos 2t) j x sin 2 t, y 1cos 2t, z 0

6 y sin 2 t, x 1cos 2t, z 0

7 x sin 2 t, y 1cos 2t, z 1 x sin 2 t, y cos 2t, z 1 Question No: 5 What are the parametric equations that correspond to the following vector equation? ^ ^ ^ r(t) 2t 1i3 t jsin 3t k z 2t 1, x 3 t, y sin 3t y 2t 1, x 3 t, z sin 3t x 2t 1, z 3 t, y sin 3t x 2t 1, y 3 t, z sin 3t Question No: 6 What is the derivative of following vector-valued function? r (t) (cos 5t, tan t, 6 sin t) r (t)

8 sin 5t t, sec t, 6 cos 5

9 r (t) ( sin 5t, sec t, 6 cos t) 5 r (t) ( 5sin 5t, sec 2 t, 6 cos t) r (t) (sin 5t, sec 2 t, 6 cos t) Question No: 7 What is the derivative of following vector-valued function? t 1, 3 r (t) t 4, t r (t) 4t 3,, t 1 t 3 1 r (t) 4t 3,, 6 2 t 1 t3 1 6 r (t) 4t 4,, 2 t 1 t 3 r (t) 4t 3 1 6,, 2 t 1 t 3 Question No: 8

10 The following differential is exact dz (x 2 y y) dx x dy

11 True False Question No: 9 Which one of the following is correct Wallis Sine formula when n is even and n 2? 2 n n1 n3 n5 page # x dx 2 n n 2 n sin 0 2 n1 n3 n sinn x dx 0 n n 2 n n n2 n sinn x dx 2 0 n 1 n 3 n n n2 n4 n sin x dx n 1 n 3 n Question No: 10 Match the following equation in polar co-ordinates with its graph.

12 r cosa where a is an arbitrary cons tant Question No: 11 If the equation of a curve, in polar co-ordinates, remains unchanged after replacing (r, ) by (r, ) then the curve is

13 said to be symmetric about which of the following?

14 Initial line Y-axis Pole Question No: 12 If the equation of a curve, in polar co-ordinates, remains unchanged after replacing (r, ) by ( r, ) then the curve is said to be symmetric about which of the following? Initial line y-axis Pole Question No: 13 What is the amplitude of a periodic function defined by f (x) sin x? 3 0 1

15 1 3 Does not exist Question No: 14 What is the period of a periodic function defined by f (x) 4 cos 3x? Question No: 15 Match the following periodic function with its graph. 3 0 x 4 f (x) 0 4 x 6

17 Page #198 Question No: 16 What is the period of periodic function whose graph is as below? 2 5 6

18 8

Question No: 17 What is the period of periodic function whose graph is as below? 0 4 6 8 Question No: 18 Let L denotes the Laplace Transform.

19 Question No: 17 What is the period of periodic function whose graph is as below? Question No: 18 Let L denotes the Laplace Transform. lim F(t) If L{F (t)} f (s) where s is a constant.and exists then t 0 t which of the following equation holds? L F (t) f (s a) t L F (t) f (s a) t

21 L F (t) f (s) ds t s L F (t) d { f (s)} t ds Question No: 19 Which of the following is Laplace inverse transform of the function f (s) defined by f (s) 3 2 s 2 s? 3te 2t 2 3e 2t 2t 3e 2t 2 None of these. Question No: 20 Let (x 1, y 1, z 1 ) and (x 2, y 2, z 2 ) be any two points in three dimensional space. What does the formula (x x ) 2 ( y y ) 2 (z z ) 2 calculates? Distance between these two points

22 Midpoint of the line joining these two points Ratio between these two points Question No: 21 Let the functions P(x, y) and Q(x, y) are finite and continuous inside and at the boundary of a closed curve C in the xy- plane. If P dx Q dy is an exact differential then Ñ P dx Q dy C Zero One Infinite Question No: 22 What is Laplace transform of the function F (t) if F (t) t? Lt 1 s Lt 1 s 2

23 L t e s L t s Question No: 23 What is the value of L{e 5t } if L denotes laplace transform? L{e 5t 1 } s 5 L{e 5t } s s 2 25 L{e 5t 5 } s 2 25 L{e 5t } 5! s 6 Question No: 24 Evaluate the line integral (3x 2 y) dx (2x y) dy C where C is the line segment from (0, 0) to (0, 2). 1

24 0

25 2-2 Question No: 25 Evaluate the line integral (2x y) dx (x2 y) dy C line segment from (0, 0) to (2, 0). where C is the Do not exist Question No: 26 Which of the following are direction ratios for the line joining the points (1, 3, 5) and (2, 1, 4)?

26 3, 2 and 9

27 1, -4 and -1 2, -3 and , -3 and 5/4 Question No: 27 If R {(x, y) / 0 x 2 and 1 y 4},then (6x2 4xy 3 )da R 4 2 (6x 2 4xy 3 )dydx (6x 2 4xy 3 )dxdy (6x 2 4xy 3 )dxdy (6x 2 4xy 3 )dxdy 2 0

28 Question No: 28 Which of the following is true for a periodic function whose graph is as below? Even function Odd function Neither even nor odd function Question No: 29 Which of the following is true for a function whose graph is given above. An odd function

29 An even function Neither even nor odd Question No: 30 At each point of domain, the function Is defined Is continuous Is infinite Has a limit Question No: 31 ( Marks: 2 ) Determine whether the following differential is exact or not. dz 4x 3 y 3 dx 3x 4 y 2 dy

30 Solution:

31 dz 4x 3 y 3 dx 3x 4 y 2 dy p 12x 3 y 2 y Q 12x 3 y 2 X p Q y X yes Question No: 32 ( Marks: 2 ) Evaluate sin nx dx where n is an integer other than zero. Solution: sin nx dx cosnx n cosn cosn n n 1 (cos ncos n) n 0 Question No: 33 ( Marks: 2 ) Find Laplce transform of the function F (t) if F (t) e 3t Solution:

32 L(e 3t ) e3t e st 0 e ( s 3)t.dt 0 ( s 3)t e { }lim 0 (s 3) 1 s 3 ( 1 ) e ( s 3)t 1 (0 1) s Ans s 3 Question No: 34 ( Marks: 3 ) Determine the Fourier co-efficient defined below: f (x) 2x 1 0 x 2 Solution: a 1 o f (x)dx f (x) (2x 1) (0, 2) 2 0 (2x 1)dx x 2 x a 0 of the periodic function Question No: 35 ( Marks: 3 )

33 Determine whether the following differential is exact or not. dz (3x 2 e 2 y 2 y 2 e 3x ) dx (2x 3 e 2 y 2 ye 3x ) dy Solution: dz = Pdx + Qdy Therefore, For dz to be an exact differential it must satisfy P y = Q x But this test fails becuase P y Q x Not Exact Question No: 36 ( Marks: 3 ) 2 Use Wallis sine formula to evaluate sin 3 x sin 5 xdx 0 Solution:

34 0 2 sin 3 xdx n1. n sin 5 xdx 0 n1. n3 n n sin 3 x sin 5 xdx Question No: 37 ( Marks: 5 ) Evaluate the following line integral which is independent of path. (3,2) (0,0) (2xe y ) dx (x 2 e y ) dy Solution:

35 p z 2e y x Q z x 2 e y y (3,2) 2 y z 2xe y x ye (0,0) z 6e 2 18e 2 z 24e 2 2ey dx x2 e y dy Question No: 38 ( Marks: 5 ) Determine the Fourier coefficients of period 2 defined by 4 (1 + t) -1 < t < 0 f (t) 0 0 < t < 1 Solution: b n for a periodic function f (t) bn 1 f (x) sin nxdx 1 1 4(1t) sin nxdx (1t)cosnx n 1 4(1t) cos n(1) cos n( 1) n 4(1t) (cos n cos n) n Question No: 39 ( Marks: 5 ) Determine whether the following vector field F or not. ^ ^ ^ F (x, y, z) (4x z) i(3y z) j( y x) k.. is conservative

36 ASSALAM O ALAIKUM All Dearz fellows ALL IN ONE MTH301 Final term PAPERS & MCQz Created BY Farhan & Ali BS (cs) 2nd sem Hackers Group From Mandi Bahauddin Remember us in your prayers Mindhacker124@gmail.com Hearthacker124@gmail.com FINALTERM EXAMINATION Spring 2010 MTH301- Calculus II (Session - 2) Time: 90 min Marks: 60 Student Info StudentID: $$ Center: OPKST

37 ExamDate: 19 Aug 2010

38 For Teacher's Use Only Q No. Marks Total Q No Marks Q No Marks Q No Marks Q No Marks

39 Question No: planes intersect at right angle to form three dimensional space. Three Four Eight Twelve Question No: 2 If the positive direction of x, y axes are known then the positive direction of z-axis. Horizontal rightward direction is Vertical upward direction is Left hand rule tells Right hand rule tells Question No: 3 What is the distance between points (3, 2, 4) and (6, 10, -1)? Question No: 4

40 The equation ax by cz d 0, where a,b,c,d are real numbers, is the general equation of which of the following? Plane Line Curve Circle Question No: 5 The spherical co-ordinates of a point are 3,, 2. What are its 3 cylinderical co-ordinates? 3 3,, cos, 3 sin, sin,, 3 cos ,, 0 3 Question No: 6 Domain of the function f (x, y) y x 2 is y x 2 y x 2 y x 2 Entire space

41 Question No: 7

42 Match the following vector-valued function with its graph. ^ ^ ^ r(t) cos t isin t j2 k and 0 t 2 Question No: 8 What are the parametric equations that correspond to the following vector equation? ^ ^ r (t) sin 2 t i(1cos 2t) j x sin 2 t, y sin 2 t,

43 y 1cos 2t x 1cos 2t, z 0, z 0

44 x sin 2 t, y 1cos 2t, z 1 x sin 2 t, y cos 2t, z 1 Question No: 9 What are the parametric equations that correspond to the following vector equation? ^ ^ ^ r(t) 2t 1i3 t jsin 3t k z 2t 1, x 3 t, y sin 3t y 2t 1, x 3 t, z sin 3t x 2t 1, z 3 t, y sin 3t x 2t 1, y 3 t, z sin 3t Question No: 10 Is the following vector-valued function r (t) continuous at t 1? If not, why? t 1 r (t), t 2, 2t t 1 r (t) is continuous at r (1) is not defined r (1) is defined but t 1 lim r (t) t 1 does not exist r (1) is defined and lim r (t) exists but these two t 1 numbers are not equal. Question No: 11 Which one of the following is correct Wallis Sine formula when n

45 is even and n 2?

46 2 n1 n3 n sinn x dx n 2 n n 2 n1 n3 n sinn x dx n 0 n 2 n sinn x dx 2 n n 2 n n 1 n 3 n n n2 n4 n sin x dx n 1 n 3 n Question No: 12 Which one of the following is correct Wallis Cosine formula when n is odd and n 3? 2 n1 n3 n cosn x dx n 2 n n 2 n n 2 n cosn x dx 2 0 n 1 n 3 n n n n2 n4 cos x dx n 1 n 3 n n1 n3 n cosn x dx 0 n n 2 n Question No: 13 If the equation of a curve, in polar co-ordinates, remains unchanged after replacing (r, ) by (r, ) then the curve is said to be symmetric about which of the following? Initial line y-axis Pole

47 Question No: 14

48 If a 0, then the equation, in polar co-ordinates, of the form r 2 a 2 cos 2 represent which of the following family of curves? Leminscate Cardiods Rose curves Spiral Question No: 15 x f x What is the period of a periodic function defined by ( ) sin? Question No: 16 What is the period of periodic function whose graph is as below? Question No: 17 What is the period of periodic function whose graph is as below?

0 2.25 3 6 Question No: 18 Let L denotes the Laplace Transform. If L{F (t)} f (s) where s is a constant, then which of the following equation holds?

49 Question No: 18 Let L denotes the Laplace Transform. If L{F (t)} f (s) where s is a constant, then which of the following equation holds? d L{t F (t)} { f (s)} L{t F (t)} L{t F (t)} ds f (s t) f (s) L{t F (t)} f (s) ds s Question No: 19 The graph of an odd function is symmetrical about x-axis y-axis origin Question No: 20 Consider the function f 0, 1, 1 f (x, y, z) 1 x 2 y 2 z 2. What is the value of 2 2 f 0, 1, f 0, 1, 1 2

50 2 2

51 f 0, 1, f 0, 1, Question No: 21 The path of integration of a line integral must be straight and single-valued continuous and single-valued straight and multiple-valued continuous and multiple-valued Question No: 22 Sign of line integral is reversed when path of integration is divided into parts. path of integration is parallel to y-axis. direction of path of integration is reversed. path of integration is parallel to x-axis. Question No: 23 Let the functions P(x, y) and Q(x, y) are finite and continuous inside and at the boundary of a closed curve C in the xy-plane. If P dx Q dy is an exact differential then Ñ P dx Q dy C Zero One Infinite

52 Question No: 24

53 What is the value of L{e 5t } if L denotes laplace transform? L{e 5t 1 } L{e 5t } s 5 s s 2 25 L{e 5t 5 } s 2 25 L{e 5t } 5! s 6 Question No: 25 What is laplace transform of the function F (t) if F (t) sin 3t? 3 L{sin 3t} L{sin 3t} s 2 9 L{sin 3t} L{sin 3t} s 2 9 s 1 s 3 3! s 4 Question No: 26 If L denotes laplace transform then L{te 5t } L{te 5t 1 } s 2 5 L{te 5t 1 } s 2 5 L{te 5t } 1 2 s 5 1 L{te 5t } s 5 2 Question No: 27

54 Evaluate the line integral C line segment from (0, 0) to (0, 2) (3x 2 y) dx (2x y) dy where C is the Question No: 28 Evaluate the line integral C line segment from (0, 0) to (0, 2) (2x y) dx (x 2 y) dy where C is the Question No: 29 Divergence of a vector function is always a Scalar Vector Question No: 30

55 Which of the following is true for a function whose graph is given above An odd function An even function Neither even nor odd Question No: 31 ( Marks: 2 ) Does the following limit exist? If yes find its value, if no give reason lim (e t 0 ^ 5) i (t 2t 2 ^ 1 k ^ 2t 3) t j Question No: 32 ( Marks: 2 ) Define the periodic function whose graph is shown below. Question No: 33 ( Marks: 2 ) Find Laplace Transform of the function F (t) if F (t) t 4 Solution:

56 The Laplace transform of the given function will be:

57 f (t) t 4 L{t 4 } 4! s 5 Question No: 34 ( Marks: 3 ) Determine whether the following differential is exact or not. dz (4x 3 y 2xy 3 ) dx (x 4 3x 2 y 2 ) dy Question No: 35 ( Marks: 3 ) 2 Use Wallis sine formula to evaluate sin 3 x sin 5 xdx 0 Solution: 2 sin8 xdx Question No: 36 ( Marks: 3 ) Find Laplace transform of the function F (t) if F (t) e 2t sin 3t Solution: Laplace transform will be

58 L(t) e 2t s 2 L(t) sin 3t... 2 a L(t) s a L(t) s 2 9 Combining, 1 L(t) ( )( a s 2 s 2 9 ) Question No: 37 ( Marks: 5 ) Using definite integral, find area of the region that is enclosed between the curves y x 2 and y x Question No: 38 ( Marks: 5 ) Determine the fourier co-efficient b n of the following function. f (x) x 2 0 x 2

59 Question No: 39 ( Marks: 5 )

60 Determine whether the following vector field F or not. ^ ^ ^ F (x, y, z) (4x z) i(3y z) j( y x) k is conservative ASSALAM O ALAIKUM All Dearz fellows ALL IN ONE MTH301 Final term PAPERS & MCQz Created BY Farhan & Ali BS (cs) 2nd sem Hackers Group From Mandi Bahauddin Remember us in your prayers Mindhacker124@gmail.com Hearthacker124@gmail.com FINALTERM EXAMINATION

61 Fall 2009

62 MTH301- Calculus II Time: 120 min Marks: 80 Question No: 1 is an example of Irrational numbers Rational numbers Integers Natural numbers Question No: 2 Straight line is a special kind of Surface Curve

63 Plane

64 Parabola Question No: 3 An ordered triple corresponds to in three dimensional space. A unique point A point in each octant Three points Infinite number of points Question No: 4 The angles which a line makes with positive x,y and z-axis are known as Direction cosines Direction ratios

65 Direction angles

66 Question No: 5 Is the function f (x, y) continuous at origin? If not, why? f (x, y) 4xy sin 3x 2 y f (x, y) is continuous at origin f (0, 0) is not defined f (0, 0) is defined but ( x, lim, 0) y) (0 f (x, y) does not exist f (0, 0) is defined and ( x, lim, 0) numbers are not equal. y) (0 f (x, y) exists but these two Question No: 6 Match the following vector-valued function with its graph. ^ ^ r(t) 3cos t i3sin t j and 0 t 2

68 Question No: 7 Match the following vector-valued function with its graph. ^ ^ ^ r(t) t it 2 jt 3 k and t 0

69 Question No: 8 What are the parametric equations that correspond to the following vector equation? ^ ^

70 r (t) sin 2 t i(1cos 2t) j

71 x sin 2 t, y 1cos 2t, z 0 y sin 2 t, x 1cos 2t, z 0 x sin 2 t, y 1cos 2t, z 1 x sin 2 t, y cos 2t, z 1 Question No: 9 Is the following vector-valued function r (t) continuous at t 0? If not, why? r (t) (4 cos t, t, 4 sin t) r (0) is not defined r (0) is defined but lim r (t) t 0 does not exist r (0) is defined and are not equal. r (t) is continuous at t 0 lim r (t) exists but these two numbers t 0

72 Question No: 10

73 What is the derivative of following vector-valued function? r (t) (cos 5t, tan t, 6 sin t) sin 5t, sec t, 6 cos t r (t) 5 r (t) ( sin 5t, sec t, 6 cos t) 5 r (t) ( 5sin 5t, sec 2 t, 6 cos t) r (t) (sin 5t, sec 2 t, 6 cos t) Question No: 11 The following differential is exact dz (3x 2 y 2) dx (x 3 y) dy True False Question No: 12

74 The following differential is exact dz (3x 2 4xy) dx (2x 2 2 y) dy

75 True False Question No: 13 Which one of the following is correct Wallis Sine formula when n is odd and n 3? 2 sinn x dx 0 n1 n3 n n n 2 n n n2 n sinn x dx 2 0 n 1 n 3 n n1 n3 n sinn x dx 0 n n 2 n n n2 n4 n sin x dx n 1 n 3 n Question No: 14 Which of the following is correct?

76 2 sin 4 x dx sin 4 x dx sin 4 x dx sin 4 x dx Question No: 15 Match the following equation in polar co-ordinates with its graph. r 4sin

77 Question No: 16 If the equation of a curve, in polar co-ordinates, remains unchanged after replacing (r, ) by (r, ) then the curve is said to be symmetric about which of the following? Initial line y-axis Pole Question No: 17 x f x What is the period of a periodic function defined by ( ) sin? 2

78 2

79 3 2 4 Question No: 18 Match the following periodic function with its graph. 3 x 4 f (x) 7 x 3 0 x 4 4 x x 13

81 Question No: 19 What is the period of periodic function whose graph is as below? Question No: 20 What is the period of periodic function whose graph is as below? 0 2

83 2 Question No: 21 3 Polar co-ordinates of a point are 2, 2. Which of the following is another possible polar co-ordinates representation of this point? 2, 4 2, 2 2, 3 3 2, 4 Question No: 22 The function f ( x) x 3 e x is Even function Odd function

84 Neither even nor odd Question No: 23 The graph of an even function is symmetrical about x-axis y-axis origin Question No: 24 At which point the vertex of parabola, represented by the y x 2 4x 3 equation, occurs? (0, 3) (2, 1) ( 2, 15) (1, 0)

85 Question No: 25

86 The equation y x2 4x 2 represents a parabola. Find a point at which the vertex of given parabola occurs? (2, 2) ( 4, 34) (0, 0) ( 2, 14) Question No: 26 Is the function f (x, y) continuous at origin? If not, why? xy f (x, y) x 2 y 2 f (x, y) is continuous at origin ( x, lim, 0) f (x, y) does not exist y ) (0 f (0, 0) is defined and ( x, lim, 0) numbers are not equal. y) (0 f (x, y) exists but these two Question No: 27 Sign of line integral is reversed when

87 path of integration is divided into parts.

88 path of integration is parallel to y-axis. direction of path of integration is reversed. path of integration is parallel to x-axis. Question No: 28 What is Laplace transform of a function F(t)? (s is a constant) s est 0 F (t ) dt est 0 F (t) dt e st F (t) dt est 0 F (t ) dt Question No: 29

89 What is the value of L{e 5t } if L denotes laplace transform?

90 L{e 5t 1 } s 5 s L{e 5t } s L{e 5t } s 2 25 L{e 5t } 5! s 6 Question No: 30 What is the Laplace Inverse Transform of 1 s 1 L 1 1 t 1 s 1 L 1 1 e t e t s 1 L 1 1 e t s 1 L 1 1 e t s 1

91 Question No: What is Laplace Inverse Transform of s L 2 sin 5t s L 2 cos 5t s L 2 sin 25t s L 2 cos 25t s 25 Question No: 32 What is L{ 6} if L denotes Laplace Transform? 1 L{ 6} s 6 L{ 6} 6 s s L{ 6} s 2 36 L{ 6}

92 6 s 2 36

93 Question No: 33 Evaluate the line integral C line segment from (0, 0) to (2, 0). (3x 2 y) dx (2x y) dy where C is the Do not exist Question No: 34 Evaluate the line integral C line segment from (0, 0) to (0, 2). (2x y) dx (x 2 y) dy where C is the

94 2

95 Question No: 35 Plane is an example of Curve Surface Sphere Cone Question No: 36 If R {(x, y) / 0 x 2 and 1 y 1},then (x 2 y2 )da R 1 2 (x 2 y 2 )dydx (x 2 y )dxdy 2

96 1 2 (x 2 y 2 )dxdy (x 2 y2 )dxdy 1 1 Question No: 37 To evaluate the line integral, the integrand is expressed in terms of x, y, z with dr dx î dy ĵ dr dx î dy ĵ dy kˆ dr dx dy dz dr dx dy Question No: 38 Match the following equation in polar co-ordinates with its graph. r a where a is an arbitrary constant.

97 Question No: 39 Which of the following is true for a periodic function whose graph is as below?

98 Even function

99 Odd function Neither even nor odd function Question No: 40 The graph of saw tooth wave given above is An odd function An even function Neither even nor odd

100

Come & Join Us at VUSTUDENTS.net

Come & Join Us at VUSTUDENTS.net For Assignment Solution, GDB, Online Quizzes, Helping Study material, Past Solved Papers, Solved MCQs, Current Papers, E-Books & more. Go to http://www.vustudents.net and