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2 ASSALAM O ALAIKUM All Dearz fellows ALL IN ONE MTH301 Final term PAPERS & MCQz Created BY Farhan & Ali BS (cs) nd sem Hackers Group From Mandi Bahauddin Remember us in your prayers Mindhacker14@gmail.com Hearthacker14@gmail.com Student Info Student ID: Center: Exam Date: FINALTERM EXAMINATION Spring 010 MTH301- Calculus II Time: 90 min Marks: 60 For Teacher's Use Only

3 Q No Total Marks Q No Marks Q No Marks Q No Marks Q No Marks

4 Question No: 1 ( Marks: 1 ) - Please choose one Intersection of two straight lines is Surface Curve Plane Point Question No: ( Marks: 1 ) - Please choose one Plane is a surface. One-dimensional Two-dimensional Three-dimensional Dimensionless

5 Question No: 3 ( Marks: 1 ) - Please choose one 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 ( Marks: 1 ) - Please choose one What are the parametric equations that correspond to the following vector equation? ^ ^ r ( t) = sin t i+ (1 cos t) j x = t y = t z = sin, 1 cos, 0 y = t x = t z = sin, 1 cos, 0

6 x = t y = t z = sin, 1 cos, 1 x = t y = t z = sin, cos, 1 Question No: 5 ( Marks: 1 ) - Please choose one What are the parametric equations that correspond to the following vector equation? ^ ^ ^ r( t) = t 1 i 3 t j+ sin 3t k z = t 1, x = 3 t, y = sin 3t y = t 1, x = 3 t, z = sin 3t x = t 1, z = 3 t, y = sin 3t x = t 1, y = 3 t, z = sin 3t Question No: 6 ( Marks: 1 ) - Please choose one What is the derivative of following vector-valued function? r ( t) = (cos5 t, tan t, 6sin t) sin 5t r ( t) =, sec t, 6cost 5

7 sin 5t r ( t) = (, sec t, 6cos t) 5 r ( t) = ( 5sin 5 t, sec t, 6cos t) r ( t) = (sin 5 t, sec t, 6cos t) Question No: 7 ( Marks: 1 ) - Please choose one What is the derivative of following vector-valued function? 4 3 r ( t) = t, t + 1, t 1 6 t + 1 t 3 r ( t) = 4 t,, r ( t) = 4 t,, 3 t 1 t t + 1 t 4 r ( t) = 4 t,, t + 1 t 3 r ( t) = 4 t,, 3 Question No: 8 ( Marks: 1 ) - Please choose one The following differential is exact dz = ( x y + y) dx x dy

8 True False Question No: 9 ( Marks: 1 ) - Please choose one Which one of the following is correct Wallis Sine formula when n is even and n? 0 page #18 n n 1 n 3 n sin x dx = n n n ( ) ( ) 0 ( ) ( ) n n 1 n 3 n sin x dx = n n n n n n n sin x dx = n 1 n 3 n ( ) ( ) n n n n sin x dx = n 1 n 3 n Question No: 10 ( Marks: 1 ) - Please choose one Match the following equation in polar co-ordinates with its graph.

9 r cosθ = a where a is an arbitrary cons tant Question No: 11 ( Marks: 1 ) - Please choose one If the equation of a curve, in polar co-ordinates, remains unchanged after replacing ( r, θ ) by ( r, θ ) then the curve is said to be symmetric about which of the following?

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

11 1 3 Does not exist Question No: 14 ( Marks: 1 ) - Please choose one What is the period of a periodic function defined by f ( x) = 4cos3x? Question No: 15 ( Marks: 1 ) - Please choose one Match the following periodic function with its graph. 3 0 < x < 4 f ( x) = 0 4 < x < 6

12 Page #198 Question No: 16 ( Marks: 1 ) - Please choose one What is the period of periodic function whose graph is as below? 5 6 8

13 Question No: 17 ( Marks: 1 ) - Please choose one What is the period of periodic function whose graph is as below? Question No: 18 ( Marks: 1 ) - Please choose one Let L denotes the Laplace Transform. If L{ F( t)} = f ( s) where s is a constant.and lim t 0 which of the following equation holds? F t t exists then F( t) L = f ( s + a) t F( t) L = f ( s a) t

14 F( t) L = t s f ( s) ds F( t) d L = { f ( s)} t ds Question No: 19 ( Marks: 1 ) - Please choose one Which of the following is Laplace inverse transform of the function f ( s) defined by 3 f ( s) = s s? t 3te t 3e t t 3e None of these. Question No: 0 ( Marks: 1 ) - Please choose one Let ( x1, y1, z1) and ( x, y, z ) space. What does the formula calculates? be any two points in three dimensional ( x x ) + ( y y ) + ( z z ) Distance between these two points

15 Midpoint of the line joining these two points Ratio between these two points Question No: 1 ( Marks: 1 ) - Please choose one 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 xyplane. If ( P dx + Q dy ) is an exact differential then Ñ C ( P dx Q dy ) + = Zero One Infinite Question No: ( Marks: 1 ) - Please choose one What is Laplace transform of the function F( t ) if F( t) { } 1 L t = s L t 1 = s { } = t?

16 L t = e { } s L{ t} = s Question No: 3 ( Marks: 1 ) - Please choose one What is the value of L e 5 { t } if L denotes laplace transform? L{ e } = s 5 5t 1 L e 5t { } L e 5t { } L e 5 { t } = = = s s 5! 6 s s Question No: 4 ( Marks: 1 ) - Please choose one Evaluate the line integral (3x + y) dx + ( x y) dy where C is C the line segment from (0, 0) to (0, ). 1 0

17 - Question No: 5 ( Marks: 1 ) - Please choose one ( x + y) dx + ( x y) dy Evaluate the line integral where C is the line segment from (0, 0) to (, 0). C Do not exist Question No: 6 ( Marks: 1 ) - Please choose one Which of the following are direction ratios for the line joining the points (1, 3, 5) and (, 1, 4)? 3, and 9

18 1, -4 and -1, -3 and 0 0.5, -3 and 5/4 Question No: 7 ( Marks: 1 ) - Please choose one If R = {( x, y) / 0 x and1 y 4}, then R 3 (6 4 ) x + xy da = (6 + 4 ) x xy dydx (6 + 4 ) x xy dxdy (6 + 4 ) x xy dxdy (6 + 4 ) x xy dxdy

19 Question No: 8 ( Marks: 1 ) - Please choose one Which of the following is true for a periodic function whose graph is as below? Even function page09 Odd function Neither even nor odd function Question No: 9 ( Marks: 1 ) - Please choose one Which of the following is true for a function whose graph is given above. An odd function pg 1

20 An even function Neither even nor odd Question No: 30 ( Marks: 1 ) - Please choose one At each point of domain, the function Is defined Is continuous Is infinite Has a limit Question No: 31 ( Marks: ) Determine whether the following differential is exact or not dz = 4x y dx + 3x y dy Solution:

21 3 3 4 dz = 4x y dx + 3x y dy p = 1x y y 3 Q = 1x y X p Q = y X yes 3 Question No: 3 ( Marks: ) Evaluate sin nx dx where n is an integer other than zero. Solution: sin nx dx cos nx = n cos n cos n = + n n 1 = ( cos n + cos n ) n = 0 Question No: 33 ( Marks: ) Find Laplce transform of the function F( t ) if F( t) = e 3t Solution:

22 0 3t 3t st L( e ) = e e = e 0 ( s 3) t. dt ( s 3) t e = { }lim 0 ( s 3) = 1 1 ( s 3) t s 3 e 1 = (0 1) s 3 1 =...Ans s 3 Question No: 34 ( Marks: 3 ) Determine the Fourier co-efficient a0 of the periodic function defined below: f ( x) = x < x < Solution: 1 a = f ( x) dx o f ( x) = (x + 1) (0,) 0 = (x + 1) dx = 6 = x + x 0 Question No: 35 ( Marks: 3 )

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

24 0 0 0 sin 3 n 1 =. n 3 1 = 3 = 3 sin 5 xdx xdx n 1 n 3 =. n n = = ( sin x + sin ) 4 = x dx Question No: 37 ( Marks: 5 ) Evaluate the following line integral which is independent of path. (3,) (0,0) y y + xe dx x e dy Solution:

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

26 ASSALAM O ALAIKUM All Dearz fellows ALL IN ONE MTH301 Final term PAPERS & MCQz Created BY Farhan & Ali BS (cs) nd sem Hackers Group From Mandi Bahauddin Remember us in your prayers Mindhacker14@gmail.com Hearthacker14@gmail.com FINALTERM EXAMINATION Spring 010 MTH301- Calculus II (Session - ) Time: 90 min Marks: 60 Student Info StudentID: $$ Center: OPKST ExamDate: 19 Aug 010

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

28 Question No: 1 ( Marks: 1 ) - Please choose one planes intersect at right angle to form three dimensional space. Three Four Eight Twelve Question No: ( Marks: 1 ) - Please choose one 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 ( Marks: 1 ) - Please choose one What is the distance between points (3,, 4) and (6, 10, -1)? Question No: 4 ( Marks: 1 ) - Please choose one

29 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 page # 1 Line Curve Circle Question No: 5 ( Marks: 1 ) - Please choose one The spherical co-ordinates of a point are 3,,. What are its 3 cylindrical co-ordinates? 3 3,, 0 3 cos, 3 sin, sin,, 3 cos 3 3 3,, 0 3 Question No: 6 ( Marks: 1 ) - Please choose one Domain of the function f ( x, y) y x = is y < x y x y x Entire space Question No: 7 ( Marks: 1 ) - Please choose one

30 Match the following vector-valued function with its graph. ^ ^ ^ r( t) = cost i+ sin t j+ k And 0 t Correct Question No: 8 ( Marks: 1 ) - Please choose one What are the parametric equations that correspond to the following vector equation? ^ ^ r ( t) = sin t i+ (1 cos t) j x = t y = t z = sin, 1 cos, 0 y = t x = t z = sin, 1 cos, 0

31 x = t y = t z = sin, 1 cos, 1 x = t y = t z = sin, cos, 1 Question No: 9 ( Marks: 1 ) - Please choose one What are the parametric equations that correspond to the following vector equation? ^ ^ ^ r( t) = t 1 i 3 t j+ sin 3t k z = t 1, x = 3 t, y = sin 3t y = t 1, x = 3 t, z = sin 3t x = t 1, z = 3 t, y = sin 3t x = t 1, y = 3 t, z = sin 3t Question No: 10 ( Marks: 1 ) - Please choose one Is the following vector-valued function r ( t) continuous at t = 1? If not, why? t + 1 r ( t), t, t = t 1 r ( t) is continuous at t = 1 r (1) is not defined r (1) is defined but r (1) is defined and are not equal. lim r ( t) t1 t1 does not exist lim r ( t) exists but these two numbers Question No: 11 ( Marks: 1 ) - Please choose one Which one of the following is correct Wallis Sine formula when n is even and n?

32 0 ( ) ( ) n n 1 n 3 n sin x dx = n n n page ( ) ( ) n n 1 n 3 n sin x dx = n n n n n n n sin x dx = n 1 n 3 n ( ) ( ) n n n n sin x dx = n 1 n 3 n Question No: 1 ( Marks: 1 ) - Please choose one Which one of the following is correct Wallis Cosine formula when n is odd and n 3? ( ) ( ) n n 1 n 3 n cos x dx = n n n n n n n cos x dx = n 1 n 3 n ( ) ( ) n n n n cos x dx = n 1 n 3 n page183 n n 1 n 3 n cos x dx = n n n ( ) ( ) Question No: 13 ( Marks: 1 ) - Please choose one If the equation of a curve, in polar co-ordinates, remains unchanged after replacing ( r, θ ) by ( r, θ ) then the curve is said to be symmetric about which of the following?

33 Initial line y-axis Pole Question No: 14 ( Marks: 1 ) - Please choose one If a > 0, then the equation, in polar co-ordinates, of the = represent which of the following family of form r a cos θ curves? Leminscate page 130 Cardiods Rose curves Spiral Question No: 15 ( Marks: 1 ) - Please choose one x What is the period of a periodic function defined by f ( x ) = sin? 3 4 Question No: 16 ( Marks: 1 ) - Please choose one What is the period of periodic function whose graph is as below?

34 5 6 8 Question No: 17 ( Marks: 1 ) - Please choose one What is the period of periodic function whose graph is as below? Question No: 18 ( Marks: 1 ) - Please choose one 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)} ds L{ t F( t)} = f ( s + t) L{ t F( t)} = f ( s) L { t F ( t )} = f ( s ) ds s Question No: 19 ( Marks: 1 ) - Please choose one

35 The graph of an odd function is symmetrical about x-axis y-axis origin Question No: 0 ( Marks: 1 ) - Please choose one Consider the function 1 1 of f 0,, f 0,, = 1 1 f 0,, = f 0,, = 1 1 f 0,, 0 = f ( x, y, z) 1 x y z =. What is the value Question No: 1 ( Marks: 1 ) - Please choose one 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: ( Marks: 1 ) - Please choose one Sign of line integral is reversed when

36 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: 3 ( Marks: 1 ) - Please choose one 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 C ( P dx Q dy ) Zero + = One Infinite Question No: 4 ( Marks: 1 ) - Please choose one What is the value of L{ e } = s 5 5t s L{ e } = s + 5 5t 5 L{ e } = s ! L{ e t } = 6 s 5t 1 L e 5 { t } if L denotes laplace transform? Question No: 5 ( Marks: 1 ) - Please choose one What is laplace transform of the function 3 L{sin 3 t} = s + 9 s L{sin 3 t} = s + 9 F t if F( t) = sin 3t?

37 1 L{sin 3 t} = s 3 3! L{sin 3 t} = 4 s Question No: 6 ( Marks: 1 ) - Please choose one If L denotes laplace transform then 5 L{ te t } = 5t 1 L{ te } = s 5 5t 1 L{ te } = s + 5 5t 1 L{ te } = s + 5 L te 5t { } = 1 ( s 5) Question No: 7 ( Marks: 1 ) - Please choose one Evaluate the line integral (3x + y) dx + ( x y) dy where C is C the line segment from (0, 0) to (0, ) Question No: 8 ( Marks: 1 ) - Please choose one

38 Evaluate the line integral ( x + y) dx + ( x y) dy where C is the C line segment from (0, 0) to (0, ) Question No: 9 ( Marks: 1 ) - Please choose one Divergence of a vector function is always a Scalar Vector Question No: 30 ( Marks: 1 ) - Please choose one Which of the following is true for a function whose graph is given above An odd function An even function Neither even nor odd

39 Question No: 31 ( Marks: ) Does the following limit exist? If yes find its value, if no give reason ^ ^ ^ t 1 lim ( e 5) i ( t t 3) j k t t Question No: 3 ( Marks: ) Define the periodic function whose graph is shown below. Question No: 33 ( Marks: ) Find Laplace Transform of the function F( t) if F( t) = t 4 Solution: The Laplace transform of the given function will be: f ( t) = t L t 4 { } 4 4! = 5 s Question No: 34 ( Marks: 3 ) Determine whether the following differential is exact or not dz = (4x y + xy ) dx + ( x + 3 x y ) dy Question No: 35 ( Marks: 3 )

40 3 5 Use Wallis sine formula to evaluate ( sin x + sin ) 0 x dx Solution: = sin xdx... Question No: 36 ( Marks: 3 ) Find Laplace transform of the function F( t) if t F( t) = e sin 3t Solution: Laplace transform will be L t t = e = s L( t) = sin 3 t... a L( t) = s + 3 a L( t) = s + 9 Combining, 1 a L( t) = s s + 9 Question No: 37 ( Marks: 5 ) Using definite integral, find area of the region that is enclosed between the curves y = x and y = x

41 Question No: 38 ( Marks: 5 ) Determine the fourier co-efficient bn of the following function. f ( x) = x 0 < x < Question No: 39 ( Marks: 5 ) Determine whether the following vector field F or not. ^ ^ ^ F( x, y, z) = (4 x z) i+ (3 y + 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) nd sem Hackers Group

42 From Mandi Bahauddin Remember us in your prayers FINALTERM EXAMINATION Fall 009 MTH301- Calculus II Time: 10 min Marks: 80 Question No: 1 ( Marks: 1 ) - Please choose one is an example of Irrational numbers Rational numbers Integers Natural numbers

43 Question No: ( Marks: 1 ) - Please choose one Straight line is a special kind of Surface Curve Plane Parabola Question No: 3 ( Marks: 1 ) - Please choose one An ordered triple corresponds to in three dimensional space. A unique point A point in each octant Three points Infinite number of points

44 Question No: 4 ( Marks: 1 ) - Please choose one The angles which a line makes with positive x,y and z-axis are known as Direction cosines Direction ratios Direction angles Question No: 5 ( Marks: 1 ) - Please choose one Is the function f ( x, y ) continuous at origin? If not, why? f ( x, y) = 4xy + sin 3x y f ( x, y ) is continuous at origin f (0, 0) is not defined f (0, 0) is defined but lim f ( x, y) ( x, y ) does not exist (0, 0) f (0, 0) is defined and lim f ( x, y) ( x, y ) exists but these two (0, 0) numbers are not equal.

45 Question No: 6 ( Marks: 1 ) - Please choose one Match the following vector-valued function with its graph. ^ ^ r( t) = 3cost i+ 3sin t j And 0 t

46 Question No: 7 ( Marks: 1 ) - Please choose one Match the following vector-valued function with its graph. ^ ^ ^ 3 r( t) = t i+ t j+ t k and t 0

47 Question No: 8 ( Marks: 1 ) - Please choose one What are the parametric equations that correspond to the following vector equation? ^ ^ r ( t) = sin t i+ (1 cos t) j x = t y = t z = sin, 1 cos, 0 y = t x = t z = sin, 1 cos, 0 x = t y = t z = sin, 1 cos, 1 x = t y = t z = sin, cos, 1 Question No: 9 ( Marks: 1 ) - Please choose one Is the following vector-valued function r ( t) continuous at t = 0? If not, why? r ( t) = (4cos t, t, 4sin t) r (0) is not defined r (0) is defined but lim r ( t) t0 does not exist r (0) is defined and are not equal. lim r ( t) exists but these two numbers t0

48 r ( t) is continuous at t = 0 Question No: 10 ( Marks: 1 ) - Please choose one What is the derivative of following vector-valued function? r ( t) = (cos5 t, tan t, 6sin t) sin 5t r ( t) =, sec t, 6cost 5 sin 5t r ( t) = (, sec t, 6cos t) 5 r ( t) = ( 5sin 5 t, sec t, 6cos t) r ( t) = (sin 5 t, sec t, 6cos t) Question No: 11 ( Marks: 1 ) - Please choose one The following differential is exact 3 dz = (3x y + ) dx + ( x + y) dy True False

49 Question No: 1 ( Marks: 1 ) - Please choose one The following differential is exact dz = (3x + 4 xy) dx + (x + y) dy True False Question No: 13 ( Marks: 1 ) - Please choose one Which one of the following is correct Wallis Sine formula when n is odd and n 3? 0 ( ) ( ) n n 1 n 3 n sin x dx = n n n n n n n sin x dx = n 1 n 3 n ( ) ( ) n n 1 n 3 n sin x dx = n n n ( ) ( ) n n n n sin x dx = n 1 n 3 n

50 Question No: 14 ( Marks: 1 ) - Please choose one Which of the following is correct? 4 0 sin 3 x dx = sin 3 x dx = sin 3 x dx = sin x dx = 3 Question No: 15 ( Marks: 1 ) - Please choose one Match the following equation in polar co-ordinates with its graph. r = 4sinθ

51 Question No: 16 ( Marks: 1 ) - Please choose one If the equation of a curve, in polar co-ordinates, remains unchanged after replacing ( r, θ ) by ( r, θ ) then the curve is said to be symmetric about which of the following? Initial line y-axis Pole Question No: 17 ( Marks: 1 ) - Please choose one

52 x What is the period of a periodic function defined by f ( x ) = sin? 3 4 Question No: 18 ( Marks: 1 ) - Please choose one Match the following periodic function with its graph. 3 x 0 < x < 4 4 f ( x) = 7 x 4 < x < < x < 13

53 Question No: 19 ( Marks: 1 ) - Please choose one What is the period of periodic function whose graph is as below? Question No: 0 ( Marks: 1 ) - Please choose one What is the period of periodic function whose graph is as below?

54 0 not sure Question No: 1 ( Marks: 1 ) - Please choose one 3 Polar co-ordinates of a point are,. Which of the following is another possible polar co-ordinates representation of this point?, 4,, 3, 3 4

55 Question No: ( Marks: 1 ) - Please choose one The function f ( x) 3 x = x e is Even function Odd function Neither even nor odd Question No: 3 ( Marks: 1 ) - Please choose one The graph of an even function is symmetrical about x-axis y-axis page 08 origin Question No: 4 ( Marks: 1 ) - Please choose one At which point the vertex of parabola, represented by the equation y = x 4x + 3, occurs?

56 (0, 3) (, 1) (, 15) (1, 0) pge 08 Question No: 5 ( Marks: 1 ) - Please choose one The equation y = x 4x + represents a parabola. Find a point at which the vertex of given parabola occurs? (, ) ( 4, 34) (0, 0) (, 14) Question No: 6 ( Marks: 1 ) - Please choose one Is the function f ( x, y ) continuous at origin? If not, why? xy f ( x, y) = x + y f ( x, y ) is continuous at origin

57 ( x, y ) (0, 0) lim f ( x, y) does not exist f (0, 0) is defined and lim f ( x, y) ( x, y ) exists but these two (0, 0) numbers are not equal. Question No: 7 ( Marks: 1 ) - Please choose one 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: 8 ( Marks: 1 ) - Please choose one What is Laplace transform of a function F(t)? (s is a constant) s st e F ( t) dt 0 st e F( t) dt 0

58 e st F ( t) dt st e F ( t) dt 0 Question No: 9 ( Marks: 1 ) - Please choose one What is the value of L e 5 { t } if L denotes laplace transform? L{ e } = s 5 5t 1 L e 5t { } L e 5t { } L e 5 { t } = = = s s 5! 6 s s Question No: 30 ( Marks: 1 ) - Please choose one What is the Laplace Inverse Transform of 1 s + 1 L = t + 1 s

59 t L = e + e s L = e s + 1 L 1 1 = e s t t t Question No: 31 ( Marks: 1 ) - Please choose one What is Laplace Inverse Transform of 5 s + 5 L L L L = s = s = s = s sin 5t cos 5t sin 5t cos 5t Question No: 3 ( Marks: 1 ) - Please choose one

60 What is L{ 6} if L denotes Laplace Transform? 1 L{ 6} = s + 6 L{ 6} = L{ 6} = 6 s s s + 36 L{ 6} = 6 s + 36 Question No: 33 ( Marks: 1 ) - Please choose one Evaluate the line integral (3x + y) dx + ( x y) dy where C is the C line segment from (0, 0) to (, 0) Do not exist Question No: 34 ( Marks: 1 ) - Please choose one

61 ( x + y) dx + ( x y) dy Evaluate the line integral where C is the line segment from (0, 0) to (0, ). C -4-0 Question No: 35 ( Marks: 1 ) - Please choose one Plane is an example of Curve Surface Sphere Cone Question No: 36 ( Marks: 1 ) - Please choose one

62 If R = {( x, y) / 0 x and 1 y 1}, then R ( ) x + y da = ( + ) x y dydx ( + ) x y dxdy ( + ) x y dxdy ( + ) x y dxdy Question No: 37 ( Marks: 1 ) - Please choose one To evaluate the line integral, the integrand is expressed in terms of x, y, z with dr = dxiˆ + dy ˆj dr = dxiˆ + dy ˆj + dy kˆ dr = dx + dy + dz dr = dx + dy

63 Question No: 38 ( Marks: 1 ) - Please choose one Match the following equation in polar co-ordinates with its graph. r = a where a is an arbitrary constant. Question No: 39 ( Marks: 1 ) - Please choose one

64 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: 40 ( Marks: 1 ) - Please choose one The graph of saw tooth wave given above is An odd function

65 An even function Neither even nor odd not sure Question No: 1 ( Marks: ) - Please choose one Laplace transform of t is 1 s 1 s e s s Question No: ( Marks: ) - Please choose one Symmetric equation for the line through (1,3,5) and (,-,3) is y + z 3 x = = 3 5 y + 3 z + 5 x + = = 5

66 y 3 z 5 x 1= = 5 y + 3 z 5 x + 1= = 5 5 Question No: 3 ( Marks: 1 ) - Please choose one The level curves of f(x, y) = y Cscx are parabolas. True. False. Question No: 4 ( Marks: 1 ) - Please choose one The equation z = r is written in Rectangular coordinates Cylindrical coordinates Spherical coordinates None of the above FINALTERM EXAMINATION Spring 010 MTH301- Calculus II (Session - 4)

67 Student Info StudentID: Ref No: Time: 90 min Marks: 60 Center: OPKST ExamDate: 07 Aug 010 For Teacher's Use Only Q Total No. Marks Q No Marks Q No Marks Q No Marks Q No Marks

68 Question No: 1 ( Marks: 1 ) - Please choose one There is one-to-one correspondence between the set of points on co-ordinate line and Set of real numbers Set of integers Set of natural numbers Set of rational numbers Question No: ( Marks: 1 ) - Please choose one Straight line is a special kind of Surface Curve Plane Parabola Question No: 3 ( Marks: 1 ) - Please choose one lim ( x, y) (0, 0) xy x + y = Question No: 4 ( Marks: 1 ) - Please choose one If then 3 f ( x, y) = x y y + ln x f = x

69 1 x 1 y + x 1 xy x 1 y x xy + Question No: 5 ( Marks: 1 ) - Please choose one Suppose f ( x, y) xy y where x 3t 1 and y t = = + =. Which one of the following is true? df 4t dt = + df 16t t dt = df 18t dt = + df 10t 8t 1 dt = + + Question No: 6 ( Marks: 1 ) - Please choose one Match the following vector-valued function with its graph. ^ ^ r( t) = 3cost i+ 3sin t j and 0 t

70 Question No: 7 ( Marks: 1 ) - Please choose one What are the parametric equations that correspond to the following vector equation? ^ ^ r ( t) = sin t i+ (1 cos t) j x = t y = t z = sin, 1 cos, 0 y = t x = t z = sin, 1 cos, 0 x = t y = t z = sin, 1 cos, 1 x = t y = t z = sin, cos, 1

71 Question No: 8 ( Marks: 1 ) - Please choose one Is the following vector-valued function r ( t) not, why? continuous at t =? If r ( t) = (tan t, sin t, cos t) r ( t) is continuous at t = r is not defined r is defined but lim r ( t) does not exist t r is defined and lim r ( t) exists but these two t numbers are not equal. Not sure Question No: 9 ( Marks: 1 ) - Please choose one What is the derivative of following vector-valued function? 4 3 r ( t) = t, t + 1, t r ( t) = 4 t,, 3 t 1 t r ( t) = 4 t,, 3 t 1 t r ( t) = 4 t,, 3 t 1 t r ( t) = 4 t,, 3 t 1 t + Question No: 10 ( Marks: 1 ) - Please choose one The following differential is exact dz = (6xy + y 5) dx + (3x + 4xy 6) dy

72 True False Question No: 11 ( Marks: 1 ) - Please choose one The following differential is exact dz = ( x y + y) dx x dy True False Question No: 1 ( Marks: 1 ) - Please choose one Which one of the following is correct Wallis Sine formula when n is even and n? sin x dx ( ) 0 sin x dx ( ) 0 sin x dx sin x dx 0 0 ( ) n n 1 n 3 n = n n n ( ) n n 1 n 3 n = n n n n n n n = n 1 n 3 n ( ) ( ) n n n n = n 1 n 3 n Question No: 13 ( Marks: 1 ) - Please choose one Match the following equation in polar co-ordinates with its graph. r = 4sinθ

73 Question No: 14 ( Marks: 1 ) - Please choose one Match the following equation in polar co-ordinates with its graph. r = 4sinθ

74 Question No: 15 ( Marks: 1 ) - Please choose one Match the following periodic function with its graph. 3 0 < x < 4 f ( x) = 0 4 < x < 6 page 198 Question No: 16 ( Marks: 1 ) - Please choose one Match the following periodic function with its graph. 4 0 < x < 5 f ( x) = 0 5 < x < 8

75 pg199 Question No: 17 ( Marks: 1 ) - Please choose one Which of the following condition must be satisfied for a vector field F to be a conservative vector field? Line integral of F along a curve, depends only on the endpoints of that curve, not on the particular route taken. Divergence of F should be zero Gradient of F should be zero. F = 0 Question No: 18 ( Marks: 1 ) - Please choose one What is the period of periodic function whose graph is as below? Question No: 19 ( Marks: 1 ) - Please choose one Let L denotes the Laplace Transform.

76 According to First Shift Theorem, if L{ F( t)} = f ( s) then which of the following equation holds? s and a are constants. at L{ e F( t)} = f ( s a) at L{ e F( t)} = f ( s + a) at L{ e F( t)} = f ( s) at L{ e F( t)} = f ( a) Question No: 0 ( Marks: 1 ) - Please choose one Polar co-ordinates of a point are 7,. Which of the following is 4 another possible polar co-ordinates representation of this point? 7, , 4 7, 4 3 7, 4 Question No: 1 ( Marks: 1 ) - Please choose one The graph of an even function is symmetrical about x-axis y-axis origin Question No: ( Marks: 1 ) - Please choose one Is the function f ( x, y ) continuous at origin? If not, why? xy f ( x, y) = x + y f ( x, y ) is continuous at origin lim f ( x, y) ( x, y ) (0, 0) does not exist

77 f (0, 0) is defined and lim f ( x, y) ( x, y ) exists but (0, 0) these two numbers are not equal. Question No: 3 ( Marks: 1 ) - Please choose one Consider the function 1 1 f 0,, f 0,, = 1 1 f 0,, = f 0,, = 1 1 f 0,, 0 = f ( x, y, z) 1 x y z =. What is the value of Question No: 4 ( Marks: 1 ) - Please choose one 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: 5 ( Marks: 1 ) - Please choose one 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 C ( P dx Q dy ) + = Zero

78 One Infinite Question No: 6 ( Marks: 1 ) - Please choose one What is laplace transform of the function F( t) if F( t) cos L{cos t} = s L{cos t} = s s L{cos t} = s + 4! L{cos t} = 3 s = t? Question No: 7 ( Marks: 1 ) - Please choose one What is L{ 6} if L denotes Laplace Transform? 1 L{ 6} = s L{ 6} = s s L{ 6} = s L{ 6} = s + 36 Question No: 8 ( Marks: 1 ) - Please choose one Curl of vector function is always a Scalar Vector Question No: 9 ( Marks: 1 ) - Please choose one

79 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: 30 ( Marks: 1 ) - Please choose one 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: )

80 x dx Evaluate the line integral where C is the line segment from (0, ) to (, 6) C Question No: 3 ( Marks: ) Use Wallis sine formula to evaluate 5 0 sin x dx Question No: 33 ( Marks: ) Find Laplace Transform of the function F( t) if F( t) sin = t. Question No: 34 ( Marks: 3 ) Find div F ^ ^ ^, if F = (3 x + y) i + xy z j + ( xz ) k Question No: 35 ( Marks: 3 ) Determine whether the following differential is exact or not dz = (4x y + xy ) dx + ( x + 3 x y ) dy Question No: 36 ( Marks: 3 ) Prove whether the following function is even, odd or neither. 3 x f ( x) = x e Question No: 37 ( Marks: 5 ) Evaluate the following line integral which is independent of path. ( 1,0) (, ) 3 + (3 ) xy dx y x dy Question No: 38 ( Marks: 5 )

81 Determine the fourier co-efficient bn of the following function. f ( x) = x 0 < x < Question No: 39 ( Marks: 5 ) Determine whether the following vector field F or not. ^ ^ ^ F( x, y, z) = (3 x + y) i+ xy z j+ xz k is conservative ASSALAM O ALAIKUM All Dearz fellows ALL IN ONE MTH301 Final term PAPERS & MCQz Created BY Farhan & Ali BS (cs) nd sem Hackers Group From Mandi Bahauddin Remember us in your prayers Mindhacker14@gmail.com Hearthacker14@gmail.com