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
Symmetric equation for the line through (1,3,5) and (2,-2,3) is x 2 y 2 z 3 3 5 x 2 y 3 z 5 5 2 x 1 y 3 z 5 5 2 x 1 y3 z 5 5 5 Question No: 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
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
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
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: 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)
sin 5t t, sec t, 6 cos 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: 7 What is the derivative of following vector-valued function? t 1, 3 r (t) t 4, t 2 1 6 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
The following differential is exact dz (x 2 y y) dx x dy
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 #182 5 3 1 x dx 2 n n 2 n 4 6 4 2 sin 0 2 n1 n3 n5 6 4 2 sinn x dx 0 n n 2 n 4 7 5 3 2 n n2 n4 6 4 2 sinn x dx 2 0 n 1 n 3 n 5 5 3 1 2 n n2 n4 n sin x dx 6 4 2 0 n 1 n 3 n 5 5 3 1 Question No: 10 Match the following equation in polar co-ordinates with its graph.
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
said to be symmetric about which of the following?
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
1 3 Does not exist Question No: 14 What is the period of a periodic function defined by f (x) 4 cos 3x? 4 3 2 3 Question No: 15 Match the following periodic function with its graph. 3 0 x 4 f (x) 0 4 x 6
Page #198 Question No: 16 What is the period of periodic function whose graph is as below? 2 5 6
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. 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
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? 2 1 2 1 2 1 Distance between these two points
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
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
0
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 0-4 4 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)?
3, 2 and 9
1, -4 and -1 2, -3 and 20 0.5, -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 1 0 2 4 (6x 2 4xy 3 )dxdy 0 1 4 2 (6x 2 4xy 3 )dxdy 1 0 4 1 (6x 2 4xy 3 )dxdy 2 0
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
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
Solution:
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:
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 3 1...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 6 2 0 a 0 of the periodic function Question No: 35 ( Marks: 3 )
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:
0 2 sin 3 xdx n1. n 31 3 2 3 2 sin 5 xdx 0 n1. n3 n n 2 51. 53 5 5 2 4. 2 5 3 2 sin 3 x sin 5 xdx 0 2 4. 2 3 5 3 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:
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 1 1 1 4(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
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
ExamDate: 19 Aug 2010
For Teacher's Use Only Q No. Marks 1 2 3 4 5 6 7 8 Total Q No. 9 10 11 12 13 14 15 16 Marks Q No. 17 18 19 20 21 22 23 24 Marks Q No. 25 26 27 28 29 30 31 32 Marks Q No. 33 34 35 36 37 38 39 Marks
Question No: 1 -------------------- 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)? 7 2 2 6 34 7 3 Question No: 4
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,, 0 2 2 3 cos, 3 sin, 0 3 3 3 sin,, 3 cos 3 2 3 3,, 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
Question No: 7
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,
y 1cos 2t x 1cos 2t, z 0, z 0
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
is even and n 2?
2 n1 n3 n5 5 3 1 sinn x dx n 2 n 4 6 4 2 0 2 n 2 n1 n3 n5 6 4 2 sinn x dx n 0 n 2 n 4 7 5 3 sinn x dx 2 n n 2 n 4 6 4 2 2 0 n 1 n 3 n 5 5 3 1 2 n n2 n4 n sin x dx 6 4 2 0 n 1 n 3 n 5 5 3 1 Question No: 12 Which one of the following is correct Wallis Cosine formula when n is odd and n 3? 2 n1 n3 n5 5 3 1 cosn x dx n 2 n 4 6 4 2 0 2 n 2 n n 2 n 4 6 4 2 cosn x dx 2 0 n 1 n 3 n 5 5 3 1 2 n n n2 n4 cos x dx 6 4 2 0 n 1 n 3 n 5 5 3 1 2 n1 n3 n5 6 4 2 cosn x dx 0 n n 2 n 4 7 5 3 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
Question No: 14
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? 2 2 3 2 4 Question No: 16 What is the period of periodic function whose graph is as below? 2 5 6 8 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? 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, 1 1 2 2 2 f 0, 1, 1 2
2 2
f 0, 1, 1 1 2 2 2 f 0, 1, 1 0 2 2 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
Question No: 24
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
Evaluate the line integral C line segment from (0, 0) to (0, 2). 1 0 2-2 (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). -4-2 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
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:
The Laplace transform of the given function will be:
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 7. 5. 3. 1 0 8 6 4 2 2 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
L(t) e 2t... 1 1 s 2 L(t) sin 3t... 2 a L(t) s 2 3 2 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
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 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
Fall 2009
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
Plane
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
Direction angles
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
Question No: 7 Match the following vector-valued function with its graph. ^ ^ ^ r(t) t it 2 jt 3 k and t 0
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 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
Question No: 10
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
The following differential is exact dz (3x 2 4xy) dx (2x 2 2 y) dy
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 n5 5 3 1 2 n n 2 n 4 6 4 2 2 n n2 n4 6 4 2 sinn x dx 2 0 n 1 n 3 n 5 5 3 1 2 n1 n3 n5 6 4 2 sinn x dx 0 n n 2 n 4 7 5 3 2 n n2 n4 n sin x dx 6 4 2 0 n 1 n 3 n 5 5 3 1 Question No: 14 Which of the following is correct?
2 sin 4 x dx 3 0 16 2 sin 4 x dx 3 16 0 2 sin 4 x dx 3 0 8 2 sin 4 x dx 2 0 3 Question No: 15 Match the following equation in polar co-ordinates with its graph. r 4sin 2 1-1 -2 1 2 3 4 4 3 2 1-2 -1 1 2
2 1-4 -3-2 -1-1 -2-2 -1 1 2-1 -2-3 -4 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
2
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 10 10 x 13
Question No: 19 What is the period of periodic function whose graph is as below? 2 3 4 5 Question No: 20 What is the period of periodic function whose graph is as below? 0 2
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
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)
Question No: 25
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 -----------
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: 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
What is the value of L{e 5t } if L denotes laplace transform?
L{e 5t 1 } s 5 s L{e 5t } s 2 25 5 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
Question No: 31 5 2 What is Laplace Inverse Transform of s 25 1 5 L 2 sin 5t s 25 1 5 L 2 cos 5t s 25 1 5 L 2 sin 25t s 25 1 5 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}
6 s 2 36
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 6-6 0 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 -4-2 0
2
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 1 0 2 1 0 1 (x 2 y )dxdy 2
1 2 (x 2 y 2 )dxdy 1 0 2 0 (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.
Question No: 39 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 The graph of saw tooth wave given above is -------------- An odd function An even function Neither even nor odd