Ordinary Differential Equations
|
|
|
- Amelia Spencer
- 5 years ago
- Views:
Transcription
1 Ordinary Differential Equations Previous year Questions from 016 to 199 Ramanasri 016 S H O P NO- 4, 1 S T F L O O R, N E A R R A P I D F L O U R M I L L S, O L D R A J E N D E R N A G A R, N E W D E L H I. W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : /6363/6464 1
2 d y / 3 Find a particular integral of y e sin (10 marks). Show that the family of parabolas y 4c 4c is self orthogonal. (10 marks) 3. Solve{ y(1 tan ) cos } 0 (10 marks) 4. Using the method of variation of parameter solve the differential equation d ( D D 1) y e log( ), D (15 marks) 3 5. Find the general solution of the equation d y 4 d y (15 marks) 6. Using Laplace transformation solves the following: y'' y' 8y 0, y(0) 3, y'(0) 6 (10 marks) Solve the differential equation: cos y( sin cos ) Solve the differential equation: ( y y e y y) ( y e y 3 ) 0 9. Find the constant a so that ( y) a is the integrating factor of (4 y 6 y) ( 9y 3 ) 0 and hence solve the differential equation (1 Marks) 10. (i) 1 s 5s Obtain Laplace Inverse transform of ln 1 e s s 5 (ii) Using Laplace transform, solve y" y t, y(0) 1, y '(0) (6+6=1 Marks) 11. Solve the differential equation py p where p (13 Marks) d y 3 d y d y Solve 6 4 4y cos(log ) 4 3 e (13 Marks) Justify that a differential equation of the form: f ( y ) y f ( y ) yf ( y ) 0 where 1 is an arbitrary function of ( y ), is not an eact differential equation and y an integrating factor for it. Hence solve this differential equation for f ( y ) ( y ) 14. Find the curve for which the part of the tangent cut-off by the aes is bisected at the point of tangency 15. Solve by the method of variation of parameters: 5y sin d y d y Solve the differential equation: 3 8y 65cos 3 log e (0 Marks) is Reputed Institute for IAS, IFoS Eams Page
3 d y 1 ( ), 17. Solve the following differential equation: y e when is a solution to its corresponding homogeneous differential equation. M, y N, y 0, to have an 18. Find the sufficient condition for the differential equation integrating factor as a function of ( ). What will be the integrating factor in that case? Hence y find the integrating factor for the differential equation of ( y) ( y y) 0 and solve it. d y t 19. Solve the initial value problem y 8e sin t, y(0) 0, y'(0) 0 by using Laplace transform. dt (0 Marks) 0. If y 013 is a function of,such that the differential coefficient is equal tocos sin e y y. Find out a relation between and y,which is free from any derivative / differential. 1. Obtain the equation of the orthogonal trajectory of the family of curves represented by n r asin n, being the plane polar coordinates. ( r, ) 3. Solve the differential equation(5 1 6 y ) 6y 0 3. Using the method of variation of parameters, solve the differential equation a y sec a 4. d y Find the general solution of the equation y ln sin(ln ) 5. By using Laplace transform method, solve the differential equation( D n ) asin( nt ), d D subject to the initial conditions 0 and 0, at t 0, in which dt dt constants. ( / y) ye 6. Solve y y (1 e ) e ( / ) ( / y) 01 anand, are (1 Marks) 7. Find the orthogonal trajectory of the family of curves y a (1 Marks) 8. Using Laplace transforms, solve the initial value problem y y y e y y '' ' t, (0) 1, '(0) 1 (1 Marks) 9. Show that the differential equation (ylog y) ( y y 1) 0 is not eact. Find an integrating factor and hence, the solution of the equation (0 Marks) 30. Find the general solution of the equation y''' y'' 1 6 (0 Marks) 31. Solve the ordinary differential equation ( 1) y'' ( 1) y' y ( 3) (0 Marks) Reputed Institute for IAS, IFoS Eams Page 3
4 Obtain the solution of the ordinary differential equation 4 y1, if 33. Determine the orthogonal trajectory of a family of curves represented by the polar equation r a(1 cos ), being the plane polar coordinates of any point. ( r, ) y(0) Obtain Clairaut s form of the differential equation y y a. Also find its general solution 35. Obtain the general solution of the second order ordinary differential equation y'' y' y e cos, where dashes denote derivatives w.r.t. 36. Using the method of variation of parameters, solve the second order differential equation 4y tan 37. Use Laplace transform method to solve the following initial value problem: d t e, (0) and 1 dt dt dt t Consider the differential equation y ', 0 where is a constant. Show that (i) If is any solution and ( ) ( ) e, then ( ) is a constant; (ii) If 0, then every solution tends to zero as (1 Marks) ( ) 39. Show that the differential equation (3 y ) y( y 3) y ' 0 admits an integrating factor which is a function of ( y ). Hence solve the equation (1 Marks) 40. Verify that 1 ( M Ny) d log 1 e( y) ( M Ny) d log e( / y) M N. Hence show that- (i) If the differential equation M N 0 is homogeneous, then( M Ny ) is an integrating factor unless M Ny 0; (ii) If the differential equation M N 0 is not eact but is of the form 1 f ( y) y f ( y) 0 then( M Ny ) is an integrating factor unless M Ny 0; 1 (0 Marks) 41. Use the method of undermined coefficients to find the particular solutions of y'' y sin (1 ) e and hence find its general solution. (0 Marks) Find the Wronskian of the set of functions: 3, 3 on the interval [ 1, 1] and determine whether the set is linearly dependent on[ 1, 1] (1 Marks) 43. Find the differential equation of the family of circles in the y- plane passing through ( 1, 1) and (1, 1) (0 Marks) Reputed Institute for IAS, IFoS Eams Page 4
5 44. Find the inverse Laplace transform of 45. Solve : y ( y) 3y y 4y 3, y(0) 1 s 1 F( s) 1n (0 Marks) s s 008 (0 Marks) Solve the differential equation y ( y ) 0 (1 Marks) 47. Use the method of variation of parameters to find the general solution of y'' 4 y' 6y sin (1 Marks) 48. Using Laplace transform, solve the initial value problem '' 3 ' 4 3 t y y y t e, y '(0) Solve the differential equation 3 y y y 50. Solve the equation y p yp 0, where 51. Solve the ordinary differential equation 5. Find the solution of the equation y(0) 1, '' 3 ' sin(ln ) 1 y p 007 cos3 3 sin3 sin6 sin 3,0 1 y (1 Marks) y 4 (1 Marks) 53. Determine the general and singular solutions of the equation y a 1, a being a constant Obtain the general solution of [ D 6D 1D 8] y 1 e e, 4 where D 55. d y 3 Solve the equation 3 3y 56. Use the method of variation of parameters to find the general solution of the equation d y 3 y e Find the family of curves whose tangents form an angle with the hyperbolas y c, c Solve the differential equation y e y 0 (1 Marks) (1 Marks) Reputed Institute for IAS, IFoS Eams Page 5
6 1 tan y 59. Solve: (1 y ) ( e ) 0 p py y y 0 using the substitution y u and y v and find its 60. Solve the equation singular solution, where p 61. Solve the differential equation 3 d y 1 d y y Solve the differential equation( D D ) y e tan, parameters. 005 D by the method of variation of 63. Find the orthogonal trajectory of the family of co-aial circles y g c 0, where g is the parameter. (1 Marks) 64. Solve: y ( y y 1) (1 Marks) 65. Solve the differential equation: ( 1) D ( 1) D ( 1) D ( 1) y ( 1) 66. Solve the differential equation: ( y )(1 p) ( y)(1 p)( yp) ( yp) 0 where p, by reducing it to Clairaut s form by using suitable substitution. 67. Solve the differential equation (sin cos ) y'' sin y' ysin 0 given that y sin is a solution of this equation. 68. Solve the differential equation y'' y' y log, 0 by variation of parameters Find the solution of the following differential equation ycos sin (1 Marks) 70. Solve: y( y y ) ( y y ) 0 (1 Marks) Solve: ( D 4D 5) y e ( cos ) 7. Reduce the equation( p y)( py ) p, where p to Clairaut s equation and hence solve it. 73. d y Solve: ( ) ( 5) y ( 1) e 74. Solve the following differential equation: (1 d y ) 4 (1 ) y Reputed Institute for IAS, IFoS Eams Page 6
7 Show that the orthogonal trajectory of a system of confocals ellipses is self orthogonal (1 Marks) 76. Solve: ylog y ye (1 Marks) Solve ( D D) 4( e cos ), where D. 78. Solve the differential equation( p y )( p y) ( P 1), where p, by reducing it to Clairaut s form using suitable substitutions 79. Solve (1 ) y'' (1 ) y' y sinlog(1 ) Solve the differential equation y'' 4 y' 6y sec by variation of parameters Solve : 3y y (1 Marks) 8. d y Find the values of for which all solutions of 3 y 0tend to zero as (1 Marks) 83. Find the value of constant such that the following differential equation becomes eact. y y (e 3 y ) (3 e ) 0. Further, for this value of 84. y 4 Solve : y 6., solve the equation. 85. Using the method of variation of parameters, find the solutions of y e sin with y(0) 0 and Solve : ( D 1)( D D ) y e where D 001 1t 1 e, 0 t A continuous function yt () satisfies the differential equation If t 3 t, 1 t 5 y(0) e find y () (1 Marks) 88. d y Solve : 3y log e (1 Marks) 89. y y(log e y) Solve : loge y 90. Find the general solution of ayp ( b) p y 0, a Solve: ( D 1) y 4 cos given that y Dy D y 0 and d y 3 Dy1 when 0 Reputed Institute for IAS, IFoS Eams Page 7
8 9. Using the method of variation of parameters, solve 4y 4tan d y Show that 3 4 8y 0. Deduce the general solution. (1 Marks) 94. d y Reduce P Qy R, are functions of,to the normal form. Hence solve d y 4 (4 1) y 3e sin 95. Solve the differential equation y ap ap. Find the singular solution and interpret it geometrically 96. Show that(4 3y 1) (3 y 1) 0 represents a family of hyperbolas with a common ais and tangent at the verte 97. d y Solve y ( 1) 1 by the method of parameters P, Q, R 98. Solve the differential equation 3 d y d y 1999 y 1 y y y 1/ (0 Marks) 99. Solve 3 4 y e cos 3 (0 Marks) 100. By the method of variation of parameters solve the differential equation a y sec( a) 1998 (0 Marks) Solve the differential equation: y y e (0 Marks) 10. Show that the equation: (4 3y 1) (3 y 1) 0 represents a family of hyperbolas having as asymptotes the lines y 0, y 1 0. (0 Marks) 103. Solve the differential equation: y p p 3 4 (0 Marks) d y y e ( 9) 104. Solve the differential equation: (0 Marks) d y 105. Solve the differential equation: y sin (0 Marks) Reputed Institute for IAS, IFoS Eams Page 8
9 106. Solve the differential equation: 3 3 d y d y 1 y (0 Marks) 107. Solve the initial value problem, y (0) 0 3 y y (0 Marks) 108. Solve( y 3 y) ( y 3 y) 0 (0 Marks) 109. Assume that a spherical rain drop evaporates at a rate proportional to its surface area. If its radius originally is 3mm, and one hour later has been reduced to mm. find an epression for the radius of the rain drop at any time. (0 Marks) d y d y d y Solve e sin 4 3 (0 Marks) 111. Make use of the transformation y( ) u( )sec to obtain the solution of y'' y'tan 5y 0, y(0) 0, y'(0) 6 (0 Marks) 11. Solve (1 ) d y 6(1 ) 16y 8(1 ), y(0) 0, y'(0) (0 Marks) 115. Solve : Find the curves for which the sum of the reciprocals of the radius vector and polar sub tangent is constant. (0 Marks) 114. Solve : ( y p) yp, p (0 Marks) ysin 1 y cos 0 (0 Marks) 116. d y 10y 37sin3 0 y when, if it is given that y 3 and 0 when 0 (0 Marks) d y d y d y Solve : 3 3e 4sin 4 3 (0 Marks) d y d y Solve : 3 y log 3 (0 Marks) Determine a family of curves for which the ratio of the y- intercept of the tangent to the radius vector is a constant. (0 Marks) 10. Solve( 3y 7) ( 3 y 8) y 0 (0 Marks) 11. Test whether the equation ( y) ( y y ) 0 is eact and hence solve it. (0 Marks) Reputed Institute for IAS, IFoS Eams Page 9
10 d y d y 1 Solve y 10 3 (0 Marks) 13. Determine all real valued solutions of the equations y''' iy'' y' iy 0, y' (0 Marks) 14. Find the solution of the equation 4y 8cos, given that 0 (0 Marks) 3 sin y cos y Solve: (0 Marks) 16. Show that if 1 P Q Q y is a function of ( ) only, say, then F( ) e f is an integration factor of P Q 0 (0 Marks) 17. Find the family of curves whose tangent from an angel 4 f( ), with the hyperbola y c Reputed Institute for IAS, IFoS Eams Page 10 (0 Marks) d y Transform the differential equation cos sin ycos cos into one having z as independent variable where and solve it. (0 Marks) d g 19. If ( ) 0 a ( aband g being positive constants) and a' and 0 when 0, dt b dt t g show that a ( a' a)cost t (0 Marks) b 130. Solve ( D 4D 4) y 8 e sin where D (0 Marks), z sin Determine the curvature for which the radius of curvature is proportional to the slope of the tangent. (0 Marks) 13. y Show that the system of co focal conics 1 is self orthogonal. a b (0 Marks) Solve y1 cos y log sin y 0 (0 Marks) 134. Solve 135. Solve d y y y 0 cos (0 Marks) dt y a t and discuss the nature of solution as (0 Marks) 4 / Solve ( D D 1) y e cos (0 Marks)
11 137. By eliminating the constants abobtain the differential equation for which y ae be is a solution (0 Marks) 138. Find the orthogonal trajectory of the family of semi cubical parabolas where a is a variable parameter. (0 Marks) 139. Show that (4 3y 1) (3 y 1) 0 represents hyperbolas having the following lines as asymptotes y 0, y 1 0 (0 Marks) 140. Solve the following differential equation y(1 y) (1 y) 0 (0 Marks) 141. Find the curves for which the portion of the cube of the abscissa of the point of contact. (0 Marks) 14. Solve the following differential equation: ( D 4) y sin, given that when and, ay 3, y- ais cut off between the origin and the tangent varies as 0,then y 0 (0 Marks) Solve : ( D 1) y e cos (0 Marks) 144. Solve : ( D D 4) y (0 Marks) Reputed Institute for IAS, IFoS Eams Page 11
Ordinary Differential Equations (ODE)
++++++++++ Ordinary Differential Equations (ODE) Previous year Questions from 07 to 99 Ramanasri Institute W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : 8 7 5 0 7 0 6 6 /
Previous Years Questions ( ) Segment-wise
Institute for IAS/ IFoS/ CSIR/ GATE Eaminations Preious Years Questions (98 ) Segment-wise Orinary Differential Equations an Laplace Transforms (Accoring to the New Syllabus Pattern) Paper - I 98 Sole
Partial Differential Equations
++++++++++ Partial Differential Equations Previous year Questions from 017 to 199 Ramanasri Institute 017 W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : 8 7 5 0 7 0 6 6 /
CALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=
APPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
PRACTICE PAPER 6 SOLUTIONS
PRACTICE PAPER 6 SOLUTIONS SECTION A I.. Find the value of k if the points (, ) and (k, 3) are conjugate points with respect to the circle + y 5 + 8y + 6. Sol. Equation of the circle is + y 5 + 8y + 6
Differential Equation. Introduction Booklet - I With the help of this booklet you can easily follow the concepts of differential equations.
Differential Equation Introduction Booklet - I With the help of this booklet you can easily follow the concepts of differential equations. Dr. Ajay Kumar Tiwari 1/16/019 Definition The equation which contains
6675/01 Edexcel GCE Pure Mathematics P5 Further Mathematics FP2 Advanced/Advanced Subsidiary
6675/1 Edecel GCE Pure Mathematics P5 Further Mathematics FP Advanced/Advanced Subsidiary Monday June 5 Morning Time: 1 hour 3 minutes 1 1. (a) Find d. (1 4 ) (b) Find, to 3 decimal places, the value of.3
Time : 3 hours 02 - Mathematics - July 2006 Marks : 100 Pg - 1 Instructions : S E CT I O N - A
Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Instructions :. Answer all questions.. Write your answers according to the instructions given below with the questions. 3. Begin each section on a new
2 nd ORDER O.D.E.s SUBSTITUTIONS
nd ORDER O.D.E.s SUBSTITUTIONS Question 1 (***+) d y y 8y + 16y = d d d, y 0, Find the general solution of the above differential equation by using the transformation equation t = y. Give the answer in
UBC-SFU-UVic-UNBC Calculus Exam Solutions 7 June 2007
This eamination has 15 pages including this cover. UBC-SFU-UVic-UNBC Calculus Eam Solutions 7 June 007 Name: School: Signature: Candidate Number: Rules and Instructions 1. Show all your work! Full marks
Drill Exercise Differential equations by abhijit kumar jha DRILL EXERCISE - 1 DRILL EXERCISE - 2. e x, where c 1
DRILL EXERCISE -. Find the order and degree (if defined) of the differential equation 5 4 3 d y d y. Find the order and degree (if defined) of the differential equation n. 3. Find the order and degree
The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.
Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.
1985 AP Calculus AB: Section I
985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of
MATHEMATICS Code No. 13 INSTRUCTIONS
DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO COMBINED COMPETITIVE (PRELIMINARY) EXAMINATION, 00 Serial No. MATHEMATICS Code No. A Time Allowed : Two Hours Maximum Marks : 00 INSTRUCTIONS.
Chapter 3: Topics in Differentiation
Chapter 3: Topics in Differentiation Summary: Having investigated the derivatives of common functions in Chapter (i.e., polynomials, rational functions, trigonometric functions, and their combinations),
(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
![(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)](/thumbs/85/92620692.jpg "(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)")
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
DIFFERENTIAL EQUATION. Contents. Theory Exercise Exercise Exercise Exercise
DIFFERENTIAL EQUATION Contents Topic Page No. Theor 0-0 Eercise - 04-0 Eercise - - Eercise - - 7 Eercise - 4 8-9 Answer Ke 0 - Sllabus Formation of ordinar differential equations, solution of homogeneous
y x is symmetric with respect to which of the following?
AP Calculus Summer Assignment Name: Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number. Part : Multiple Choice Solve
2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y), 2 3
. Find an equation for the line that contains the points (, -) and (6, 9).. Find the value of y for which the line through A and B has the given slope m: A(-, ), B(4, y), m.. Find an equation for the line
Reg. No. : Question Paper Code : B.E./B.Tech. DEGREE EXAMINATION, JANUARY First Semester. Marine Engineering
WK Reg No : Question Paper Code : 78 BE/BTech DEGREE EXAMINATION, JANUARY 4 First Semester Marine Engineering MA 65 MATHEMATICS FOR MARINE ENGINEERING I (Regulation ) Time : Three hours Maimum : marks
MATH Final Review
MATH 1592 - Final Review 1 Chapter 7 1.1 Main Topics 1. Integration techniques: Fitting integrands to basic rules on page 485. Integration by parts, Theorem 7.1 on page 488. Guidelines for trigonometric
Chapter 27 AB Calculus Practice Test
Chapter 7 AB Calculus Practice Test The Eam AP Calculus AB Eam SECTION I: Multiple-Choice Questions DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour and 45 minutes Number
Math 190 (Calculus II) Final Review
Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the
AP Calculus (BC) Summer Assignment (169 points)
AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
AP Calculus (BC) Summer Assignment (104 points)
AP Calculus (BC) Summer Assignment (0 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A
Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -), find M (3. 5, 3) (1.
FILL THE ANSWER HERE
HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP. If A, B & C are matrices of order such that A =, B = 9, C =, then (AC) is equal to - (A) 8 6. The length of the sub-tangent to the curve y = (A) 8 0 0 8 ( ) 5 5
Conic section. Ans: c. Ans: a. Ans: c. Episode:43 Faculty: Prof. A. NAGARAJ. 1. A circle
Episode:43 Faculty: Prof. A. NAGARAJ Conic section 1. A circle gx fy c 0 is said to be imaginary circle if a) g + f = c b) g + f > c c) g + f < c d) g = f. If (1,-3) is the centre of the circle x y ax
ENGI 3424 First Order ODEs Page 1-01
ENGI 344 First Order ODEs Page 1-01 1. Ordinary Differential Equations Equations involving only one independent variable and one or more dependent variables, together with their derivatives with respect
Algebra 2 Unit 9 (Chapter 9)
Algebra Unit 9 (Chapter 9) 0. Spiral Review Worksheet 0. Find verte, line of symmetry, focus and directri of a parabola. (Section 9.) Worksheet 5. Find the center and radius of a circle. (Section 9.3)
MULTIPLE CHOICE QUESTIONS SUBJECT : MATHEMATICS Duration : Two Hours Maximum Marks : 100. [ Q. 1 to 60 carry one mark each ] A. 0 B. 1 C. 2 D.
![MULTIPLE CHOICE QUESTIONS SUBJECT : MATHEMATICS Duration : Two Hours Maximum Marks : 100. [ Q. 1 to 60 carry one mark each ] A. 0 B. 1 C. 2 D.](/thumbs/95/125385791.jpg "MULTIPLE CHOICE QUESTIONS SUBJECT : MATHEMATICS Duration : Two Hours Maximum Marks : 100. [ Q. 1 to 60 carry one mark each ] A. 0 B. 1 C. 2 D.")
M 68 MULTIPLE CHOICE QUESTIONS SUBJECT : MATHEMATICS Duration : Two Hours Maimum Marks : [ Q. to 6 carry one mark each ]. If sin sin sin y z, then the value of 9 y 9 z 9 9 y 9 z 9 A. B. C. D. is equal
TMTA Calculus and Advanced Topics Test 2010
. Evaluate lim Does not eist - - 0 TMTA Calculus and Advanced Topics Test 00. Find the period of A 6D B B y Acos 4B 6D, where A 0, B 0, D 0. Solve the given equation for : ln = ln 4 4 ln { } {-} {0} {}
2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2
29 April PreCalculus Final Review 1. Find the slope and y-intercept (if possible) of the equation of the line. Sketch the line: y = 3x + 13 2. Determine the domain of the function. Verify your result with
QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)
QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents
KCATM 2013 Algebra Team Test. E) No Solution. C x By. E) None of the Above are correct C) 9,19
KCTM 03 lgebra Team Test School ) Solve the inequality: 6 3 4 5 5 3,,3 3, 3 E) No Solution Both and B are correct. ) Solve for : By C C B y C By B C y C By E) None of the bove are correct 3) Which of the
( ) 7 ( 5x 5 + 3) 9 b) y = x x
New York City College of Technology, CUNY Mathematics Department Fall 0 MAT 75 Final Eam Review Problems Revised by Professor Kostadinov, Fall 0, Fall 0, Fall 00. Evaluate the following its, if they eist:
10.1 Review of Parametric Equations
10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations
ALGEBRA 2 X. Final Exam. Review Packet
ALGEBRA X Final Exam Review Packet Multiple Choice Match: 1) x + y = r a) equation of a line ) x = 5y 4y+ b) equation of a hyperbola ) 4) x y + = 1 64 9 c) equation of a parabola x y = 1 4 49 d) equation
Math 1B Final Exam, Solution. Prof. Mina Aganagic Lecture 2, Spring (6 points) Use substitution and integration by parts to find:
Math B Final Eam, Solution Prof. Mina Aganagic Lecture 2, Spring 20 The eam is closed book, apart from a sheet of notes 8. Calculators are not allowed. It is your responsibility to write your answers clearly..
Welcome to Advanced Placement Calculus!! Summer Math
Welcome to Advanced Placement Calculus!! Summer Math - 017 As Advanced placement students, your first assignment for the 017-018 school year is to come to class the very first day in top mathematical form.
3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A
Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(4, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -9), find M 3. A( 2,0)
Math Exam 1a. c) lim tan( 3x. 2) Calculate the derivatives of the following. DON'T SIMPLIFY! d) s = t t 3t
Math 111 - Eam 1a 1) Evaluate the following limits: 7 3 1 4 36 a) lim b) lim 5 1 3 6 + 4 c) lim tan( 3 ) + d) lim ( ) 100 1+ h 1 h 0 h ) Calculate the derivatives of the following. DON'T SIMPLIFY! a) y
1. Find the area enclosed by the curve y = arctan x, the x-axis and the line x = 3. (Total 6 marks)
1. Find the area enclosed by the curve y = arctan, the -ais and the line = 3. (Total 6 marks). Show that the points (0, 0) and ( π, π) on the curve e ( + y) = cos (y) have a common tangent. 3. Consider
A. Correct! These are the corresponding rectangular coordinates.
Precalculus - Problem Drill 20: Polar Coordinates No. 1 of 10 1. Find the rectangular coordinates given the point (0, π) in polar (A) (0, 0) (B) (2, 0) (C) (0, 2) (D) (2, 2) (E) (0, -2) A. Correct! These
Pure Core 2. Revision Notes
Pure Core Revision Notes June 06 Pure Core Algebra... Polynomials... Factorising... Standard results... Long division... Remainder theorem... 4 Factor theorem... 5 Choosing a suitable factor... 6 Cubic
Lecture 17. Implicit differentiation. Making y the subject: If xy =1,y= x 1 & dy. changed to the subject of y. Note: Example 1.
Implicit differentiation. Lecture 17 Making y the subject: If xy 1,y x 1 & dy dx x 2. But xy y 2 1 is harder to be changed to the subject of y. Note: d dx (f(y)) f (y) dy dx Example 1. Find dy dx given
C H A P T E R 9 Topics in Analytic Geometry
C H A P T E R Topics in Analtic Geometr Section. Circles and Parabolas.................... 77 Section. Ellipses........................... 7 Section. Hperbolas......................... 7 Section. Rotation
Math Bank - 6. What is the area of the triangle on the Argand diagram formed by the complex number z, -iz, z iz? (a) z 2 (b) 2 z 2
Math Bank - 6 Q.) Suppose A represents the symbol, B represents the symbol 0, C represents the symbol, D represents the symbol 0 and so on. If we divide INDIA by AGRA, then which one of the following is
Department of Mathematical and Statistical Sciences University of Alberta
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #5 Completion Date: Frida Februar 5, 8 Department of Mathematical and Statistical Sciences Universit of Alberta Question. [Sec.., #
Integration Techniques for the AB exam
For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior
Integration Techniques for the AB exam
For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior
MATHEMATICS PAPER IB COORDINATE GEOMETRY(2D &3D) AND CALCULUS. Note: This question paper consists of three sections A,B and C.
MATHEMATICS PAPER IB COORDINATE GEOMETRY(D &3D) AND CALCULUS. TIME : 3hrs Ma. Marks.75 Note: This question paper consists of three sections A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. 0X =0.
Math 322. Spring 2015 Review Problems for Midterm 2
Linear Algebra: Topic: Linear Independence of vectors. Question. Math 3. Spring Review Problems for Midterm Explain why if A is not square, then either the row vectors or the column vectors of A are linearly
Sample Questions to the Final Exam in Math 1111 Chapter 2 Section 2.1: Basics of Functions and Their Graphs
Sample Questions to the Final Eam in Math 1111 Chapter Section.1: Basics of Functions and Their Graphs 1. Find the range of the function: y 16. a.[-4,4] b.(, 4],[4, ) c.[0, ) d.(, ) e.. Find the domain
ENGI 2422 First Order ODEs - Separable Page 3-01
ENGI 4 First Order ODEs - Separable Page 3-0 3. Ordinary Differential Equations Equations involving only one independent variable and one or more dependent variables, together with their derivatives with
Objective Mathematics
Chapter No - ( Area Bounded by Curves ). Normal at (, ) is given by : y y. f ( ) or f ( ). Area d ()() 7 Square units. Area (8)() 6 dy. ( ) d y c or f ( ) c f () c f ( ) As shown in figure, point P is
Partial Differential Equations
++++++++++ Partial Differential Equations Previous ear Questions from 016 to 199 Ramanasri 017 S H O P NO- 4, 1 S T F L O O R, N E A R R A P I D F L O U R M I L L S, O L D R A J E N D E R N A G A R, N
DIFFERENTIAL EQUATION
MDE DIFFERENTIAL EQUATION NCERT Solved eamples upto the section 9. (Introduction) and 9. (Basic Concepts) : Eample : Find the order and degree, if defined, of each of the following differential equations
(ii) y = ln 1 ] t 3 t x x2 9
![(ii) y = ln 1 ] t 3 t x x2 9](/thumbs/89/100579300.jpg "(ii) y = ln 1 ] t 3 t x x2 9")
Study Guide for Eam 1 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its epression to be well-defined. Some eamples of the conditions are: What is inside
Conic Sections. Geometry - Conics ~1~ NJCTL.org. Write the following equations in standard form.
Conic Sections Midpoint and Distance Formula M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -), find M 2. A(5, 7) and B( -2, -), find M 3. A( 2,0)
MATH 2 - PROBLEM SETS
MATH - PROBLEM SETS Problem Set 1: 1. Simplify and write without negative eponents or radicals: a. c d p 5 y cd b. 5p 1 y. Joe is standing at the top of a 100-foot tall building. Mike eits the building
Differentiating Functions & Expressions - Edexcel Past Exam Questions
- Edecel Past Eam Questions. (a) Differentiate with respect to (i) sin + sec, (ii) { + ln ()}. 5-0 + 9 Given that y =, ¹, ( -) 8 (b) show that = ( -). (6) June 05 Q. f() = e ln, > 0. (a) Differentiate
BE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: Unlimited and Continuous! (21 points)
BE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: United and Continuous! ( points) For #- below, find the its, if they eist.(#- are pt each) ) 7 ) 9 9 ) 5 ) 8 For #5-7, eplain why
y mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent
Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()
IB ANS. -30 = -10 k k = 3
IB ANS.. Find the value of k, if the straight lines 6 0y + 3 0 and k 5y + 8 0 are parallel. Sol. Given lines are 6 0y + 3 0 and k 5y + 8 0 a b lines are parallel a b -30-0 k k 3. Find the condition for
Circles. Example 2: Write an equation for a circle if the enpoints of a diameter are at ( 4,5) and (6, 3).
Conics Unit Ch. 8 Circles Equations of Circles The equation of a circle with center ( hk, ) and radius r units is ( x h) ( y k) r. Example 1: Write an equation of circle with center (8, 3) and radius 6.
1 (C) 1 e. Q.3 The angle between the tangent lines to the graph of the function f (x) = ( 2t 5)dt at the points where (C) (A) 0 (B) 1/2 (C) 1 (D) 3
[STRAIGHT OBJECTIVE TYPE] Q. Point 'A' lies on the curve y e and has the coordinate (, ) where > 0. Point B has the coordinates (, 0). If 'O' is the origin then the maimum area of the triangle AOB is (A)
Math 1160 Final Review (Sponsored by The Learning Center) cos xcsc tan. 2 x. . Make the trigonometric substitution into
Math 60 Final Review (Sponsored by The Learning Center). Simplify cot csc csc. Prove the following identities: cos csc csc sin. Let 7sin simplify.. Prove: tan y csc y cos y sec y cos y cos sin y cos csc
Math 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have
Math 10 Final Eam Review 1. 4 5 6 5 4 4 4 7 5 Worked out solutions. In this problem, we are subtracting one polynomial from another. When adding or subtracting polynomials, we combine like terms. Remember
a Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
CURVATURE AND RADIUS OF CURVATURE
CHAPTER 5 CURVATURE AND RADIUS OF CURVATURE 5.1 Introduction: Curvature is a numerical measure of bending of the curve. At a particular point on the curve, a tangent can be drawn. Let this line makes an
( ) 9 b) y = x x c) y = (sin x) 7 x d) y = ( x ) cos x
NYC College of Technology, CUNY Mathematics Department Spring 05 MAT 75 Final Eam Review Problems Revised by Professor Africk Spring 05, Prof. Kostadinov, Fall 0, Fall 0, Fall 0, Fall 0, Fall 00 # Evaluate
Get Solution of These Packages & Learn by Video Tutorials on SHORT REVISION
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com SHORT REVISION DIFFERENTIAL EQUATIONS OF FIRST
The American School of Marrakesh. AP Calculus AB Summer Preparation Packet
The American School of Marrakesh AP Calculus AB Summer Preparation Packet Summer 2016 SKILLS NEEDED FOR CALCULUS I. Algebra: *A. Exponents (operations with integer, fractional, and negative exponents)
GOVERNMENT OF KARNATAKA KARNATAKA STATE PRE-UNIVERSITY EDUCATION EXAMINATION BOARD SCHEME OF VALUATION. Subject : MATHEMATICS Subject Code : 35
GOVERNMENT OF KARNATAKA KARNATAKA STATE PRE-UNIVERSITY EDUCATION EXAMINATION BOARD II YEAR PUC EXAMINATION MARCH APRIL 0 SCHEME OF VALUATION Subject : MATHEMATICS Subject Code : 5 PART A Write the prime
The Princeton Review AP Calculus BC Practice Test 1
8 The Princeton Review AP Calculus BC Practice Test CALCULUS BC SECTION I, Part A Time 55 Minutes Number of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each
n and C and D be positive constants so that nn 1
Math Activity 0 (Due by end of class August 6). The graph of the equation y is called an astroid. a) Find the length of this curve. {Hint: One-fourth of the curve is given by the graph of y for 0.} b)
AP Calculus BC : The Fundamental Theorem of Calculus
AP Calculus BC 415 5.3: The Fundamental Theorem of Calculus Tuesday, November 5, 008 Homework Answers 6. (a) approimately 0.5 (b) approimately 1 (c) approimately 1.75 38. 4 40. 5 50. 17 Introduction In
CIRCLES: #1. What is an equation of the circle at the origin and radius 12?
1 Pre-AP Algebra II Chapter 10 Test Review Standards/Goals: E.3.a.: I can identify conic sections (parabola, circle, ellipse, hyperbola) from their equations in standard form. E.3.b.: I can graph circles
Warmup for AP Calculus BC
Nichols School Mathematics Department Summer Work Packet Warmup for AP Calculus BC Who should complete this packet? Students who have completed Advanced Functions or and will be taking AP Calculus BC in
Algebra Review. Unit 7 Polynomials
Algebra Review Below is a list of topics and practice problems you have covered so far this semester. You do not need to work out every question on the review. Skip around and work the types of questions
All work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
2018 Pre-Cal Spring Semester Review Name: Per:
08 Pre-Cal Spring Semester Review Name: Per: For # 4, find the domain of each function. USE INTERVAL NOTATION!!. 4 f ( ) 5. f ( ) 6 5. f( ) 5 4. f( ) 4 For #5-6, find the domain and range of each graph.
First Degree Programme in Mathematics. Semester 1 MM 1141 Methods of Mathematics
UNIVERSITY OF KERALA First Degree Programme in Mathematics Model Question Paper Semester 1 MM 1141 Methods of Mathematics Time: Three hours Maximum marks : 80 Section I All the first ten questions are
2. Use your graphing calculator to graph each of the functions below over the interval 2, 2
MSLC Math 0 Final Eam Review Disclaimer: This should NOT be used as your only guide for what to study. if 0. Use the piece-wise defined function f( ) to answer the following: if a. Compute f(0), f(), f(-),
HEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436)
HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA TIME-(HRS) Select the correct alternative : (Only one is correct) MAX-MARKS-(()+0(5)=6) Q. Suppose & are the point of maimum and the point of minimum
e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks)
Chapter 0 Application of differential calculus 014 GDC required 1. Consider the curve with equation f () = e for 0. Find the coordinates of the point of infleion and justify that it is a point of infleion.
abc Mathematics Further Pure General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES
abc General Certificate of Education Mathematics Further Pure SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER
and y c from x 0 to x 1
Math 44 Activity 9 (Due by end of class August 6). Find the value of c, c, that minimizes the volume of the solid generated by revolving the region between the graphs of y 4 and y c from to about the line
Math 1050 Final Exam Form A College Algebra Fall Semester Student ID ID Verification Section Number
Math 1050 Final Eam Form A College Algebra Fall Semester 2010 Instructor Name Student ID ID Verification Section Number Time Limit: 120 minutes Graphing calculators without CAS are allowed. All problems
Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.
Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.
Solution: As x approaches 3, (x 2) approaches 1, so ln(x 2) approaches ln(1) = 0. Therefore we have a limit of the form 0/0 and can apply the.
MATH, solutions to practice problems for the final eam. Compute the it: a) 3 e / Answer: e /3. b) 3 ln( 3) Answer:. c) 3 ln( ) 3 Solution: As approaches 3, ( ) approaches, so ln( ) approaches ln() =. Therefore
Math 1431 Final Exam Review. 1. Find the following limits (if they exist): lim. lim. lim. lim. sin. lim. cos. lim. lim. lim. n n.
. Find the following its (if they eist: sin 7 a. 0 9 5 b. 0 tan( 8 c. 4 d. e. f. sin h0 h h cos h0 h h Math 4 Final Eam Review g. h. i. j. k. cos 0 n nn e 0 n arctan( 0 4 l. 0 sin(4 m. cot 0 = n. = o.
1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: and. So the slopes of the tangent lines to the curve
MAT 11 Solutions TH Eam 3 1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: Therefore, d 5 5 d d 5 5 d 1 5 1 3 51 5 5 and 5 5 5 ( ) 3 d 1 3 5 ( ) So the
Chapter 4: Higher Order Linear Equations
Chapter 4: Higher Order Linear Equations MATH 351 California State University, Northridge April 7, 2014 MATH 351 (Differential Equations) Ch 4 April 7, 2014 1 / 11 Sec. 4.1: General Theory of nth Order
Coordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general
A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate
IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB
` KUKATPALLY CENTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 017-18 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500
A BRIEF REVIEW OF ALGEBRA AND TRIGONOMETRY
A BRIEF REVIEW OF ALGEBRA AND TRIGONOMETR Some Key Concepts:. The slope and the equation of a straight line. Functions and functional notation. The average rate of change of a function and the DIFFERENCE-
SECTION A Time allowed: 20 minutes Marks: 20
Mathcity.org Merging man and maths Federal Board HSSC-II Eamination Mathematics Model Question Paper Roll No: Answer Sheet No: FBISE WE WORK FOR EXCELLENCE Signature of Candidate: Signature of Invigilator: