Important Instructions to the Examiners:
|
|
- Maurice Horn
- 5 years ago
- Views:
Transcription
1 Winter 0 Eamination Subject & Code: Basic Maths (70) Model Answer Page No: / Important Instructions to the Eaminers: ) The Answers should be eamined by key words and not as word-to-word as given in the model answer scheme. ) The model answer and the answer written by candidate may vary but the eaminer may try to assess the understanding level of the candidate. ) The language errors such as grammatical, spelling errors should not be given more importance. (Not applicable for subject English and Communication Skills.) ) While assessing figures, eaminer may give credit for principal components indicated in the figure. The figures drawn by the candidate and those in the model answer may vary. The eaminer may give credit for any equivalent figure drawn. 5) Credits may be given step wise for numerical problems. In some cases, the assumed constant values may vary and there may be some difference in the candidate s Answers and the model answer. 6) In case of some questions credit may be given by judgment on part of eaminer of relevant answer based on candidate s understanding. 7) For programming language papers, credit may be given to any other program based on equivalent concept.
2 Subject & Code: Basic Maths (70) Page No: / ) a) b) Attempt any TEN of the following: 5 Find the value of If A, find the matri B such that 0 A + B 6 A 8 6 B A 8 6 B 8 A + B 0 B A B 8 c) Find the value of a and b, if a b a b a b a + b 5 b 6 a b a + b 6 5 b a
3 Subject & Code: Basic Maths (70) Page No: / ) d) e) 6 Find the adjoint of matri 7 6 Let A 7 7 C ( A) adj ( A) 6 Let A 7 A 7 A A 6 A 7 C ( A) adj ( A) Resolve into partial fractions: A B A + B Put 0 i. e., + A + 0 A A Put + 0 i. e., B B 0 + B + +
4 Subject & Code: Basic Maths (70) Page No: / ) A B A + + B A B + + Note for partial fraction problems: The problems of partial fractions could also be solved by the method of equating equal power coefficients. This method is also applicable. Give appropriate marks in accordance with the scheme of marking in the later problems as the solution by this method is not discussed. For the sake of convenience, the solution of the above problem with the help of this method is illustrated hereunder. A B A + B A + A + B B + 0 A + B + A B A + B A B 0 A + B A B 0 A A B A B + +
5 Subject & Code: Basic Maths (70) Page No: 5/ ) f) Show that π tanθ tan θ + tan θ g) π tan tanθ π tan θ π + tan tanθ tanθ + tanθ Prove that cos A cos A cos A cos A + A cos Acos A sin Asin A cos A sin ( A) cos A cos cos A + cos cos A cos cos sin A A A A A ( A) cos A cos cos A h) If sin A 0., find the value of sin A. sin sin sin A A A * Note (*): Due to the use of advance scientific calculator, writing directly the step (*) is allowed. No marks to be deducted. Given that sin A 0.. A sin º sin A sin.578º 0.9
6 Subject & Code: Basic Maths (70) Page No: 6/ ) i) Prove that cos θ sin θ + cos θ cosθ sinθ cos θ sin θ sinθ cos θ + cosθ sin θ + cosθ sinθ cosθ sinθ sin ( θ + θ ) cosθ sinθ sin θ cosθ sinθ sin ( θ ) cosθ sinθ sin θ cos θ cosθ sinθ sinθ cosθ cos θ cosθ sinθ cos θ cos sin cos cos sin sin + + cosθ sinθ cosθ sinθ cos θ + sin θ θ θ θ θ θ θ cos θ sin θ ( θ θ ) cos sin cos θ j) Evaluate without using calculator tan 66º + tan 69º tan 66º tan 69º tan 66º + tan 69º tan 66º + 69º tan 66º tan 69º tan5º tan 90º + 5º tan 80º 5º cot 5º tan 5º
7 Subject & Code: Basic Maths (70) Page No: 7/ ) k) y Find the slope and y-intercept of the line y 0 a b c a slope m b or 0.75 c y int 6 b y y 0 a b c a slope m b or 0.75 c y int 6 b y y 6 slope m or 0.75 y int 6 l) Find the range of the following:,,, 0, 6,, 7, 0, L S Range L S 0
8 Subject & Code: Basic Maths (70) Page No: 8/ ) Attempt any FOUR of the following: a) Solve the equations for y and z y z y z y z + 5, +, by using Cramer s rule. y z + 5 y z + 5 y z D D D y z or or or. D y.8 y 08.5 D Dz. z.970 D (Please refer note on the net page)
9 Subject & Code: Basic Maths (70) Page No: 9/ ) Note: As the use of the advance scientific calculator is permissible, calculating directly the values of fractional quantities e.g., is allowed. The same is also applicable in the net alternative method. No marks to be deducted for such direct calculations. y + 6z y 6z 0 8 y + z 5 6 D y D b) z D Dy 5586 y D 95 Dz 5590 z.970 D 95 If A, find A. A A A (Please check note on net page)
10 Subject & Code: Basic Maths (70) Page No: 0/ ) Note: In the answer matri of A², if to elements are wrong either in sign or value, deduct marks; if to 6 elements are wrong, you may deduct mark; other deduct all marks. c) If A, B, C 0, verify that A B + C AB + AC. B + C + 0 A( B + C ) AB AC AB + AC A B + C AB + AC
11 Subject & Code: Basic Maths (70) Page No: / ) d) If A, find A A 9I +, where I is the unit matri A A A A I A A + 9I Note: The above problem could also be solved by taking all the terms simultaneously as follows: A A + 9I
12 Subject & Code: Basic Maths (70) Page No: / ) e) Resolve into partial fractions: y + y + 5y + ( Put y) y + A B + y + y + y + y + y + y + A + y + B Put y + 0 or y + + A + 0 A A Put y + 0 or y B B B
13 Subject & Code: Basic Maths (70) Page No: / ) y + + y + 5y + y + y f) Resolve into partial fractions: A B A + B Put 0 i. e., 0 6A A Put + 0 i. e., 0 0 6B 5 B ) Attempt any FOUR of the following: a) Solve the equations + y + z, + y + z, + y + z by using matri inversion method. + y + z + y + z + y + z
14 Subject & Code: Basic Maths (70) Page No: / ) A X y K z A C ( A) The minor matri of A is M ( A) the mati of cofactors is, 8 5 C ( A) 5 5 The minors of matri A are A 8 A A 5 A A 5 A A 5 A A
15 Subject & Code: Basic Maths (70) Page No: 5/ ) the mati of cofactors is, 8 5 C ( A) adj ( A) A X A K y z b) Resolve into partial fractions: ( + ) + + A B + C + ( )( + ) + A B + C A B C
16 Subject & Code: Basic Maths (70) Page No: 6/ ) Put + + A A 9 A 5 Put 0 A ( C ) A C 9 0 C 5 9 C 5 C 5 Put A ( B C ) A B C 9 B B B 5 B ( + ) Note for Partial Fraction Methods: The above Q. (e) & (f) problems of partial fractions could be solved by the method of equating equal power coefficients also. This method, illustrated in the solution of Q. (e), is also applicable. Give appropriate marks in accordance with the scheme of marking. As this method is very tedious and complicated, hardly someone use this method in such cases. So such solution methods for partial fraction problems are not illustrated herein.
17 Subject & Code: Basic Maths (70) Page No: 7/ ) c) e + Resolve into partial fractions: e + 7e + 5 e + e + 7e + 5 y + y + 7 y + 5 y + ( y + 5)( y + ) y + 5 e + 5 ( Put e y) e + e + 7e + 5 y + y + 7y + 5 ( Put e y) y + A B + y + 5 y + y + 5 y + y + y + A + y + 5 B 5 Put y y A + 0 A A Put y + 0 y B 0 B 0 B y y + y + y + y e + e + e + e
18 Subject & Code: Basic Maths (70) Page No: 8/ ) d) Prove that sin( A+ B) sin A.cos B + cos A.sin B QN sin( A + B) OQ QR + RN OQ QR + PM OQ QR PM + OQ OQ QR PQ PM OP + PQ OQ OP OQ cos A.sin B + sin A.cos B e) Note: The above is proved by different ways in several books. Consider all these proof but check whether the method is falling within the scope of curriculum and give appropriate marks in accordance with the scheme of marking. In accordance with the Teacher s Manual published by MSBTE, the result is treated as Fundamental Result which is not proved by the help of any another result. If the above result is proved by students using any another result, suppose using cos (A+B), then this result i.e., cos (A+B) must have been proved first. 5 π Prove that cot ( ) + cos ec cot ( ) tan tan tan +
19 Subject & Code: Basic Maths (70) Page No: 9/ ) cot ( ) tan + tan tan tan + Let A cos ec 5 cos eca 5 5 cot ( ) + cos ec tan + tan tan tan π + 5 cot ( ) + cos ec tan + cot π Note that the result tan cot π + can be used directly
20 Subject & Code: Basic Maths (70) Page No: 0/ ) f) tan + tan + tan π Prove that π π + π π π + + tan ( ) π π + π π tan tan tan tan tan + + tan π + + tan tan tan tan tan π + + tan tan π + π tan tan + tan + tan + tan ( ) tan + π + tan π π + + tan tan + tan tan π ) Attempt any FOUR of the following. a) Without using the calculator, find the value of + sin 0º cos ec 0º + cos 70º tan 0º cot 0º tan 0º sin0º sin 90º + 0º cos 0º
21 Subject & Code: Basic Maths (70) Page No: / ) sin 0º cos ec0º cos 0º ec cot 0º cos 70º 0 cot0º cot 90º + 0º tan 0º cos 70º cos 90º + 0 sin 0 0 But given that + sin 0º cos ec 0º + cos 70º tan 0º cot 0º or.5 b) Prove that cos A + cos5 A + cos7 A cos A sin Atan A cos A + cosa + cos5a cosa + cos5a + cos7a cosa + cos 7A + cos5a cos A + cosa + cos5a cos A + cos5a + cosa cos5acos A + cos5a cosacos A + cosa ( A) ( A) cos5a cos + cosa cos + cos5a cosa cos( A + A) cosa
22 Subject & Code: Basic Maths (70) Page No: / ) cos AcosA sin Asin A cosa cos A sin Atan A c) Prove that (in ABC ), tan A + tan B + tan C tan A tan B tan C We have, A + B + C 80º or π A + B 80º C ( A B) ( C) tan + tan 80º tan A + tan B tan C tan Atan B tan A + tan B tan C tan Atan B [ ] tan A + tan B tan C + tan Atan B tan C tan A + tan B + tan C tan Atan B tan C d) Prove that tanθ tan θ tan θ tan θ tan θ tan θ + θ tanθ + tan θ tanθ tan θ tanθ tanθ + tan θ tanθ tan tan θ θ tanθ tan θ + tanθ tan θ tan θ tanθ tanθ tan θ tanθ tan θ + tanθ tan θ tan θ tanθ tan θ tan θ
23 Subject & Code: Basic Maths (70) Page No: / ) e) Prove that cos + cos cos 5 65 A cos B cos 5 cos A cos B 5 cos A + B cos Acos B sin Asin B A + B cos 65 cos + cos cos 5 65 f) If 5 tan, tan y, show that 6 π + y 5 tan, tan y 6 5 tan, y tan y tan + tan 6 tan tan π
24 Subject & Code: Basic Maths (70) Page No: / ) 5 tan, tan y 6 tan + tan y tan ( + y) tan tan y π + y tan 5) Attempt any FOUR of the following. a) Without using calculator prove that cos 0º cos 0º cos 60º cos80º 6 cos 0º cos 0º cos 60º cos 80º cos 0º cos 0º cos 80º ( cos 0º cos 0º ) cos 80º ( cos 60º + cos 0º ) cos 80º + cos 0º cos 80º cos 80º + cos 80º cos 0º cos 80º + cos 80º cos 0º cos 80º + ( c ) os00º + cos 60º cos 80º cos00º cos 90º cos ( 0º )
25 Subject & Code: Basic Maths (70) Page No: 5/ 5) Note The above problem may also be solved by making various combinations of cosine ratios. Consequently the solutions vary in accordance with the combinations. Please give the appropriate marks in accordance with the scheme of marking. For the sake of convenience one of the solutions is illustrated hereunder. cos 0º cos 0º cos60º cos80º cos 0º cos 0º cos80º ( cos 0º cos80º ) cos 0º ( cos0º + cos 0º ) cos 0º ( cos ( 90º + 0º ) + cos 0º ) cos 0º ( sin 0º + cos 0º ) cos 0º + cos 0º cos 0º cos 0º + cos 0º cos 0º cos 0º + cos 0º cos 0º cos 0º cos 60º cos ( 0º ) + + cos 0º cos 0º
26 Subject & Code: Basic Maths (70) Page No: 6/ 5) b) Prove that sin + sin 5 + sin 6 tan 5 cos + cos5 + cos6 sin + sin 5 + sin 6 sin + sin 6 + sin 5 cos + cos5 + cos 6 cos + cos 6 + cos5 sin 5 cos + sin 5 cos5 cos + cos5 sin 5 cos + cos5 cos + tan 5 + c) Prove that + y y tan + tan y tan, > 0, y > 0, y < Put tan A and tan y B tan A and y tan B tan A + tan B tan ( A + B) tan A tan B + y y + y y A + B tan + y y tan + tan y tan d) Find the equation of a straight line passing through (, 5) and the point of intersection of the lines + y 0, y 9. + y 0 y 9 9 y Point of intersection,
27 Subject & Code: Basic Maths (70) Page No: 7/ 5) equation is, y y y y y y 0 ( ) Point of intersection, y y Slope m equation is, y y m y y 0 e) Find the equation of the straight line passing through (-, 0) and sum of their intercepts is 8. Let int a y int b a + b 8 equation is y y + or + a b a 8 a b + ay ab ( 8 a) ay a( 8 a) ( ) ( 8 a) 0a a ( 8 a) + But passing through, a + 0a 8a a 5a 0 + a a, 8 y y + or
28 Subject & Code: Basic Maths (70) Page No: 8/ 5) f) Find the acute angle between the lines + y, 5y For + y, a slope m b For 5y + 7 0, a slope m b 5 5 m m tanθ + m m or.55 6 θ tan or tan (.55) 6) Attempt any FOUR of the following. a) Find the equation of straight line passing through (5, 6) and making an angle 50º with -ais. Given θ 50º slope m tanθ tan50º equation is ( ) y y m y 6 5 y y 6 5 0
29 Subject & Code: Basic Maths (70) Page No: 9/ 6) equation is θ ( ) y y tan y 6 tan50º 5 y 6 5 ( ) y y b) If the length of perpendicular from (5, ) on the straight line + y + k 0 is 5 units. Find the value of k. p 5 5 a + by + c a k + k b k 0 + k 0 + k or 0 + k + 6 k or k + c) The scores of two batsmen A and B in ten innings during a certain season are as under: A B Find which of the two batsmen is more consisting in scoring (use coefficient of variance).
30 Subject & Code: Basic Maths (70) Page No: 0/ 6) For Batsman A: i σ CV ( A) i For Batsman B: i σ CV ( B) i
31 Subject & Code: Basic Maths (70) Page No: / 6) CV ( A) CV B < B is more consistent. d) Find the range and the coefficient of range for the following: of Students L 99 S 0 Difference between two sets D Range L S + D L S + D Coeff. of Range L + S or Class Cont. Class L 99.5 S 9.5 Range L S L S Coeff. of Range L + S or
32 Subject & Code: Basic Maths (70) Page No: / 6) e) Calculate the mean deviation for the following data: Class intervals of families Class i f i fi i Di i fid i fii N 0 fidi M. D. N f) Find the variance and coefficient of variance for the following distribution: Class Intervals Frequency Class i f i fi i i fi i
33 Subject & Code: Basic Maths (70) Page No: / 6) fii N 05 fii fii S. D. N N Variance ( S. D. ) S. D. Coeff. of Variance fii fii Variance N N variance Coeff. of Variance Class i f i d i i i f d d i f d i i
34 Subject & Code: Basic Maths (70) Page No: / 6) i A A.5, h 5, di h fidi A + h N fidi fidi S. D. h N N Variance ( S. D. ) S. D. Coeff. of Variance f d f d i i i i Variance h N N variance Coeff. of Variance Important Note In the solution of the question paper, wherever possible all the possible alternative methods of solution are given for the sake of convenience. Still student may follow a method other than the given herein. In such case, FIRST SEE whether the method falls within the scope of the curriculum, and THEN ONLY give appropriate marks in accordance with the scheme of marking.
MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION (Autonomous) (ISO/IEC Certified)
SUMMER 8 EXAMINATION Important Instructions to eaminers: ) The answers should be eamined by key words and not as word-to-word as given in the model answer scheme. ) The model answer and the answer written
More informationWINTER 16 EXAMINATION
(ISO/IEC - 700-005 Certified) WINTER 6 EXAMINATION Model wer ject Code: Important Instructions to examiners: ) The answers should be examined by key words and not as word-to-word as given in the model
More informationImportant Instructions to the Examiners:
(ISO/IEC - 7 - Certified) Winter Eamination Subject & Code: Applied Maths (7) Model Answer Page No: /6 Important Instructions to the Eaminers: ) The answers should be eamined by key words and not as word-to-word
More informationImportant Instructions to the Examiners:
Summer 0 Examnaton Subject & Code: asc Maths (70) Model Answer Page No: / Important Instructons to the Examners: ) The Answers should be examned by key words and not as word-to-word as gven n the model
More informationAnswer. Find the gradient of the curve y x at x 4
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: Important Instructions to eaminers: ) The answers should be eamined by key words and not as word-to-word
More informationADDITIONAL MATHEMATICS
005-CE A MATH HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 005 ADDITIONAL MATHEMATICS :00 pm 5:0 pm (½ hours) This paper must be answered in English 1. Answer ALL questions in Section A and any FOUR
More informationNATIONAL QUALIFICATIONS
Mathematics Higher Prelim Eamination 04/05 Paper Assessing Units & + Vectors NATIONAL QUALIFICATIONS Time allowed - hour 0 minutes Read carefully Calculators may NOT be used in this paper. Section A -
More informationMATH 127 SAMPLE FINAL EXAM I II III TOTAL
MATH 17 SAMPLE FINAL EXAM Name: Section: Do not write on this page below this line Part I II III TOTAL Score Part I. Multiple choice answer exercises with exactly one correct answer. Each correct answer
More informationGOVERNMENT 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
More informationPublic Assessment of the HKDSE Mathematics Examination
Public Assessment of the HKDSE Mathematics Examination. Exam Format (a) The examination consists of one paper. (b) All questions are conventional questions. (c) The duration is hours and 30 minutes. Section
More informationTHE COMPOUND ANGLE IDENTITIES
TRIGONOMETRY THE COMPOUND ANGLE IDENTITIES Question 1 Prove the validity of each of the following trigonometric identities. a) sin x + cos x 4 4 b) cos x + + 3 sin x + 2cos x 3 3 c) cos 2x + + cos 2x cos
More informationWritten as per the revised G Scheme syllabus prescribed by the Maharashtra State Board of Technical Education (MSBTE) w.e.f. academic year
Written as per the revised G Scheme syllabus prescribed by the Maharashtra State Board of Technical Education (MSBTE) w.e.f. academic year 2012-2013 Basic MATHEMATICS First Year Diploma Semester - I First
More informationFIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, TECHNICAL MATHEMATICS- I (Common Except DCP and CABM)
TED (10)-1002 (REVISION-2010) Reg. No.. Signature. FIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, 2010 TECHNICAL MATHEMATICS- I (Common Except DCP and CABM) (Maximum marks: 100)
More informationACS MATHEMATICS GRADE 10 WARM UP EXERCISES FOR IB HIGHER LEVEL MATHEMATICS
ACS MATHEMATICS GRADE 0 WARM UP EXERCISES FOR IB HIGHER LEVEL MATHEMATICS DO AS MANY OF THESE AS POSSIBLE BEFORE THE START OF YOUR FIRST YEAR IB HIGHER LEVEL MATH CLASS NEXT SEPTEMBER Write as a single
More informationChapter 6. Trigonometric Functions of Angles. 6.1 Angle Measure. 1 radians = 180º. π 1. To convert degrees to radians, multiply by.
Chapter 6. Trigonometric Functions of Angles 6.1 Angle Measure Radian Measure 1 radians = 180º Therefore, o 180 π 1 rad =, or π 1º = 180 rad Angle Measure Conversions π 1. To convert degrees to radians,
More informationabc 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
More informationContact hour per week: 04 Contact hour per Semester: 64 ALGEBRA 1 DETERMINANTS 2 2 MATRICES 4 3 BINOMIAL THEOREM 3 4 LOGARITHMS 2 5 VECTOR ALGEBRA 6
BOARD OF TECHNICAL EXAMINATION KARNATAKA SUBJECT: APPLIED MATHEMATICS I For I- semester DIPLOMA COURSES OF ALL BRANCHES Contact hour per week: 04 Contact hour per Semester: 64 UNIT NO. CHAPTER TITLE CONTACT
More informationPreliminary Mathematics
NORTH SYDNEY GIRLS HIGH SCHOOL 010 YEARLY EXAMINATION Preliminary Mathematics General Instructions Reading Time 5 minutes Working Time hours Write using black or blue pen Board-approved calculators may
More informationMA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically
1 MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically Definition Trigonometric identity Investigate 1. Using the diagram
More informationC3 Exam Workshop 2. Workbook. 1. (a) Express 7 cos x 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2
C3 Exam Workshop 2 Workbook 1. (a) Express 7 cos x 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2 π. Give the value of α to 3 decimal places. (b) Hence write down the minimum value of 7 cos
More informationReview of Topics in Algebra and Pre-Calculus I. Introduction to Functions function Characteristics of a function from set A to set B
Review of Topics in Algebra and Pre-Calculus I. Introduction to Functions A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in set B.
More information0606 ADDITIONAL MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the October/November 0 series 0606 ADDITIONAL MATHEMATICS 0606/ Paper, maimum raw
More informationMathematics Trigonometry: Unit Circle
a place of mind F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagog Mathematics Trigonometr: Unit Circle Science and Mathematics Education Research Group Supported b UBC Teaching and
More information*n23494b0220* C3 past-paper questions on trigonometry. 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (2)
C3 past-paper questions on trigonometry physicsandmathstutor.com June 005 1. (a) Given that sin θ + cos θ 1, show that 1 + tan θ sec θ. (b) Solve, for 0 θ < 360, the equation tan θ + secθ = 1, giving your
More informationThese items need to be included in the notebook. Follow the order listed.
* Use the provided sheets. * This notebook should be your best written work. Quality counts in this project. Proper notation and terminology is important. We will follow the order used in class. Anyone
More informationMATHEMATICS. Time allowed : 3 hours Maximum Marks : 100
MATHEMATICS Time allowed : hours Maimum Marks : General Instructions:. All questions are compulsory.. The question paper consists of 9 questions divided into three sections, A, B and C. Section A comprises
More informationTime : 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
More informationoo ks. co m w w w.s ur ab For Order : orders@surabooks.com Ph: 960075757 / 84000 http://www.trbtnpsc.com/07/08/th-eam-model-question-papers-download.html Model Question Papers Based on Scheme of Eamination
More informationLesson-3 TRIGONOMETRIC RATIOS AND IDENTITIES
Lesson- TRIGONOMETRIC RATIOS AND IDENTITIES Angle in trigonometry In trigonometry, the measure of an angle is the amount of rotation from B the direction of one ray of the angle to the other ray. Angle
More informationWritten as per the revised G Scheme syllabus prescribed by the Maharashtra State Board of Technical Education (MSBTE) w.e.f. academic year
Written as per the revised G Scheme syllabus prescribed by the Maharashtra State Board of Technical Education (MSBTE) w.e.f. academic year 2012-2013 Basic MATHEMATICS First Year Diploma Semester - I First
More informationProblems with an # after the number are the only ones that a calculator is required for in the solving process.
Instructions: Make sure all problems are numbered in order. (Level : If the problem had an *please skip that number) All work is in pencil, and is shown completely. Graphs are drawn out by hand. If you
More informationMATHEMATICS CLASS : XI. 1. Trigonometric ratio identities & Equations Exercise Fundamentals of Mathematics - II Exercise 28-38
CONTENT Preface MATHEMATICS CLASS : XI Page No.. Trigonometric ratio identities & Equations Eercise 0-7. Fundamentals of Mathematics - II Eercise 8-8. Straight Line Eercise 9-70 4. Circle Eercise 70-9
More informationabc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES
abc General Certificate of Education Mathematics Pure Core SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS
More informationPhysicsAndMathsTutor.com
PhysicsAndMathsTutor.com physicsandmathstutor.com June 2005 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (b) Solve, for 0 θ < 360, the equation 2 tan 2 θ + secθ = 1, giving your
More information0606 ADDITIONAL MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the March 06 series 0606 ADDITIONAL MATHEMATICS 0606/ Paper, maimum raw mark 80 This
More informationMATH 52 MIDTERM 1. April 23, 2004
MATH 5 MIDTERM April 3, Student ID: Signature: Instructions: Print your name and student ID number and write your signature to indicate that you accept the honor code. During the test, you may not use
More informationIntroduction Assignment
FOUNDATIONS OF MATHEMATICS 11 Welcome to FOM 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying this year.
More informationVersion 1.0: abc. General Certificate of Education. Mathematics MPC2 Pure Core 2. Mark Scheme examination - June series
Version.0: 0608 abc General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 008 eamination - June series Mark schemes are prepared by the Principal Eaminer and considered, together with
More informationHKUST. MATH1014 Calculus II. Directions:
HKUST MATH114 Calculus II Midterm Eamination (Sample Version) Name: Student ID: Lecture Section: Directions: This is a closed book eamination. No Calculator is allowed in this eamination. DO NOT open the
More informationCambridge Assessment International Education Cambridge Pre-U Certificate. Published
Cambridge Assessment International Education Cambridge Pre-U Certificate MATHEMATICS 979/0 Paper Pure Mathematics MARK SCHEME Maimum Mark: 80 Published This mark scheme is published as an aid to teachers
More informationDO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO
DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO T.B.C. : P-AQNA-L-ZNGU Serial No.- TEST BOOKLET MATHEMATICS Test Booklet Series Time Allowed : Two Hours and Thirty Minutes Maximum Marks : 00
More informationCambridge Assessment International Education Cambridge Ordinary Level. Published
Cambridge Assessment International Education Cambridge Ordinary Level ADDITIONAL MATHEMATICS 07/ Paper May/June 08 MARK SCHEME Maimum Mark: 80 Published This mark scheme is published as an aid to teachers
More informationTrigonometry: Graphs of trig functions (Grade 10) *
OpenStax-CNX module: m39414 1 Trigonometry: Graphs of trig functions (Grade 10) * Free High School Science Texts Project This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution
More informationMath 132 Exam 3 Fall 2016
Math 3 Exam 3 Fall 06 multiple choice questions worth points each. hand graded questions worth and 3 points each. Exam covers sections.-.6: Sequences, Series, Integral, Comparison, Alternating, Absolute
More informationLearning Objectives These show clearly the purpose and extent of coverage for each topic.
Preface This book is prepared for students embarking on the study of Additional Mathematics. Topical Approach Examinable topics for Upper Secondary Mathematics are discussed in detail so students can focus
More informationMATH 100 REVIEW PACKAGE
SCHOOL OF UNIVERSITY ARTS AND SCIENCES MATH 00 REVIEW PACKAGE Gearing up for calculus and preparing for the Assessment Test that everybody writes on at. You are strongly encouraged not to use a calculator
More informationwww.onlineeamhelp.com www.onlineeamhelp.com UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Level MARK SCHEME for the May/June question paper for the guidance of teachers FURTHER MATHEMATICS
More informationBasic Maths(M-I) I SCHEME. UNIT-I Algebra
Basic Maths(M-I) I SCHEME UNIT-I Algebra Prepared By : Sameer V. shaikh {Engr.sameer@gmail.com} {9765158158} Website : www.mechdiploma.com, www.diplomamaths.com, msbte.engg info.website Shaikh sir s Reliance
More informationNorth Seattle Community College Computer Based Mathematics Instruction Math 102 Test Reviews
North Seattle Community College Computer Based Mathematics Instruction Math 10 Test Reviews Click on a bookmarked heading on the left to access individual reviews. To print a review, choose print and the
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)
N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions.
More informationDuVal High School Summer Review Packet AP Calculus
DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and
More informationSPECIALIST MATHEMATICS
Victorian Certificate of Education 00 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Words SPECIALIST MATHEMATICS Written eamination Monday November 00 Reading time:.00 pm to.5
More informationGeneral Certificate of Education examination - January series
Version.: 6 General Certificate of Education abc Mathematics 66 MFP Further Pure Mark Scheme 6 eamination - January series Mark schemes are prepared by the Principal Eaminer and considered, together with
More informationDIRECTORATE OF EDUCATION GOVT. OF NCT OF DELHI
456789045678904567890456789045678904567890456789045678904567890456789045678904567890 456789045678904567890456789045678904567890456789045678904567890456789045678904567890 QUESTION BANK 456789045678904567890456789045678904567890456789045678904567890456789045678904567890
More information(Section 4.7: Inverse Trig Functions) 4.82 PART F: EVALUATING INVERSE TRIG FUNCTIONS. Think:
PART F: EVALUATING INVERSE TRIG FUNCTIONS Think: (Section 4.7: Inverse Trig Functions) 4.82 A trig function such as sin takes in angles (i.e., real numbers in its domain) as inputs and spits out outputs
More informationDESIGN OF THE QUESTION PAPER
DESIGN OF THE QUESTION PAPER MATHEMATICS - CLASS XI Time : 3 Hours Max. Marks : 00 The weightage of marks over different dimensions of the question paper shall be as follows:. Weigtage of Type of Questions
More information22 (Write this number on your Answer Sheet)
Question Booklet Version (Write this number on your Answer Sheet) Day and Date : Thursday, 0th May, 08 QUESTION BOOKLET (MHT-CET - 08) Subjects : Paper I : Mathematics MH-CET 08 Roll No. Question Booklet
More informationFIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- MARCH, 2013
TED (10)-1002 (REVISION-2010) Reg. No.. Signature. FIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- MARCH, 2013 TECHNICAL MATHEMATICS- I (Common Except DCP and CABM) (Maximum marks: 100)
More informationWEDNESDAY, 18 MAY 9.00 AM AM. 1 Full credit will be given only where the solution contains appropriate working.
X00/0 NATINAL QUALIFICATINS 0 WEDNESDAY, 8 MAY 9.00 AM 0.0 AM MATHEMATICS HIGHER Paper (Non-calculator) Read carefull Calculators ma NT be used in this paper. Section A Questions 0 (40 marks) Instructions
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION COURSE III. Friday, January 25, :15 a.m. to 12:15 p.m.
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION THREE-YEAR SEQUENCE FOR HIGH SCHOOL MATHEMATICS COURSE III Friday, January 5, 00 9:5 a.m. to :5 p.m., only Notice... Scientific calculators
More informationMATHEMATICS. r Statement I Statement II p q ~p ~q ~p q q p ~(p ~q) F F T T F F T F T T F T T F T F F T T T F T T F F F T T
MATHEMATICS Directions : Questions number to 5 are Assertion-Reason type questions. Each of these questions contains two statements : Statement- (Assertion) and Statement- (Reason). Each of these questions
More informationSECTION 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:
More information2010 HSC Mathematics Extension 1 Sample Answers
010 HSC Mathematics Extension 1 Sample Answers This document contains sample answers, or, in the case of some questions, answers could include. These are developed by the examination committee for two
More informationChapter 6: Extending Periodic Functions
Chapter 6: Etending Periodic Functions Lesson 6.. 6-. a. The graphs of y = sin and y = intersect at many points, so there must be more than one solution to the equation. b. There are two solutions. From
More informationPhysicsAndMathsTutor.com
. The end A of a uniform rod AB, of length a and mass 4m, is smoothly hinged to a fixed point. The end B is attached to one end of a light inextensible string which passes over a small smooth pulley, fixed
More informationMT EDUCARE LTD. SUMMATIVE ASSESSMENT Roll No. Code No. 31/1
CBSE - X MT EDUCARE LTD. SUMMATIVE ASSESSMENT - 03-4 Roll No. Code No. 3/ Series RLH Please check that this question paper contains 6 printed pages. Code number given on the right hand side of the question
More information10 th MATHS SPECIAL TEST I. Geometry, Graph and One Mark (Unit: 2,3,5,6,7) , then the 13th term of the A.P is A) = 3 2 C) 0 D) 1
Time: Hour ] 0 th MATHS SPECIAL TEST I Geometry, Graph and One Mark (Unit:,3,5,6,7) [ Marks: 50 I. Answer all the questions: ( 30 x = 30). If a, b, c, l, m are in A.P. then the value of a b + 6c l + m
More informationD. 6. Correct to the nearest tenth, the perimeter of the shaded portion of the rectangle is:
Trigonometry PART 1 Machine Scored Answers are on the back page Full, worked out solutions can be found at MATH 0-1 PRACTICE EXAM 1. An angle in standard position θ has reference angle of 0 with sinθ
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More informationCambridge International Examinations Cambridge International General Certificate of Secondary Education. Published
Cambridge International Examinations Cambridge International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/ Paper March 07 MARK SCHEME Maximum Mark: 80 Published This mark scheme
More informationModel Answers Attempt any TEN of the following :
(ISO/IEC - 70-005 Certified) Model Answer: Winter 7 Sub. Code: 17 Important Instructions to Examiners: 1) The answers should be examined by key words and not as word-to-word as given in the model answer
More informationSECTION 6.3: VECTORS IN THE PLANE
(Section 6.3: Vectors in the Plane) 6.18 SECTION 6.3: VECTORS IN THE PLANE Assume a, b, c, and d are real numbers. PART A: INTRO A scalar has magnitude but not direction. We think of real numbers as scalars,
More informationWhere, m = slope of line = constant c = Intercept on y axis = effort required to start the machine
(ISO/IEC - 700-005 Certified) Model Answer: Summer 07 Code: 70 Important Instructions to examiners: ) The answers should be examined by key words and not as word-to-word as given in the model answer scheme.
More informationSec 4 Maths SET D PAPER 2
S4MA Set D Paper Sec 4 Maths Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e Answer all questions. Write your answers and working on the separate Answer Paper provided.
More information2014 Summer Review for Students Entering Algebra 2. TI-84 Plus Graphing Calculator is required for this course.
1. Solving Linear Equations 2. Solving Linear Systems of Equations 3. Multiplying Polynomials and Solving Quadratics 4. Writing the Equation of a Line 5. Laws of Exponents and Scientific Notation 6. Solving
More informationCLASS XII CBSE MATHEMATICS CONTINUITY AND DIFFERENTIATION
CLASS XII CBSE MATHEMATICS CONTINUITY AND DIFFERENTIATION sin5 + cos, if 0 ) For what value of k is the function f() = { 3 k, if = 0 ) For what value of k is the following function continuous at =? ; f
More information(c) cos Arctan ( 3) ( ) PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER
PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER Work the following on notebook paper ecept for the graphs. Do not use our calculator unless the problem tells ou to use it. Give three decimal places
More informationMockTime.com. (b) (c) (d)
373 NDA Mathematics Practice Set 1. If A, B and C are any three arbitrary events then which one of the following expressions shows that both A and B occur but not C? 2. Which one of the following is an
More informationLesson 7.6 Exercises, pages
Lesson 7.6 Exercises, pages 658 665 A. Write each expression as a single trigonometric ratio. a) sin (u u) b) sin u sin u c) sin u sin u d) cos u cos u sin U cos U e) sin u sin u f) sin u sin u sin U 5.
More information9231 FURTHER MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International Advanced Level MARK SCHEME for the October/November series 9 FURTHER MATHEMATICS 9/ Paper, maimum raw mar This mar scheme is published as an
More informationADDITIONAL MATHEMATICS
00-CE MTH HONG KONG EXMINTIONS UTHORITY HONG KONG CERTIFICTE OF EDUCTION EXMINTION 00 DDITIONL MTHEMTICS 8.0 am.00 am (½ hours) This paper must be answered in English. nswer LL questions in Section and
More informationInverse Trigonometric Functions. inverse sine, inverse cosine, and inverse tangent are given below. where tan = a and º π 2 < < π 2 (or º90 < < 90 ).
Page 1 of 7 1. Inverse Trigonometric Functions What ou should learn GOAL 1 Evaluate inverse trigonometric functions. GOAL Use inverse trigonometric functions to solve real-life problems, such as finding
More informationMathematics Paper 1 (Non-Calculator)
H National Qualifications CFE Higher Mathematics - Specimen Paper F Duration hour and 0 minutes Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full
More informationMessiah College Calculus I Placement Exam Topics and Review
Messiah College Calculus I Placement Exam Topics and Review The placement exam is designed to test a student s knowledge of material that is essential to the Calculus I course. Students who score less
More informationAS and A-level Mathematics Teaching Guidance
ΑΒ AS and A-level Mathematics Teaching Guidance AS 7356 and A-level 7357 For teaching from September 017 For AS and A-level exams from June 018 Version 1.0, May 017 Our specification is published on our
More informationMath 132 Exam 3 Fall 2016
Math 3 Exam 3 Fall 06 multiple choice questions worth points each. hand graded questions worth and 3 points each. Exam covers sections.-.6: Sequences, Series, Integral, Comparison, Alternating, Absolute
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION COURSE III. Wednesday, June 21, :15 to 4:15 p.m.
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION THREE-YEAR SEQUENCE FOR HIGH SCHOOL MATHEMATICS COURSE III Wednesday, June, 000 :5 to 4:5 p.m., only Notice... Scientific calculators
More informationWINTER 2017 EXAMINATION
(ISO/IEC - 700-00 Certfed) WINTER 07 EXAMINATION Model wer ject Code: Important Instructons to Eamners: ) The answers should be eamned by key words and not as word-to-word as gven n the model answer scheme.
More information0606 ADDITIONAL MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 0 series 0606 ADDITIONAL MATHEMATICS 0606/ Paper, maximum raw mark 80
More informationMATHEMATICS Higher Grade - Paper I (Non~calculator)
Higher Mathematics - Practice Eamination G Please note the format of this practice eamination is the same as the current format. The paper timings are the same, however, there are some differences in the
More informationREFRESHER. William Stallings
BASIC MATH REFRESHER William Stallings Trigonometric Identities...2 Logarithms and Exponentials...4 Log Scales...5 Vectors, Matrices, and Determinants...7 Arithmetic...7 Determinants...8 Inverse of a Matrix...9
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION THREE-YEAR SEQUENCE FOR HIGH SCHOOL MATHEMATICS COURSE III
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION THREE-YEAR SEQUENCE FOR HIGH SCHOOL MATHEMATICS COURSE III Friday, June 0, 00 1:15 to 4:15 p.m., only Notice... Scientific calculators
More information6.1: Reciprocal, Quotient & Pythagorean Identities
Math Pre-Calculus 6.: Reciprocal, Quotient & Pythagorean Identities A trigonometric identity is an equation that is valid for all values of the variable(s) for which the equation is defined. In this chapter
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION THREE-YEAR SEQUENCE FOR HIGH SCHOOL MATHEMATICS COURSE III
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION THREE-YEAR SEQUENCE FOR HIGH SCHOOL MATHEMATICS COURSE III Thursday, June 20, 2002 :5 to 4:5 p.m., only Notice... Scientific calculators
More informationVersion 1.0. General Certificate of Education (A-level) June 2012 MPC2. Mathematics. (Specification 6360) Pure Core 2. Mark Scheme
Version.0 General Certificate of Education (A-level) June 0 Mathematics MPC (Specification 660) Pure Core Mark Scheme Mark schemes are prepared by the Principal Eaminer and considered, together with the
More informationContents. 1 Vectors, Lines and Planes 1. 2 Gaussian Elimination Matrices Vector Spaces and Subspaces 124
Matrices Math 220 Copyright 2016 Pinaki Das This document is freely redistributable under the terms of the GNU Free Documentation License For more information, visit http://wwwgnuorg/copyleft/fdlhtml Contents
More informationFormulae and Summary
Appendix A Formulae and Summary Note to student: It is not useful to memorise all the formulae, partly because many of the complicated formulae may be obtained from the simpler ones. Rather, you should
More informationAmherst College, DEPARTMENT OF MATHEMATICS Math 11, Final Examination, May 14, Answer Key. x 1 x 1 = 8. x 7 = lim. 5(x + 4) x x(x + 4) = lim
Amherst College, DEPARTMENT OF MATHEMATICS Math, Final Eamination, May 4, Answer Key. [ Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value,
More informationSYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS
SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be one -hour paper consisting of 4 questions..
More informationSPECIALIST MATHEMATICS
Victorian Certificate of Education 06 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Section Written examination Monday 7 November 06 Reading time:.5 am to.00 noon
More information