KENDRIYA VIDYALAYA SANGATHAN, CHENNAI REGION CLASS XII-COMMON PRE-BOARD EXAMINATION. Answer key (Mathematics) Section A
|
|
- Randolph Curtis
- 6 years ago
- Views:
Transcription
1 KENDRIYA VIDYALAYA SANGATHAN, CHENNAI REGION CLASS XII-COMMON PRE-BOARD EXAMINATION Answer key (Mathematics) Section A. x =. x + y = 6. degree =. π x + y + z = Section B. Proving Reflexive Proving Symmetric Proving Transitive Conclusion. tan + tan 5 + tan 7 + tan 8 tan + 5 Χ 5 tan 7 + tan tan Χ + tan Χ 8 tan π Page of 5
2 OR tan x x + tan x+ x+ = π tan x x + tan x+ x+ = tan x tan x = tan x+ tan x+ By applying formula on the R.H.S. x tan x = tan x+ Applying tan both sides and solving x = ±. x y 7 = x + y mark log x y 7 = log x + y logx + 7 log y = log x + y Differentiating with respect to x y 7x = y 7x x x+y y x+y dy = y x dy mark Let x = cosθ cos x = θ Let y = tan +x x +x + x +cos θ cos θ y = tan +cos θ+ cos θ OR Page of 5
3 y = tan ( tanθ +tanθ ) y = (π θ) y = ( π cos x ) Let z = cos x y = π z dy dz =. For calculating LHL = 8 For calculating RHL = 8 For calculating K = 8 5. ex sin x cosx = ex sin xcos x cos x = e x [tanx sec x] = e x tanx + c 6. y x + = ln x + x Differentiating with respect to x y x + x + x + dy = x + x x + x xy + x + x + dy = x x + x + x + x xy + x + x + dy = x + x + dy = (+xy ) x + x + dy + xy + = Page of 5
4 7. x sinπx = x sinπx = x sinπx + x sinπx x sinπx On integrating both integrals on right-hand side, we get = xcosπx π = π + π + sinπx xcosπx π π + sinπx π 8. sin x sin x+α = sin x sinxcosα +cosxsinα = sin x cosα +cotxsinα = cose c x cosα +cotxsinα marks On substitution of cosα + cotxsinα = t = sinα t dt = t sinα + C On substitution of t t = cosα + cotxsinα = sinα sin x+α sinx + C Page of 5
5 9. a = i + j + k, b = i + j 5k, c = λi + j + k aχ b+c b+c = (i) b + c = + λ i + 6j k b + c = λ + λ (ii) aχ b + c = i j k + λ 6 By equation (i), (ii) & (iii) = 8i + + λ j + λ k (iii) 8i + +λ j + λ k λ +λ+ = On solving we will get λ = OR a + b + c = a + b = c a + b. a + b = c. c a + b + a. b = c By Substitution of values a. b = 9 a. b = 5 Page 5 of 5
6 a b cosθ = 5 By Substitution of values cosθ = θ = 6 mark. x+ = y+ = z+ 7 6 a = i j k a = i + 5j + 7k b = 7i 6j + k b = i j + k and x = y 5 = z 7 a a = i + 6j + 8k b Χ b = i 6j 8k b Χ b = 6 Shortest distance = a a. b Χ b b Χ b = 6 6 = 6 OR Equation of plane passing through,, is a x + b y + c z + = (i) (i) Passes through,, Page 6 of 5
7 a + b + 5c = (ii) (i) is perpendicular to x y + z = a b + c = (iii) On solving (ii) and (iii) a = 8, b = 7, c = From (i) equation of plane 8x + 7y + z 9 =. Let X is the random variable denoting the number of selected scouts, X takes values,,. P X = = C 5 C = 8 5 P X = = C Χ C 5 C = 5 P X = = C 5 C = 87 5 Now mean = P i X i = 9 Relevant Value 5 Page 7 of 5
8 . x x + y x + y x + y x x + y x + y x + y x Applying R R + R + R = (x + y) (x + y) (x + y) x + y x x + y x + y x + y x = x + y x + y x x + y x + y x + y x Applying C C C, C C C = (x + y) y y x + y y y x Expanding along R we get = 9y x + y Y x. Diagram Volume of the tank = 8m = xy----- (i) Let C be the cost of making the tank C = 7xy + 5 Χ (x + y) C = 7xy + 8(x + y) Page 8 of 5
9 From equation (i) C = 7x. x + 8(x + x ) C = x + x (ii) dc = 8 x For maxima and minima, dc = 8 x = x = as x -, x= Now d C Dx = 8 Χ 8 x d C Dx x= = 8 > C is minimum at x = By equation (ii) C x= = Rs. OR Area of ABC = Χ AB Χ DC = Χ bsinθ(a acosθ) = absinθ( cosθ)-----(i) Page 9 of 5
10 da dθ = ab(sin θ + cos θ cos θ) For maxima and minima da dθ = ab(sin θ + cos θ cos θ) = cosθ cosθ = cosθ = cosθ θ = nπ ± θ θ = nπ ± θ ----(ii) As θ (, π) by equation (ii) θ = π θ θ = π d A = ab(sinθ sinθ) dθ d A dθ θ= π < A is maximum. By equation (i) A max = ab sq.unit. Let x, y, z be the amount of prize to be awarded in the field of agriculture, education & social science respectively. The given situation Can be written in the matrix form as: AX = B Page of 5
11 Where A = , X = x y z, B = AX = B X = A B A = 75 adj A = A = adj A A = marks x y z = x y z = x =, y =, z = Value based relevant answer 5. Page of 5
12 Point of intersection of given curves is x = Required Area = x + 9 x =. x + x 9 x sin x = 9π sin OR I = x + x + x ---- (i) I = x + x + x Let I = x = ( x) + x = x + x = 5 I = x = ( x) + x = x + x = I = x = ( x) + x = x + x = 5 By equation (i) I = I + I + I = = Page of 5
13 6. Let x and y be the number of pieces of type A and B manufactured per week respectively. If z is the profit then, z = 8x + y Maximize z subject to the constraint 9x + y 8 x + y 6----(i) x + y ----(ii) x, y ----(iii) Mark mark Page of 5
14 Corner Points (,) A(,) B(,6) C(,) Relevant answer z = 8x + y 6 68 maximum 7. Let the event b defined as E = The examinee guesses the answer E = The examinee copies the answer E = The examinee knows the answer A = The examinee answers correctly P E = 6, P E = 9, P E = = 8 s P A E = P A E = 8 (Out of choices is correct) s s P A E = (If the answer is known it is always correct) s P E A = Required P E A = P E.P A E P E.P A E +P E.P A E +P E.P A E On substitution P E A = Yes the probability of copying is less than other probability. 8. The given planes are x + y 6z = ---- (i) 5x y + z + 9 = ---- (ii) Equation of the plane passing through the intersection of (i) and (ii) x + y 6z + λ 5x y + z + 9 = ---- (iii) mark Given that plane (iii) is parallel to x = y = z 5 + 5λ. + λ. + λ 6. = On solving λ = 8 7 On substitution of λ in (iii) equation of plane 5x 7y z + 5 = Page of 5
15 9. x dy + y x + y = dy xy +y = (i) x Let y = vx dy dv = v + x By equation (i) v + x dv = vx +v x x dv = v +v x dv = v +v x dv = v v+ x log v log v + = log x + C log vx vx v+ = log k = k v+ Putting v = y we get x x y = k y + x For Particular solution k = Therefore Particular Solution is x y = y + x Page 5 of 5
CLASS XII-COMMON PRE-BOARD EXAMINATION
CLASS XII-COMMON PRE-BOARD EXAMINATION Subject: Mathematics General Instructions: Time Allotted: 3 Hours. Max.Marks:100 a. All the questions are compulsory b. The question paper consists of 29 questions
More informationSAMPLE QUESTION PAPER MATHEMATICS CLASS XII ( ) BLUE PRINT. Unit VSA (1) SA (4) LA (6) Total. I. Relations and Functions 1 (1) 4 (1) 5 (2)
SAMPLE QUESTION PAPER MATHEMATICS CLASS XII (2013-14) BLUE PRINT Unit VSA (1) SA (4) LA (6) Total I. Relations and Functions 1 (1) 4 (1) 5 (2) Inverse Trigonometric Functions 1 (1) *4 (1) 5 (2) II. Matrices
More informationSaturday, March 27, :59 PM Annexure 'F' Unfiled Notes Page 1
Annexure 'F' CLASS-XII SAMPLE QUESTION PAPER MATHEMATICS CLASS XII (2013-14) BLUE PRINT Unit VSA (1) SA (4) LA (6) Total I. Relations and Functions 1 (1) 4 (1) 5 (2) Inverse Trigonometric Functions
More informationCLASS 12 SUBJECT : MATHEMATICS
CLASS 2 SUBJECT : MATHEMATICS CBSE QUESTION PAPER 27(FOREIGN) General Instructions: (i) All questions are compulsory. (ii) Questions 4 in Section A carrying mark each (iii) Questions 5 2 in Section B carrying
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 informationStudy Material Class XII - Mathematics
Study Material Class XII - Mathematics 2016-17 1 & 2 MARKS QUESTIONS PREPARED BY KENDRIYA VIDYALAYA SANGATHAN TINSUKIA REGION Study Material Class XII Mathematics 2016-17 1 & 2 MARKS QUESTIONS CHIEF PATRON
More information(x 3)(x + 5) = (x 3)(x 1) = x + 5. sin 2 x e ax bx 1 = 1 2. lim
SMT Calculus Test Solutions February, x + x 5 Compute x x x + Answer: Solution: Note that x + x 5 x x + x )x + 5) = x )x ) = x + 5 x x + 5 Then x x = + 5 = Compute all real values of b such that, for fx)
More informationMath 21B - Homework Set 8
Math B - Homework Set 8 Section 8.:. t cos t dt Let u t, du t dt and v sin t, dv cos t dt Let u t, du dt and v cos t, dv sin t dt t cos t dt u v v du t sin t t sin t dt [ t sin t u v ] v du [ ] t sin t
More informationSOLUTIONS FOR PRACTICE FINAL EXAM
SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable
More informationy= sin3 x+sin6x x 1 1 cos(2x + 4 ) = cos x + 2 = C(x) (M2) Therefore, C(x) is periodic with period 2.
. (a).5 0.5 y sin x+sin6x 0.5.5 (A) (C) (b) Period (C) []. (a) y x 0 x O x Notes: Award for end points Award for a maximum of.5 Award for a local maximum of 0.5 Award for a minimum of 0.75 Award for the
More information2013 HSC Mathematics Extension 2 Marking Guidelines
3 HSC Mathematics Extension Marking Guidelines Section I Multiple-choice Answer Key Question Answer B A 3 D 4 A 5 B 6 D 7 C 8 C 9 B A 3 HSC Mathematics Extension Marking Guidelines Section II Question
More informationSeries SC/SP Code No. SP-16. Mathematics. Time Allowed: 3 hours Maximum : 100
Sample Paper (CBSE) Series SC/SP Code No. SP-16 Mathematics Time Allowed: 3 hours Maximum : 100 General Instructions: (i) (ii) (iii) (iv) (v) (vi) There are 26 questions in all. All questions are compulsory.
More informationMath 223 Final. July 24, 2014
Math 223 Final July 24, 2014 Name Directions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Total 1. No books, notes, or evil looks. You may use a calculator to do routine arithmetic computations. You may not use your
More informationMathematics 1052, Calculus II Exam 1, April 3rd, 2010
Mathematics 5, Calculus II Exam, April 3rd,. (8 points) If an unknown function y satisfies the equation y = x 3 x + 4 with the condition that y()=, then what is y? Solution: We must integrate y against
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
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 informationQUESTION BOOKLET 2016 Subject : Paper III : Mathematics
QUESTION BOOKLET 06 Subject : Paper III : Mathematics ** Question Booklet Version Roll No. Question Booklet Sr. No. (Write this number on your Answer Sheet) Answer Sheet No. (Write this number on your
More informationIIT JEE (2011) PAPER-B
L.K.Gupta (Mathematic Classes) www.pioneermathematics.com. MOBILE: 985577, 4677 IIT JEE () (Integral calculus) TOWARDS IIT- JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE TIME: 6 MINS
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program
King Fahd University of Petroleum and Minerals Prep-Year Math Program Math 00 Class Test II Textbook Sections: 6. to 7.5 Term 17 Time Allowed: 90 Minutes Student s Name: ID #:. Section:. Serial Number:.
More information( )( 2 ) n n! n! n! 1 1 = + + C = +
Subject : Mathematics MARKING SCHEME (For Sample Question Paper) Class : Senior Secondary. ( )( )( 9 )( 9 ) L.H.S. ω ω ω. ω ω. ω ( ω)( ω )( ω)( ω ) [ ω ] ω ω ( )( ) ( ) 4 ω +ω +ω. [ 4 ] + + 49 R.H.S (Since
More informationAB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2
AB CALCULUS Page 1 of 6 NAME DATE 1. Evaluate each it: AB CALCULUS Show all work on separate paper. x 3 x 9 x 5x + 6 x 0 5x 3sin x x 7 x 3 x 3 5x (d) 5x 3 x +1 x x 4 (e) x x 9 3x 4 6x (f) h 0 sin( π 6
More information12 th Class Mathematics Paper
th Class Mathematics Paper Maimum Time: hours Maimum Marks: 00 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 9 questions divided into four sections A, B, C
More informationInverse Trig Functions
6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)
More informationSAMPLE QUESTION PAPER MATHEMATICS CLASS XII:
SAMPLE QUESTION PAPER MATHEMATICS CLASS XII: 05-6 Question numbers to 6 carry mark each. SAMPLE PAPER II Section A Q. Evaluate: - 3 sin(cos (- )). 5 Q. State the reason for the following Binary Operation
More informationQUESTION BOOKLET 2016 Subject : Paper III : Mathematics
QUESTION BOOKLET 06 Subject : Paper III : Mathematics ** Question Booklet Version Roll No. Question Booklet Sr. No. (Write this number on your Answer Sheet) Answer Sheet No. (Write this number on your
More informationQUESTION BOOKLET 2016 Subject : Paper III : Mathematics
QUESTION BOOKLET 06 Subject : Paper III : Mathematics ** Question Booklet Version Roll No. Question Booklet Sr. No. (Write this number on your Answer Sheet) Answer Sheet No. (Write this number on your
More informationPRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209
PRELIM 2 REVIEW QUESTIONS Math 9 Section 25/29 () Calculate the following integrals. (a) (b) x 2 dx SOLUTION: This is just the area under a semicircle of radius, so π/2. sin 2 (x) cos (x) dx SOLUTION:
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 informationDifferential Equations: Homework 2
Differential Equations: Homework Alvin Lin January 08 - May 08 Section.3 Exercise The direction field for provided x 0. dx = 4x y is shown. Verify that the straight lines y = ±x are solution curves, y
More informationAssignment 6 Solution. Please do not copy and paste my answer. You will get similar questions but with different numbers!
Assignment 6 Solution Please do not copy and paste my answer. You will get similar questions but with different numbers! This question tests you the following points: Integration by Parts: Let u = x, dv
More informationSECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
More information4754A A A * * MATHEMATICS (MEI) Applications of Advanced Mathematics (C4) Paper A ADVANCED GCE. Tuesday 13 January 2009 Morning
ADVANCED GCE MATHEMATICS (MEI) Applications of Advanced Mathematics (C) Paper A 75A Candidates answer on the Answer Booklet OCR Supplied Materials: 8 page Answer Booklet Graph paper MEI Examination Formulae
More information9. The x axis is a horizontal line so a 1 1 function can touch the x axis in at most one place.
O Answers: Chapter 7 Contemporary Calculus PROBLEM ANSWERS Chapter Seven Section 7.0. f is one to one ( ), y is, g is not, h is not.. f is not, y is, g is, h is not. 5. I think SS numbers are supposeo
More informationM152: Calculus II Midterm Exam Review
M52: Calculus II Midterm Exam Review Chapter 4. 4.2 : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance
More informationLesson 33 - Trigonometric Identities. Pre-Calculus
Lesson 33 - Trigonometric Identities Pre-Calculus 1 (A) Review of Equations An equation is an algebraic statement that is true for only several values of the variable The linear equation 5 = 2x 3 is only
More informationFinal Exam Review Quesitons
Final Exam Review Quesitons. Compute the following integrals. (a) x x 4 (x ) (x + 4) dx. The appropriate partial fraction form is which simplifies to x x 4 (x ) (x + 4) = A x + B (x ) + C x + 4 + Dx x
More information= π + sin π = π + 0 = π, so the object is moving at a speed of π feet per second after π seconds. (c) How far does it go in π seconds?
Mathematics 115 Professor Alan H. Stein April 18, 005 SOLUTIONS 1. Define what is meant by an antiderivative or indefinite integral of a function f(x). Solution: An antiderivative or indefinite integral
More informationGreen s Theorem. Fundamental Theorem for Conservative Vector Fields
Assignment - Mathematics 4(Model Answer) onservative vector field and Green theorem onservative Vector Fields If F = φ, for some differentiable function φ in a domaind, then we say that F is conservative
More informationMcGill University April Calculus 3. Tuesday April 29, 2014 Solutions
McGill University April 4 Faculty of Science Final Examination Calculus 3 Math Tuesday April 9, 4 Solutions Problem (6 points) Let r(t) = (t, cos t, sin t). i. Find the velocity r (t) and the acceleration
More information3.4 Conic sections. Such type of curves are called conics, because they arise from different slices through a cone
3.4 Conic sections Next we consider the objects resulting from ax 2 + bxy + cy 2 + + ey + f = 0. Such type of curves are called conics, because they arise from different slices through a cone Circles belong
More informationQUESTION BOOKLET 2016 Subject : Paper III : Mathematics
QUESTION BOOKLET 06 Subject : Paper III : Mathematics ** Question Booklet Version Roll No. Question Booklet Sr. No. (Write this number on your Answer Sheet) Answer Sheet No. (Write this number on your
More informationMath Calculus II Homework # Due Date Solutions
Math 35 - Calculus II Homework # - 007.08.3 Due Date - 007.09.07 Solutions Part : Problems from sections 7.3 and 7.4. Section 7.3: 9. + d We will use the substitution cot(θ, d csc (θ. This gives + + cot
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 informationReview Problems for the Final
Review Problems for the Final Math -3 5 7 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the
More informationMath 222 Spring 2013 Exam 3 Review Problem Answers
. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w
More information3. On the grid below, sketch and label graphs of the following functions: y = sin x, y = cos x, and y = sin(x π/2). π/2 π 3π/2 2π 5π/2
AP Physics C Calculus C.1 Name Trigonometric Functions 1. Consider the right triangle to the right. In terms of a, b, and c, write the expressions for the following: c a sin θ = cos θ = tan θ =. Using
More informationC3 A Booster Course. Workbook. 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. (3) b) Hence, or otherwise, solve the equation
C3 A Booster Course Workbook 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. b) Hence, or otherwise, solve the equation x = 2x 3 (3) (4) BlueStar Mathematics Workshops (2011) 1
More informationFinal Exam SOLUTIONS MAT 131 Fall 2011
1. Compute the following its. (a) Final Exam SOLUTIONS MAT 131 Fall 11 x + 1 x 1 x 1 The numerator is always positive, whereas the denominator is negative for numbers slightly smaller than 1. Also, as
More informationy = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx
Millersville University Name Answer Key Mathematics Department MATH 2, Calculus II, Final Examination May 4, 2, 8:AM-:AM Please answer the following questions. Your answers will be evaluated on their correctness,
More informationBLUE PRINT: CLASS XII MATHS
BLUE PRINT: CLASS XII MATHS CHAPTER S NAME MARK 4 MARKS 6 MARKS TOTAL. RELATIONS AND FUNCTIONS 5. INVERSE TRIGONOMETRIC 5 FUNCTIONS 3. MATRICES 7 4. DETERMINANTS 6 5. 8 DIFFERENTIATION 6 APPLICATION OF
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Extension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Board-approved calculators may be used A table
More informationMath Final Exam Review
Math - Final Exam Review. Find dx x + 6x +. Name: Solution: We complete the square to see if this function has a nice form. Note we have: x + 6x + (x + + dx x + 6x + dx (x + + Note that this looks a lot
More informationAP Calculus BC Chapter 4 AP Exam Problems. Answers
AP Calculus BC Chapter 4 AP Exam Problems Answers. A 988 AB # 48%. D 998 AB #4 5%. E 998 BC # % 5. C 99 AB # % 6. B 998 AB #80 48% 7. C 99 AB #7 65% 8. C 998 AB # 69% 9. B 99 BC # 75% 0. C 998 BC # 80%.
More informationThere are some trigonometric identities given on the last page.
MA 114 Calculus II Fall 2015 Exam 4 December 15, 2015 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during
More information1 are perpendicular to each other then, find. Q06. If the lines x 1 z 3 and x 2 y 5 z
Useful for CBSE Board Examination of Math (XII) for 6 For more stuffs on Maths, please visit : www.theopgupta.com Time Allowed : 8 Minutes Max. Marks : SECTION A 3 Q. Evaluate : sin cos 5. Q. State the
More informationBC Exam 2 - Part I 28 questions No Calculator Allowed. C. 1 x n D. e x n E. 0
1. If f x ( ) = ln e A. n x x n BC Exam - Part I 8 questions No Calculator Allowed, and n is a constant, then f ( x) = B. x n e C. 1 x n D. e x n E.. Let f be the function defined below. Which of the following
More informationTaking Derivatives. Exam II Review - Worksheet Name: Math 1131 Class #31 Section: 1. Compute the derivative of f(x) = sin(x 2 + x + 1)
Taking Derivatives 1. Compute the derivative of f(x) = sin(x 2 + x + 1) 2. Compute the derivative of f(x) = cos(x 2 ) sin(x 2 ) 3. Compute the derivative of f(x) = sin(x e x ) 4. Compute the derivative
More informationAll Rights Reserved Wiley India Pvt. Ltd. 1
Question numbers to carry mark each. CBSE MATHEMATICS SECTION A. If R = {(, y) : + y = 8} is a relation of N, write the range of R. R = {(, y)! + y = 8} a relation of N. y = 8 y must be Integer So Can
More informationINVERSE TRIGONOMETRY: SA 4 MARKS
INVERSE TRIGONOMETRY: SA MARKS To prove Q. Prove that sin - tan - 7 = π 5 Ans L.H.S = Sin - tan - 7 5 = A- tan - 7 = tan - 7 tan- let A = Sin - 5 Sin A = 5 = tan - ( ( ) ) tan - 7 9 6 tan A = A = tan-
More informationName: AK-Nummer: Ergänzungsprüfung January 29, 2016
INSTRUCTIONS: The test has a total of 32 pages including this title page and 9 questions which are marked out of 10 points; ensure that you do not omit a page by mistake. Please write your name and AK-Nummer
More informationC3 papers June 2007 to 2008
physicsandmathstutor.com June 007 C3 papers June 007 to 008 1. Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e x + 3e x = 4. *N6109A04* physicsandmathstutor.com June 007 x + 3 9+
More informationMath 147 Exam II Practice Problems
Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
More informationb n x n + b n 1 x n b 1 x + b 0
Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)
More informationMark Scheme. Mathematics General Certificate of Education examination - June series. MFP3 Further Pure 3
Version.0: 0606 abc General Certificate of Education Mathematics 660 MFP Further Pure Mark Scheme 006 examination - June series Mark schemes are prepared by the Principal Examiner and considered, together
More informationand in each case give the range of values of x for which the expansion is valid.
α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω Mathematics is indeed dangerous in that it absorbs students to such a degree that it dulls their senses to everything else P Kraft Further Maths A (MFPD)
More informationTrig Practice 08 and Specimen Papers
IB Math High Level Year : Trig: Practice 08 and Spec Papers Trig Practice 08 and Specimen Papers. In triangle ABC, AB = 9 cm, AC = cm, and Bˆ is twice the size of Ĉ. Find the cosine of Ĉ.. In the diagram
More informationH I G H E R S T I L L. Extended Unit Tests Higher Still Higher Mathematics. (more demanding tests covering all levels)
M A T H E M A T I C S H I G H E R S T I L L Higher Still Higher Mathematics Extended Unit Tests 00-0 (more demanding tests covering all levels) Contents Unit Tests (at levels A, B and C) Detailed marking
More informationLesson 22 - Trigonometric Identities
POP QUIZ Lesson - Trigonometric Identities IB Math HL () Solve 5 = x 3 () Solve 0 = x x 6 (3) Solve = /x (4) Solve 4 = x (5) Solve sin(θ) = (6) Solve x x x x (6) Solve x + = (x + ) (7) Solve 4(x ) = (x
More informationStudy Guide/Practice Exam 3
Study Guide/Practice Exam 3 This study guide/practice exam covers only the material since exam. The final exam, however, is cumulative so you should be sure to thoroughly study earlier material. The distribution
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More information4 Ab C4 Integration 1 ln. 12 Bc C4 Parametric differentiation dy
C4 Consolidation C3 Consol. Mechanics Drill Done BP Ready A Assignment Phi Cover Sheet Name: Question Topic Comment Aa C4 Integration 1 tan(4 x 1) c 4 Ab C4 Integration 1 ln x x 5 c Ac C4 Integration 1
More informationFind the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x
Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc
More informationTopic 3 Part 1 [449 marks]
Topic 3 Part [449 marks] a. Find all values of x for 0. x such that sin( x ) = 0. b. Find n n+ x sin( x )dx, showing that it takes different integer values when n is even and when n is odd. c. Evaluate
More informationMAT137 - Week 8, lecture 1
MAT137 - Week 8, lecture 1 Reminder: Problem Set 3 is due this Thursday, November 1, at 11:59pm. Don t leave the submission process until the last minute! In today s lecture we ll talk about implicit differentiation,
More informationSolutions to Sample Questions for Final Exam
olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the
More informationMathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx.
Mathematics 14 Fall Term 26 Solutions to Final Exam 1. Evaluate sin(ln t) dt. Solution. We first make the substitution t = e x, for which dt = e x. This gives sin(ln t) dt = e x sin(x). To evaluate the
More informationChapter 5 Notes. 5.1 Using Fundamental Identities
Chapter 5 Notes 5.1 Using Fundamental Identities 1. Simplify each expression to its lowest terms. Write the answer to part as the product of factors. (a) sin x csc x cot x ( 1+ sinσ + cosσ ) (c) 1 tanx
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 informationMATH 1080 Test 2 -Version A-SOLUTIONS Fall a. (8 pts) Find the exact length of the curve on the given interval.
MATH 8 Test -Version A-SOLUTIONS Fall 4. Consider the curve defined by y = ln( sec x), x. a. (8 pts) Find the exact length of the curve on the given interval. sec x tan x = = tan x sec x L = + tan x =
More informationMIDTERM 2. Section: Signature:
MIDTERM 2 Math 3A 11/17/2010 Name: Section: Signature: Read all of the following information before starting the exam: Check your exam to make sure all pages are present. When you use a major theorem (like
More informationwhich can also be written as: 21 Basic differentiation - Chain rule: (f g) (b) = f (g(b)) g (b) dg dx for f(y) and g(x), = df dy
2 Basic differentiation - Chain rule: Chain rule: Suppose f and g are differentiable functions so that we can form their composition f g. The derivative of f g at input b can be computed in terms of the
More informationMath 101 Fall 2006 Exam 1 Solutions Instructor: S. Cautis/M. Simpson/R. Stong Thursday, October 5, 2006
Math 101 Fall 2006 Exam 1 Solutions Instructor: S. Cautis/M. Simpson/R. Stong Thursday, October 5, 2006 Instructions: This is a closed book, closed notes exam. Use of calculators is not permitted. You
More information2005 Mathematics. Advanced Higher. Finalised Marking Instructions
2005 Mathematics Advanced Higher Finalised Marking Instructions These Marking Instructions have been prepared by Examination Teams for use by SQA Appointed Markers when marking External Course Assessments.
More informationMth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.
For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin
More informationMark Scheme (Results) June GCE Core Mathematics C4 (6666) Paper 1
Mark Scheme (Results) June 0 GCE Core Mathematics C4 (6666) Paper Edexcel is one of the leading examining and awarding bodies in the UK and throughout the world. We provide a wide range of qualifications
More informationMATHEMATICS Paper & Solutions
CBSE-XII-8 EXAMINATION Series SGN MATHEMATICS Paper & Solutions SET- Code : 6/ Time : Hrs. Ma. Marks : General Instruction : (i) All questions are compulsor. (ii) The question paper consists of 9 questions
More informationPower Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n.
.8 Power Series. n x n x n n Using the ratio test. lim x n+ n n + lim x n n + so r and I (, ). By the ratio test. n Then r and I (, ). n x < ( ) n x n < x < n lim x n+ n (n + ) x n lim xn n (n + ) x
More informationMATH 152, Fall 2017 COMMON EXAM II - VERSION A
MATH 15, Fall 17 COMMON EXAM II - VERSION A LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited.. TURN OFF cell phones
More information2) ( 8 points) The point 1/4 of the way from (1, 3, 1) and (7, 9, 9) is
MATH 6 FALL 6 FIRST EXAM SEPTEMBER 8, 6 SOLUTIONS ) ( points) The center and the radius of the sphere given by x + y + z = x + 3y are A) Center (, 3/, ) and radius 3/ B) Center (, 3/, ) and radius 3/ C)
More informationFinal Exam 2011 Winter Term 2 Solutions
. (a Find the radius of convergence of the series: ( k k+ x k. Solution: Using the Ratio Test, we get: L = lim a k+ a k = lim ( k+ k+ x k+ ( k k+ x k = lim x = x. Note that the series converges for L
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall 2017, WEEK 14 JoungDong Kim Week 14 Section 5.4, 5.5, 6.1, Indefinite Integrals and the Net Change Theorem, The Substitution Rule, Areas Between Curves. Section
More informationEXAM. Practice for Second Exam. Math , Fall Nov 4, 2003 ANSWERS
EXAM Practice for Second Eam Math 135-006, Fall 003 Nov 4, 003 ANSWERS i Problem 1. In each part, find the integral. A. d (4 ) 3/ Make the substitution sin(θ). d cos(θ) dθ. We also have Then, we have d/dθ
More information1. (16pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist. lim
Spring 10/MAT 250/Exam 1 Name: Show all your work. 1. (16pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist. lim x 1 +f(x) = lim x 3 f(x) = lim x
More informationSAMPLE QUESTION PAPER MATHEMATICS CLASS XII :
SAMPLE QUESTION PAPER MATHEMATICS CLASS XII : 05-6 TYPOLOGY VSA ( M) L A I (4M) L A II (6M) MARKS %WEIGHTAGE Remembering 3, 6, 8, 9, 5 0 0% Understanding, 9, 0 4, 6 % Applications 4 3, 5, 6, 7, 0 9 9%
More informationIntegration 1/10. Integration. Student Guidance Centre Learning Development Service
Integration / Integration Student Guidance Centre Learning Development Service lds@qub.ac.uk Integration / Contents Introduction. Indefinite Integration....................... Definite Integration.......................
More informationNOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system.
NOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system. Duplicating, selling, or otherwise distributing this product
More information1. The accumulated net change function or area-so-far function
Name: Section: Names of collaborators: Main Points: 1. The accumulated net change function ( area-so-far function) 2. Connection to antiderivative functions: the Fundamental Theorem of Calculus 3. Evaluating
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 informationUNIT-IV DIFFERENTIATION
UNIT-IV DIFFERENTIATION BASIC CONCEPTS OF DIFFERTIATION Consider a function yf(x) of a variable x. Suppose x changes from an initial value x 0 to a final value x 1. Then the increment in x defined to be
More informationWBJEE Answer Keys by Aakash Institute, Kolkata Centre
WBJEE - 08 Answer Keys by, Kolkata Centre MATHEMATICS Q.No. 0 A B C D 0 C D A B 0 B D A C 04 C B A B 05 C A C C 06 A C D C 07 B A C C 08 B *C,D C A 09 C D D B 0 D A C D B A B C C D A B B A A C 4 C C B
More information