ANNUAL EXAMINATION - ANSWER KEY II PUC - MATHEMATICS PART - A
|
|
- Amy Dorsey
- 6 years ago
- Views:
Transcription
1 . LCM of and cosec ( ) -. π a a A a a. A A A A sin d ANNUAL EXAMINATION - ANSWER KEY -7 + d + + C II PUC - MATHEMATICS PART - A 7. or more vectors are said to be collinear vectors if they are parallel to the same line, irrespective of their magnitudes and directions. 8. Let the direction cosines be cos α, cosα, cosα cos α + cos α + cos α cos α ± Direction cosines are ±, ±, ± 9. The common region determined by all the constrains including non negative constraints of a linear programming problem is called the feasible region.. P(A B) P(A) P(B) f() + -,, f() f(-) PART - B - but f(-) f() f is not one one Only the +ve real numbers have pre images hence f is not onto.. put cos θ θ cos - Sin - ( cos θ sin θ) sin - (sin θ) θ cos -. use the formula tan - tan -
2 tan tan ( ) ( + ) tan + tan - ± >. K ± ± 8 K + 8 K, 8 5. a + (by + sin y) d a d by + sin y 6. f is continuous in [ -, ] f is differentiable in ( -, ) and f(-) f() There eists a real number c (-, ) such that f (c) c + c - (-, ) Rolle s theorem is verified with C - dv 7. V % of. d Change in volume 8. put t tan dv V..9 m d dt sec d f ( ) + 8 f '( ) + f '( c) c + sec d dt 5 t 5 t dt + c tan + c 5 5
3 9. / / d tan tan + π 6. order degree. m : n : internally uuur ( iˆ + ˆj + kˆ ) + ( iˆ + ˆj kˆ ) iˆ ˆj kˆ OR R,, m : n : eternally uuur ( iˆ + ˆj + kˆ ) ( iˆ + ˆj kˆ ) OR iˆ + kˆ R ( -,, ) r. A a b r iˆ ˆj kˆ a b iˆ + 5 ˆj 5kˆ 7 a b ( 5) 5. a i j 5 k, b i j + 6k Vector equation is a + λ ( b a). (i j 5k) + λ(k) Cartesian form X p() y y z z y y z z y + z + 5
4 PART - C 5. Refleive: Let a R. Then a a is true ( a, a) R R is refleive Symmetric: Let a, b R and ( a, b) R a b b a may not be true Eg: (, ) R But is not true (, ) R R is not symmetric. Transitive: Let a, b, c R and ( a, b), ( b, c) R a b and b c a c is true ( a, c) R R is transitive. 6. Put tan θ θ tan - + tan θ secθ tan tan tan cosθ tanθ tan θ sinθ cosθ θ sin cosθ θ tan tan tan tan sinθ θ θ sin cos θ tan. 7. Let A and B are symmetric matrices of same order then A T A and B T B Consider AB BA. To prove AB is symmetric Now (AB) T B T A T B.A AB AB is symmetric Consider AB is symmetric. To prove AB BA AB (AB) T B T A T B A 8. Let y (log) cos log y cos log(log ) cos + log (log ) ( sin ) y d log cos d log cos (log ) sin log (log ) 9. Let u sin, V e cos du d dv d cos sin cos, e ( sin )
5 du sin cos cos cos cos dv e sin e. Given + y 6 6 y Let P y ( 6 y) y 6y y dp 8y y d p 6y y dp At maimum, y ( 5 y) When d p y or y 5 y, when y 5, d P is ve At y 5 P is maimum Numbers are 5 and 5. A B ( + ) ( + ) + + A( + ) + B( + ) When -, B - When -, A d d log ( + ) log ( + ) + c log ( + ) ( + ). Let I e sin I sin + c d e e cos d I e sin cos e e ( sin ) d e sin e cos e sin d + c e (sin cos) I + c I e (sin cos) + c e I (sin cos ) + c. y y A 5
6 9 [ 7 ] sq. units.. Equation of circle is + ( y k) - () ( y k) 9 Differentiate () + ( y k) d d d uuur uuur uuur 5. Now AB OB OA i + ( ) j + k uuur uuur uuur AC OC OA i + j k uuur uuur uuur AD OD OA i + j k Now uuur uuur uuur AB. ( AC AD) ( + 9) ( ) ( - + 9) + ( ) a + b + c a + b + c. a + b + c. Now ( ) ( ) r a + a. b + a. c + b. a + b + b. c + c. a + c. b + c r a + b + c + ( a. b + b. c + c. a) (+ 6 + ) a. b + b. c + c. a 7. a i + j + k, b i j + k a i j k b i + j + k a a i j k ( a a ). ( b b ) d b b d 6
7 8. ( ) + ( ) + ( ) ( ) ( ) ( + ) d (,),(,), (,),(,5),(,6) (,),(,)...(,6) (,)... (,6) E (,)... (,6) (5,)...(5,6) (6,)......(6,6) F { (, ), (, ), (, )} E F {, ), (, )} P(E), P( F) P( E F) P( E F) 6 P( F E) P( E) 5 6 PART -D 9. Let y be an arbitrary element of range f. Then y for some in N, which implies that y ( + ) + 6 ( y 6) This gives, as y 6 ( y ) Let us define g : s N by g(y) 6 Now gof() g(f()) g( + + 5) g( +) + 6) ( ) ( + ) and fog(y) (( y ) ) ( y 6) ( y 6) f ( ) y 6 y y 7
8 . Hence gof I N and fog I S This implies that f is invertible with f - g. A A A A A A A A A I A y B z A - B A - Adj A T 9 6 A 9 6 adj A A , y 5, z.. y (tan - ) Differentiating w.r.t ; y tan - + ( + )y tan - Differentiating again w.r.t ; ( + )y + y () + 8
9 ( + ) y + (+ )y. a) P perimeter ( + y). Let dp d + dt dt dt d Given 5 cm / min ( is decreasing) dt cm / min dt di ( 5 ) cm / min dt + b) Area y da dt 8() + 6( 5) cm /min. I a d. da d + y when 8 cm, y 6 cm dt dt dt a I a du I a d a a + a a d a a a d a a I a a a d a d a d a a I a d a log + a + C () I d ( ) d ( -) - Using (); ( ) log 8 7 I c 9
10 5. Given equation of sides of the triangle are y +, y + and on solving these equations, we obtain the vertices of triangle a A(, ), B(, ) and C(, 9) Required area (shown in shaded region) Area ( OLBAO) Area (OLCAO) ( + ) d ( + ) d sq unit. 6. Diveide the D.E by os. we have tan d + cos cos d + This is of the form Py Q d +. y i. e sec. y tan sec Here P sec and Q tan. sec. Both are functions of. Linear in y and solution is y ( I. F) Q( I. F) d + c pol I. F e e tan sec d e t tan tan tan y. e tan.sec. e d + c t t. e dt + c t d( e t ) + c + + t t t t t e e dt c te e c i.e. ye tan tan. e tan - e tan + c y tan + C e -tan 7. Let a plane pass through a point A with position vector a r and perpendicular to the vector N r. Let be position vector of any point P (, y, z) (,) A dt sec in the plane. Then the point P lies in plane if and only if AP N. 8. Let represents the number of bulbs that will fuse after 5 days of use in an eperiment X of s trials. The trials are Bernoulli trials. P P(success).5 and q p has a binomial distribution with n 5, p.5 and q.95. ' y O y' d y + y + Z a r O B (, ) A L C (, 9) N r P(, y, z) Y
11 (i) Required probability P( ) 5 C P q 5 q 5 (.5) 5 (ii) Required probability P( ) P() + P() 5 C P q C p q q (q + 5P) (.95) ( (.5) (.95. (vi) Required probability P( > ) {P () + P()} (.95). 9. (a) a a f ( ) du f ( a ) du PART E Proof : Put u a on R.H.S. Then du -du when, u a and when, u a f ( u)( du) a b a f ( u) du ( Q f ( ) d ( f ( ) d) a a b a b b f ( u) du Q f ( u) du f ( ) d a a (Let π / π / I (log sin logsin ) d ( log sin log sin ) I d π / sin log d sin π / π / sin tan log d log d...() cos π / tan( π / log d. π / cot log d...() + π / tan cot I log + log d
12 π / tan cot I log. d π / I log d I log [ ] π / π I log π I log. (b) yz y y z y y z z y yz z z z y ( ) ( ) ( ) ( + + ) yz LHS y y yz yz z z yz Using, R R R yr R zr yz yz y z y z y y z z Epanding corresponding to c y y z z (y ) (z ) (z ) (y )] ( y) ( y z) (z ) (y + yz + z) RHS. (using R R R and R R R ) 5. + y.() + y () + 5y..(), y 6, y y y, y, 6 y, 5
13 8 + y + 5y 6 C X + y Cornor points Z 6 + y A(, ) B(, 6) C(, ) D(6, ) E(5, ) 6 maimum 6 Maimum value of z and it occurs at and y Minimum value of z 6 and it occurs at and y. E 6 D 8
Rao IIT Academy/ 2015/ XII - CBSE - Board Mathematics Code(65 /2 /MT) Set-2 / Solutions XII - CBSE BOARD CODE (65/2/MT) SET - 2
Rao IIT Academ/ 5/ XII - CBSE - Board Mathematics Code(65 / /MT) Set- / Solutions XII - CBSE BOARD CODE (65//MT) SET - Date: 8.3.5 MATHEMATICS - SOLUTIONS. Let a iˆ 3iˆ kˆ b iˆ ˆj and a b 3 5, b a b Projection
More informationCBSE 2018 ANNUAL EXAMINATION DELHI
CBSE 08 ANNUAL EXAMINATION DELHI (Series SGN Code No 65/ : Delhi Region) Ma Marks : 00 Time Allowed : Hours SECTION A Q0 Find the value of tan cot ( ) Sol 5 5 tan cot ( ) tan tan cot cot 6 6 6 0 a Q0 If
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 informationMarking Scheme. Section A 3. 2 [1] l m n 1 n 1 cos [1] Direction ratios of the given line are 2, 1, 2.
Marking Scheme Section A. B. AB 6 A B 6. sin( ) cos( ) or sin( ). 4. l m n n cos 45 or 6 4 OR Direction ratios of the given line are,,. [/] Hence, direction cosines of the line are:,, or,, [/] Section
More informationRao IIT Academy/ ISC - Board 2018_Std XII_Mathematics_QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS. XII - ISC Board
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS XII - ISC Board MATHEMATICS - QP + SOLUTIONS Date: 6..8 Ma. Marks : Question SECTION - A (8 Marks)
More informationMATHEMATICS (SET -1)
8 Class th (SET ) BD PPER -7 M T H E M T I C S (). adj 8 I 8 I 8I 8 SECTION - I. f () is continuous at f () lim f () ( ) 6 k lim ( )( 6) k lim ( ) k sin cos d tan cot d sin cos ln sec ln sin C.. P : z
More informationDIRECTORATE OF EDUCATION GOVT. OF NCT OF DELHI
456789045678904567890456789045678904567890456789045678904567890456789045678904567890 456789045678904567890456789045678904567890456789045678904567890456789045678904567890 QUESTION BANK 456789045678904567890456789045678904567890456789045678904567890456789045678904567890
More informationSample Paper-05 Mathematics Class XII. Time allowed: 3 hours Answers Maximum Marks: 100. Section A. Section B
Sample Paper-05 Mathematics Class XII Time allowed: hours Answers Maimum Marks: 00. No. (, ) R but (, ) R r. a () + ( ) + ( 5) 8 5 l, m, n 8 8 8. [0, ]. A A ( 8) 8 ( 6) A 8 Hence Prove tan cos sin sin
More informationMATHEMATICS (SET -3) Labour cost Z 300x 400y (to be minimized) The constraints are: SECTION - A 1. f (x) is continuous at x 3 f (3) lim f (x)
8 Class th (SET -) BD PPER -7 M T H E M T I C S () SECTION -. f () is continuous at f () lim f () ( ) 6 k lim ( )( 6) k lim ( ) k. adj I 8 I 8 I 8I 8. P : z 5 5 P : 5 5z z 8 Distance between P & P sin
More informationMarking Scheme (Mathematics XII )
Sr. No. Marking Scheme (Mathematics XII 07-8) Answer Section A., (, ) A A: (, ) A A: (,),(,) Mark(s). -5. a iˆ, b ˆj. (or an other correct answer). 6 6 ( ), () ( ) ( ). Hence, is not associative. Section
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 informationOperating C 1 C 1 C 2 and C 2 C 2 C 3, we get = 0, as R 1 and R 3 are identical. Ans: 0
Q. Write the value of MATHEMATICS y y z z z y y y z z z y Operating R R + R, we get y z y z z y z y ( y z) z y Operating C C C and C C C, we get 0 0 0 0 ( y z) z y y ( y z)( ) z y y 0 0 0 0 = 0, as R and
More informationSET-I SECTION A SECTION B. General Instructions. Time : 3 hours Max. Marks : 100
General Instructions. All questions are compulsor.. This question paper contains 9 questions.. Questions - in Section A are ver short answer tpe questions carring mark each.. Questions 5- in Section B
More informationHALF SYLLABUS TEST. Topics : Ch 01 to Ch 08
Ma Marks : Topics : Ch to Ch 8 HALF SYLLABUS TEST Time : Minutes General instructions : (i) All questions are compulsory (ii) Please check that this question paper contains 9 questions (iii) Questions
More informationANSWERS EXERCISE 7.1 EXERCISE C. 1 sin 3 x 3. 1 e cos 2x 1 ( ) ax+ b. ax bx. x + C. + x
5 MATHEMATIS ANSWERS EXERISE.. cos. ( ) a+ b a 5. sin. cos e e e + + +. a b + + c + + e + 0. + log +. + 5+ +. 7 7 + + +. + +. 5 + 5. 5 7 5 + + + 7 5 sin + e + 0 + cos+ +. tan + sec + tan + 0. tan sec +..
More informationC.B.S.E Class XII Delhi & Outside Delhi Sets
SOLVED PAPER With CBSE Marking Scheme C.B.S.E. 8 Class XII Delhi & Outside Delhi Sets Mathematics Time : Hours Ma. Marks : General Instructions : (i) All questions are compulsory. (ii) The question paper
More informationMATHEMATICS Class: XII Mock Paper 1 Solutions
MATHEMATICS Class: XII Mock Paper Solutions Section A. If A { a, b, c } and B {,, } and a function f : A B is given by f { (a, ), ( b, ), ( c, ) } Every element of set A is mapped to the unique element
More informationCBSE Board Paper Class-XII. Time allowed : 3 hours Maximum Marks : 100
L.K.Gupta (Mathematic Classes) www.poineermathematics.com. MOBILE: 98155771, 461771 CBSE Board Paper -011 Class-XII (SET-1) Time allowed : hours Maimum Marks : 100 General Instructions: (i) All questions
More informationCLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. (ii) Algebra 13. (iii) Calculus 44
CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (iii) Calculus 44 (iv) Vector and Three Dimensional Geometry 7 (v) Linear Programming 06 (vi) Probability 0 Total
More informationMODEL PAPER - I MATHEMATICS. Time allowed : 3 hours Maximum marks : 100
MODEL PAPER - I MATHEMATICS Time allowed : 3 hours Maimum marks : General Instructions. All questions are compulsy.. The question paper consists of 9 questions divided into three sections A, B and C. Section
More informationCLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. Type of Questions Weightage of Number of Total Marks each question questions
CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (Matrices and Determinants) (iii) Calculus 44 (iv) Vector and Three dimensional Geometry 7 (v) Linear Programming
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 informationBoard Answer Paper: MARCH 2014
Board Answer Paper: MARCH 04 and Statistics SECTION I Q.. (A) Select and write the correct answer from the given alternatives in each of the following: i. (C) Let l 0, m 3, n be the direction cosines of
More informationSCIENCE ENTRANCE ACADEMY PREPARATORY EXAMINATION-3 (II P.U.C) SCHEME 0F EVALUATION Marks:150 Date: duration:4hours MATHEMATICS-35
JNANASUDHA SCIENCE ENTRANCE ACADEMY PREPARATORY EXAMINATION- (II P.U.C) SCHEME F EVALUATION Mrks:5 Dte:5-9-8 durtion:4hours MATHEMATICS-5 Q.NO Answer Description Mrk(s)!6 Zero mtri. + sin 4det A, 4 R 4
More informationQUESTION PAPER CODE 65/2/2/F EXPECTED ANSWER/VALUE POINTS
QUESTION PAPER CODE EXPECTED ANSWER/VALUE POINTS SECTION A. P 6 (A A ) P 6 9. (a b c) (a b c) 0 a b c (a b b c c a) 0 a b b c c a. a b sin θ a b cos θ 400 b 4 4. x z 5 or x z 5 mark for dc's of normal
More informationMATHEMATICS. Time allowed : 3 hours Maximum Marks: 100
MATHEMATICS Time allowed : 3 hours Maimum Marks: 00 General Instructions:. All questions are compulsory.. This question paper contains 9 questions. 3. Questions 4 in Section A are very short-answer type
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 informationCBSE Examination Paper, Foreign-2014
CBSE Eamination Paper, Foreign-4 Time allowed: hours Maimum marks: General Instructions: As per given in CBSE Eamination Paper Delhi-4. SET I SECTION A Question numbers to carr mark each.. Let R = {(a,
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 informationSTD. XII ISC - BOARD MATHEMATICS - SOLUTIONS
Rao IIT Academ/XII-ISC-Board -05_Mathematics_SOLUTIONS Date: 6.0.05 Question STD. XII ISC - BOARD MATHEMATICS - SOLUTIONS SECTION A (i) 4 6 M 6 4 9 5 8 M 8 km k k k k M km I 0 5 8 k k 0 0 0 8 k k 0 0 0
More informationMathematics. Guess Paper: 2014 Class: XII. Time Allowed: 3Hours Maximum Marks: 70. Section A
Mathematics Guess Paper: 04 Class: XII Time llowed: Hours Maimum Marks: 70 General Instructions:. The question paper consists of 9 questions divided into three sections, B and C.. Section comprises of
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 informationANSWER KEY 1. [A] 2. [C] 3. [B] 4. [B] 5. [C] 6. [A] 7. [B] 8. [C] 9. [A] 10. [A] 11. [D] 12. [A] 13. [D] 14. [C] 15. [B] 16. [C] 17. [D] 18.
ANSWER KEY. [A]. [C]. [B] 4. [B] 5. [C] 6. [A] 7. [B] 8. [C] 9. [A]. [A]. [D]. [A]. [D] 4. [C] 5. [B] 6. [C] 7. [D] 8. [B] 9. [C]. [C]. [D]. [A]. [B] 4. [D] 5. [A] 6. [D] 7. [B] 8. [D] 9. [D]. [B]. [A].
More informationANSWERS 1.3 EXERCISE. 7. 2, 1 8. (i) represents function which is surjective but not injective (ii) does not represent function.
ANSWERS 87 ANSWERS. EXERCISE. (b,b), (,), (a,). [-5,5]. 4 4 4. f 5. f { ( ba, ),( db, ),( a, ),( d, ) } 4 6. f f 6 0 7., 8. (i) represents funtion whih is surjetive but not injetive (ii) does not represent
More informationHSC - BOARD MATHEMATICS (40) - SOLUTIONS
Date: 8..5 Q. (A) SECTION - I (i) (d) A () (ii) (c) A A I 6 6 6 A I 64 I I A A 6 (iii) (a) fg cos A cos HSC - BOARD - 5 MATHEMATICS (4) - SOLUTIONS cos cos ch () hy g fy c...(i) Comparing with A Hy By
More informationMATHEMATICS SOLUTION
MATHEMATICS SOLUTION MHT-CET 6 (MATHEMATICS). (A) 5 0 55 5 9 6 5 9. (A) If the school bus does not come) (I will not go to school) ( I shall meet my friend) (I shall go out for a movie) ~ p ~ q r s ~ p
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 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 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 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 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 information2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time is
. If P(A) = x, P = 2x, P(A B) = 2, P ( A B) = 2 3, then the value of x is (A) 5 8 5 36 6 36 36 2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time
More information3. Total number of functions from the set A to set B is n. 4. Total number of one-one functions from the set A to set B is n Pm
ASSIGNMENT CLASS XII RELATIONS AND FUNCTIONS Important Formulas If A and B are finite sets containing m and n elements, then Total number of relations from the set A to set B is mn Total number of relations
More informationAnswers for NSSH exam paper 2 type of questions, based on the syllabus part 2 (includes 16)
Answers for NSSH eam paper type of questions, based on the syllabus part (includes 6) Section Integration dy 6 6. (a) Integrate with respect to : d y c ( )d or d The curve passes through P(,) so = 6/ +
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 informationVectors for Physics. AP Physics C
Vectors for Physics AP Physics C A Vector is a quantity that has a magnitude (size) AND a direction. can be in one-dimension, two-dimensions, or even three-dimensions can be represented using a magnitude
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 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 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 informationPart r A A A 1 Mark Part r B B B 2 Marks Mark P t ar t t C C 5 Mar M ks Part r E 4 Marks Mark Tot To a t l
Part Part P t Part Part Total A B C E 1 Mark 2 Marks 5 Marks M k 4 Marks CIRCLES 12 Marks approximately Definition ; A circle is defined as the locus of a point which moves such that its distance from
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 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 informationTABLE OF CONTENTS 2 CHAPTER 1
TABLE OF CONTENTS CHAPTER 1 Quadratics CHAPTER Functions 3 CHAPTER 3 Coordinate Geometry 3 CHAPTER 4 Circular Measure 4 CHAPTER 5 Trigonometry 4 CHAPTER 6 Vectors 5 CHAPTER 7 Series 6 CHAPTER 8 Differentiation
More information02. If (x, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) x + y = 0 (B) bx ay = 0 (C) ax by = 0 (D) bx + ay = 0 (E) ax + by =
0. π/ sin d 0 sin + cos (A) 0 (B) π (C) 3 π / (D) π / (E) π /4 0. If (, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) + y = 0 (B) b ay = 0 (C) a by = 0 (D) b + ay = 0 (E) a + by = 0 03.
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 information(c) n (d) n 2. (a) (b) (c) (d) (a) Null set (b) {P} (c) {P, Q, R} (d) {Q, R} (a) 2k (b) 7 (c) 2 (d) K (a) 1 (b) 3 (c) 3xyz (d) 27xyz
318 NDA Mathematics Practice Set 1. (1001)2 (101)2 (110)2 (100)2 2. z 1/z 2z z/2 3. The multiplication of the number (10101)2 by (1101)2 yields which one of the following? (100011001)2 (100010001)2 (110010011)2
More informationLIST OF MEMBERS WHO PREPARED QUESTION BANK FOR MATHEMATICS FOR CLASS XII TEAM MEMBERS. Sl. No. Name Designation
LIST OF MEMBERS WHO PREPARED QUESTION BANK FOR MATHEMATICS FOR CLASS XII TEAM MEMBERS Sl. No. Name Designation. Sh. S.B. Tripathi R.S.B.V., Jheel Khuranja (Group Leader) Delhi. (M. 98086). Sh. Sanjeev
More informationSolutionbank Edexcel AS and A Level Modular Mathematics
Page of Exercise A, Question Use the binomial theorem to expand, x
More informationDESIGN OF THE QUESTION PAPER Mathematics Class X NCERT. Time : 3 Hours Maximum Marks : 80
DESIGN OF THE QUESTION PAPER Mathematics Class X Weightage and the distribution of marks over different dimensions of the question shall be as follows: (A) Weightage to Content/ Subject Units : S.No. Content
More informationAMB121F Trigonometry Notes
AMB11F Trigonometry Notes Trigonometry is a study of measurements of sides of triangles linked to the angles, and the application of this theory. Let ABC be right-angled so that angles A and B are acute
More informationSpecial Mathematics Notes
Special Mathematics Notes Tetbook: Classroom Mathematics Stds 9 & 10 CHAPTER 6 Trigonometr Trigonometr is a stud of measurements of sides of triangles as related to the angles, and the application of this
More informationSection 6.2 Trigonometric Functions: Unit Circle Approach
Section. Trigonometric Functions: Unit Circle Approach The unit circle is a circle of radius centered at the origin. If we have an angle in standard position superimposed on the unit circle, the terminal
More informationLesson 28 Working with Special Triangles
Lesson 28 Working with Special Triangles Pre-Calculus 3/3/14 Pre-Calculus 1 Review Where We ve Been We have a new understanding of angles as we have now placed angles in a circle on a coordinate plane
More informationSect 7.4 Trigonometric Functions of Any Angles
Sect 7.4 Trigonometric Functions of Any Angles Objective #: Extending the definition to find the trigonometric function of any angle. Before we can extend the definition our trigonometric functions, we
More information63487 [Q. Booklet Number]
WBJEE - 0 (Answers & Hints) 687 [Q. Booklet Number] Regd. Office : Aakash Tower, Plot No., Sector-, Dwarka, New Delhi-0075 Ph. : 0-7656 Fa : 0-767 ANSWERS & HINTS for WBJEE - 0 by & Aakash IIT-JEE MULTIPLE
More information2 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
More informationMATH section 3.1 Maximum and Minimum Values Page 1 of 7
MATH section. Maimum and Minimum Values Page of 7 Definition : Let c be a number in the domain D of a function f. Then c ) is the Absolute maimum value of f on D if ) c f() for all in D. Absolute minimum
More informationSOLUTION CLASS-XII / (CBSE)
CBSE XII EXAMINATION-6 SOLUTION--6 CBSE th Board MATHEMATICS SET- CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar,
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 informationInverse Trigonometric Functions
Inverse Trigonometric Functions. Inverse of a function f eists, if function is one-one and onto, i.e., bijective.. Trignometric functions are many-one functions but these become one-one, onto, if we restrict
More informationMATHEMATICS. MINIMUM LEVEL MATERIAL for CLASS XII Project Planned By. Honourable Shri D. Manivannan Deputy Commissioner,KVS RO Hyderabad
MATHEMATICS MINIMUM LEVEL MATERIAL for CLASS XII 06 7 Project Planned By Honourable Shri D. Manivannan Deputy Commissioner,KVS RO Hyderabad Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist
More informationINVERSE TRIGONOMETRIC FUNCTIONS
Inverse Trigonometric MODULE - IV 8 INVERSE TRIGONOMETRIC FUNCTIONS In the previous lesson, you have studied the definition of a function and different kinds of functions. We have defined inverse function.
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 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 informationSolutionbank C2 Edexcel Modular Mathematics for AS and A-Level
file://c:\users\buba\kaz\ouba\c_rev_a_.html Eercise A, Question Epand and simplify ( ) 5. ( ) 5 = + 5 ( ) + 0 ( ) + 0 ( ) + 5 ( ) + ( ) 5 = 5 + 0 0 + 5 5 Compare ( + ) n with ( ) n. Replace n by 5 and
More informationWBJEEM Answer Keys by Aakash Institute, Kolkata Centre MATHEMATICS
WBJEEM - 05 Answer Keys by, Kolkata Centre MATHEMATICS Q.No. μ β γ δ 0 B A A D 0 B A C A 0 B C A * 04 C B B C 05 D D B A 06 A A B C 07 A * C A 08 D C D A 09 C C A * 0 C B D D B C A A D A A B A C A B 4
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 informationAnalytic Trigonometry. Copyright Cengage Learning. All rights reserved.
Analytic Trigonometry Copyright Cengage Learning. All rights reserved. 7.1 Trigonometric Identities Copyright Cengage Learning. All rights reserved. Objectives Simplifying Trigonometric Expressions Proving
More informationUsing the Definitions of the Trigonometric Functions
1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities Signs and Ranges of Function Values Pythagorean Identities Quotient Identities February 1, 2013 Mrs. Poland Objectives Objective
More information(iii) For each question in Section III, you will be awarded 4 Marks if you darken only the bubble corresponding to the
FIITJEE Solutions to IIT - JEE 8 (Paper, Code 4) Time: hours M. Marks: 4 Note: (i) The question paper consists of parts (Part I : Mathematics, Part II : Physics, Part III : Chemistry). Each part has 4
More informationIB 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
More informationPhysics 101 Lecture 2 Vectors Dr. Ali ÖVGÜN
Phsics 101 Lecture 2 Vectors Dr. Ali ÖVGÜN EMU Phsics Department www.aovgun.com Coordinate Sstems qcartesian coordinate sstem qpolar coordinate sstem Januar 21, 2015 qfrom Cartesian to Polar coordinate
More informationCode : N. Mathematics ( ) ( ) ( ) Q c, a and b are coplanar. x 2 = λ µ... (ii) 1. If (2, 3, 5) is one end of a diameter of the sphere
Mathematics. If (, 3, ) is one end of a diameter of the sphere x + y + z 6x y z + 0 = 0, then the coordinates of the other end of the diameter are () (4, 3, 3) () (4, 9, 3) (3) (4, 3, 3) (4) (4, 3, ) Sol.
More informationMore with Angles Reference Angles
More with Angles Reference Angles A reference angle is the angle formed by the terminal side of an angle θ, and the (closest) x axis. A reference angle, θ', is always 0 o
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 informationObjective 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
More informationContents PART II. Foreword
Contents PART II Foreword v Preface vii 7. Integrals 87 7. Introduction 88 7. Integration as an Inverse Process of Differentiation 88 7. Methods of Integration 00 7.4 Integrals of some Particular Functions
More informationSolutions to RSPL/1. Mathematics 10
Solutions to RSPL/. It is given that 3 is a zero of f(x) x 3x + p. \ (x 3) is a factor of f(x). So, (3) 3(3) + p 0 8 9 + p 0 p 9 Thus, the polynomial is x 3x 9. Now, x 3x 9 x 6x + 3x 9 x(x 3) + 3(x 3)
More informationMathematics. Class 12th. CBSE Examination Paper 2015 (All India Set) (Detailed Solutions)
CBSE Eamination Paer (All India Set) (Detailed Solutions) Mathematics Class th z z. We have, z On aling R R R, we get z z z z (/) Taking common ( z) from R common from R, we get ( z)( ) z ( z)( ) [ R R
More informationMATHEMATICS. metres (D) metres (C)
MATHEMATICS. If is the root of the equation + k = 0, then what is the value of k? 9. Two striaght lines y = 0 and 6y 6 = 0 never intersect intersect at a single point intersect at infinite number of points
More informationTotal marks 70. Section I. 10 marks. Section II. 60 marks
THE KING S SCHOOL 03 Higher School Certificate Trial Eamination Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Board-approved calculators
More informationAPPM 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
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 informationIB Practice - Calculus - Differentiation Applications (V2 Legacy)
IB Math High Level Year - Calc Practice: Differentiation Applications IB Practice - Calculus - Differentiation Applications (V Legacy). A particle moves along a straight line. When it is a distance s from
More information1. Which of the following defines a function f for which f ( x) = f( x) 2. ln(4 2 x) < 0 if and only if
. Which of the following defines a function f for which f ( ) = f( )? a. f ( ) = + 4 b. f ( ) = sin( ) f ( ) = cos( ) f ( ) = e f ( ) = log. ln(4 ) < 0 if and only if a. < b. < < < < > >. If f ( ) = (
More informationReview (2) Calculus II (201-nyb-05/05,06) Winter 2019
Review () Calculus II (-nyb-5/5,6) Winter 9 Note. You should also review the integrals eercises on the first review sheet. Eercise. Evaluate each of the following integrals. a. sin 3 ( )cos ( csc 3 (log)cot
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 informationTime allowed : 3 hours Maximum Marks : 100
CBSE XII EXAMINATION-8 Series SGN SET- Roll No. Candiates must write code on the title page of the answer book Please check that this question paper contains printed pages. Code number given on the right
More information1993 AP Calculus AB: Section I
99 AP Calculus AB: Section I 90 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among
More information