ASSIGNMENT ON MATRICES AND DETERMINANTS (CBSE/NCERT/OTHERSTATE BOARDS). Write the orders of AB and BA. x y 2z w 5 3
|
|
- Katherine Ferguson
- 5 years ago
- Views:
Transcription
1 1 If A = [a ij ] = is a matrix given by A [a ] ij Write the order of A and find the elements a 4, a 34 Also, show that a 3 = a 3 + a 4 If A = and B = Write the orders of AB and BA Find x, y, z and w such that x y z w 5 3 x y x w For what values of x and y are the following matrices equal x 1 3y x 3 y A,B 0 y 5y Construct a 4 3 matrix whose elements are (i) a ij = i + i (ii) a ij = i j j i j 7 3x 4y x y 4 Find x, y, a and b if a b a b The sales figure of two car dealers during January 007 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars, while dealer B sold 7 deluxe, premium and 3 standard cars Total sales over the month period of January-February revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars In the same month period, dealer B sold 10 deluxe, 5 premium and 7 standard cars Write 3 matrices summarizing sales data for January and -month period for each dealer 9 Two farmers Ram Kishan and Gurcharan singh cultivate only three varities of rice namely Basmati, Permal and Naura The sale (in Rs) of these varities of rice by both the farmers in the month of September and October are given by the following matrices A and B September sales (in Rs) Basmati Permal Naura 10, 000 0, , 000 Ram Kishan A 50,000 30,000 10,000 Gurcharan Singh
2 Basmati Permal Naura 5, , 000 6, 000 Ram Kishan B 0, , , 000 Gurcharan Singh Find : (i) What were the combined sales in September and October for each farmer in each variety (ii) What was the change in sales from September to October? (iii) If both farmers receive % profit on gross rupees sales, compute the profit for each farmer and for each variety sold in October Prove that product of matrices cos cos sin cos cos sin and cos sin sin cos sin sin is the null matrix, when and differ by an odd multiple of 10 If A and B , find the values of for which A = B tan( / ) Let A tan( / ) 0 and I be the identity matrix of order Show that I + A = (I A) cos sin sin cos 0 1 Let f(x) = x 5x + 6 Find f(a) if A = Evaluate the following : (i) (iii) (ii) Compute the elements a 43 and a of the matrix :
3 A If w is a complex cube root of unity, show that 16 w 1 w w w 1 w 0 If x 4 1 x , find x w w w w w w 1 w 1 w w 0 17 If 3 A 1 0, show that A - A + 3I = O If A 1 show that A -5A + 7I = O use this to find A 4 3 If A 4, find k such that A = ka - I 0 7 x 14x 7x Find the value of x for which the matrix product x 4x x equal an identity matrix 0 1 If If 0 0 A 4 0, find A16 cos sin sin A, prove that sin cos sin n cos n sin n sin n A for all n N sin n cos n sin n Let A Use the principle of mathematical induction to show that every positive integer n 1 n n(n 1) / for n A 0 1 n 3 A matrix X has a + b rows and a + columns while the matrix Y has b + 1 rows and a + 3 columns Both matrices XY and YX exist Find a and b Can you say XY and YX are of the same type? Are they equal
4 4 Give examples of matrices (i) A and B such that AB BA (ii) A and B such that AB = O but A 0, B 0 (iii) A and B such that AB = O but BA O (iv) A, B and C such that AB = AC but B C, A 0 5 Three shopkeepers A, B and C go to a store to buy stationary A purchases 1 dozen notebooks, 5 dozen pens and 6 dozen pencils B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils C purchases 11 dozen notebooks, 13 dozen pens and 8 dozen pencils A notebook costs 40 paise, a pen costs Rs 15 and a pencil costs 35 paise Use matrix multiplication to calculate each individual's bill 6 In a legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways: telephone, house calls and letters The cost per contact (in paise) is given matrix A as Cost per contact 40 Telephone A 100 House call 50 Letter The number of contacts of each type made in two cities X and Y is given in matrix B as Telephone House call Letter X B Y Find the total amount spent by the group in the two cities X and Y 7 Find the inverse of the matrix INVERSE BY USING ELEMEMENTRY OPERATIONS 1 3 A 7, using elementary row transformations 8 Using elementary row transformation find eh inverse of the matrix 3 1 A Find the inverse of the matrix A by using elementary row transformations Using elementary row transformation find The inverse of the following matrix
5 1 4 (i) (ii) (iii) TRANSPOSE OF A MATRIX 1 If A and B = 1 4, verify that (AB) T = B T A T 3 cos x sin x If A= sin x cos x, find x satisfying 0 < x < when A + A T =I If 1 If A =, write AA T 3 1 A 1 is a matrix satisfying AA T = 9I 3, then find the values of a and b a b 35 Express the matrix 3 3 A as the sum of a symmetric and a skew-symmetric matrix y z Find the values of x, y, z if the matrix A x y z satisfy the equation A T A = I 3 x y z 37 Prove that every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skewsymmetric matrix 38 Show that the matrix B T AB is symmetric or skew-symmetric according as A is symmetric or skewsymmetric 39 If A and B are symmetric matrices, then show that AB is symmetric iff AB = BA ie A and B commute 40 Show that sin10 sin 80 o o cos10 cos80 o o 1 DETERMINANTS 41 log 51 log Evaluate the determinant log 8 log 9 3 4
6 4 If [] denotes the greatest integer less than or equal to the real number under consideration, and -1 x < 0, 0 y < 1, 1 z <, then find the value of the determinant 43 Evaluate the determinant 1 sin 1 sin 1 sin 1 sin 1 Also, prove that 4 44 Find the integral value of x, if x x = 8 45 x For what value of x the matrix A = 1 x 1 1 is singular? 1 1 x 1 46 If x 3 3, find the values of x 3x x 47 Evaluate PROPERTIES OF DETERMINANTS 48 Without expanding evaluate the determinant If w is a complex cube root of unity Show that 50 Show that 1 w w w w 1 0 w 1 w 1 a b c 1 b c a 0 1 c a b
7 51 b c c a a b Show that c a a b b c 0 a b b c c a 5 1 bc a(b c) Show that 1 ca b(c a) 0 1 ab c(a b) 53 b c bc b c Without expanding show that c a ca c a 0 a b ab a b 54 Without expanding evaluate the determinant sin cos sin( ) sin cos sin( ) sin cos sin( ) 55 Without expanding evaluate the determinant x x x x (a a ) (a a ) 1 y y y y (a a ) (a a ) 1 z z z z (a a ) (a a ) 1, where a, > 0 and x, y, z R 56 If a, b, c are in AP find the value of y 4 5y 7 8y a 3y 5 6y 8 9y b 4y 6 7y 9 10y c Find the value of the determinant = x 9x 1x 58 Without expanding evaluate the determinant = If 1 = x y z x y z and = yz zx xy, without expanding prove that 1 = x y z
8 60 Without expanding, prove that 61 Without expanding, prove that a bx c dx p qx a c p ax b cx d px q (1 x ) b d q u v w u v w a b c x y z y b q x y z p q r x a p p q r a b c z c r 6 a ab ac Prove that ba b bc 4a b c ac bc c Prove that 1 1x 1 xy y 64 If a, b, c are all positive and are pth, qth and rth terms of a GP, then show that log a p 1 = log b q 1 0 log c r 1 65 Evaluate : 66 Show that 67 Prove that 1 a a 1 b b 1 c c x y z x y z xyz(x y)(y z)(z x) x y z ( )( )( )( ) 68 Prove that 1 a a b b (a b)(b c)(c a)(a b c) 1 c c 3
9 69 Prove that 70 Prove that 71 Show that a b c c c a b c a a 4abc b b c a b a(b c a ) b c 3 3 a b(c a b ) c abc(a b c ) a b c(a b c ) a b c a b c a b c (a b)(b c)(c a)(ab bc ca) bc ca ab a b c 7 For any scalar p prove that = 73 If x y z and 74 x x 1 px y y 1 py (1 pxyz)(x y)(y z)(z x) z z 1 pz 3 x x 1 x 3 3 y y 1y 0, z z 1 y 3 then prove that xyz = If m N and m, prove that m m C 1 C m C m m1 m C C C 75 Evaluate : 10! 11! 1! 11! 1! 13! 1! 13! 14! Prove that Show that x y x x 5x 4y 4x x x 10x 8y 8x 3x a a b a b c a 3a b 4a 3b c = a 3 3a 6a 3b 10a 6b 3c 3
10 78 Show that 79 Prove that 80 Prove that 81 Solve 8 Solve 83 Show that 84 Show that 85 Prove that b c c a a b a b c q r r p p q p q r y z z x x y x y z 1 a b 1 abc1 abc bc ca ab a b c c (b c) a a b (c a) b c c (a b) a x a x a x a x a x a x 0 a x a x a x x x 3 3x 4 x 4 x 9 3x 16 0 x 8 x 7 3x 64 1 a b ab b ab 1a b a (1 a b ) 3 b a 1 a b b c ab ac ba c a bc 4a b c ca cb a b a b ax by b c bx cy (b ac)(ax bxy cy ) ax by bx cy 0 86 If, a, b, c are positive and unequal, show that the value of the determinant a b c b c a c a b is always negative 87 Let r = r x n(n 1) r 1 y n 3r z n(3n 1) Show that r 0 n r1
11 88 If m is a positive integer and D r = m r 1 Cr 1 m m 1 m 1 sin (m ) sin (m) sin (m 1) Prove that m Dr 0 r0 89 Solve 1 1 x p 1 p 1 p x 0 3 x 1 x AREA OF A TRIANGLE 90 Find the area of the triangle with vertices A(5, 4), B(-, 4) and C(, -6) 91 If A(x 1, y 1 ), B(x, y ) and C(x 3, y 3 ) are vertices of an equilateral triangle whose each side is equal to a, then prove that x y 1 1 x y 3a x y Using determinants, find the value of k so that the points (k, -k),(-k + 1, k) and (- 4 - k, 6 - k) may be collinear 93 Find the value of so that the points (1, - 5), (- 4,5) and (, 7) are collinear 94 Show that the points (a, b + c), (b, c + a) and (c, a + b) are collinear 95 Find the equation of the line joining A(1, 3) and B(0, 0) using determinants and find k if D(k, 0) is a point such that area of ABD is 3 sq units 96 Find the inverse of A and verify that A -1 A = I tan x If A tan x 1, show that A T A -1 cos x sin x = sin x cos x 3 Show that A 3 4 satisfies the equation x 6x + 17 = 0 Hence, find A -1 3 For the matrix A 1 1, find the numbers a and b such that A + aa + bi = 0 Hence, find A -1
12 Find the matrix X for which X For the matrix A = 1 3 Show that A 3-6A + 5A + 11I 3 = O Hence, find A Find the matrix X satisfying the matrix equation X Find the matrix X satisfying the equation X Use matrix method to examine the following system of equations for consistency or inconsistency : 4x y = 3, 6x 3y = Show that the following system of equations is consistent x y + 3x = 5, 3x + y z = 7, 4x + 5y 5z = Solve the following system of equations, using matrix method : [CBSE 00, 003, 005] x + y + z = 7, x + 3z = 11, x 3y = If A 1 3, find A -1 and hence solve the system of linear equations x + y + z = 4, -x + y + z = 0, x 3y + z = Determine the product and use it to solve the system of equations : x y + z = 4, x y x = 9, x + y + 3z = If A = and B = 4 4 are two square matrices, find AB and hence solve the system of linear equations: x - y = 3, x + 3y + 4z = 17,y + z = 7
13 Find A -1, where A 3 Hence solve the system of equations x + y 3z = -4, x + 3y + z =, 3x 3y 4z = An amount of Rs 5000 is put into three investments at the rate of interest of 6%, 7% and 8% per annum respectively The total annual income is Rs 358 If the combined income from the first two investments is Rs 70 more than the income from the third, find the amount of each investment by matrix method 11 solve the system of equations by matrix method : , 1, =; x, y, z 0 x y z x y z x y z 113 A company produces three products every day Their production on a certain day is 45 tons, It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product Determine the production level of each product using matrix method 114 The prices of three commodities P, Q and R are Rs x, y and z per unit respective A purchases 4 units of R and sells 3 units of P and 5 units of Q B purchases 3 units of Q and sells units of P and 1 unit of R C purchases 1 unit of P and sells 4 units of Q and 6 units of R In the process A, B and C earn Rs 6000, Rs 5000 and Rs respectively If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method
MATRICES. Chapter Introduction. 3.2 Matrix. The essence of Mathematics lies in its freedom. CANTOR
56 Chapter 3 MATRICES The essence of Mathematics lies in its freedom. CANTOR 3.1 Introduction The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful
More informationCLASS XII CBSE MATHEMATICS MATRICES 1 Mark/2 Marks Questions
CLASS XII CBSE MATHEMATICS MATRICES 1 Mark/ Marks Questions 1) How many matrices of order 3 x 3 are possible with each entry as 0 or 1? ) Write a square matrix of order, which is both symmetric and skewsymmetric.
More informationRatio and Proportion, Indices and Logarithm
CHAPTER 1 Ratio and Proportion, Indices and Logarithm 2006 November [1] Two numbers are in the ratio 2 : 3 and the difference of their squares is 320. The numbers are : 12, 18 16, 24 14, 21 None. 649 [2]
More informationis equal to = 3 2 x, if x < 0 f (0) = lim h = 0. Therefore f exists and is continuous at 0.
Madhava Mathematics Competition January 6, 2013 Solutions and scheme of marking Part I N.B. Each question in Part I carries 2 marks. p(k + 1) 1. If p(x) is a non-constant polynomial, then lim k p(k) (a)
More informationRATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER RATIO AND PROPORTION, INDICES, LOGARITHMS UNIT I: RATIO LEARNING OBJECTIVES After reading this unit a student will learn How to compute and compare two ratios; Effect of increase or decrease of
More information1. Matrices and Determinants
Important Questions 1. Matrices and Determinants Ex.1.1 (2) x 3x y Find the values of x, y, z if 2x + z 3y w = 0 7 3 2a Ex 1.1 (3) 2x 3x y If 2x + z 3y w = 3 2 find x, y, z, w 4 7 Ex 1.1 (13) 3 7 3 2 Find
More informationDISCUSSION CLASS OF DAX IS ON 22ND MARCH, TIME : 9-12 BRING ALL YOUR DOUBTS [STRAIGHT OBJECTIVE TYPE]
DISCUSSION CLASS OF DAX IS ON ND MARCH, TIME : 9- BRING ALL YOUR DOUBTS [STRAIGHT OBJECTIVE TYPE] Q. Let y = cos x (cos x cos x). Then y is (A) 0 only when x 0 (B) 0 for all real x (C) 0 for all real x
More informationJEE/BITSAT LEVEL TEST
JEE/BITSAT LEVEL TEST Booklet Code A/B/C/D Test Code : 00 Matrices & Determinants Answer Key/Hints Q. i 0 A =, then A A is equal to 0 i (a.) I (b.) -ia (c.) -I (d.) ia i 0 i 0 0 Sol. We have AA I 0 i 0
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 informationMatrix operations Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the
More informationCHAPTER - 2 EQUATIONS. Copyright -The Institute of Chartered Accountants of India
CHAPTER - EQUATIONS EQUATIONS LEARNING OBJECTIVES After studying this chapter, you will be able to: u Understand the concept of equations and its various degrees linear, simultaneous, quadratic and cubic
More informationContents CONTENTS 1. 1 Straight Lines and Linear Equations 1. 2 Systems of Equations 6. 3 Applications to Business Analysis 11.
CONTENTS 1 Contents 1 Straight Lines and Linear Equations 1 2 Systems of Equations 6 3 Applications to Business Analysis 11 4 Functions 16 5 Quadratic Functions and Parabolas 21 6 More Simple Functions
More informationCLASS 12 ALGEBRA OF MATRICES
CLASS 12 ALGEBRA OF MATRICES Deepak Sir 9811291604 SHRI SAI MASTERS TUITION CENTER CLASS 12 A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements
More informationLINEAR SYSTEMS AND MATRICES
CHAPTER 3 LINEAR SYSTEMS AND MATRICES SECTION 3. INTRODUCTION TO LINEAR SYSTEMS This initial section takes account of the fact that some students remember only hazily the method of elimination for and
More informationEXERCISE - 01 CHECK YOUR GRASP
XRCIS - 1 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR) 1. If A B = 1 5 3 7 and A 3B = 5 7, then matrix B is equal to - 4 5 6 7 (B) 6 3 7 1 3 6 1. If A = cos sin sin cos, then A A is
More informationCBSE Class-12 Mathematics Sample Paper (By CBSE)
CBSE Class-12 Mathematics Sample Paper (By CBSE) General Instructions: All questions are compulsory. This question paper contains 29 questions. Question 1-4 in Section A are very short-answer type 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 informationNATIONAL INSTITUTE OF HOTEL MANAGEMENT, KOLKATA BUSINESS MATHEMATICS 3 rd Semester
NATIONAL INSTITUTE OF HOTEL MANAGEMENT, KOLKATA BUSINESS MATHEMATICS 3 rd Semester Choose (tick) the appropriate from the following options given below. 1. Find the number of subsets of a set {x : x is
More information7.2 Matrix Algebra. DEFINITION Matrix. D 21 a 22 Á a 2n. EXAMPLE 1 Determining the Order of a Matrix d. (b) The matrix D T has order 4 * 2.
530 CHAPTER 7 Systems and Matrices 7.2 Matrix Algebra What you ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix
More informationSCTT The pqr-method august 2016
SCTT The pqr-method august 2016 A. Doledenok, M. Fadin, À. Menshchikov, A. Semchankau Almost all inequalities considered in our project are symmetric. Hence if plugging (a 0, b 0, c 0 ) into our inequality
More information2 Functions and Their
CHAPTER Functions and Their Applications Chapter Outline Introduction The Concept of a Function Types of Functions Roots (Zeros) of a Function Some Useful Functions in Business and Economics Equilibrium
More information13 = m m = (C) The symmetry of the figure implies that ABH, BCE, CDF, and DAG are congruent right triangles. So
Solutions 2005 56 th AMC 2 A 2. (D It is given that 0.x = 2 and 0.2y = 2, so x = 20 and y = 0. Thus x y = 0. 2. (B Since 2x + 7 = 3 we have x = 2. Hence 2 = bx 0 = 2b 0, so 2b = 8, and b = 4. 3. (B Let
More informationFall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop
Fall 2017 Inverse of a matrix Authors: Alexander Knop Institute: UC San Diego Row-Column Rule If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding
More information* is a row matrix * An mxn matrix is a square matrix if and only if m=n * A= is a diagonal matrix if = 0 i
CET MATRICES *A matrix is an order rectangular array of numbers * A matrix having m rows and n columns is called mxn matrix of order * is a column matrix * is a row matrix * An mxn matrix is a square matrix
More informationMATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Final Revision CLASS XII CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION.
MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Final Revision CLASS XII 2016 17 CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.),
More informationThe product is 8a 11 b 6. Simplify. 3a 3 b 12 27a 3 b 12 64x 6 b 3. y 10 n 28 x (y 5 ) 2 2. (n 7 ) 4 3. (x 2 ) 5 (x 3 )
Answers (Lesson 8-) Lesson 8-8- 6- Study Guide and Intervention Multiplying Monomials Multiply Monomials A monomial is a number, a variable, or a product of a number and one or more variables. An expression
More informationHomework 1/Solutions. Graded Exercises
MTH 310-3 Abstract Algebra I and Number Theory S18 Homework 1/Solutions Graded Exercises Exercise 1. Below are parts of the addition table and parts of the multiplication table of a ring. Complete both
More informationWe could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2
Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1
More informationPRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES PRADEEP SHARMA. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 1
PRADEEP SHARMA PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page Chapter Relation and Functions Mark Questions A relation R in a Set A is called..., if each element of A is related to every element
More informationTABLE OF CONTENTS. Our aim is to give people math skillss in a very simple way Raymond J. Page 2 of 29
TABLE OF CONTENTS Topic.Page# 1. Numbers..04 2. Ratio, Profit & Loss 06 3. Angles......06 4. Interest..06 5. Algebra..07 6. Quadratic Equations.08 7. Logarithms...09 8. Series..10 9. Sphere.11 10. Coordinate
More informationPrepared by: M. S. KumarSwamy, TGT(Maths) Page
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 50 - CHAPTER 3: MATRICES QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 03 marks Matrix A matrix is an ordered rectangular array of numbers
More information1. The unit vector perpendicular to both the lines. Ans:, (2)
1. The unit vector perpendicular to both the lines x 1 y 2 z 1 x 2 y 2 z 3 and 3 1 2 1 2 3 i 7j 7k i 7j 5k 99 5 3 1) 2) i 7j 5k 7i 7j k 3) 4) 5 3 99 i 7j 5k Ans:, (2) 5 3 is Solution: Consider i j k a
More informationChapter 1: Systems of Linear Equations and Matrices
: Systems of Linear Equations and Matrices Multiple Choice Questions. Which of the following equations is linear? (A) x + 3x 3 + 4x 4 3 = 5 (B) 3x x + x 3 = 5 (C) 5x + 5 x x 3 = x + cos (x ) + 4x 3 = 7.
More informationTEST CODE: MMA (Objective type) 2015 SYLLABUS
TEST CODE: MMA (Objective type) 2015 SYLLABUS Analytical Reasoning Algebra Arithmetic, geometric and harmonic progression. Continued fractions. Elementary combinatorics: Permutations and combinations,
More informationPhys 201. Matrices and Determinants
Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1
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 informationTransweb Educational Services Pvt. Ltd Tel:
. An aeroplane flying at a constant speed, parallel to the horizontal ground, km above it, is observed at an elevation of 6º from a point on the ground. If, after five seconds, its elevation from the same
More informationMathematics Paper 2 (Calculator)
Write your name here Surname Other names Pearson Edexcel Level 1/Level 2 GCSE (9-1) Centre Number Candidate Number Mathematics Paper 2 (Calculator) Specimen Papers Set 1 Time: 1 hour 30 minutes Higher
More informationMatrices and Determinants for Undergraduates. By Dr. Anju Gupta. Ms. Reena Yadav
Matrices and Determinants for Undergraduates By Dr. Anju Gupta Director, NCWEB, University of Delhi Ms. Reena Yadav Assistant Professor, NCWEB, University of Delhi Matrices A rectangular arrangement consisting
More informationMath Bank - 6. What is the area of the triangle on the Argand diagram formed by the complex number z, -iz, z iz? (a) z 2 (b) 2 z 2
Math Bank - 6 Q.) Suppose A represents the symbol, B represents the symbol 0, C represents the symbol, D represents the symbol 0 and so on. If we divide INDIA by AGRA, then which one of the following is
More informationThe P/Q Mathematics Study Guide
The P/Q Mathematics Study Guide Copyright 007 by Lawrence Perez and Patrick Quigley All Rights Reserved Table of Contents Ch. Numerical Operations - Integers... - Fractions... - Proportion and Percent...
More informationSYLLABUS. MATHEMATICS (041) CLASS XII One Paper Three Hours Marks: 100
SYLLABUS MATHEMATICS (041) CLASS XII 2012-13 One Paper Three Hours Marks: 100 Units Marks I. RELATIONS AND FUNCTIONS 10 II. ALGEBRA 13 III. CALCULUS 44 IV. VECTS AND THREE - DIMENSIONAL GEOMETRY 17 V.
More informationPrepared by: M. S. KumarSwamy, TGT(Maths) Page
Prepared b: M. S. KumarSwam, TGT(Maths) Page - 77 - CHAPTER 4: DETERMINANTS QUICK REVISION (Important Concepts & Formulae) Determinant a b If A = c d, then determinant of A is written as A = a b = det
More informationCARIBBEAN SECONDARY EDUCATION CERTIFICATE EXAMINATION (MAY 2015) MATHEMATICS Paper 02 General Proficiency. 2 hours and 40 minutes
CARIBBEAN SECONDARY EDUCATION CERTIFICATE EXAMINATION (MAY 2015) MATHEMATICS Paper 02 General Proficiency. 2 hours and 40 minutes Section I (Answer ALL questions in this section) 1. (a) Using a calculator,
More informationUNIT CSEC Multiple Choice Items Sample Paper 01
UNIT 0 Sample CSEC Multiple Choice Items and Revision Questions UNIT 0.. CSEC Multiple Choice Items Sample Paper 0 This paper consists of 60 Multiple Choice items from the Core Syllabus according to the
More informationModel Answer Paper 24(24 1) 2 12
ICSE X SUBJECT : MATHEMATICS Marks : 80 Exam No. : MT/ICSE/Semi Prelim II- Set-B-00 Model Answer Paper Time : ½ hrs. SECTION I (40 Marks) A. (a) Maturity value ` 0,000 No. of months (n) 4 months Rate of
More informationPRMO Solution
PRMO Solution 0.08.07. How many positive integers less than 000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3?. Suppose a, b
More informationGrade: 10 Mathematics Olympiad Qualifier Set: 2
Grade: 10 Mathematics Olympiad Qualifier Set: 2 ----------------------------------------------------------------------------------------------- Max Marks: 60 Test ID: 22101 Time Allotted: 40 Mins -----------------------------------------------------------------------------------------------
More informationLinear Algebra Homework and Study Guide
Linear Algebra Homework and Study Guide Phil R. Smith, Ph.D. February 28, 20 Homework Problem Sets Organized by Learning Outcomes Test I: Systems of Linear Equations; Matrices Lesson. Give examples of
More informationLinear Algebra. Workbook
Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx
More informationGURU GOBIND SINGH PUBLIC SCHOOL SECTOR V/B, BOKARO STEEL CITY
GURU GOBIND SINGH PUBLIC SCHOOL SECTOR V/B, BOKARO STEEL CITY Class :- XII ASSIGNMENT Subject :- MATHEMATICS Q1. If A = 0 1 0 0 Prove that (ai + ba)n = a n I + na n-1 ba. (+) Q2. Prove that (+) = 2abc
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
More informationQuantitative Aptitude
Quantitative Aptitude 1. Find the value of (a+b) (b-a) if 0.976767676...= Answer: 45011 Solution: 0.976767676 = + +... = + (1+ + +...) = + ( ) = + So, a = 967 and b = 990 (a+b) (b-a) = (967+990)(990-967)
More informationx 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
More informationFunction: State whether the following examples are functions. Then state the domain and range. Use interval notation.
Name Period Date MIDTERM REVIEW Algebra 31 1. What is the definition of a function? Functions 2. How can you determine whether a GRAPH is a function? State whether the following examples are functions.
More informationPre-Regional Mathematical Olympiad Solution 2017
Pre-Regional Mathematical Olympiad Solution 07 Time:.5 hours. Maximum Marks: 50 [Each Question carries 5 marks]. How many positive integers less than 000 have the property that the sum of the digits of
More informationMatrices and Determinants
Chapter1 Matrices and Determinants 11 INTRODUCTION Matrix means an arrangement or array Matrices (plural of matrix) were introduced by Cayley in 1860 A matrix A is rectangular array of m n numbers (or
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 informationThis operation is - associative A + (B + C) = (A + B) + C; - commutative A + B = B + A; - has a neutral element O + A = A, here O is the null matrix
1 Matrix Algebra Reading [SB] 81-85, pp 153-180 11 Matrix Operations 1 Addition a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn + b 11 b 12 b 1n b 21 b 22 b 2n b m1 b m2 b mn a 11 + b 11 a 12 + b 12 a 1n
More informationPossible C2 questions from past papers P1 P3
Possible C2 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P1 January 2001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
More informationIntroduction. Vectors and Matrices. Vectors [1] Vectors [2]
Introduction Vectors and Matrices Dr. TGI Fernando 1 2 Data is frequently arranged in arrays, that is, sets whose elements are indexed by one or more subscripts. Vector - one dimensional array Matrix -
More informationMathematics Class X Board Paper 2011
Mathematics Class X Board Paper Solution Section - A (4 Marks) Soln.. (a). Here, p(x) = x + x kx + For (x-) to be the factor of p(x) = x + x kx + P () = Thus, () + () k() + = 8 + 8 - k + = k = Thus p(x)
More informationVarsity Meet 3 January 8, 2014 Round 1: Similarity and Pythagorean Theorem
Round 1: Similarity and Pythagorean Theorem 1. A baseball diamond is a square 90 feet on each side. The right fielder picks up the ball 30 feet behind first base (at point R) and throws it to the third
More informationThe matrix: "Is an organization of some elements written in rows
st term st Sec. Algebra. The matrix: "Is an organization of some elements written in rows Ex: and columns between brackets in the form ( ) ". st column nd rd - st row - nd row 7 rd row * The order of any
More informationPermutations and Polynomials Sarah Kitchen February 7, 2006
Permutations and Polynomials Sarah Kitchen February 7, 2006 Suppose you are given the equations x + y + z = a and 1 x + 1 y + 1 z = 1 a, and are asked to prove that one of x,y, and z is equal to a. We
More informationRAJASTHAN P.E.T. MATHS 1997
RAJASTHAN P.E.T. MATHS 1997 1. The value of k for which the points (0,0), (2,0), (0,1) and (0,k) lies on a circle is : (1) 1,2 (2) -1,2 (3) 0,2 (4) 0, 1 2. The area of the triangle formed by the tangent
More informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
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 information( ) Chapter 7 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3.
Chapter 7 Exercise Set 7.1 1. A prime number is an integer greater than 1 that has exactly two factors, itself and 1. 3. To factor an expression means to write the expression as the product of factors.
More informationFinite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero.
Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form
More information50 Must Solve Algebra Questions
50 Must Solve Algebra Questions 50 Must Solve Algebra Questions 1. Classic furniture gallery employs male and female carpenters to create designer chairs for their stores. 5 males and 3 females can create
More informationTEST CODE: MIII (Objective type) 2010 SYLLABUS
TEST CODE: MIII (Objective type) 200 SYLLABUS Algebra Permutations and combinations. Binomial theorem. Theory of equations. Inequalities. Complex numbers and De Moivre s theorem. Elementary set theory.
More informationOn a class of three-variable inequalities. Vo Quoc Ba Can
On a class of three-variable inequalities Vo Quoc Ba Can 1 Theem Let a, b, c be real numbers satisfying a + b + c = 1 By the AM - GM inequality, we have ab + bc + ca 1, therefe setting ab + bc + ca = 1
More information81-E If set A = { 2, 3, 4, 5 } and set B = { 4, 5 }, then which of the following is a null set? (A) A B (B) B A (C) A U B (D) A I B.
81-E 2 General Instructions : i) The question-cum-answer booklet contains two Parts, Part A & Part B. ii) iii) iv) Part A consists of 60 questions and Part B consists of 16 questions. Space has been provided
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More information1 Linear Algebra Problems
Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and
More informationCSL361 Problem set 4: Basic linear algebra
CSL361 Problem set 4: Basic linear algebra February 21, 2017 [Note:] If the numerical matrix computations turn out to be tedious, you may use the function rref in Matlab. 1 Row-reduced echelon matrices
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationChapter 3. Q1. Show that x = 2, if = 1 is a solution of the system of simultaneous linear equations.
Chapter 3 Q1. Show that x = 2, if = 1 is a solution of the system of simultaneous linear equations. Q2. Show that x = 2, Y = 1 is not a solution of the system of simultaneous linear equations. Q3. Show
More informationAdditional Practice Lessons 2.02 and 2.03
Additional Practice Lessons 2.02 and 2.03 1. There are two numbers n that satisfy the following equations. Find both numbers. a. n(n 1) 306 b. n(n 1) 462 c. (n 1)(n) 182 2. The following function is defined
More information[Type the document subtitle] Math 0310
[Typethe document subtitle] Math 010 [Typethe document subtitle] Cartesian Coordinate System, Domain and Range, Function Notation, Lines, Linear Inequalities Notes-- Cartesian Coordinate System Page
More informationDigital Workbook for GRA 6035 Mathematics
Eivind Eriksen Digital Workbook for GRA 6035 Mathematics November 10, 2014 BI Norwegian Business School Contents Part I Lectures in GRA6035 Mathematics 1 Linear Systems and Gaussian Elimination........................
More informationName Advanced Math Functions & Statistics. Non- Graphing Calculator Section A. B. C.
1. Compare and contrast the following graphs. Non- Graphing Calculator Section A. B. C. 2. For R, S, and T as defined below, which of the following products is undefined? A. RT B. TR C. TS D. ST E. RS
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationPolynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:
Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a
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 informationSLCSE Math 1050, Spring, 2013 Lesson 1, Monday, January 7, 2013: Quadratic Functions
SLCSE Math 1050, Spring, 2013 Lesson 1, Monday, January 7, 2013: Quadratic Functions Note: The activities are to be done and discussed in class. Homework, due at 4 pm Monday, Jan 14, 2013 consists of all
More informationMath 313 Chapter 1 Review
Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations
More informationTHE UNITED REPUBLIC OF TANZANIA NATIONAL EXAMINATIONS COUNCIL CERTIFICATE OF SECONDARY EDUCATION EXAMINATION. Instructions
THE UNITED REPUBLIC OF TANZANIA NATIONAL EXAMINATIONS COUNCIL CERTIFICATE OF SECONDARY EDUCATION EXAMINATION 041 BASIC MATHEMATICS (For Both School and Private Candidates) Time: 3 Hours Tuesday, 03 rd
More informationTopic: Chap 3 & 4 (Matrices & Determinants) Basic Concepts
Topic: Chap 3 & 4 (Matrices & Determinants) Basic Concepts Matrices: A matrix is rectangular arrangement of elements(or functions) in order. For example = 2 3 3 0 1 Elements and Order of Matrix: In above
More informationCERT Grade 11 Mathematics Test 2 60 Minutes 60 Questions
DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer sheet. Do not linger over problems that take too much time. Solve as many as you can; then
More informationCore Mathematics C12
Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C12 Advanced Subsidiary Tuesday 12 January 2016 Morning Time: 2 hours
More informationCore Mathematics C12
Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C12 Advanced Subsidiary Tuesday 12 January 2016 Morning Time: 2 hours
More informationTo Find the Product of Monomials. ax m bx n abx m n. Let s look at an example in which we multiply two monomials. (3x 2 y)(2x 3 y 5 )
5.4 E x a m p l e 1 362SECTION 5.4 OBJECTIVES 1. Find the product of a monomial and a polynomial 2. Find the product of two polynomials 3. Square a polynomial 4. Find the product of two binomials that
More informationReview of Vectors and Matrices
A P P E N D I X D Review of Vectors and Matrices D. VECTORS D.. Definition of a Vector Let p, p, Á, p n be any n real numbers and P an ordered set of these real numbers that is, P = p, p, Á, p n Then P
More informationChapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)
Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J.. 7. 8 5. 6 6. 7. y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5
More informationLesson 6. Diana Pell. Monday, March 17. Section 4.1: Solve Linear Inequalities Using Properties of Inequality
Lesson 6 Diana Pell Monday, March 17 Section 4.1: Solve Linear Inequalities Using Properties of Inequality Example 1. Solve each inequality. Graph the solution set and write it using interval notation.
More informationCDS-I 2019 Elementary Mathematics (Set-C)
1 CDS-I 019 Elementary Mathematics (Set-C) Direction: Consider the following for the next three (03) items : A cube is inscribed in a sphere. A right circular cylinder is within the cube touching all the
More information4016 MATHEMATICS TOPIC 1: NUMBERS AND ALGEBRA SUB-TOPIC 1.11 MATRICES
MATHEMATICS TOPIC : NUMBERS AND ALGEBRA SUB-TOPIC. MATRICES CONTENT OUTLINE. Display of information in the form of a matrix of any order. Interpreting the data in a given matrix. Product of a scalar quantity
More information