## AIMCA - Mathematics for 2020-21 - A

### TWPT Metrices - 02

### Inverse trigonometry DPT - 01

### Inverse trigonometry DPT - 02

### Zero Level Function - 01

### DPT Function - 01

### DPT Function - 02

### TWPT Function - 01

### All Materials (Total 4)

### AIMCA - 2021_B_Batch

Internet connection required

500+ Hours Online Live classes

Interactive Live classes are scheduled everyday

Live Classes & Videos can be played on Web & App both

During the term of the course entire syllabus will be covered

Live doubt clearing sessions to resolve every single doubt you may have

Recording of every session available for revision

A Video can be played for approx. 270 minutes (Max. video play time = 3 times the length of the video)

Delivery by very experienced teachers

Weekly Discussion with students for study-plan, queries and more

Detailed theory & assignment material

Study Material and Live Test series provided online

Test series with 100+ Mock Papers

Topic wise practice tests (TWPT) on all topics and Performance Appraisal Tests (PAT) prepared by experts

Detailed Analysis of all the Test Paper with problem solving techniques

Doubts resolved in Real Time during class

Will cover Mathematics, Computer, QA&LR

Special sessions are provided for exams like NIMCET, BHU, JNU…etc

Content will be active for till exam

Enrollment closed on 27th Nov 2020

### AIMCA - Mathematics for 2020-21 - A

Internet connection required

350+ Hours Online Live classes

Interactive Live classes are scheduled everyday

Live Classes & Videos can be played on Web & App both

During the term of the course entire syllabus will be covered

Live doubt clearing sessions to resolve every single doubt you may have

Recording of every session available for revision

A Video can be played for approx. 270 minutes (Max. video play time = 3 times the length of the video)

Delivery by very experienced teachers

Weekly Discussion with students for study-plan, queries and more

Detailed theory & assignment material

Study Material and Live Test series provided online

Test series with 60+ Mock Papers

Topic wise practice tests (TWPT) on all topics and Performance Appraisal Tests (PAT) prepared by experts

## Editing Exercises for Class 9 CBSE With Answers PDF

This grammar section explains English Grammar in a clear and simple way. There are example sentences to show how the language is used. You can also visit the most accurate and elaborate NCERT Solutions for Class 9 English. Every question of the textbook has been answered here.

### Editing Solved Exercise With Answers for Class 9 CBSE

Question 1.

The following passage has not been edited. There is one error in each of the lines. Write the incorrect word and the correction in your notebooks as given below against the correct blank numbers. Remember to underline the word that you have supplied.

Incorrect | Correct | |

All of us enjoy the excitement of the cinema. | An | the |

We don’t think about how a film was created | ……… | ……… |

while we’re watching them. However, behind every | ……… | ……… |

success film there is a lengthy, creative process | ……… | ……… |

who involves a variety of different activities. | ……… | ……… |

Usually, the process begins by what is called the | ……… | ……… |

‘treatment’. This is the basic outline off the film story. | ……… | ……… |

Error | Correction | ||

A one-eyed man was traveling | e.g | An | A |

through a bus one day. He was | (a) | ______________ | ______________ |

carrying a huge bag on him | (b) | ______________ | ______________ |

shoulder. Anyone sitting next to him | (c) | ______________ | ______________ |

said, “Why didn’t you keep your bag | (d) | ______________ | ______________ |

beneath the seat?” The man smiled | (e) | ______________ | ______________ |

and said, “It is too big to be kept their”. | (f) | ______________ | ______________ |

Answer:

(a) through – by

(b) him – his

(c) Anyone – Somebody

(d) didn’t – don’t

(e) beneath – under

(f) their – there

Error | Correction | ||

Kashmir is right called the e.g. right rightly | e.g | right | rightly |

“Paradise in Earth”, Its beauty | (a) | ______________ | ______________ |

is to be see to be believed. | (b) | ______________ | ______________ |

The rivers, hill, mountains and | (c) | ______________ | ______________ |

gardens are the tourist attract. | (d) | ______________ | ______________ |

The snow-clad mountains is | (e) | ______________ | ______________ |

a treat for.the eyes. | (f) | ______________ | ______________ |

Answer:

(a) in – on

(b) see – seen

(c) hill – hills

(d) attract – attractions

(e) is – are

(f) for – to

Error | Correction | ||

The more important advantage | e.g | more | most |

of a nuclear family are that | (a) | ______________ | ______________ |

people get his privacy. | (b) | ______________ | ______________ |

In a joint family, their are | (c) | ______________ | ______________ |

restrictions which todays | (d) | ______________ | ______________ |

generation do not like. | (e) | ______________ | ______________ |

They disregard the disadvantages of a joint family. | (f) | ______________ | ______________ |

Answer:

(a) are – is

(b) his – their

(c) their – there

(d) todays – today’s

(e) do – does

(f) disadvantages – advantages

Question 5.

There is an error in each line. Underline the incorrect word and write the correct word in the blank given. The first one has been done for you as an example. I entered the manager’s office and sat down, (entered)

I have just lost five hundred rupees and I felt very upset, (a) __________ “I leave the money in my desk,” I said, (b) __________ “and it is not there now”. The manger was very sympathetic but he can do nothing, (c) __________ “Everyone loses money theses days,” (d) __________ he said. He start to complain about this wicked world, (e) __________ but is interrupted by a knock at the door, (f) __________.

Answer:

(a) was feeling

(b) left

(c) could

(d) is losing, these

(e) started

(f) was

## 17: 09 Pre-Class Assignment - Determinants - Mathematics

**How you will be graded:**

We will use a points-based system to grade assignments. The point values for each assignment will vary.

Tests (2-3 per marking period) - *45% of marking period grade*

Quizzes (3-4 per marking period) - *40% of marking period grade*

Quarterly - *10% of marking period grade*

Extended Learning Activity ( Frequency at the discretion of the teacher) - 5 *% of marking period grade*

(Extended Learning Activities should not take longer than 20-30 minutes to complete)

Extra Credit will **not** be provided by anyone in the mathematics department.

**What you need to do each day:**

2. Come prepared with the following materials:

- A graphing calculator is recommended for this course and future math courses. (I recommend TI-84+ Silver Edition.)

- Pencils (All work should be done using a pencil.)
- MacBook (with access to Google Classroom and eTextbook)

3. Show respect to the teacher, classmates, and yourself.

When you return to school, check the class folder to get assignments or worksheets. Work is expected to be completed the following day. If you miss a test or quiz, you have 2 weeks to make it up or it becomes a zero.

## Course Schedule: (subject to change)

Sep 03 Introduction, Linear Systems

Sep 05 Linear Systems (Section 1.1)

Sep 08 Row Echelon Form (Section 1.2)

Sep 10 Applications (Section 1.2 applications)

Sep 12 Matrix Algebra (Section 1.3)

Sep 15 Applications (Section 1.3 applications)

Sep 17 Elementary Matrices (Section 1.4)

Sep 19 Partitioned Matrices (Section 1.5)

Sep 22 Determinants I (Section 2.1)

Sep 24 Determinants II (Section 2.2)

Sep 26 Cramer's Rule (Section 2.3)

Sep 29 Vector Spaces (Section 3.1)

Oct 01 Subspaces (Section 3.2, first half)

Oct 03 Spans (Section 3.2, second half)

Oct 06 Linear Independence (Section 3.3)

Oct 08 Basis and Dimension (Section 3.4)

Oct 10 Change of Basis (Section 3.5)

Oct 12 Row and Column Space (Section 3.6)

Oct 15 Linear Transformations (Section 4.1)

Oct 17 Matrix Representations I (Section 4.2, first half)

Oct 20 Matrix Representations II (Section 4.2, second half)

Oct 22 Similarity (Section 4.3)

Oct 23 Review Session: 5:30 PM - 7:00 PM JFSB B002

Oct 24 Inner Products I (Section 5.1)

Oct 27 Inner Products II (Section 5.4)

Oct 29 Inner Products III (Section 5.4)

Oct 31 Orthogonal Subspaces I (Section 5.2)

Nov 03 Orthogonal Subspaces II (Section 5.5, first half)

Nov 05 Least Squares I (Section 5.3)

Nov 07 Least Squares II (Sections 5.3 and 5.5, second half)

Nov 10 Gram Schmidt (Section 5.6)

Nov 12 Orthogonal Polynomials (Section 5.7)

Nov 14 Eigenvalues and Eigenvectors (Section 6.1)

Nov 17 Diagonalization (Section 6.3, first half)

Nov 19 Pep talk, Diagonalization (Section 6.3, second half)

Nov 21 Schur's Lemma (Section 6.4, first half)

Nov 24 Spectral Theorem (Section 6.4, second half)

Dec 01 Positive Definite Matrices (Section 6.7)

Dec 03 Quadratic Forms (Section 6.6)

Dec 05 SVD I (Section 6.5, first half)

Dec 08 SVD II (Section 6.5, second half)

Dec 10 Review

## Math 601-602 — Spring 2009 (Narcowich)

- The matrix [A|b] below is the augmented matrix for a system of linear equations. Find the reduced echelon form of [A|b], then find rank(A) and rank([A|b]). Is the system consistent? If so, does the system have a unique solution or are there many solutions? State the leading columns of A. Solve the corresponding homogeneous system put the solution in parametric form.

**Assignment 3**

- Read section 2.1 in Heffron's text.
- Do the following problems.

- For the matrices A and B below, find det(A) and det(B) using the method employing row operations. For each matrix, answer the following questions.
- Is the matrix invertible?
- Are the columns of the matrix LI or LD?
- Are the rows of the matrix LI or LD?
- Is there a nonzero vector
**x**such that the matrix times the vector is**0**? - Is the rank of the matrix equal to 4?

**Assignment 4**- Read section 2.1 in Heffron's text.
- Do the following problems.

- Show that P
_{2}is a subspace of P_{3} - Let U be the subset of P
_{n}comprising all polynomials such that p(1)+p'(1)=0 and p(2)=0. Is U a subspace? What happens if the condition is changed to p(1)+p'(1)=0*or*p(2)=0?

**Assignment 5**- Read sections III.1-III.3 of Chapter 2 in Heffron's text and my notes on Methods for Finding Bases and on Coordinate Vectors and Examples.
- Do the following problems.

- B= <1, 2x, 4x 2 -1>and D=<1, x, x 2 >, where
**v**is the polynomial p(x)=x(3-2x). - B= <(1,0,0) T , (0,1,0) T , (0,0,1) T >and D=<(1,0,-1) T , (1,1,1) T , (-1,2,1) T >, where
**v**is the column vector (in B coordinates) (1,-2,1) T .

- Show that

A_{1}[**v**]_{B}= A_{2}[**v**]_{D},

where**v**is an arbitrary vector in V. - Use the result from the previous problem to find the matrix that takes coordinates relative to B into ones relative to D if B = <1,2x-1,x 2 +x>and D = <1-x,x 2 , x+1>. Hint: take E=<1, x, x 2 >.

**Assignment 6**- Do the following problems.

- Find the matrix A

_{B}that represents L relative to B. In addition, find the p for which L[p] = x 2 -x +1.**Assignment 7**- Read Chapters 14 and 15 in the Cain and Herod online text,
**Multivariable Calculus**. (Be aware that in chapter 14, the text DOES NOT distinguish between the arclength differential, ds, and the vector differential, d**r**.) - Do the following problems.

**Assignment 8**- Read my notes on Surfaces and Chapters 16 and 17 in the Cain and Herod online text,
**Multivariable Calculus**. - Do the following problems.

- Let S be the spiral surface parameterized by
**r**= u cos(v)**i**+ u sin(v)**j**+ v**k**.- Calculate the standard normal
**N**, the unit normal**n**, and the area element dS - Suppose that S is covered with material that has a mass density ([M/L 2 ]) of &rho(
**r**) = (x 2 + y 2 ) 1/2 . Find the total mass M on the surface, given that 0 &le u &le 2, &pi &le v &le 3&pi.

**F**(**r**) = 3z**i**+ 2x**j**+ y**k**, where the surface S is the upper half of the hemisphere x 2 +y 2 + z 2 = 9 the normal direction is upward.**F**(**r**) = zx**i**- 2xyz**j**+ z 2**k**, where the surface is the curved part of the cylinder x 2 +y 2 = 25, with 0 &le z &le 2. Take the normal as pointing away from the z-axis.

**Assignment 9**- Read my notes on Surfaces and Chapters 17 and 18 in the Cain and Herod online text,
**Multivariable Calculus**. - Do the following problems.

for

**F**(**x**) = (3x-y)**i**+ (x+y)**j**, where C is the circle x 2 +y 2 = 4 traversed once in the positive (counterclockwise) direction.for

**F**(**x**) = 2y**i**- 3z**j**+ x**k**, with S being the part of the sphere x 2 +y 2 + z 2 = 4 in the first octant. The normal to S points away from the origin, and C is the positively oriented curve that serves as the boundary of S.for

**F**(**x**) = 2y**i**+ 3x**j**- z 3**k**, with S being the surface of the closed cylinder (top, bottom, and curved side) 0 &le z &le 2, x 2 + y 2 = 25, with outward drawn normal. (This is the same cylinder as in Problem 2(b), Assignment 8.)**Assignment 10**- Read chapters 1 and 2 in the online text, Matthias Beck, Gerald Marchesi, and Dennis Pixton,
**A First Course in Complex Analysis**. - Do the following problems in the Beck-Marchesi-Pixton text.

**Assignment 11**- Read sections 8.1, 8.3, 9.1, and 9.2 in the online text, Matthias Beck, Gerald Marchesi, and Dennis Pixton,
**A First Course in Complex Analysis**. - Do the following problems in the Beck-Marchesi-Pixton text.

## Allen Study Material For IIT JEE Free Download | Allen Study Material For NEET 2021

The content of the study material is deeply researched and developed from time to time to stay pace with the changing syllabus and therefore the resulting needs of the scholars . The advance level problems along side each topic are specially designed to offer the scholars an in-depth understanding of the actual subject and prepare them to remain ahead within the competition.

#### CONTENTS:

##### ALLEN DLP Chemistry Modules Content | ALLEN Modules Solution

- PERIODIC TABLE
- CHEMICAL BONDING
- HYDROGEN
- S-BLOCK
- ORGANIC NOMENCLATURE
- HYDROCARBONS
- ISOMERISM
- REACTION MECHANSIM I & II
- ENVIRONMENTAL CHEMISTRY
- SOLID STATE
- SURFACE CHEMISTRY
- ELECTROCHEMISTRY
- CHEMICAL KINETICS
- SOLUTIONS
- D & F BLOCK
- METALLURGY
- COORDINATION COMPOUNDS
- HALOGEN
- OXYGEN CONTAINING COMPOUNDS
- NITROGEN CONTAINING COMPOUNDS
- BIMOLOECULES
- POLYMERS
- CHEMISTRY IN EVERYDAY LIFE

##### ALLEN DLP Physics Modules Content | ALLEN Modules Solution

- MECHANICS
- ELECTRICITY & MAGNETISM
- OPTICS
- THERMAL PHYSICS
- MODERN PHYSICS
- SEMICONDUCTOR

##### ALLEN DLP Mathematics Modules Content | ALLEN Modules Solution

- SETS, RELATION & FUNCTIONS
- COMPLEX NUMBER & QUADRATIC EQUATIONS
- MATRICES & DETERMINANTS
- PERMUTATIONS & COMBINATION
- MATHEMATICS INDUCTION
- SEQUENCE & SERIES
- LIMIT
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- DIFFERENTIABILITY
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##### ALLEN DLP Biology Modules Content | ALLEN Module Solution

- DIVERSITY OF LIVING ORGANISM
- STRUCTURAL ORGANISATION IN PLANTS & ANIMALS
- CELL: STRUCTURE AND FUNCTION
- PLANT PHYSIOLOGY
- HUMAN PHYSIOLOGY
- REPRODUCTION
- GENETICS AND EVOLUTION
- BIOLOGY AND HUMAN WELFARE
- BIOTECHNOLOGY AND ITS APPLICATIONS
- ECOLOGY AND ENVIRONMENT

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## Math 3C, Differential Equations and Linear Algebra 1

Assignments and Grading Policy: Webwork will be used for homework. Your permnumber will be used for both your username and password. You can change your password after your first login. Homework will be assigned on Fridays, and will be due the following Friday. Note that selected homework problems (or similar) may be given on tests. This is why it is crucial for you to do the homework before each class and, moreover, remember the ideas and techniques used in your solutions. Late homework will not be accepted. More practice problems can be found Practice problems.

**Quizzes:**There will be pop-quizes at the lecture and regular quizzes during the disscusion section. The lowest quiz score will be dropped for the final grade. Quizzes cannot be made up.**Tests:**There will be a midterm and a final exam. Make-up exams will only be given in exceptional circumstances, and then only when notice is given to me before hand and a suitable written excuse forthcoming.

## Class 12 Linear Programming Mathematics Extra Questions

**Chapter 12****Linear Programming**- Maximum Z = 16 at (0, 4)
- Maximum Z = 19 at (1, 5)
- Maximum Z = 18 at (1, 4)
- Maximum Z = 17 at (0, 5)

- finding the optimal value (maximum or minimum) of a linear function of several variables
- finding the limiting values of a linear function of several variables
- finding the lower limit of a linear function of several variables
- finding the upper limits of a linear function of several variables

- 5 bags of brand P and 6 bags of brand Q Minimum cost of the mixture = Rs 2250
- 3 bags of brand P and 6 bags of brand Q Minimum cost of the mixture = Rs 1950
- 6 bags of brand P and 6 bags of brand Q Minimum cost of the mixture = Rs 2350
- 4 bags of brand P and 6 bags of brand Q Minimum cost of the mixture = Rs 2150

In order to supplement daily diet, a person wishes to take some X and some wishes Y tablets. The contents of iron, calcium and vitamins in X and Y (in milligram per tablet) are given as below:

Tablets Iron Calcium Vitamin X 6 3 2 Y 2 3 4 The person needs at least 18 milligram of iron, 21 milligram of calcium and 16 milligram of vitamins. The price of each tablet of X and Y is Rs 2 and Re 1 respectively. How many tablets of each should the person take in order to satisfy the above requirement at the minimum cost?

A factory owner purchases two types of machine A and B for his factory. The requirements and the limitations for the machines are as follows,

Machine Area Occupied Labour force Daily on each machine output (in units) A 1000 m 2 12 men 60 B 1200 m 2 8 men 40 He has maximum area 9000 m 2 available and 72 skilled labours who can operate both the machines. How many machines of each type should he buy to maximize the daily out put?

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hour of machine time an 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.

- What number of rackets and bats must be made if the factory is t work at full capacity?
- If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.

**Chapter 12****Linear Programming**- finding the optimal value (maximum or minimum) of a linear function of several variables
**Explanation:**A linear programming problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables.

- Maximum Z = 16 at (0, 4)

**Explanation:**Objective function is Z = 3x + 4 y ……(1).

The given constraints are : x + y ≤ 4, x ≥ 0, y ≥ 0.

The corner points obtained by constructing the line x+ y= 4, are (0,0),(0,4) and (4,0).**Corner points****Z = 3x +4y**O ( 0 ,0 ) Z = 3(0)+4(0) = 0 A ( 4 , 0 ) Z = 3(4) + 4 (0) = 12 B ( 0 , 4 ) Z = 3(0) + 4 ( 4) = 16 …( Max. ) **Corner points****Z = x + y**P( 0 , 0 ) 0 Q(3 , 0) 3 R( 0, 2 ) 2 S(28/11 , 15/11 ) 43/11.(Max.) **Corner points****Z =250 x +200 y**C( 0,12 ) 2400 B (18,0) 4500 D(3,6 ) 1950 (Min.) A(9,2) 2650 **Corner points****Z = 3x – 4y**(0, 0) 0 (5,0) 15 (6,8) -14 (6 ,5) -2 (4,10) -28 (0,8) -32……………..(Min.) **Corner Points****Values of Z = 2x + y**(8, 0) 16 (6, 1) 13 (1, 6) 8 (minimum) (0, 9) 9 Corner Points Z = 60x + 40y O(0, 0) Z = 0 A(6, 0) Z = 360 Maximummathrm < B >left( frac < 9 > < 4 >, frac < 45 > < 8 > ight) Z = 360 Maximummathrm < C >left( 0 , frac < 15 > < 2 > ight) Z = 300 Type A(x) Type B(y) Maximum stock Resistors 20 10 200 Transistors 10 20 120 Capacitors 10 30 150 Profit Rs 50 Rs 60 Corner Points Corresponding value of Z (4, 0) 16 (2, 1) 9 (0, 3) 3 (minimum) Z = x + y

and also P = 20x + 10yfrac<3><2>x + 3y leqslant 42

Rightarrow x + 2y leqslant 28 3x + y leqslant 24 x geqslant 0,y geqslant 0

Solving ,x + 2y = 28 and 3x + y = 24,we get,x = 4,y = 12**Tailor A****Tailor B****Required**Shirts per day 6 10 at least 60 Pants per day 4 4 at least 32 Wages per day 150 200 herefore The required linear programming problem is to minimize the wages per day. Let Z represent the objective function which represent the sum of the wages.Hence the equation of the objective function is given as (Z) = 150x + 200y

Subject to constraints6 x + 10 y geq 60 ( constraints for tailor A) ( dividing throughout by 2 we get)Rightarrow quad 3 x + 5 y geq 30 4 x + 4 y geq 32 ( constraints for tailor B) ( dividing throughout by 4 we get)Rightarrow quad x + y geq 8

andx geq 0 , y geq 0 ( non negative constraints ,which will restrict the solution of the given inequalities in the first quadrant only)

On considering the inequalities as equations, we get

3x + 5y = 50 …(i)

x + y = 8 …(ii)

Table for line 3x + 5y = 30 isSo, it passes through the points with coordinates (0, 6) and (10, 0).

On replacing the coordinates of the origin O (0, 0) is,3 x + 5 y geq 30 we get0 geq 30 [which is false)

So, the half plane for the inequality of the line ( i) is away from the origin, which means that the origin is not a point in the feasible region.

Again, table for line ( ii) x + y = 8 is given below.So, it passes through the points with coordinates (0, 8) and (8, 0).

On replacing the origin O (0, 0) inx + y geq 8 , we get0 geq 8 (which is false)

So, the half plane for the inequation of the line ( ii) is away from origin, which means that the point O( 0,0) is not a point in the feasible region of the inequality of the line (ii).

On solving Eqs. (i) and (ii), we get

x = 5 and y = 3

so, the point of intersection is P(5, 3).from the above graph, APB is the feasible region and it is unbounded. The corner points are A(0, 8), P(5, 3) and B(10, 0).

The values of Z at corner points are as follows:

## Watch the video: Business Statistics Ch 14 Pre-class Assignment Question 1 (January 2022).

- Calculate the standard normal