Sections in the Chapter

1.Introduction

2 Description of real functions

3 Intervals (Closed and open)

4 Domains and ranges of real functions

5 Real functions - Examples

6 Operations on real functions

7 Even and odd functions

8 Extension of a function

9 Periodic function

1 Description of real functions

2 Intervals (Closed and open)

3 Domains and ranges of real functions

4 Real functions - Examples

5 Operations on real functions

6 Even and odd functions

7 Extension of a function

8 Periodic function

If the domain and co-domain of a function are subsets of R (set of all real numbers), it is called a real valued function or in short a real function.

Updated 17 Jan 2016, 2 Dec 2008

# Learning Mathematics for IIT JEE

## Sunday, January 17, 2016

### 27. Definite Integrals - Revision Facilitator

**Revision Facilitator**

**Try to recollect relevant points on the topic.**

If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back.

If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back.

1 The definite integral

2 Evaluation of definite integrals

3 Geometric interpretation of definite integral

4 Evaluation of integrals by substitution

5 Properties of definite integrals

6 Integral function

7 Summation of series using definite integral as the limit of a sum

8 Gamma function

### 26. Indefinite Integrals - Revision Facilitator

1. Indefinite Integral - Antiderivative – Primitive

2. Integrals of some standard functions

3. Integration – Some standard formulas

a. ∫kf(x)dx =

b. ∫[f(x)± g(x)]dx =

c. d/dx [∫f(x)dx] =

4. Integration by substitution

5. Integrals of the form [f '(x)/f(x)]dx

6. Integrals of the form sin ^m x cos ^n x dx

7. Integrals of the functional form 1/(x²±a²)

8. Integrals of the form [1/(ax²+bx+c)]dx

9. Integrals of the form [1/√(ax²+bx+c)]dx

10. Integrals of the form [(px+q)/(ax²+bx+c)]dx

11. Integrals of the functional form [P(x)/(ax²+bx+c)]dx

12. Integrals of the form [(px+q)/√(ax²+bx+c)]dx

13. Integrals of the functional form [1/(a sin²x + b cos²x +c)]dx

14. Integrals of the functional form [1/(a sin x + b cos x +c)]dx

15. Integrals of the functional form [(a sin x + b cos x)/(c sin x + d cos x)]dx

16. Integrals of [(a sin x+b cos x +c)/(p sin x + q cos x +r)] dx

17. Integration by parts

18. Integral of e^x [f(x)+f'(x)]dx

19. Integrals of e^ax sinbx dx,e^ax cos bx dx

20. Integrals of √(a²±x²) and √(x²-a²)

21.Integrals of the functions of the form √(ax²+bx+c)dx

22. Integrals of the functions of the form (px+q)[√(ax²+bx+c)]dx

23. Integration of Rational Algebraic Functions by Using Partial Fractions

24. Integration of [(x²+1)/(x^4+λx²+1)]dx

25. Integration of Function [G(x)/(P√Q)]dx

### 24. Increasing and Decreasing Functions - Study Plan

Sections in the Chapter

1 Increasing and Decreasing Functions - Definitions

2 Necessary and sufficient conditions for monotonicity of functions

3 Properties of monotonic functions

Day 1

1 Increasing and Decreasing Functions - Definitions

2 Necessary and sufficient conditions for monotonicity of functions

3 Properties of monotonic functions

Day 2

Illustrative Objective Type Examples: 1 to 15

Day 3

Objective Type Exercise: 1 to 20

Day 4

O.T.E.: 21 to 38

Day 5 Fill in the blanks type exercise 1 to 8

Practice Exercise 1 to 20

Day 6

Concept review

Day 7

Formula review

Day 8

Difficult problem review

Day 9 and 10

Test paper problems

1 Increasing and Decreasing Functions - Definitions

2 Necessary and sufficient conditions for monotonicity of functions

3 Properties of monotonic functions

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1 Increasing and Decreasing Functions - Definitions

2 Necessary and sufficient conditions for monotonicity of functions

3 Properties of monotonic functions

**Study Plan**Day 1

1 Increasing and Decreasing Functions - Definitions

2 Necessary and sufficient conditions for monotonicity of functions

3 Properties of monotonic functions

Day 2

Illustrative Objective Type Examples: 1 to 15

Day 3

Objective Type Exercise: 1 to 20

Day 4

O.T.E.: 21 to 38

Day 5 Fill in the blanks type exercise 1 to 8

Practice Exercise 1 to 20

**Revision period**Day 6

Concept review

Day 7

Formula review

Day 8

Difficult problem review

Day 9 and 10

Test paper problems

**Revision Facilitator****Try to recollect relevant points on the topic.**

If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back.If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back.

1 Increasing and Decreasing Functions - Definitions

2 Necessary and sufficient conditions for monotonicity of functions

3 Properties of monotonic functions

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### Ch. 12 Determinants - 1

**12.1 Definition**

Every square matrix can be associated to an expression or a number which is known as determinant.

If the matrix has only one element a

_{11}then a

_{11}is the determinant.

If the matrix is of order 2 that 2 by 2 matrix

|A| =

|a11 a12|

|a21 a22| =

a11*a22 – a12*a21

**Determinant of a square matrix of order 3**

Determinant of a square matrix of order 3 is the sum of the product of elements a1j in the first row with (-1)

^{1+j }times the determinant of a 2×2 sub-matrix obtained by leaving the first row and column passing through the element.

(i) Only square matrices have determinants.

(ii) The determinant of a square matrix of order three can be expanded along any row or column.

Determinant of a square matrix of order 4 or more

(iii) Determinant of a square matrix of order 4 or more can be determined following the procedure of finding the determinant of a square matrix of order 3. But in this case, especially in the case of 4×4 matrix, when we omit the rows and columns containing the elements of a row, we get 3×3 sub-matrices and we have to find determinants for them.

**12.2 Singular matrix**

A square matrix is a singular matrix if its determinant is zero.

Otherwise it is a non-singular matrix.

**12.3 Minors and cofactors**

Minor: For a square matrix [a

_{ij}] or order n, the minor M

_{ij}of a

_{ij}, in A is the determinant of the square sub-matrix of order (n-1), obtained by leaving (or striking off) i

^{th}row and j

^{th}column of A.

Cofactor: Cofactor of an element a

_{ij}in a square matrix [a

_{ij}] is termed C

_{ij}.

C

_{ij}= (-1)

^{i+j }M

_{ij}

M

_{ij}is the minor of element a

_{ij}in a square matrix [a

_{ij}].

Minors and cofactors are defined for elements of a square matrix only. They are not defined for determinants.

**12.4 Properties of determinants**

1. For a square matrix, the sum of the product of elements of any row (or column) with their cofactors is always equal to determinants of the matrix.

2. For a square matrix, the sum of the product of elements of any row (or column) with the cofactors of corresponding elements of some other row (or column) is zero.

3. The value of a determinant remains unchanged if its rows and columns are interchanged.

4. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant changes by minus sign only.

5. If any two rows or columns of a determinant are identical then its value is zero.

6. If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the new determinant is ‘k’ times the value of the original determinant.

7. If each element of a row (or a column) of a determinant is expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants of the same order.

8. If each element of a row (or a column) of a determinant is multiplied by the same constant and then added to the corresponding elements of some other row (column) then the value of the determinant remains same.

9. If each element of a row (or column) in a determinant is zero, then its value is zero.

10. If the matrix is a diagonal square matrix then its determinant is the product of all the diagonal elements.

11. If A and B are square matrices of the same order, then

|AB| = |A| |B|

12. If a matrix is a triangular matrix of order n, then its determinant is the product of all the diagonal elements.

**12.5 Evaluation of Determinants**

To evaluate determinants or large matrices, we use the properties of determinants given in section 12.4 above, to create many zeroes in the elements of a row or column and then expand the determinant using elements and cofactors of that row or column.

**12.6 Evaluation of Determinants by using Factor Theorem**

If f(x) is a polynomial and f(α) = 0 the, (x- α) is a factor of f(x).

If a determinant is a polynomial in x, then (x- α) is factor of the determinant if its value is zero when we put x = α.

Using this rule we can find determinant as a product of its factors.

**12.7 Product of Determinants**

A definition of product of determinants is similar to the rule of multiplication of matrices.

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Updated 17 Jan 2016, 7 June 2008

### Limits - Chapter Revision Points

Sections in the Chapter

20.1 Informal approach to limit

20.2 Formal approach to limit

20.3 Evaluation of left hand and right hand limits

20.4 Difference between the value of a function at a point and the limit at a point

20.5 The algebra of limits

20.6 Evaluation of limits

We can approach a given number ‘a’ on the real line from its left hand side by increasing numbers which are less than ‘a’. It means starting from a- δ and increasing to reach a.

We can also approach a given number ‘a’ on the real line from its right hand side by decreasing numbers which are greater than ‘a’. It means starting from a+δ and decreasing to reach a.

Hence there are two types of limits – left hand limit and right hand limit.

For some functions both these limits are equal at a point and for some functions they are not equal.

If both are equal we say lim (x→a) f(x) exists. Otherwise it does not exist.

Updated 17 Jan 2016, 2 Dec 2008

20.1 Informal approach to limit

20.2 Formal approach to limit

20.3 Evaluation of left hand and right hand limits

20.4 Difference between the value of a function at a point and the limit at a point

20.5 The algebra of limits

20.6 Evaluation of limits

We can approach a given number ‘a’ on the real line from its left hand side by increasing numbers which are less than ‘a’. It means starting from a- δ and increasing to reach a.

We can also approach a given number ‘a’ on the real line from its right hand side by decreasing numbers which are greater than ‘a’. It means starting from a+δ and decreasing to reach a.

Hence there are two types of limits – left hand limit and right hand limit.

For some functions both these limits are equal at a point and for some functions they are not equal.

If both are equal we say lim (x→a) f(x) exists. Otherwise it does not exist.

Updated 17 Jan 2016, 2 Dec 2008

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