Pgc Math

05/07/2023

Mjhe Composition of function asan words mai smjhna hai?

Bilkul! Main aapko composition of functions ko asan shabdon mein samjhata hoon.

Composition of functions ek mathematical concept hai jismein do functions ko ek saath combine karke naya function banaya jata hai. Ye process functions ko ek sequence mein execute karne ka tarika hai.

Sochiye ki aapke paas do functions hain: function A aur function B. Function A ko "f" se represent kiya jata hai aur function B ko "g" se represent kiya jata hai. Ab aapko f(g(x)) calculate karna hai.

Composition of functions ka matalab hota hai ki function B ke output ko function A ke andar daal kar calculate karna. Yani, pehle function B ko x ke liye execute karein aur uska result lein, phir us result ko function A ke andar daal kar uska output calculate karein.

Is process ko asan tareeke se samjha sakte hain. Sochiye ki function B ek machine hai jo kuch input leti hai aur uska output deta hai. Function A dusri machine hai, jo pehli machine ke output ko input ke roop mein leti hai aur apna output deta hai.

Toh composition of functions ko aap aise samajh sakte hain:

Li jiye koi input value x.
Is input value ko function B (g) mein daal kar uska output nikalien. Is output ko f(g(x)) likhein.
Ab is output value ko function A (f) mein daal kar final output calculate karein.
Mujhe ummeed hai ki yeh explanation aapko composition of functions ko samajhne mein madad karega.

11/06/2023

First Year Math Guess 2023

Math Guess
1St year

Unit 1
Important Definitions: 1. complex numbers 2. rational numbers 3. irrational numbers
Example #6 (Page #10),
Ex #1.1 Q.1 (ii, iii. iv) Q.4(i,ii) Q.5, Q.6(i)
Ex #1.2 Q.4(i, ii, iii) Q.5(ii,iii) Q.11, Q.14(ii, iii) Q.5(ii) Q.16(i, ii)
Ex # 1.3: Q.2(ii, iii, iv), Q.4, Q.5 (i, ii, iii), Q.6(i, ii)
Theorms: (iii, iv) (page #21)

-> Also learn associative laws of addition and multiplicatioin
Unit 2
Important Definitions: equal and equivalent sets, Singleton set, null set, finite and infinite sets
Exercise 2.1 Q.2(i, v, vi,ix,xii,xvi),
Ex # 2.2 Q.4(ii,vii),Q.5(iii.iv)
De Morgan’s Laws,
Ex #2.3 (Q # 9[i], [iv]),
Example #4 (page #53),
Ex # 2.4 (Q #5),
Ex # 2.5 (Q #4),
Example #4 (page #53,
Solution of Linear Equations(pg.76), Reversal Law of Inverses(pg.77), Exercise 2.8: Q.5, Q.6

Unit 3
Ex #3.1: Q.2, Q.3(i,ii), Q.5,8, Q.12(i, ii),
Ex # 3.2 Q.3(ii), Q.5(i), Q.6(iii), Q.8(ii), Q.2(ii), Q.4(iv), Q.7(i)
Ex # 3.3 Q.1(i), Q.2(i,ii,iii), Q.3(ii,iii,iv), Q.4(ii), Q.5(i,iii,v), Q.6(i,iii),
Ex # 3.4: Q.6(i), Q.8, Q.10(ii, iii), Example 3: (pg.137), Q.5
Ex # 3.5: Q.1(iii), Q.2(ii), Q.4(ii)

Unit 4
Ex # 4.1 (Q. 8),Example #6 (page #406)..
Example #4 (page #145)
Ex # 4.2 (Q. 10, 14, 24),
Ex # 4.3 (Q. 5, 12),
Three cube roots of Unity,
Ex # 4.4 (Q. 3[i], [iii], Q. 5),
Ex # 4.5 (Q. 16),
Ex # 4.6 (Q. 2, 3[ii], 4, 6, 9),
Nature Of The Roots of a
Quadratic Equation.,
Ex # 4.7 (Q. 5, 8),
Ex # 4.8 (Q. 5),
Ex # 4.9 (Q. 5, 8),
Ex # 4.10 (Q. 13,17)

Unit 5
Ex # 5.1 Q. 4,7,10, Q.5, Example 1: (pg.184), Example 2: (pg.184),
Ex # 5.3 Q.1,10, Example 1: (pg.188) Q.3,6,8

Unit 6
Ex # 6.1 Q.1(ii,iii, v,vi,vii, viii)
Ex # 6.2 Q.4,7,8,9,12, 2,6,13,
Ex # 6.8 (Q. 3[ii, iv], 8),
Ex # 6.10 Q.1(i), Q.2(ii), Q.7,8,12,13, Q.14(i), Q.15(i), Q.17

Unit 7
Ex # 7.2 Q.1(i,iii,v), Q.2(i,ii,iii), Q.3,4,6,10,11, Example #1 (page #237
Ex # 7.3 Q.1(ii, iii), Q.4,12, Q.3,11,
Ex # 7.4: Q.1(i,ii,iii), Q.2(i,ii,iii), Q.3(i) Q.4,10,
Ex # 7.5 Q.3(i, ii)
Ex # 7.7 (Q. 2, 5),
Example 3: (pg.238)

Unit 8
Ex # 8.1 Q.1,3,5,7,15,22,
Ex # 8.2 (Q. 7[i], 9[i], 10[i, ii])
Ex # 8.3 (Q. 3[iii], 4[iv,vi],12, Q.5,7,11

Unit 9
Ex # 9.1 Q.1(ix,xii,xiii), Q.2(ii,vi,xv), Q.3, Q.4(i), Q.5(ii), Q.6(i), Q.7
Ex # 9.2 Q.3(iv,v,vi), Q.4(ii)
Ex # 9.3 Q.1(i,ii, iii. iv), Q.2(i), Q.3(i,ii), Q.4, Q.5(iv,vii), Q.6(v,ix)
Ex # 9.4 Q.5,6,7,9,10

Unit 10
Ex # 10.1 (Q. 3[iii]),
Ex # 10.2 Q.1(iii,vi), Q.3(ii), Q.7(ii), Q.11, Example 1: (pg.330),
Ex # 10.4 Q.1(i,iii,iv,v), Q.2(v,vi), Q.3(ii), Q.4

Unit 11
Ex # 11.1 (Q. 5, 9)
Unit 12
Ex # 12.3 Q.1,5,3,9
Ex #12.6 Q.2,6,7,10 Q.1,8,
Ex # 12.8 Q.1(i), Q.3(iii), Q.5(iv), Q.6(i), Q.7(i), Q.12

Unit 13
Ex # 13.1: Q.1(iv,v,vi), Q.2(i,iii), Q.3(iii,iv,vii),
Ex # 13.2: Q.3,11,17,18,19
Unit 14
Example 1-3: (pg.401&402), Example 1,2,4,5: (pg.403,405 & 406),
Ex # 14: Q.1(ii,iv), Q.2(i,iii), Q.3,5

18/05/2023

11th Math Full Book Test Click on Link

View this file, and add comments too.

15/05/2023

First Year Math Full Book Mcqs Test

13/05/2023

2nd Year Math Full Book Mcqs Test

30/04/2023

Comments your answer

23/04/2023

https://pgcmath381.blogspot.com/2023/04/fsc-math-part2-chapter-1-exercise-11.html

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11/04/2023

Integration by parts formula

The integration by parts formula is a technique used to find the integral of a product of two functions. It is based on the product rule of differentiation and can be written as:

∫u dv = uv - ∫v du

where u and v are functions of x and du/dx and dv/dx are their respective derivatives with respect to x.

To use this formula, one function is chosen as u and the other as dv. The choice of u is typically based on a hierarchy of functions, known as ILATE rule, where I stands for Inverse trigonometric functions, L for Logarithmic functions, A for Algebraic functions, T for Trigonometric functions, and E for Exponential functions.

Once u and dv have been chosen, we differentiate u to get du/dx and integrate dv to get v. We can then substitute these values into the integration by parts formula to get the final answer. It is important to note that integration by parts can be used multiple times to simplify integrals, but care must be taken to choose the functions in the correct order.

28/03/2023

History of Algebra

Algebra is a branch of mathematics that deals with mathematical operations and equations using letters and symbols instead of numbers. The history of algebra can be traced back to ancient times, where different civilizations developed their own methods of solving mathematical problems.

Babylonian Algebra: The Babylonians, who lived in Mesopotamia around 2000 BC, were the first to develop algebraic equations. They used algebra to solve problems related to trade, agriculture, and surveying. They developed methods for solving linear and quadratic equations, and even used algebraic methods to solve geometric problems.

Greek Algebra: The Greeks were the first to develop a systematic approach to algebra. They used geometric methods to solve algebraic equations, and developed algebraic methods for solving problems in geometry. The famous Greek mathematician Diophantus is considered the "father of algebra" for his pioneering work in algebraic equations.

Arabic Algebra: During the Islamic Golden Age, which lasted from the 8th to the 13th centuries, Arabic mathematicians made significant contributions to the development of algebra. They developed algebraic notation and introduced the concept of variables. The famous Persian mathematician Al-Khwarizmi wrote a book on algebra called "Al-Jabr wa-al-Muqabilah" (The Compendious Book on Calculation by Completion and Balancing), which is considered a landmark in the history of algebra.

Renaissance Algebra: During the Renaissance, algebra underwent a major transformation. Mathematicians such as François Viète and John Wallis developed algebraic notation, which made it easier to write and solve equations. They also developed new methods for solving equations, such as the method of Descartes and the method of Cardano.

Modern Algebra: In the 19th century, algebra underwent a revolution with the development of abstract algebra. Mathematicians such as Évariste Galois and Augustin-Louis Cauchy developed new concepts in algebra, such as groups, rings, and fields. These concepts are now fundamental to modern algebra and have applications in many areas of mathematics, science, and engineering.

Today, algebra remains a critical branch of mathematics that is used in a wide range of fields, from computer science to physics and engineering.

17/02/2023

Limit of a function

We find the limit of a function to determine what value the function approaches as the input approaches a certain value or as the input becomes infinitely large or small.

Limits are an important concept in calculus and are used to study the behavior of functions, particularly as they approach certain points or as they approach infinity. By analyzing the limits of a function, we can determine if the function is continuous at a certain point, or if it has any discontinuities or asymptotes.

In practical terms, finding the limit of a function can help us to solve real-world problems that involve variables that approach certain values or limits. For example, if we are calculating the trajectory of a moving object, we might use limits to determine the velocity or acceleration of the object as it approaches a certain point.

Overall, finding the limit of a function is a fundamental tool in mathematics and has many applications in various fields, including physics, engineering

15/02/2023

12/02/2023

Definite Integrals, Properties of Definite Integral
https://youtu.be/djDsIyrvZnU

04/02/2023

Integration by Partial Fraction,12Th Class Math

02/02/2023

Integration by Partial Fraction -12Th Class Math
Chapter:3 Case:3 Non Repeated Quadratic Factors
Exercise 3.5, QNo:21
For complete Exercise Subscribe YouTube Channel
https://youtu.be/r_diwkTE-Iw

01/02/2023

Integration by Partial Fraction -12Th Class Math
Case:2 Non Repeated Linear Factors in resolving Partial Fraction into its components
Chapter 3: Exercise 3.5 ,QNo:16

31/01/2023

Integration by Partial Fractions
Exercise 3.5, QNo:08

30/01/2023

Integration - integration by Partial Fraction
https://youtu.be/0ZBNr8D6Cjc

26/01/2023

Fsc Math Part2 -Integration
Chapter 3: Exercise 3.4, QNo:5 (x), (xi)
For complete Exercise Visit YouTube Channel
https://youtu.be/hHakIk7Wk1k

25/01/2023

FSC Math Part2 Chapter 3|Ex 3.4 QNo:5 (viii), (ix) |12Th Class Math

25/01/2023

What is slope?

The slope of a line is a measure of how steep it is. It is often represented by the letter "m" and is calculated as the ratio of the change in the y-coordinate (rise) to the change in the x-coordinate (run) between two points on the line. The formula for slope is:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line.

For example, if a line has a slope of 2, it means that for every unit increase in the x-coordinate, the y-coordinate increases by 2 units. If a line has a slope of -2, it means that for every unit increase in the x-coordinate, the y-coordinate decreases by 2 units.

A line with a slope of 0 is horizontal and a line with an undefined slope is vertical.

Slope is a fundamental concept in algebra and is used in many areas of mathematics and science, including calculus, physics, and engineering.

24/01/2023

FSC Math Part2 -Integration
Chapter 3: Exercise 3.4 QNo:5 (v) to (vii)

21/01/2023

12th Class Math- Integration
Chapter 3: Exercise 3.4, QNo:5 (ii) to (iv)

19/01/2023

12th Maths- Integration
Chapter 3: Exercise 3.4 QNo:05 (i)

Subscribe YouTube Channel 👇👇👇

https://youtu.be/rM5Mn2-0Yyo

19/01/2023

Fsc Math Part2 - Integration by Parts
Chapter 3: Exercise 3.4 QNo:04 (iii), (v)

17/01/2023

12th Class Math- Integration by Parts
Chapter:03, Exercise 3.4, QNo:04 (ii)
Subscribe YouTube Channel For Complete Lectures of Chapter 3 👇👇👇👇
https://youtu.be/begriMQdW5c

09/01/2023

Coordinate geometry is a branch of mathematics that deals with the study of points, lines, and curves in two-dimensional and three-dimensional space. The origins of coordinate geometry can be traced back to ancient civilizations, such as the Greeks and the Indians, who used geometric techniques to solve problems in mathematics and physics.

The modern form of coordinate geometry, however, began to take shape in the 17th and 18th centuries with the work of mathematicians such as René Descartes, Pierre de Fermat, and others. Descartes is often credited with the development of the Cartesian coordinate system, which is a system of coordinates that uses a pair of perpendicular lines, called the x-axis and y-axis, to locate points on a plane.

In the 19th and early 20th centuries, the study of coordinate geometry was further advanced by mathematicians such as August Ferdinand Möbius, George Peacock, and others, who developed new techniques for analyzing and manipulating geometric objects using coordinates. Today, coordinate geometry is an important tool in a wide range of fields, including physics, engineering, and computer science.

08/01/2023

History of Fundamental law of Trigonometry

The fundamental laws of trigonometry are a set of relationships between the angles and sides of a right triangle (a triangle with one 90 degree angle). These laws are the basis for many other important results and techniques in trigonometry and have been known and used for thousands of years.

The three most well-known fundamental laws of trigonometry are:

The Pythagorean theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

The sine function: The ratio of the length of the side opposite an angle to the length of the hypotenuse is equal to the sine of the angle.

The cosine function: The ratio of the length of the side adjacent to an angle to the length of the hypotenuse is equal to the cosine of the angle.

These laws were first formally stated and proved by the ancient Greek mathematician Euclid in his book "Elements," which was written around 300 BC. However, they were likely known and used by earlier civilizations such as the Babylonians and the Egyptians.

08/01/2023

The binomial theorem is a fundamental result in algebra that describes the expansion of powers of a binomial (a polynomial with two terms). It was first formally stated by Isaac Newton in his 1665 book "Arithmetica Universalis," although similar results had been known and used much earlier.

The binomial theorem states that:

(a + b)^n = a^n + n*a^{n-1}b + n(n-1)*a^{n-2}*b^2 + ... + b^n

for any positive integer n.

The theorem gives a formula for expanding any power of a binomial into a sum of terms, each of which is the product of a coefficient and a power of either a or b. It has many important applications in mathematics and science, including in calculus, probability, and physics.

08/01/2023

The concept of angle has a long history dating back to ancient civilizations. The earliest known recorded use of the concept of angle was by the ancient Egyptians, who used the term "jnt" to describe the corner of a right triangle. The ancient Greeks also made significant contributions to the study of angles, with Euclid's "Elements" providing a systematic treatment of the subject. The term "angle" itself is derived from the Latin word "angulus," which means "corner." In mathematics, an angle is defined as the figure formed by two rays with a common endpoint, called the vertex. Angles are usually measured in degrees, with a full circle being 360 degrees.