Math 1st Year Notes

Introduction of Math 1st Year Notes

       Are you searching for 11th class/ Fsc part 1 notes? Then you’re welcome to our website because we are providing Math 1st year notes in PDF to download. Our notes or guidebook is prepared by qualified teachers and subject specialists. We are also providing Pakistan Studies 10 class notes Punjab board.

Number Systems first chapter of 11 class math notes. You will read in this chapter about Rational Numbers and Irrational Numbers. 

      A rational number is a number that can be put in the form p q where p, q belongs to Z / q ≠ 0. The numbers 3.7,4, etc., are rational numbers. Square root of 16  can be reduced to the form p, q where p, q belongs to Z, and q ≠ 0 because square root of 16 = 4 = 4/1 . 

     Irrational numbers are those numbers that cannot be put into the form p/q where p, q belongs to Z, and q ≠ 0. The numbers square root of 2 , 3,  5/16 , 7/5 are irrational numbers.

      You can also define a decimal representation of rational and irrational numbers. Properties of Real Numbers, Complex Numbers, and every real number is a complex number with 0 as its imaginary part. A real number is self-conjugate.

      You will read the geometrical representation of complex numbers and the complex plane.

 Sets Functions and Groups second chapter of 11 class notes of the Punjab board. You will read in this chapter about three different ways to describe a set descriptive method, tabular method, and set-builder method. 

        Explain the order of a set, singleton set, empty set, null set, finite and infinite sets, subset, proper subset, improper subset, power set, and universal set. You will read about operations on sets like the union of two sets, the intersection of two sets, disjoint sets, overlapping sets, the complement of a set, and the difference between two sets.

      Explain operations on three sets, properties of union and intersection, inductive and deductive logic, implication or conditional, truth sets, and a link between set theory and logic. 

        Also describe the relations, functions, inverse of a function, binary operations, groups, solution of linear equations, and reversal law of inverses.

Matrices And Determinants third chapter of Math 1st year notes PDF format. A rectangular array of numbers enclosed by a pair of brackets is called a matrix. The matrix A is called real if all of its elements are real.

         You will be able to explain row matrix or row vector, column matrix or column vector, rectangular matrix, square matrix, diagonal matrix, scalar matrix, unit matrix or identity matrix, null matrix or zero matrix, and equal matrices.

           Define addition of matrices, transpose of a matrix, scalar multiplication, subtraction of matrices, multiplication of two matrices, determinant of a 2*2 matrix, properties of matrix addition, and scalar multiplication and matrix multiplication. 

 You can explain determinants, properties of determinants that help in their evaluation, adjoint, and inverse of a square matrix of Order n3, and elementary row and column operations on a matrix. 

           The systems of linear equations involving the same variables are equivalent if they have the same solution. Also define echelon and reduced echelon forms of matrices, the system of linear equations, and Cramer’s rule.

Quadratic Equations fourth chapter of math 1st year notes Punjab board. You can describe a quadratic equation in x as an equation that can be written in the form ax2+bx+c=0; 

       There are three basic techniques for solving a quadratic equation:

 i) by factorization.

 ii) by completing squares, and extracting square roots.

 iii) by applying the quadratic formula.

       You can define the solution of equations reducible to the quadratic equation, three cube roots of unity, four fourth roots of unity, polynomial function, synthetic division, and relations between the roots and the coefficients of a quadratic equation.

       Explain the formation of an equation whose roots are given, the nature of the roots of a quadratic equation, the system of two equations involving two variables, and problems on quadratic equations.

 Partial Fractions fifth chapter of Math 1st year notes. In this chapter we shall learn how to reverse the order in 

 

        that is to express a single rational function as a sum of two or more single rational functions which are called Partial Fractions. Expressing a rational function as a sum of partial fractions is called Partial Fraction Resolution. 

       It is an extremely valuable tool in the study of calculus. An open sentence formed by using the sign of equality ‘=’ is called an equation. The equations can be divided into the following two kinds. Conditional equation: It is an equation in which two algebraic expressions are equal for particular value/s of the variable.

Sequences and Series sixth chapter of Math 1st year notes. Sequences also called Progressions, are used to represent ordered lists of numbers. As the members of a sequence are in a definite order, a correspondence can be established by matching them one by one with the numbers 1, 2, 3, 4,…..

        For example, if the sequence is 1, 4, 7, 10, …., nth member, then such a correspondence can be set up as shown in the diagram below:

Thus a sequence is a function whose domain is a subset of the set of natural numbers. A sequence is a special type of function from a subset of N to R or C.

        Sometimes, the domain of a sequence is taken to be a subset of the set {0, 1, 2, 3,…}, i.e., the set of non-negative

integers. If all members of a sequence are real numbers, then it is called a real sequence.

        You will read in this chapter about types of sequences, arithmetic progression, arithmetic mean, and series. The sum of an indicated number of terms in a sequence is called a series. For example, the sum of the first seven terms of the sequence {n2} is the series, 1 + 4 + 9 + 16 + 25 + 36 + 49. 

The above series is also named the 7th partial sum of the sequence {n2}. If the number of terms in a series is finite, then the series is called a finite series, while a series consisting of an unlimited number of terms is termed an infinite series.

Permutation Combination and Probability seventh chapter of Math 1st year notes. You will read in this chapter the factorial notation was introduced by Christian Kramp (1760 – 1826) in 1808. This notation will be frequently used in this chapter as well as in finding the Binomial coefficients in a later chapter. Let us have an introduction to factorial notation. Let n be a positive integer. Then the product n(n – 1)(n – 2). . . . 3 . 2 . 1 is denoted by n! or In and read as n factorial.   

        That is, n!=n(n-1)(n-2)……3.2.1

       Thus for a positive integer n, we define n factorial as

 n! =n(n – 1)! where 0!= 1 

       Explain permutation, Suppose we like to find the number of different ways to name the triangle with vertices A, B, and C. The various possible ways are obtained by constructing a tree diagram as follows:      

    To determine the possible ways, we count the paths of the tree, beginning from the start to the end of each branch. So, we get 6 different names of triangles. ABC, ACB, BCA, BAC, CAB, CBA.  

     You can define a combination, While counting the number of possible permutations of a set of objects, the order is important. But there are situations where the order is immaterial. For example i) ABC, ACB, BAC, BCA, CAB, and CBA are the six names of the triangle whose vertices are A, B, and C.

      We notice that in spite of the different arrangements of the vertices of the triangle, they represent one and the same triangle. ii) The 11 players of a cricket team can be arranged in 11! ways, but they are players of the same single team. 

      So, we are interested in the membership of the committee (group) and not in the way the members are listed (arranged). Therefore, a combination of n different objects taken r at a time is a set of r objects. The number of combinations of n different objects taken r at a time is denoted by nCr.

   Mathematical Inductions and Binomial Theorem eighth chapter of Math 1st year notes. Francesco Mourolico (1494-1575) devised the method of induction and applied this device first to prove that the sum of the first n odd positive integers equals n2. 

        He presented many properties of integers and proved some of these properties using the method of mathematical induction. We are aware of the fact that even one exception or case to a mathematical formula is enough to prove it to be false. 

      Such a case or exception which fails the mathematical formula or statement is called a counter-example. The validity of a formula or statement depending on a variable belonging to a certain set is established if it is true for each element of the set under consideration.

        You will read about Binomial Theorem. An algebraic expression consisting of two terms such as a + x, x – 2y, ax + b, etc., is called a binomial or a binomial expression.

We know by actual multiplication that

     The right sides of (i) and (ii) are called binomial expansions of the binomial a + x for the

indices 2 and 3 respectively.

    In general, the rule or formula for the expansion of a binomial raised to any positive integral power n is called the binomial theorem for positive integral index n. For any positive integer n,

 Fundamentals of Trigonometry ninth chapter of Math 1st year notes. Trigonometry is an important branch of Mathematics. The word Trigonometry has been derived from three Greek words: Trei (three), Goni (angles), and Metron (measurement). Literally, it means measurement of the triangle.

       For the study of calculus, it is essential to have a sound knowledge of trigonometry. It is extensively used in Business, Engineering, Surveying, Navigation, Astronomy, Physical and Social Sciences.

       You can learn in this chapter units of measures of angles, the relation between the length of an arc of a circle and the

the circular measure of its central angle, general angle (coterminal angles), angle in the standard position, trigonometric functions, and trigonometric functions of any angle.

 Trigonometric Identities tenth chapter of Math 1st year notes. In this chapter, we shall first establish the fundamental law of trigonometry before discussing the Trigonometric Identities. For this, we should know the formula to find the distance between two points in a plane.

        Let P (x1, y1) and Q (x2, y2) be two points. If “ d “ denotes the distance between them,

             

       You will explain deductions from fundamental law, trigonometric ratios of allied angles, further application of basic identities, double angle identities, half angle identities, triple angle identities, sum, difference and product of sines and

cosines.

Trigonometric Functions and their Graphs eleventh chapter of Math 1st year notes. We have already defined trigonometric functions

sin, cos, tan, csc, sec, and cot. We know that

if P(x, y) is any point on the unit circle with center at the origin

O such that ∠XOP = is the standard position, then

cos  = x and sin = y

     ⇒ for any real number q there is one and only  

one value of each x and y. i.e., of each cos q

and sin q.

Hence sin q and cos q are the functions of q and their domain is R a set of real numbers. Since P(x, y) is a point on the unit circle with a center at the origin O.

        You can define domains and ranges of tangent and cotangent functions, domain and range of secant function, domain and range of cosecant function, period of trigonometric functions, values of trigonometric functions, and graphs of trigonometric functions.

 Application of Trigonometry twelfth chapter of Math 1st year notes in PDF. A triangle has six important elements; three angles and three sides. In a triangle ABC, the measures of the three angles are usually denoted by a, b, and c and the measures of the three sides opposite to them are denoted by a, b, and c respectively.

         You can define tables of trigonometric ratios, solutions of right triangles, (a) heights and distances, (b) angles of elevation and depression, engineering and heights and distances, oblique triangles, solution of oblique triangles, area of the triangle, circles connected with triangle, engineering, and circles connected with triangles.

 Inverse Trigonometric Functions thirteenth chapter of Math 1st year notes. We have been finding the values of trigonometric functions for given measures of the angles.

         But in the application of trigonometry, the problem has also been the other way around and we are required to find a measure of the angle when the value of its trigonometric function is given. 

          For this purpose, we need to have the knowledge of inverse trigonometric functions. You can describe the inverse sine, cosine, tangent function, and inverse cotangent, secant, and cosecant functions.

          Explain domains and ranges of principal trigonometric functions and Inverse trigonometric functions. Addition and subtraction formulas.

 Solutions of Trigonometric Equation fourteenth chapter of Math 1st year notes. The Equations, containing at least one trigonometric function, are called trigonometric equations, e.g., each of the following is a trigonometric equation:

        Trigonometric equations have an infinite number of solutions due to the periodicity of the trigonometric functions. For example,

         You can define the solution of general trigonometric equations when a trigonometric equation contains more than one trigonometric function, trigonometric identities, and algebraic formulae are used to transform such trigonometric equation to an equivalent equation that contains only one trigonometric function.

1. You can download 1st year notes free of cost from Freeilmi.com.
2. Expertise in math expands your academic horizons.
3. Math enhances your quantitative abilities.   

1. Math subjects often emphasize accuracy, and errors can be seen as failures. 
2. Due to the demanding nature of math subjects, some students may experience frustration and discouragement.

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