11th Class Math Notes

Very Important Handwritten 11th Class Math Notes written by Professor Asad Khalid Suib. These notes are very helpful in the preparation of Mathematics for first-year students and these are according to the paper patterns of all Colleges in the World.

Summary and Contents:
 Topics which are discussed in the notes are given below:
  • Here are the detailed 11th Class Math Notes to help you prepare for your exams.
  • All solved exercises of  all the chapters in the 1st year math of the Punjab Textbook Boards.
  • Rational Numbers and Irrational Numbers
  • We know that a rational number is a number which can be put in the form p/q where p, q belong to Z / q ≠ 0.
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  • The numbers 3.7, 4 etc., are rational numbers. squareroot of 16 can be reduced to the form p/q where p, qUZ, and q ≠ 0 because 16 = 4 = 4/1.
  • Irrational numbers are those numbers which cannot be put into the form p/q where p, q belongs to Z and q ≠ 0.
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  • Terminating decimals: A decimal which has only a inite number of digits in its decimal part, is called a terminating decimal. Thus 202.04, 0.0000415, 100000.41237895 are examples of terminating decimals.
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  • Since a terminating decimal can be converted into a common fraction, so every terminating decimal represents a rational number.
  • Recurring Decimals: This is another type of rational numbers. In general, a recurring or periodic decimal is a decimal in which one or more digits repeat indeinitely.
  • By Reading 11th class math notes pdf download explore detailed explanations and examples to enhance your understanding.
  • It will be shown (in the chapter on sequences and series) that a recurring decimal can be converted into a common fraction. So every recurring decimal represents a rational number:
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  • A non-terminating, non-recurring decimal is a decimal which neither terminates nor it is recurring. It is not possible to convert such a decimal into a common fraction. Thus a non-terminating, non-recurring decimal represents an irrational number.
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  • Polar form of a Complex number: Consider adjoining diagram representing the complex number z = x + iy. From the diagram, we see that x = rcosq and y = rsinq where r = |z| and q is called argument of z. Hence x + iy = rcosθ + rsinθ.
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  • The Descriptive Method: A set may be described in words. For instance, the set of all vowels of the English alphabets.
  • The Tabular Method: A set may be described by listing its elements within brackets. If A is the set mentioned above, then we may write: A = {a,e,i,o,u}.
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  • Union of two sets: The Union of two sets A and B, denoted by A~B, is the set of all elements, which belong to A or B. Symbolically; AUB {x|x A ∈ B ∪ x∈B }, Thus if A = {1, 2, 3}, B = { 2, 3, 4, 5}, then AUB ={1,2,3,4,5}.
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  • Note: The elements of the set of this example are the fourth roots of unity.
  • Note: For performing multiplication of residue classes 0 is generally omitted.
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  • Note: The rule can obviously be extended to the product of three or more elements of a group.
  • Note: Since the inverse (left or right) of any element a of a group is unique, from the abovenprocedure, it follows that the above solution is also unique.
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  • Note: The matrix A is called real if all of its elements are real.
  • Rectangular Matrix : If m ≠ n, then the matrix is called a rectangular matrix of order m % n, that is, the matrix in which the number of rows is not equal to the number of columns, is said to be a rectangular matrix.
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  • Note: The systems of linear equations involving the same variables, are equivalent if they have the same solution.
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  • Note: Matrices A and B are equivalent if B can be obtained by applying in turn a inite number of row operations on A.
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