Ayush Agarwal Maths

💯👩‍🏫      💯👩‍🏫            💯👩‍🏫         💯👩‍🏫      💯👩‍🏫  Hello friends welcome to ayushagarwal_maths page. Are you preparing for board exams or competitive exams than you are at the right place to enhance your knowledge.                             REAL NUMBERS  Real numbers - Real numbers are the numbers which can be obtained on the number line. Real numbers are denoted by 'R'. They includes Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers.  Natural numbers - Natural numbers are the counting and ordering numbers (excluding 0) i.e. N = { 1, 2, 3, 4, 5, .......... } Whole numbers- Whole numbere are collection of natural numbers including zero (0) i.e. W = {0, 1, 2, 3, 4, 5, ........ }  Integers - Integers consist of natural numbers, their negatives and zero i.e. Z = { ....... -3, -2, -1, 0, 1, 2, 3, 4, 5, ........... }  Prime numbers - A prime number is defined as a number that has no factor other than 1 and its

                        Key Points and Useful Formulas for Exams       Important Formulas For Board Exams                                                            Number system  Prime numbers - A prime number is defined as a number that has no factor other than 1 and itself. Co-prime numbers - co-primes are considered in pairs and two numbers are co-prime if they have no common factors other than 1. Twin prime numbers - Twin prime numbers are the prime numbers whose difference is always equal to 2. For example, the difference between 5 and 7 is 2, and hence 5 and 7 are twin prime numbers. Composite numbers -  All the natural numbers which are not prime numbers are composite numbers as they can be divided by more than two numbers.  HCF and LCM  About Fraction  ° HCF =       HCF of Numerators /LCM of Denominators  ° LCM =  LCM of Numerators / HCF of Denominators • Product of two Numbers = L.C.M. × H.C.F  Keys/ Formulas  (a + b) 2  = a 2  + 2ab + b 2 (a – b) 2   = a 2  – 2ab + b 2 (a

Monomial  - Polynomial having one term.  Binomial  - Polynomial having two terms.  Trinomial  - Polynomial having three terms.  Zero Polynomial  - Polynomial having all its coefficient zero.  Constant Polynomial  - A polynomial having only a single term ( of real number).  Acute angle -  An angle whose measure is less than 90° is called a acute angle.  Adjacent angles  - Two angles are called adjacent angles if :-  i) they have the same vertex.  ii) they have a common arm, and  iii) Uncommon arms and an either side of the common arm.  Arithmatic Mean -  The arithmatic mean of a set of Observations is equal to their sum divided by the total number of ovservations.  Axioms -  The basic facts which are taken for granted, without proof are called axioms.  Bi -  Quadratic polynomial -  A fourth degree polynomial is called a bi-quadratic polynomial.  Chord-  A line segment joining any two points on a circle is called a chord of the circle. Circle-  A circle is a collection of tho

                       Quadrilaterals ° What is quadrilateral?      A plane, closed figure bounded by four line segments is called quadrilateral. There are different types of quadrilaterals, they are trapezium, parallelogram, rhombus, rectangle, square.  • What is the sum of all the interior angles of a quadrilateral?  The sum of all the interior angles of a quadrilateral is 360 degree.              Important Formulae                      Special quadrilaterals Parallelogram - A parallelogram is a quadrilateral with both pairs of opposite sides parallel.  Important points - Opposite sides of a parallelogram are equal Opposite angles of a parallelogram are equal The diagonals of a parallelogram bisect one another.  Each diagonal divides it into two congruent triangles.  The opposite sides and angles of a parallelogram are equal. The diagonals of a parallelogram bisect each other.  Rectangle - a parallelogram in which all four angles are right angles is a rectangle.  Important points - 

      NTSE EXAM   -----  SCHOLARSHIP  NTSE or  National Talent Search Examination  conducted by the National Council of Educational Research and Training (NCERT) for granting the scholarships to the exemplary students studying in class 10th .  The exam is conducted in two different stages: Stage I – at the State level and Stage II – at the National level. The first stage exam which is at a state-level exam and is conducted by all the Indian states and UTs. On the other hand, NTSE stage II is conducted for those students who qualifies the first stage.                          NTSE Terminology  NTSE scholarship will be awarded on the basis of the overall performance of students in both stages of the examination. NTSE Scholarship Awardee can pursue there career in two streams science or social studies. They also get many benefits in their higher studies.  . NCERT Conducts NTSE Exam once in a year in a offline mode while applications are to be submitted online only.  .NTSE exam

              POLYNOMIALS Monomial - Polynomial having one term.  Binomial - Polynomial having two terms.  Trinomial - Polynomial having three terms Zero Polynomial - Polynomial having all its coefficient zero.  Constant Polynomial - A polynomial having only a single term ( of real number).  Standard form - Powers of x are either in increasing or decreasing order.  1. A parallelogram whose all sides are equal is called a rhombus.  2. A quadrilateral is called a square if all of its sides are equal and all of its angles are equal.  3. A quadrilateral which has one pair of opposite sides parallel is called a trapezium.  4. A square is a rectangle.  5. A square is a rhombus.  6. A parallelogram is a trapezium but converse is not true.  7. A rectangle ( Or rhombus) is not a square.                   Thanks for visiting 

Tables from 1 to 25 Tables from 26 to 50

Q. 1. If a : b = 7 : 9 and b : c =15 : 7, then what is a : c A) 5 : 3 B) 3 : 5 C) 7 : 21 D) 7 : 15 Q. 2. ₹ 6200 is divided into three parts proportional to 1/2 : 1/3 : 1/5 are respectively A) ₹3000, ₹2000, ₹1200 B) ₹3500, ₹1500, ₹1200 C) ₹2500, ₹2000, ₹1700 D) ₹2200,  ₹3000,  ₹1000   Q. 3. The ratio of A to B is 4 : 5 and that of B to C is 2 : 3.If A equals 800,C equals A) 1000 B) 1200 C) 1500 D) 2000 Q. 4. If A = 4/5 of B and B = 5/2 of C, then the ratio of A : C is  A) 1 : 2 B) 2 : 1 C) 2 : 3 D) 1 : 3 Q. 5. If 2A=3B=4C, then A:B:C is A) 2 : 3 : 4 B) 4 : 3 : 2 C) 6 : 4 : 3 D) 3 : 4 : 6 Q. 6. If (3x + 5)/(5x - 2) = 2/3, then the  value of x is  A) 11 B) 19 C) 23 D) 7 Q. 7. What must be added to each term of the ratio 2 : 5 so that it may equal to 5 : 6 ? A) 65 B) 78 C) 13 D) 12 Q. 8. If x/y = 3/4, the ratio of (2x +3y) and (3y - 2x) is A) 2 : 1 B) 3 : 2 C) 1 : 1 D) 3 : 1 Q. 9. Two numbers are in the ratio 4 : 5 and their L.C.M. is 180. The smaller number is  A) 9 B) 15 C) 36 D) 45 Q. 1

SSC CGL....  Q. 1. L.C.M. of 2/3, 4/9, 5/6 is  A) 8/27 B) 20/3 C) 10/3 D) 20/27 Q. 2. H.C.F. of 2/3, 4/5 and 6/7 is  A) 48/105 B) 2/105 C) 1/105 D) 24/105 Q.3. The HCF of two numbers is 16 and their LCM is 160. If one of the number is 32, then the other number is  A) 48 B) 80 C) 96 D) 112 Q. 4. The least number which when divided by 4, 6, 8, 12 and 16 leaves a remainder of 2 in each case is:  A) 46 B) 48 C) 50 D) 56 Q. 5. Four bells ring at the intervals of 5, 6, 8 and 9 seconds. All the bells ring simultaneously at same time. They will again ring simultaneously after A) 6 minutes B) 12 minutes C) 18 minutes D) 24 minutes Q. 6.The LCM of two numbers is 48. The numbers are in the ratio 2:3. The sum of the numbers is  A) 28 B) 32 C) 40 D) 64 Q. 7. The H.C.F. and L.C.M. of two numbers are 21 and 84 respectively. If the ratio of two numbers is 1:4, then the larger of the two numbers is  A) 12 B) 108 C) 48 D) 84 Q. 8. The sum of two numbers is 36 and their H.C.F. is

                      Number System Natural Numbers - Counting numbers are called natural numbers.  •  The least natural number is 1.  • There are infinitely many natural numbers.  Whole Numbers - All natural numbers together with 0 form the collection of all whole numbers.  • The least whole number is 0.  • There are infinitely many whole numbers.  • Every natural number is a whole number.  • All whole numbers are not natural numbers, as 0 is a whole number which is not a natural number.  Integers -   All natural numbers, 0 and negatives of natural numbers form the collection of all integers.  • 0 is neither negative nor positive.  • There are infinitely many integers.  • Every natural number is an integer.  • Every whole number is an integer.  SSC previous year questions (Important)  Q. 1. Which of the following fraction is the greatest? 2/3 , 5/6 , 11/15 , and 7/8 A). 7/8 B). 11/15 C).  5/6 D). 2/3 Q. 2. If a number is as much greater than 31 as it is less than 75, then

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Q.1. लुप्त संख्या ज्ञात कीजिए?             8,  10, 23,?,  297 A) 83     B) 82      C) 22        D)73 Q. 2.लुप्त संख्या ज्ञात कीजिए?               20, 28, 37, ?, 58, 70 A) 47       B) 46           C) 45             D) 48 Q. 3.लुप्त संख्या ज्ञात कीजिए?                 150, 135, 120, ?, 90, 75 A) 100       B) 110           C) 105         D) 95 Q. 4.लुप्त संख्या ज्ञात कीजिए?                 520, 510, 490, 460, ? , 360, 280 A) 420        B) 480            C) 470        D) 450 Q. 5. लुप्त संख्या ज्ञात कीजिए?                  20,  40,  63,  90,   122, ?  A) 130          B) 138            C) 150          D)160 Q. 6. लुप्त संख्या ज्ञात कीजिए?              9,  16,  25,   36,  49,  64,  ?  A) 61              B) 81               C) 68             D) 52 Q. 7. लुप्त संख्या ज्ञात कीजिए?                 8, 27, 64,  ?,  216, 343, 512 A) 225            B) 125              C) 115           D) 215 Q. 8. लुप्त संख्या ज्ञात कीजिए?              4, 7, 12, 19, 28, ?  A) 39            B) 42           C) 71     

                          Number Series Reasoning previous year questions  Q.1. Find the missing number?  10, 15, 20, 25, ?, 35, 40 A) 33         B) 31       C) 30          D)38 Q. 2. Find the missing number?  20, 28, 37, ?, 58, 70 A) 47      B) 46          C) 45            D) 48 Q. 3. Find the missing number?  150, 135, 120, ?, 90, 75 A) 100       B) 110          C) 105        D) 95 Q. 4.Find the missing number?  520, 510, 490, 460, ? , 360, 280 A) 420       B) 480           C) 470       D) 450 Q. 5. Find the missing number?  20, 40, 63, 90, 122, ?  A) 130         B) 138           C) 150         D)160 Q. 6. Find the missing number?  9, 16, 25, 36, 49, 64, ?  A) 61             B) 81              C) 68            D) 52 Q. 7. Find the missing number?  8, 27, 64,  ?,  216, 343, 512 A) 225           B) 125             C) 115          D) 215 Q. 8. Find the missing number?  4, 7, 12, 19, 28, ?  A) 39           B) 42          C) 71         D) 36 Q. 9.Find the missing n

“Kishore Vaigyanik Protsahan Yojana”  ( KVPY) - is a scholarship program started by the Department of Science and Technology (DST) of the Government of India to encourage the students to take up research careers in the areas of basic sciences without facing financial problems. Students who are in 11th standard must apply for this scheme if they are from science side or aimed to become a scientist. From 11th you may come across the experience of different govt exams which helps you a lot in future. Secure your future by taking a right decision at right time. Eligibility  - Students enrolled in XI Standard (Science Subjects) during the academic year 2020-21 and having secured a minimum of 75% (65% for SC/ST/PWD) marks in aggregate in MATHEMATICS and SCIENCE subjects in the X Standard Board examination immediately in the preceding academic year are eligible to appear for Aptitude test. Likely fellowships are provided up to the pre-Ph.D. level to the selected KVPY Fellows. For more details

              Matrix In class 12th matrix plays an important role in board exams.Learn deeply with simple method and gain full marks in maths.  Matrix Definition - A matrix is an array of real numbers (or other suitable entities), arranged in rows and columns. Types of matrices Rectangular matrix-  Any m×n matrix, where m≠n, is called a rectangular matrix. Row matrix - A matrix having only one row is called a row matrix. Column matrix - A matrix which has only one column is called a column matrix. Square matrix-  A matrix in which the number of rows is equal to the number of columns is called a square matrix. Diagonal matrix-  It is a square matrix all of whose elements except those in the leading diagonal, are zero. Scalar matrix -  A square matrix in which the diagonal elements are all equal, all other elements being zeros, is called a scalar matrix. Unit matrix or Identity matrix- A square matrix in which each diagonal element is unity, all other elements being zeros,

                              Statistics  Meaning-  Statistics means numerical presentation of facts. Statistics is the science of collecting, classifying or organising, interpreting numerical facts(data) from the logical analysis.  Use of Statistics in Education 1. It helps the teacher to provide the accurate description of the child.  2. It enables the teacher to extract general conclusions.  3. Statistics forms the basis of scientific approach to problems of Educational Psychology. 4. It proves useful in number of fields viz. railways, Banks, Army etc.   Collection  - Data collection is a process of gathering information from all the relevant sources to find a solution of the problem.  Sources of collecting of data :- 1. Primary data - Directly from field or experiment. 2. Secondary data -Obtained from pre saved data or review.      Methods of Collection of data 1. Direct personal investigation 2. Observation method 3. Interview method 4. Information through Questionnai

                               EXPONENTS Exponents - An exponent refers to a number that how many times to use the number in a mathematical operation like multiplication, division etc.  Exponents is also termed as power.  Power - An expression that represents repeated multiplication of the same number is called a power. Special cases when the exponent is 0 or 1 :- Case 1.  When the power is 0 a^0 = 1 67^0 = 1 Case 2. When the power is 1  a^1 = a 9^1 = 9                     Laws of Exponents The following identities hold for all integer exponents :- Practice Questions:- 1.Find out the value of (2/5) 3  ?   2.What is the power of a number when squared? 3. Simplify: (3/2) 2  + (2/3) 2  + (5/4) 2 4.  If 4 x  = 64, then 3 2x  = ? 5. Divide x 5  by x 2     Thank you for visiting  Keep sharing  Join us now :-  Instagram

Percentage -   According to mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted by using the percent sign, "%" .  Percentage formula -  Percentage = (Obtained marks/Total                                marks)×100 Example :-  3/5 ×100 = 0.60×100 = 60% =                                  60percent About Percentage - 1. 10% is equal to 1/10 fraction 2.  20% is equivalent to ⅕ fraction 3.  25% is equivalent to ¼ fraction 4.  50% is equivalent to ½ fraction 5.  75% is equivalent to ¾ fraction 6.  90% is equivalent to 9/10 fraction Points to remember :- 1. A percentage is a dimensionless number, it has no unit of measurement.  Practice Questions - Q1.   Convert as a fraction and as a decimal :   40% Q2. Express the ratio 4:5 as a percentage. Q3. Find 30% of ₹800. Q4. If 10% of a number is 45, then find the number? Thank you for visiting  Keep sharing  Join us now :-  Instagram

Compound Interest - compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. Applications of Compound Interest  1. In increase or decrease in population.  2. The growth of bacteria when the rate of growth is known.  3. Depreciation in the values of machines.  Important Facts :  1. When time is given in days, we convert it to year by dividing it by 365. 2. When time is given in months, we convert it to year by dividing it by 12. 3. When dates are given, the day on which the sum is borrowed is not included but the day on which money is returned is included, while counting the number of days. Remark - negative sign of rate indicates the depreciation.  Practice Questions  1.Find the compound interest on ₹ 5000 at 8% per annum for 2 years, compounded annually?  2.Find the amount of ₹18000 for 3 years, compounded annually at 10% per annum. Also, find the compound interest.  3.The compound interest on Rs. 30,000 at 7% pe

Simple interest -  Simple interest  is  interest  calculated on the principal portion of a loan  to a savings account.

Direct proportion - Two quantities x and y are said to be in direct proportion if whenever the value of x increases or decreases, then the value of y increases or decreases in such a way that the ratio x/y remains constant. 

Factorisation - The process of finding two or more expressions whose product is the given expression is called factorisarion. 

Algebra-  Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.  ALGEBRAIC EXPRESSIONS- A combination of constants and variables, connected by some or all of the operations +, -, ×, and  ÷ is known as an algebraic expression.                          Polynomials  Monomial  - Polynomial having one term.  Binomial  - Polynomial having two terms.  Trinomial  - Polynomial having three terms Zero Polynomial  - Polynomial having all its coefficient zero.  Constant Polynomial  - A polynomial having only a single term ( of real number).  Standard form  - Powers of x are either in increasing or decreasing order.  Addition of Algebraic expressions - In the addition of algebraic expression we collect the like terms and add them.  Practice Questions  1. Solve the following equation for P  : 2P/3 = 8 + 4P 2.If 8x + 5x + 2x + 4x = 114, then 7x + 8 = ?  3. If a = 3, then a 3 (a 3 -a)= ?  4. 5x + 2(x + 7) = 14x – 7. Find x ?  5. 

Rational numbers :- The numbers of the form p/q , where p and q are integers and q is not equal to zero, q≠ 0 , are called rational numbers.  0 is neither positive nor negative.  Example-  Numbers 3/5 , -1/14 is a rational number.  - Every integer is a rational number.  - Every whole number is a rational number because every whole number can be expressed as a fraction. Positive rationals - A rational number is said to be positive if its numerator and denominator are either both positive or both negative.  Example - 3/7 or -6/11 are positive rationals.  Negative rationals - A rational number is said to be negative if its numerator and denominator are of opposite signs.  Example- -7/9, 4/-11 are negative rationals.  Some important facts :- 1. The product of a rational number and its reciprocal is 1. 2.  Zero '0' has no reciprocal.  3. The reciprocal of a positive rational number is positive.  4. The reciprocal of a negative rational number is negative.  5. Every natural number N

                   CUBES AND CUBE ROOTS Cube number- A cube number is a number multiplied by itself twice, or we can say the cube of a number m is its third power.  For example- m³ =  m  ×  m 2  =  m  ×  m  ×  m . 8³ = 8 × 8 2  = 8  × 8  × 8 = 512 .  15³ = 15  ×  15 2  =  15  ×  15  ×  15 = 3375 .   Perfect cube - A natural number is said to be a perfect cube if it is the cube of some natural numbers.  Examples-   1 ³ = 1, 2³ = 8, 3³ = 27, 4³ = 64,  etc.            Thus 1, 8, 27, 64 etc.., are perfect cubes.  Facts About cube numbers:- 1. The cube of a number is that number raised to the power 3 .  2. The cube of an even natural number is even.  3. The cube of an odd natural number is odd.  4.  Numbers ending in 1, 2, 3, 4, 6, 7, 8, 9, 000, 125, 375, 625 or 875 may be cube numbers.  Practice Questions 1. Find the cube of 21 .  2. Find the least number by which 750 should be multiplied, so that it becomes a perfect cube. 3. Solve :- (i) ∛4096  (ii) ∛(729/1000 ) 4. By w

          SQUARES AND SQUARE ROOTS   Square number - The square of a number is the product of the number with the number itself.  A given number is a perfect square if its prime factors can be expressed in pairs of equal factors.  A perfect square number is never negative. For example - 8^2 = (8×8) = 64 Facts About Squares -  1. A number ending in 2, 3, 7 or 8 is never a perfect square.  2. A number ending in an odd number of zeros is never a perfect square.  3. The square of an even number is always even.  4. The square of an odd number is always odd.  5. The ending digits of a square number is 0, 1, 4, 5, 6, or 9 only.  Note:- It is not necessary that all numbers ending with digits 0, 1, 4, 5, 6, or 9 are square numbers.  Important Facts-  1. Numbers ending in an even number of zeros are not necessarily perfect squares. Eg- 3700, 290000, 47000000.   Practice Questions :-  1. Find the square of 54 ?  2. What least number must be added to 5607 to mak