maths tricks

  Chalana-Kalanābhyām " is a Vedic mathematics sutra that translates to "By addition and by subtraction." This sutra is used for solving algebraic equations and simplifying expressions involving addition and subtraction. When applied to algebraic equations, this sutra suggests that you can add or subtract terms from both sides of an equation to solve for an unknown variable. For example, if you have the equation  3x+7=19, you can use "Chalana-Kalanābhyām" by subtracting 7 from both sides to isolate  3x+7−7=19-7 3×=2 This technique makes solving equations more efficient by manipulating both sides of the equation simultaneously. Similarly, "Chalana-Kalanābhyām" can be used to simplify expressions by breaking down complex terms using addition and subtraction rules.

 " Yavadunam " is a Vedic mathematics sutra that translates to "whatever the extent of its deficiency." This sutra is particularly useful for squaring numbers that are close to a certain base. Here's how it works: Find the deficiency of the given number from a base that's convenient (often a power of 10). Square the deficiency. Subtract the squared deficiency from the square of the base. Add the result to the original number to get the final square. For example, let's say you want to square 97 using the "Yavadunam" sutra and use a base of 100: Deficiency = 100 - 97 = 3. Square of deficiency = 3 * 3 = 9. Square of base = 100 * 100 = 10000. Final square = 10000 + 9 = 10009. This sutra simplifies squaring numbers close to a base, making calculation s faster and more efficient.

 " Gunakasamuccayah " is a Vedic mathematics term that translates to "The sum of the product." This term refers to a mathematical concept in which you add the products of numbers that have a particular relationship. For example, suppose you have a multiplication problem like  23 × 27 23×27. Using "Gunakasamuccayah," you would add the products of the numbers with a difference of 2: (23−2)×(27+2)=21×29=609. This technique is based on the observation that when the numbers are close and have a small difference, the sum of their products and the product of their sum can be used interchangeably. This approach can simplify mental calculations for multiplication.

  Gunitasamuccayah is a Vedic mathematics sutra that translates to "The product of the sum is equal to the sum of the product." This sutra is used for solving quadratic equations and finding two numbers whose sum and product are given. For example, if you have a quadratic equation x^2 - 7x + 12 = 0, and you want to find the two numbers whose sum is -7 and product is 12: Think of two numbers whose sum is -7 and product is 12. These numbers are -4 and -3. Rewrite the equation using these numbers: (x - 4)(x - 3) = 0. Solve for x: x = 4 or x = 3. This sutra provides a method to quickly find solutions to quadratic equations and factorize them using the given sum and product conditions.

  Sopantyadvayam Antyam " is a Vedic mathematics sutra that translates to "The sum of the products of the exterior and interior." This sutra is used for multiplying two numbers when the sum of their digits is 10, and the numbers have a difference of 2. For example, if you want to multiply 48 and 42 using the "Sopantyadvayam Antyam" sutra: Add the last digits of both numbers: 8 + 2 = 10. Multiply the first digits of both numbers: 4 * 4 = 16. The product is obtained by placing the results from step 2 and step 1 side by side: 1610. This sutra provides a quick method for multiplying numbers with specific properties, making mental calculations more efficient.

  Antyayor Dasake'pi " is a multiplication technique that's particularly useful when multiplying numbers that are close to each other and close to a base of 10 or its multiples. The sutra suggests breaking down the numbers into components that are close to 10 and then compensating for the deviation from 10. For example, let's say you want to multiply 12 by 18. Using this sutra, you would break down 12 into 10 + 2 and 18 into 10 + 8. Then you multiply the components: (10 + 2) × (10 + 8) = 10 × 10 + 10 × 8 + 2 × 10 + 2 × 8 Simplifying this, you get: 100 + 80 + 20 + 16 = 216 This method can help simplify calculations involving numbers that are near 10 or its multiples and streamline the multiplication process.

Probability is a fundamental concept in mathematics and statistics that deals with uncertainty and randomness. Here are some key points about probability: Basic Notions: Probability measures the likelihood of an event occurring. An event can be anything from rolling a specific number on a dice to a coin landing heads-up. Probability Scale: The probability of an event ranges from 0 (impossible) to 1 (certain). An event with a probability of 0 will never happen, while an event with a probability of 1 will always happen. Probability Calculation: The probability of an event A happening is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. This is often denoted as P(A) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes). Mutually Exclusive and Independent Events: Mutually Exclusive Events: These are events that cannot happen at the same time. For example, if you flip a coin, it can't be both heads and tails simultaneo...

Power Rule:     ∫x^n dx = (x^(n+1))/(n+1) + C, where "n" is a constant different from -1, and "C" is the constant of integration. Exponential Rule: ∫e^x dx = e^x + C. Trigonometric Integrals: ∫sin(x) dx = -cos(x) + C. ∫cos(x) dx = sin(x) + C. ∫sec^2(x) dx = tan(x) + C. ∫csc^2(x) dx = -cot(x) + C. Inverse Trigonometric Integrals: ∫(1/√(1-x^2)) dx = arcsin(x) + C. ∫(1/(1+x^2)) dx = arctan(x) + C. Integration by Parts:   ∫u dv = uv - ∫v du, where "u" and "v" are differentiable functions of "x." Integration of Rational Functions:   Integrals of the form ∫(P(x)/Q(x)) dx, where P(x) and Q(x) are polynomials, can be solved using partial fraction decomposition. Integration of Trigonometric Products : Integrals involving products of trigonometric functions can be solved using trigonometric identities and manipulation. Substitution Rule: Also known as the u-substitution method, this involves substituting a new variable to simplify the integral. ∫...

 A definite integral is a mathematical concept used to calculate the accumulated area under a curve between two specified limits. It's denoted by ∫[a to b] f(x) dx, where "a" and "b" are the lower and upper limits of integration, "f(x)" is the function being integrated, and "dx" represents the infinitesimally small change in the variable "x." The result of evaluating the definite integral gives you the net area between the curve and the x-axis within the given interval. Definite integrals have several important properties and applications: Area Under a Curve : As mentioned earlier, the definite integral calculates the area under a curve between two points on the x-axis. It can be used to find the area of irregular shapes bounded by the curve and the x-axis. Net Accumulation: The definite integral can also represent the net accumulation of a quantity over an interval. For instance, if "f(x)" represents the rate of change o...