Mathematics

The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers, irrespective of their order. For example, 28 can be expressed as 2 × 2 × 7. This theorem is foundational for understanding the properties of numbers.

To find the HCF, multiply the smallest power of all common prime factors in the numbers. For LCM, multiply the highest power of all prime factors present in either number. For example, for 12 (2²×3) and 18 (2×3²), HCF is 2×3=6 and LCM is 2²×3²=36.

A linear equation is of the form ax + b = 0, where the highest power of the variable is 1. A quadratic equation is of the form ax² + bx + c = 0, where the highest power is 2. Linear equations have one solution, while quadratic equations can have two real solutions.

To solve ax² + bx + c = 0 by completing the square, first divide by a if a ≠ 1. Move c to the other side, add (b/2a)² to both sides to complete the square, then take the square root of both sides and solve for x. For example, x² + 6x + 5 = 0 becomes (x+3)² = 4, so x = -3 ± 2.

For a quadratic polynomial ax² + bx + c, the sum of zeroes (α+β) is -b/a, and the product (αβ) is c/a. These relationships help in finding the polynomial when zeroes are known or vice versa.

Two lines are parallel if their slopes are equal (m1 = m2). They are perpendicular if the product of their slopes is -1 (m1 × m2 = -1). For example, lines y=2x+3 and y=2x-4 are parallel, while y=2x+3 and y=-0.5x+1 are perpendicular.

The distance d between two points is given by d = √[(x2 - x1)² + (y2 - y1)²]. This formula is derived from the Pythagorean theorem and is used to calculate the straight-line distance between any two points in a plane.

The area of a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) is |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| / 2. This formula is useful when coordinates are known but base and height are not directly available.

The identity sin²θ + cos²θ = 1 is fundamental in trigonometry. It is derived from the Pythagorean theorem applied to a right-angled triangle with hypotenuse 1. This identity is used to simplify trigonometric expressions and solve equations.

To convert a terminating decimal to a fraction, write the decimal without the decimal point as the numerator and 10^n as the denominator, where n is the number of decimal places. Simplify the fraction. For example, 0.75 becomes 75/100, which simplifies to 3/4.

The volume V of a sphere with radius r is given by V = (4/3)πr³. This formula is essential for calculating the space occupied by spherical objects in real-world applications like sports balls or planets.

The surface area A of a cylinder with radius r and height h is A = 2πr(h + r). This includes the lateral surface area (2πrh) and the areas of the two circular bases (2πr²). It's useful for determining material needed to make cylindrical objects.

The probability of an event is a measure of the likelihood that the event will occur, calculated as the number of favorable outcomes divided by the total number of possible outcomes. For example, the probability of rolling a 3 on a fair die is 1/6.

The mean is the average of the numbers, calculated by summing all the numbers and dividing by the count of numbers. For example, the mean of 2, 4, 6 is (2+4+6)/3 = 4. It's a measure of central tendency in statistics.

The median is the middle value when data is ordered, dividing the dataset into two equal halves. The mode is the most frequently occurring value in the dataset. Both are measures of central tendency but provide different insights into the data distribution.