NCERT 10th Math - Part 1 || Learn Concepts with solutions

Learn how to express a number as a product of its prime factors. Use 140 as an example to illustrate step-by-step division and identify primes 2, 5, and 7.

Find the prime factorization of 5005 by testing small primes, showing divisibility by 5, 7, 11, and 13 to express it as 5 × 7 × 11 × 13.

Learn to express 7429 as a product of prime factors by testing divisibility from small primes. Uncover that 7429 equals 17 × 19 × 23, illustrating a step-by-step prime factorization.

Compute the least common multiple of 255 and 92 via prime-factorization; extract the highest powers of primes 2, 3, 5, 17, and 23, then verify the result.

Compute the highest common factor and the least common multiple of prime numbers 17, 23, and 29 using prime factorization; the HCF is 1 and the LCM is 11339.

Apply prime factorization to 8, 9, and 25 to find that the greatest common factor is 1 and the least common multiple is 1800.

The lecture explains that a number ends with zero only if it is divisible by 10, and demonstrates that lacking five as a factor prevents such a zero ending.

In NCERT 10th math, determine whether numbers are composite by factoring them into a product of two numbers greater than one, and recall the definitions of prime and composite numbers.

Demonstrate a contradiction-based proof that three plus two root five cannot be rational, using algebraic manipulation and the relationship with root five to show inconsistency.

Presents a contradiction-based proof that one or two is an iteration in limbo, by assuming a counterexample and deriving inconsistency.

Determine whether a rational number's decimal expansion terminates by checking if the denominator's prime factors are only 2 and 5; for 13/3125 the decimal terminates.

Determine whether the decimal expansion terminates or repeats by analyzing the denominator. Analyze the caption's concept of non terminating repeating expansion to guide problem solving.

Without performing long division, determine whether fractions have a terminating decimal expansion or a repeating expansion when the denominator is a power of 2, as in 23/2.

Learn how decimal expansions terminate only when the denominator has the form 2^a 5^b, otherwise they are non-terminating repeating decimals.

Explore how decimal expansions terminate when the denominator has only factors 2 and 5, illustrated by simplifying 6/50 to 3/25 and recognizing terminating versus non-terminating decimals.

Identify whether a rational number has a terminating or repeating decimal by examining the denominator's prime factors, and show that simplifying 35/50 to 7/10 yields a terminating decimal.

Learn to convert fractions to decimal expansions by simplifying denominators with powers of ten, and explore examples with denominators 2, 8, 16, and 100.

solve the quadratic by factoring to find its zeros, then verify the relation between zeros and coefficients: the sum of roots equals -b/a and the product equals c/a.

Find the zeros of a quadratic polynomial and verify the relationship between zeros and coefficients by comparing their sum and product to -B/A and C/A.

Identify the zeros of the quadratic, showing roots ±√15, and verify that the sum and product of zeros match the coefficients: sum equals -b/a and product equals c/a.

Apply Vieta's formulas to relate the sum and product of zeros to a quadratic polynomial. Choose convenient coefficients, such as a=3, b=-3, c=1, to obtain 3x^2 - 3x + 1, illustrating multiple valid quadratics.

Explore how the sum and product of zeros define a quadratic polynomial and how coefficients a, b, c yield infinite solutions, such as x^2 - x + 1.

Construct a quadratic from given sum and product of zeros using sum = -b/a and product = c/a; sum 4 and product 1 yield x^2-4x+1.

Use the long division method to divide polynomials, determine the quotient and remainder, and apply it to a sample, yielding quotient x^2 + x - 3 and remainder 8.

Apply long division to test if p squared minus three is a factor of the given polynomial. A zero remainder confirms divisibility and reveals the quotient.

Learn how to check if a quadratic x^2 + 3x + 1 is a factor of a quartic polynomial by performing long division, noting that zero remainder confirms a factor.

Identify the given zeros of a fourth-degree polynomial, form the corresponding factors, use long division to reduce to a quadratic, and solve for the remaining zeros.

the lecture demonstrates polynomial division by long division, identifying the dividend, divisor, and quotient, and shows a zero remainder with the dividend equaling the divisor times the quotient.

Apply the division algorithm for polynomials to achieve a zero remainder, and construct explicit examples using quotients and divisors such as x-1 and x+1.

Solve age problem using two equations: seven years ago one age was seven times the other, and three years from now the elder is thrice the younger; graph the lines.

The lecture defines x as the cost of one apple and y as the cost of one grape, derives two equations 2x+y=160 and 2x+y=150, and shows algebraic and graphical representations.

Graphically solve a pair of linear equations to find the cost of a pencil and a pen, then verify the solution.

Compare the ratios of coefficients for two linear equations to determine whether they intersect at a point, are coincident, or are parallel, indicating unique, infinite, or no solution.

Compare the ratios of coefficients in two linear equations to decide if the lines intersect at a point, are coincident, or are parallel, yielding unique, infinite, or no solutions.

Compare ratios of coefficients in two linear equations to determine consistency or inconsistency. Identify cases of intersecting, coincident, and parallel lines, and illustrate unique, infinite, or no solutions.

Identify whether linear equations are consistent or inconsistent by comparing ratios; determine if lines intersect with a unique solution, are coincident with infinite solutions, or are parallel with no solution.

Determine consistency of two linear equations by comparing coefficient ratios; intersecting lines yield a unique solution, coincident lines yield infinite solutions, and parallel lines yield no solution, as illustrated.

Assess the consistency of linear equations by coefficient-ratio analysis across three cases—intersecting lines with a unique solution, coincident lines with infinite solutions, or no solution—then solve graphically.

Compare the ratios of coefficients to assess consistency, identify whether lines intersect, coincide, or are parallel, and conclude the given equations are inconsistent with no solution.

Solve the pair of linear equations S - B = 3 and S/3 + B/2 = 6 by substitution, find S = 9 and B = 6, and verify.

Solve two numbers using substitution: with x minus y equals 26 and x equals 3y, find x and y as 39 and 13, then verify the solution.

Use the substitution method to solve two linear equations for supplementary angles, where x+y=180 and the larger angle exceeds the smaller by 18 degrees, yielding 99 and 81 degrees.

Determine the fraction x/y by substitution from two conditions: adding the same amount to numerator and denominator yields 9/11, and adding three yields 5/6; obtain x=7, y=9, i.e., 7/9.

Set up two equations from five years later and five years ago conditions, then solve by substitution to find Jakob's current age 40 and his son 10, with verification.