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.

Learn how to express a given number as a product of prime factors using prime factorization, checking divisibility by primes, and solving exercise 1.2 question 1(iii).

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.

Find the highest common factor and the least common multiple of 336 and 54 by prime factorization, then verify that their product equals the product of the two numbers.

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.

Explore how gcd and lcm relate for the numbers 306 and 657, with gcd given as 9, and apply the identity lcm(a,b) = a*b/gcd(a,b) to find the lcm.

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.

Find the least common multiple of 18 and 12 to determine when they meet at the starting point. The method uses prime factorization and the LCM, yielding 36 minutes.

This lecture walks through exercise 1.3 from NCERT 10th math part 1, using squaring both sides and common-factor arguments to derive a contradiction and justify the result.

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.

Prove that the seventh root of five is irrational by contradiction; assume it equals a/b, infer a^7 and b^7 lead to a contradiction.

Explore a contradiction-based proof that sqrt(2) is irrational, by assuming it's rational and expressing it as a ratio of integers, which leads to a contradiction.

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.

Determine if a rational number's decimal expansion terminates or repeats by factoring the denominator; the example 15/1600 shows a terminating expansion since the denominator is 2^a 5^b.

Explore when a rational number has a terminating or non-terminating decimal expansion by analyzing the denominator's factors, using 343 as an example.

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 how the denominator's form determines dominating versus non dominating repeating expansions, with examples like 77/30 and the simplification steps.

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

The lecture demonstrates that a non-terminating repeating decimal represents a rational number, expressible as p/q. It also notes that terminating decimals have denominators of the form 2^a 5^b.

Learn to find zeros of x^2 - 2x - 8 by splitting the middle term, factor to (x+2)(x-4), get zeros -2 and 4, and verify sum and product of roots.

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.

Learn to factor quadratics and split the middle term to find zeros of 3x^2 - x - 4, and verify the sum and product of roots from the coefficients.

Learn how to form a quadratic polynomial from the given sum and product of roots, using sum equals -b/a and product equals c/a, as in 4x^2 - x - 4.

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.

Construct a quadratic from the given sum and product of zeros, show infinite solutions, and illustrate with x^2+5 where the sum is 0 and the product is 5.

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.

the lecture demonstrates checking whether the first polynomial is a factor of the second by long division, and shows the remainder is nonzero, so the divisor is not 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 linear equations by forming x + y = 10 and x − y = 4, locate their intersection at x = 7, y = 3, and verify.

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 A1/A2, B1/B2, and C1/C2 for two linear equations to decide if lines intersect, are coincident, or are parallel.

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.

Analyze two linear equations to determine consistency by comparing coefficient ratios. Identify intersecting, coincident, or parallel lines to conclude whether the system has a unique, infinite, or no solution.

Compare ratios of coefficients in two linear equations to decide consistency; intersecting lines yield one solution, coincident lines yield infinite solutions, and parallel lines yield no solution.

Determine whether two linear equations are consistent or inconsistent by comparing ratios of their coefficients. Learn to distinguish intersecting, coincident, and parallel lines with a worked example.

Analyze the consistency of two linear equations by comparing coefficient ratios, identifying three cases: unique solution, infinite solutions, or no solution; then solve a sample problem in standard form.

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 a garden rectangle problem by graphical method: use L - W = 4 and L W = 36, plot lines, intersect at L = 20, W = 16, verify.

solve a system of two linear equations using substitution, deriving x=9 and y=5 from x+y=14 and x-y=4, and verify the 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.

Use the substitution method to solve the linear equations, derive x in terms of y, then find y = 0 and x = 0, verifying that (0,0) satisfies the system.

Solve a pair of linear equations using the substitution method, derive x and y, and verify the solution through substitution.

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.