Here you will learn concept of LCM and how to find least common multiple (LCM) of numbers and fractions with examples. Letโs begin โ Concept of LCM Let \(n_1\) and \(n_2\) be two natural numbers distinct from each other. The smallest natural number n that is exactly divisible by \(n_1\) and \(n_2\) is called Least โฆ
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Here you will learn what are rational and irrational numbers with examples. Letโs begin โ What are Rational and Irrational Numbers ? Rational Number A rational number is defined as number of the form a/b where a and b are integers and b \(\ne\) 0. The set of rational numbers encloses the set of integers โฆ
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Here you will learn what are prime numbers in math, its properties and method to check whether a number is prime or not. Letโs begin โ What are Prime Numbers in Math ? A natural number larger than unity is a prime number if it does not have other divisors except for itself and unity. โฆ
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Here you will learn how to find square root and cube root of a number and properties of squares and cubes. Letโs begin โ Square Root and Cube Root (a) Square and Square Roots When any number multiplied by itself, it is called as the square of the number. Thus, 3 \(\times\) 3 = \(3^2\) โฆ
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We will will looking at the types of numbers like natural numbers, whole numbers, integers, prime and composite numbers etc. and basic concepts of number system.ย Letโs begin โ Concepts of Number System Natural Numbers and Whole Numbers Natural numbers, also called counting numbers, are the first numbers we learnt. They are the number set โฆ
Solution : We know that the vector equation of line passing through two points with position vectors \(\vec{a}\) and \(\vec{b}\) is, \(\vec{r}\) = \(\lambda\) \((\vec{b} โ \vec{a})\) Here \(\vec{a}\) = \(3\hat{i} + 4\hat{j} โ 7\hat{k}\) and \(\vec{b}\) = \(\hat{i} โ \hat{j} + 6\hat{k}\). So, the vector equation of the required line is \(\vec{r}\) = (\(3\hat{i} โฆ
Solution : We have, \(\vec{a}\) = \(4\hat{i} โ 3\hat{j} + 5\hat{k}\) and \(\vec{b}\) = \(3\hat{i} + 4\hat{j} + 5\hat{k}\) Let \(\theta\) is the angle between the given vectors. Then, cos\(\theta\) = \(\vec{a}.\vec{b}\over |\vec{a}||\vec{b}|\) \(\implies\) cos\(\theta\) = \(12 โ 12 + 25\over \sqrt{16 + 9 + 25} \sqrt{16 + 9 + 25}\) = \(1\over 2\) \(\implies\) โฆ
Solution : We have \(\vec{a}\) = \(2\hat{i}+2\hat{j}-\hat{k}\) and \(\vec{b}\) = \(6\hat{i}-3\hat{j}+2\hat{k}\) \(\vec{a}\).\(\vec{b}\) = (\(2\hat{i}+2\hat{j}-\hat{k}\)).(\(6\hat{i}-3\hat{j}+2\hat{k}\)) = (2)(6) + (2)(-3) + (-1)(2) = 12 โ 6 โ 2 = 4 Similar Questions Find the angle between the vectors with the direction ratios proportional to 4, -3, 5 and 3, 4, 5. Find the vector equation of a โฆ
Solution : We have 1 + \(sin({\pi\over 4} + \theta)\) + \(2cos({\pi\over 4} โ \theta)\) = 1 + \(1\over sqrt{2}\)\((cos\theta + cos\theta)\) + \(\sqrt{2}\)\((cos\theta + cos\theta)\) = 1 + \(({1\over \sqrt{2}} + \sqrt{2})\) + \((cos\theta + cos\theta)\) = 1 + \(({1\over \sqrt{2}} + \sqrt{2})\).\(\sqrt{2}cos(\theta โ {pi\over 4})\) \(\therefore\)ย ย Maximum Value = 1 + \(({1\over \sqrt{2}} โฆ
Solution : The expression = (sin78 โ sin42) โ (sin66 โ sin6) = 2cos(60)sin(18) โ 2cos36.sin30 = sin18 โ cos36 = \(({\sqrt{5} โ 1\over 4})\) โ \(({\sqrt{5} + 1\over 4})\) = -\(1\over 2\) Similar Questions Find the maximum value of 1 + \(sin({\pi\over 4} + \theta)\) + \(2cos({\pi\over 4} โ \theta)\) If A + B โฆ
