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Real Numbers Exercise 1.2 Class 10 Solution. Introduction:           यह अभ्यास (Exercise 1.2) कक्षा 10 गणित के Real Numbers अध्याय का एक महत्वपूर्ण भाग है, जिसमें rational और irrational numbers से जुड़े सिद्धांतों को समझाया गया है। इस अभ्यास में विद्यार्थियों को यह सिखाया जाता है कि किन संख्याओं को rational और किन्हें irrational कहा जाता है। सभी प्रश्नों को सरल और step-by-step तरीके से हल किया गया है, ताकि हर विद्यार्थी इसे आसानी से समझ सके। 1. Prove that  √5 is irrational Solution: Assume that √5 is rational. Then it can be written in the form 5 = a b \sqrt{5} = \frac{a}{b} ​ where a and b are integers , b ≠ 0 b \neq 0 b  = 0 and HCF(a, b) = 1 . Squaring both sides, 5 = a 2 b 2 5 = \frac{a^2}{b^2} ​ a 2 = 5 b 2 a^2 = 5b^2 This means a 2 a^2 a 2 is divisible by 5 , so a must also be divisible by 5 . Let a = 5 k a = 5k Substitute in the equation: ( 5 k ) 2 = 5 b 2 (5k)^2 = 5b^2 25 k 2 = 5 b 2 25k^2 = 5b^2 b 2 = 5 k 2 b^2 = 5k^2 So b is also ...

 Real Numbers Exercise 1.1 class 10 Solution. Introduction:           यह अध्याय कक्षा 10 गणित का एक महत्वपूर्ण भाग है, जिसमें Real Numbers से जुड़े विभिन्न सिद्धांतों को समझाया गया है। इस पोस्ट में दिए गए प्रश्नों को सरल और आसान भाषा में step-by-step हल किया गया है, ताकि विद्यार्थियों को प्रत्येक प्रश्न का हल अच्छे से समझ में आ सके। यह समाधान परीक्षा की तैयारी के लिए भी बहुत उपयोगी है। EXERCISE 1.1 1. Express each number as a product of its prime factors: (i)  140 140 = 2 × 2 × 5 × 7        =  2² × 5   × 7 (ii)  156 156 = 2 × 2 × 3 × 13        =  2² × 3 × 13 (iii)  3825 3825 = 3 × 3 × 5 × 5 × 17          =  3² × 5² × 17 (iv)  5005 5005 = 5 × 7 × 11 × 13 (v)  7429 7429 =  17 × 19 × 23 2. Find LCM and HCF (i) 26 and 91 26 = 2 × 13 91 = 7 × 13 HCF =   13 LCM =   182 Verification: LCM  × HCF = Product of two numbers   ...

  Real Numbers class 10 Notes Introduction: Real numbers include both rational numbers and irrational numbers . This chapter explains Euclid ’s Division Algorithm and the Fundamental Theorem of Arithmetic . These concepts are used to find HCF and understand the decimal expansion of rational numbers. The Fundamental Theorem of Arithmetic The Fundamental Theorem of Arithmetic states that " every composite number can be written as a product of prime numbers, and this factorisation is unique (except for the order of the prime factors). Composite Numbers  A composite number is a positive integer greater than 1 that has more than two factors . It means the number can be divided exactly by 1, itself, and at least one more number . Examples 4 → factors: 1, 2, 4 6 → factors: 1, 2, 3, 6 8 → factors: 1, 2, 4, 8 9 → factors: 1, 3, 9 - All these numbers have more than two factors , so they are composite numbers . Difference Between Prime and Composite 1) Prime number: Only...

Matter in our Surroundings class 9 Notes Introduction: Matter is anything that has mass and occupies space . Ancient philosophers explained matter using five elements (Panch Tatva): air, earth, fire, sky, and water . Modern scientists classify matter based on physical and chemical properties . Physical Nature of Matter 1. Matter is made up of particles 1) All matter consists of very tiny particles. 2) Particles of matter have space between them. 3) Particles of matter are constantly moving. 4) Particles of matter attract each other. Example (Diffusion): When a drop of ink is put in water, it slowly spreads throughout the water without stirring. This happens because the particles of matter keep moving and mixing. 2. How small are these particles of matter? Matter is made up of very tiny particles . These particles are so small that we cannot see them with our naked eyes.  Scientists proved their small size through different experiments. 1. Experiment with Potassium P...

Why do we fall Ill? class 9 Notes Introduction: Health means physical, mental, and social well-being, while disease means disturbance in normal body functions. We fall ill due to infection by microorganisms, poor nutrition, or unhealthy surroundings . Understanding diseases helps us prevent them through hygiene, vaccination, and a healthy lifestyle . Health and its Failure The significance of "HEALTH" Health is very important for individuals as well as for society.  Good health helps a person live a happy, active, and productive life.  The significance of health can be understood in the following points: 1. Efficient Working A healthy person can perform daily activities and work efficiently without feeling tired or weak. 2. Proper Growth and Development Good health is necessary for the physical and mental development of children and adults. 3. Better Quality of Life Healthy people can enjoy life, participate in social activities, and live with happiness and satisfactio...