Posts

Real Numbers Exercise 1.2 Class 10

-
Image
  Real Numbers Exercise 1.2 Class 10 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 divisible by 5 . Thus a and b are both divisible by 5 , which contradicts the fact that HCF(a, b) = 1 . Therefore our assumption is wrong. Hence, √5 is irrational. 2. Prove that 3 + 2 5  is irrational Solution: Assume that 3 + 2 5 3 + 2\sqrt5 is rational. Then 2 5 = ( 3 + 2 5 ) − 3 2\sqrt5 = (3 + 2\sqrt5) - 3 The right side is rational, so 2 5 2\sqrt5 becomes rational . Dividing by 2, 5 = 2 5 2​ This means √5 is rati...
Post a Comment
Read more

Real Numbers Exercise 1.1 class 10

-
Image
  Real Numbers Exercise 1.1 class 10 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    182 × 13     =   26 × 91                2366 =  2366 (ii) 510 and 92 510 = 2 × 3 × 5 × 17 92 = 2 × 2 × 23 HCF =  2 LCM = 2 × 2 × 3 × 5 × 17 × 23 LCM = 23460 Verification: LCM  × HCF = Product of two numbers 23460 × 2      = 510 × 92          46920...
Post a Comment
Read more

Real Numbers class 10 Notes

-
Image
  Real Numbers class 10 Notes 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 2 factors (1...
Post a Comment
Read more

Matter in our Surroundings class 9 Notes

-
Image
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 Pe...
Post a Comment
Read more

Why do we fall Ill? class 9 Notes

-
Image
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 satisfaction...
Post a Comment
Read more