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1  (i)Prove that   √ 2 is an irrational number by            the method of contraction.   [AP SCERT MP] [CBSE]   (ii)Prove that   √ 3 is an irrational number by            the method of contraction.   [AP SCERT MP, May 2022 ]     (iii)  Prove that   √ 5 is an irrational number by       the method of contraction.    [AP SSC 2015M]   (iv) Show that    √11 is an irrational number.   (v) Show that    √7 is an irrational number.    [ AP SCERT ] 2 (i) Show  that  5- √ 3 is an irrational number [ AP  Aug 2022 ]    (ii)Prove that 3+2 √ 5 is an irrational number  [AP SCERT MP]   (iii)Prove that 3-2 √ 5 is an irrational number  [AP SCERT MP] 3(i)Prove that  √5 + √7 is irrational     [AP SSC 2017J]   (ii)Prove that  ...

Euclid’s Division Lemma (Division Algorithm) : Given positive integers a , b; there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b. This result was  first recorded in Book VII of Euclid ’s Elements. Find q and r for the following pairs of positive integers a and b, satisfying a = bq + r. (i) a = 13, b = 3 (ii) a = 80, b = 8 (iii) a = 125, b = 5 (iv) a = 132, b = 11 Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. [Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can b...

  1.  ∆ ABC is an isosceles triangle right angled at ‘C’ and AB ² = n.AC ² then, n = ............ 2.Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.  3. Write the statement of converse of Basic Proportionality theorem. 4.Draw a triangle ABC in which AB=5 cm, BC=6 cm and Ð ABC= 60 ° . Construct another triangle similar to ∆ ABC   with scale factor 5/ 7 .

For Un-grouped Data Note: Mean of first n natural numbers is (n+1)/2 Practice Questions 1. Find the arithmetic mean of the data 5, 6, 9, 10, 6, 6, 7. 2.The mean of 30 numbers is 150. It was detected on rechecking that one value of 165 was wrongly copied as 135. Find the correct mean?   [151]  3.Ʃx = 100, n = 10 then find the mean of the data.[10] 4.Mean of a given data is 10.   5 is added to each of the observation.      I. Mean do no change.      II. New Mean is  15.        Which of the statement is TRUE? Support your answer. 5.Find the mean of first 'n' natural numbers. 6.Find the average of the following set of scores 567, 434,323,290,401. [ JNV 2017 ]  7.Find the average of the following numbers. 06,0,12,14,13        [ JNV 2012 ] 8.The average of the height of 5 students having height 30,40,50,60,70 is      [ JNV 201 1] 9.Find the average...