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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...

 MCQs 1.Kavitha has a lamp placed at the centre of her square yard, each side measuring 20 m. The light of lamp covers a circle of radius 10 m on yard. What area of the yard is NOT lit by the lamp?           A. 400π sq. m              B. 100π sq. m           C. (40 - 10π ) sq. m    D. (400 - 100π ) sq. m  SAQs & LAQs 1.Find the area of the minor segment of a circle of radius 14 cm, when the angle of the corresponding sector is 60°.   2. A circular pond is 21 m in diameter. It is surrounded by 3.5 m wide path. Find the cost of constructing the path at the rate of Rs. 25 per sq.m 3.Find the area of a sector of circle of radius 21 cm and central angle 120°. [Use p = 22/7 ] 4.The following figure consists of a rectangle and a semi-circle. Find its area and perimeter      

  2.Solve for n:  3 ⁿ = 5 ⁿ ⁻ ²  [ AP SSC ] 3.If x= log  2 3,    y=log  2  5 then express    log  2  15 in terms of x and y 4.Show that log 1000 = 3 log 2 + 3 log 5 5.Write 2 log 3 + 3 log 5 – 5 log 2 as a single logarithm 6.If l og[ (x+y)/3]  = ½ l og x + l og y then prove that     (x/y) + (y/x)   = 7 7. If  log 2√3 1728 = x, then find the value of  x. 8. Expand log(343/125)  [ AP SSC 2016M ]  9. Find the value of  log 81 3   [ AP SSC 2016M ]   Objective Questions  Exponential Form to Log form  1.The logarithmic form of  8 x = 2   is   log x 8 =2   log  = x log x 2 =8  log 2 x = 8 1.The value of log 10  10 = 2.Log 10, log 100, log 1000……. are in    …………………….. progression.  Log form to  Exponential Form 1.The exponential form of log 4 8 = x  is    [ AP SS...