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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      

 MCQs 1. Assertion : x + 2y - 30 = 0, 2x + 4y - 66 = 0 equations are inconsistent equations.      Reason : a ₁ x + b ₁ y + c ₁ = 0, a ₂ x + b ₂ y + c ₂ = 0 represent parallel lines if  ( a ₁ /a ₂ )  = ( b ₁ /b ₂ ) ≠ ( c ₁ /c ₂ )      A) Both Assertion and Reason are true. Reason is supporting the Assertion.      B) Both Assertion and Reason are true. Reason does not support the Assertion.      C) Assertion is true. Reason is False      D) Assertion is False. Reason is True If the sum of two numbers is 14 and their difference is 10, find the product of these two numbers. [UPSSC LS(P) 2016] 18  20  22  24  SAQs & LAQs 1.Solve 2x-3y =12, x+3y =6. 2.Check whether the equations 2x + 3y = 1, 3x - y = 7 have a unique solution, infinitely many solutions or no solution  3.For what positive value of ‘p’, the following pair of linear equations have infinitely many solutions. ...

1.The hypotenuse of a right triangle is 6m more than twice of the shortest side. If the third side is 2 m. less than the hypotenuse, find the sides of the triangle. 2.A train, traveling at a uniform speed for 360 km would have taken 48 minutes less to travel the same distance, if its speed were 5 km/h more. Find the original speed of the train. 3. Anil takes 6 days less than the time taken by Varun to finish a piece of work. If both Anil and Varun together can finish that work in 4 days, find the time taken by Varun to finish the work independently.   4.Solve the following equation for z        4/(z-1)   -  5/(z+2)    =   3/z     (z ≠ 1,-2,0) 5.Find the roots of the quadratic equation    3x ² - 2 √ 6x + 2 = 0 6. Sum of a number and its reciprocal is 10/3. Find the number.  7. Find the discriminant of quadratic equation   x ²   - 5 x +  6 = 0  [ SCERT MP 2022] 5. F...