##### Mathematics
###### Content
• Introduction.

• Euclid’s division algorithm.

• The Fundamental theorem of arithmetic.

• Revisiting irrational numbers.

• Revisiting rational numbers and their decimal expansions.

• Exercises- 1.1, 1.2 , 1.3 and 1.4.

• Summary.

###### Questions on the real numbers

Important questions :

1. Use Euclid’s division algorithm to find the HCF of 135 and 225.

2. Use Euclid’s division lemma  to show that the square of any positive integer is either of the form of 3m or 3m + 1 for some integer m.

3. Express 156 as a product of its prime factors.

4. Find the HCF and LCM of 96 and 404 by prime factorisation method.

5. Without actually performing the long division State whether the given rational number 13/3125 will have a terminating decimal expansion or not.

6. Prove that square root of 5 is irrational.

7. Use Euclid’s algorithm to find H CF of 135 and 225.

8. Find the H CF and LCM of 510 and 92 by prime factorisation method.

9. The H CF of two numbers is 145 and their LCM  is 2175. If one of the number is 275 and the other.

10. Find the largest number which divides 320 and 457 leaving remainder 5 and 7 respectively.

11. Find the H CF of 81 and 237 and express it as a linear combination of 81 and 237.

12. Show that one and only one out of n ,n+2 , n+4 Is divisible by 3, where n is a positive integer.

13. State whether the following rational number will have a terminating or non terminating decimal expansion  15/1600

###### Content
• Introduction.

• Geometrical meaning of the series of a polynomial.

• Relationship between series and coefficients of a polynomial.

• Division algorithm for polynomials.

• Summary.

• Exercise problems

###### Questions from chapter 2 polynomials:

1) Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.

2) Find a quadratic polynomial, the sum and product of whose zeroes are -3 and 2 respectively.

3) Find all the zeroes of 2x4 – 3x3 -3x2 +6x-2, if you know that two of its zeroes are √2 and -√2.

4) Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial: x2 + 3x +1, 3x4 +5x3 – 7x2 + 2x +2.

5) Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder:  P(x) =x4 – 5x +6, g(x) = 2-x2.

6) Divide 2x2 + 3x +1 by x+2.

7) Find the zeroes of the cubic polynomial 2x3 – 5x2 – 14x + 8, and verify the relationship between the zeroes and the coefficients

###### Contents
• Introduction.

• Pair of linear equations in two variables.

• Graphical method of solution of a pair of linear equations.

• Algebraic methods of solving a pair of linear equations

• Substitution method , elimination method  and cross multiplication method.

• Equations reducible to a pair of linear equations in two variables.

• Summary.

• Exercise problems and worked problems.