NCERT Class 9th solution of Exercise 2.1 NCERT Class 9th solution of Exercise 2.2 NCERT Class 9th solution of Exercise 2.3   NCERT Class 9th solution of Exercise 2.5 NCERT Class 9th Maths Projects Exercise 2.4 Q1. Determine each of the following polynomials has `(x+1)` a factor : i) `x^3 + x^2 + x + 1`   ii)`x^4 + x^3 + x^2 + x + 1`   iii) `x^4 + 3x^3 + 3x^2 + x + 1`  iv) `x^3 - x^2 - (2 + sqrt2)x + sqrt2` Sol. : i) Let `p(x) = x^3 + x^2 + x + 1`  and zero of `x + 1` is `- 1` `p(-1) = (-1)^3 + (-1)^2 + (-1) + 1` `p(-1) = - 1 + 1 - 1 + 1` `p(-1) = 0` Answer : `(x+1)` is a factor. ii) Let `p(x) = x^4 + x^3 + x^2+ x + 1` and zero of `x+1` is `- 1` `p(-1) = (-1)^4 + (-1)^3 + (-1)^2 + (-1) + 1` `p(-1) = 1 - 1 + 1 - 1 + 1` `p(-1) = 1 ≠ 0 ` Answer : `(x+1)` is not a factor. iii) Let `p(x) = x^4 + 3x^3 + 3x^2+ x + 1` and zero of `x+1` is `- 1` `p(-1) =  (-1)^4 + 3(-1)^3 + 3(-1)^2+ (-1) + 1` `p(-1) = 1 - 3 + 3 - 1 + 1` `p(-1) = 1 ≠ 0` Answer : `(x+1)` is not a factor. iv) Let `p(x) = x^3 - x^2

NCERT Class 9th solution of Exercise 2.1 NCERT Class 9th solution of Exercise 2.2 NCERT Class 9th solution of Exercise 2.4 NCERT Class 9th solution of Exercise 2.5 NCERT Class 9th Maths Projects Exercise 2.3 Q1. Find the remainder  when `x^3 + 3x^2 + 3x + 1` is divided by : i) `x+1`  ii) `x- frac{1}{2}`  iii) `x`  iv) `x+π`   v) `5+2x` Sol. : i) Let `p(x) = x^3 + 3x^2 + 3x + 1` and zero of `x+1` is `-1` `p(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1` `p(-1) = - 1 + 3 - 3 + 1` `p(-1) = - 4 + 4` `p(-1) = 0` Answer : The remainder is `0`. ii) Let  `p(x) = x^3 + 3x^2 + 3x + 1`  and zero of  `x - frac{1}{2}` is `frac{1}{2}` `p(frac{1}{2}) = (frac{1}{2})^3 + 3(frac{1}{2})^2 + 3(frac{1}{2}) + 1` `p(frac{1}{2}) = frac{1}{8} + frac{3}{4} + frac{3}{2} + 1` `p(frac{1}{2}) = frac{1+6+12+8}{8}` `p(frac{1}{2}) = frac{27}{8}` Answer: The remainder is `frac{27}{8}` iii) Let `p(x) = x^3 + 3x^2 + 3x + 1` and zero of `x` is `0`. `p(0) = (0)^3 + 3(0)^2 + 3(0) + 1` `p(0) = 0 + 0 + 0 + 1` `p(0) = 1` Answer : The rem

NCERT Class 9th solution of Exercise 2.1 NCERT Class 9th solution of Exercise 2.3 NCERT Class 9th solution of Exercise 2.4 NCERT Class 9th solution of Exercise 2.5 NCERT Class 9th Maths Projects Exercise 2.2 Q1. Find the value of the polynomial `5x - 4x^2 + 3` at i) `x = 0`  ii) `x = - 1`  iii) `x = 2` Sol. : i) Let `p(x) = 5x - 4x^2 + 3` `p(0) = 5(0) - 4(0)^2 + 3` `p(0) = 0 - 0 + 3` `p(0) = 3` Answer : `p(0) = 3` ii) Let `p(x) = 5x - 4x^2 + 3` `p(-1) = 5(-1) - 4(-1)^2 + 3` `p(-1) = - 5 - 4 + 3` `p(-1) = - 9 + 3` `p(-1) = - 6` Answer : `p(-1) = - 6` iii) Let `p(x) = 5x - 4x^2 + 3` `p(2) = 5(2) - 4(2)^2 + 3` `p(2) = 10 - 16 + 3` `p(2) = 13 - 16` `p(2) = - 3` Answer : `p(2) = - 3` Q2. Find `p(0),` `p(1)` and `p(2)` for eac of the following polynomials: i) `p(y) = y^2 - y + 1` ii) `p(t) = 2 + t + 2t^2 - t^3` iii) `p(x) = x^3`  iv) `p(x) = (x-1)(x+1)` Sol. : i) `p(y) = y^2 - y + 1` `p(0) = (0)^2 - (0) + 1` `p(0) = 0 - 0 + 1` `p(0) = 1` `p(1) = (1)^2 - (1) + 1` `p(1) = 1 - 1 +1` `p(1) = 0 +

NCERT Class 9th solution of Exercise 2.2 NCERT Class 9th solution of Exercise 2.3 NCERT Class 9th solution of Exercise 2.4 NCERT Class 9th solution of Exercise 2.5 NCERT Class 9th solution of Exercise 1.1 NCERT Class 9th Projects Exercise 2.1 Q1. Which of the following expression are polynomials in one variable and which are not? State reason for your answer : i) `4x^2 - 3x + 7`   ii) `y^2 + sqrt2` iii)  `3sqrtt + tsqrt2`   iv)  `y + frac{2}{y}` v)  `x^(10) + y^3 + t^(50)` Answer : i) A polynomial in one variable because there is only one variable with each exponent a whole number. ii) A polynomial in one variable because there is only one variable with each exponent a whole number. iii) Not a polynomial because every exponent of the variable is not a whole number. iv) Not a polynomial because every exponent of the variable is not a whole number. v) A polynomial in three variables because there are three variables with each exponent a whole number. Q2. Write the coefficient of `x^2` in

NCERT Class 9th solution of Exercise 1.1 NCERT Class 9th solution of Exercise 1.2 NCERT Class 9th solution of Exercise 1.3 NCERT Class 9th solution of Exercise 1.4 NCERT Class 9th solution of Exercise 1.5 NCERT Class 9th Maths Projects Exercise 1.6 Q1. Find : i) `64^frac{1}{2}` ii) `32^frac{1}{5}` iii) `125^frac{1}{3}` Sol. :  i) `64^frac{1}{2}`  `= (8^2)^(frac{1}{2})`  `= 8^(frac{1}{2}×2)`  `= 8^1`  `= 8` Answer  `= 8` ii) `32^frac{1}{5}`  `= (2^5)^frac{1}{5}`  `= 2^(5×frac{1}{5})`  `= 2`. Answer  `= 2` iii) `125^frac{1}{3}` `= (5^3)^frac{1}{3}`  `= 5^(3×frac{1}{3})`  `= 5` Answer  `= 5`   Q2. Find : i) `9^frac{3}{2}` ii) `32^frac{2}{5}`  iii) `16^frac{3}{4}`  iv) `125^frac{-1}{   3}` Sol. : i) `9^frac{3}{2}`  `= (3^2)^frac{3}{2}`  `= 3^(2×frac{3}{2})`  `= 3^3`  `= 27`. Answer  `= 27` ii)  `32^frac{2}{5}`  `= (2^5)^frac{2}{5}`  `= 2^(5×frac{2}{5})`  `= 2^2`  `= 4` Answer  `= 4` iii)  `16^frac{3}{4}`  `= (2^4)^frac{3}{4}`  `= 2^(4×frac{3}{4})`  `= 2^3`  `= 8` Answer   `= 8` iv)  `125^f

NCERT Class 9th solution of Exercise 1.1 NCERT Class 9th solution of Exercise 1.2 NCERT Class 9th solution of Exercise 1.3 NCERT Class 9th solution of Exercise 1.4 NCERT Class 9th solution of Exercise 1.6 NCERT Class 9th Maths Projects Exercise 1.5  Q1. Classify the following numbers as rational or irrational : i) `2 - \sqrt5` ii) (`3 - \sqrt23)  - \sqrt23`  iii) `\frac{2\sqrt7}{7\sqrt7}`  iv) `\frac{1}{\sqrt2}` v) `2π` Answer : Rational numbers : (ii) and (iii) Irrational numbers : (i), (iv) and (v) Q2. Simplify each of the following expression : i)  (`3 + \sqrt3`)(`2 + \sqrt2`) ii) (`3 + \sqrt3`)(`3 - \sqrt3`) iii) `\(sqrt5 - sqrt2)^2`     iv) (`\sqrt5 - \sqrt2`)(`\sqrt5 + \sqrt2`) Sol. : i) (`3 + \sqrt3`)(`2 + \sqrt2`)   = `3 × 2 + 3\sqrt3 + 2\sqrt2`     `+ \sqrt3\sqrt2` = `6 + 3\sqrt3 + 2\sqrt2`     `+ \sqrt6` Answer : = `6 + 3\sqrt3 + 2\sqrt2 + \sqrt6`   ii) (`3 + \sqrt3`)(`3 - \sqrt3`) =`\(3)^2-\(sqrt3)^2` = `9 - 3` = `6`   Answer :  `6` iii) `\(sqrt5 - sqrt2)^2`   = `(\sqrt5)^2

NCERT Class 9th solution of Exercise 1.1 NCERT Class 9th solution of Exercise 1.2 NCERT Class 9th solution of Exercise 1.3 NCERT Class 9th solution of Exercise 1.5 NCERT Class 9th solution of Exercise 1.6 NCERT Class 9th Maths Projects Exercise 1.4 Q1. Visualise `3.765` on the number line using successive magnification. Sol. : 1) We take 3 and 4 on the number line, after magnifying it we get 3.7 and 3.8.   2) We take 3.7 and 3.8 on the number line, after magnifying it we get 3.76 and 3.77.   3) We take 3.76 and 3.77 on the number line, after magnifying it we get 3.765.   Thus, the required number 3.765.   Q2. Visualise  `4.\overline{26}` on the number line, up to `4` decimal places. Sol. : 1) We take 4 and 5 on the number line, after magnifying it we get 4.2 and 4.3.   2) We take 4.2 and 4.3 on the number line, after magnifying it we get 4.26 and 4.27.   3) We take 4.26 and 4.27 on the number line, after magnifying it we get 4.262 and 4.263.   4) We take 4.262 and 4.263 on the number l

NCERT Class 9th solution of Exercise 1.1 NCERT Class 9th solution of Exercise 1.2 NCERT Class 9th solution of Exercise 1.4 NCERT Class 9th solution of Exercise 1.5 NCERT Class 9th solution of Exercise 1.6 NCERT Class 9th Maths Projects Exercise 1.3 Q1. Write the following in decimal form and say what kind of decimal expansion each has : (i)  `\frac{36}{100}`  (ii) `\frac{1}{11}`  (iii) `4\frac{1}{8}` (iv) `\frac{3}{13}` (v) `\frac{2}{11}` (vi) `\frac{329}{200}` Answer : (i) `\0.36`, terminating. (ii) `0.\overline{09}`, non-terminating repating. (iii) `4.\125`, terminating. (iv)`0.\overline{230769}`, non-terminating repeating. (v) `0.\overline{18}`, non-terminating repeating. (vi)`0.\8225`, terminating. Q2. You Know that  `\frac{1}{7}` = `0.\overline{142857}`.Can you predict what the decimal expansions of  `\frac{2}{7}`,`\frac{3}{7}`,`\frac{4}{7}`,`\frac{5}{7}`,`\frac{6}{7}` are, without actually doing the long division? If so, how? [Hint: Study the remainders while finding the value of

NCERT Class 9th solution of Exercise 1.1 NCERT Class 9th solution of Exercise 1.3 NCERT Class 9th solution of Exercise 1.4 NCERT Class 9th solution of Exercise 1.5 NCERT Class 9th solution of Exercise 1.6 NCERT Class 9th Maths Projects Exercise 1.2 Q1. State whether the following statements are true or false. Justify your answer. (i) Every irrational number is a real number. (ii) Every point on the number line is of the form √m, where m is a natural number. (iii) Every real number is an irrational number. Answer : (i) True , since the collection of real numbers is made up of rational and irrational numbers. (ii) False , no negative number can be the square root of any natural number. (iii) False , for example, 2 is a rational number. Q2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number. Answer : No. For example, `sqrt4` = 2 is a rational number.  Q3. Show how `sqrt5` can be represented on the numb

NCERT Class 9th solution of Exercise 1.2 NCERT Class 9th solution of Exercise 1.3 NCERT Class 9th solution of Exercise 1.4 NCERT Class 9th solution of Exercise 1.5 NCERT Class 9th solution of Exercise 1.6 NCERT Class 9th Maths Projects Exercise 1.1 Q1. Is zero a rational number? Can you write it in the form of `frac{p}{q}`: where p and q are integers and q≠0.   Ans. : Yes,  `0` =`frac{0}{1}`=`frac{0}{2}`= `frac{0}{3}` etc., denominator q can also be taken as a negative integer. Q2. Find six rational numbers between 3 and 4. Sol.  The denominator of these rational number will 6+1 = 7, We can write given numbers 3 and 4 in `frac{p}{q}` form where q ≠ 0. `frac{3×7}{1×7}` =`frac{21}{7}`  and `frac{4×7}{1×7}` = `frac{28}{7}` So that required 6 rational numbers are: `frac{22}{7}`, `frac{23}{7}`, `frac{24}{7}`, `frac{25}{7}`, `frac{26}{7}`, `frac{27}{7}` Answer : `frac{22}{7}`, `frac{23}{7}`, `frac{24}{7}`, `frac{25}{7}`, `frac{26}{7}`, `frac{27}{7}` Q3. Find five rational  numbers between `f