Represent geometrically the number 8 . 1 on the number line.

To represent 8 . 1 on the number line, we proceed as under: (i) Draw a line and mark a point A on it. Take a point O on this line such that the length of segment AO = 8 . 1 units . (ii) From O , mark a point B on the line such that OB = 1 unit . (iii) Find the mid-point, say M , of the segment AB . With M as centre and radius = AM , draw a semicircle. (iv) At O , draw a perpendicular to AB meeting the semicircle at C, then OC = 8 . 1 units . (v) Treat the line AB as the number line with O representing 0 and B represent 1 and so on. Draw an arc with O as centre and radius = OC to meet the number line AB at P , then the point P represents the number 8 . 1 . Hence, the required answer is as shown above.

Represent geometrically the number 8 . 1 on the number line.

To represent 8 . 1 on the number line, we proceed as under: (i) Draw a line and mark a point A on it. Take a point O on this line such that the length of segment AO = 8 . 1 units . (ii) From O , mark a point B on the line such that OB = 1 unit . (iii) Find the mid-point, say M , of the segment AB . With M as centre and radius = AM , draw a semicircle. (iv) At O , draw a perpendicular to AB meeting the semicircle at C, then OC = 8 . 1 units . (v) Treat the line AB as the number line with O representing 0 and B represent 1 and so on. Draw an arc with O as centre and radius = OC to meet the number line AB at P , then the point P represents the number 8 . 1 . Hence, the required answer is as shown above.

Express 0 . 6 + 0 . 7 + 0 . 4 7 in the form p q , where p and q are integers, q ≠ 0 .

Given, 0 . 6 + 0 . 7 + 0 . 4 7 Let x = 0 . 6 Multiplying and dividing by 10 with 0 . 6 , we get x = 6 10 - - - - - i Let y = 0 . 7 = 0 . 7777 . . . . - - - - - i i Multiplying equation i i by 10 on both sides, we get 10 y = 7 . 777 . . . = 7 . 7 - - - - - - i i i Subtracting equation i i from equation i i i , we get 10 y - y = 7 . 7 - 0 . 7 9 y = 7 y = 7 9 Let z = 0 . 4 7 = 0 . 4777 . . . - - - - - i v Multiplying equation i v by 10 on both sides, we get 10 z = 4 . 7 = 4 . 7777 . . . - - - - v Further, multiplying equation v by 10 on both sides, we get 100 z = 47 . 7 = 47 . 777 . . . - - - - v i Subtracting equation v from v i , we get 100 z - 10 z = 47 . 7 - 4 . 7 90 z = 43 z = 43 90 Now, we add x + y + z , we get 6 10 + 7 9 + 43 90 L.C.M of all denominator is 90 . ⇒ 6 10 × 9 9 + 7 9 × 10 10 + 43 90 ⇒ 54 90 + 70 90 + 43 90 = 167 90 Hence, 0 . 6 , 0 . 7 , 0 . 4 7 = 167 90

If a and b are rational numbers and 6 3 2 - 2 3 = a 2 - b 3 , then find the values of a and b .

Given, 6 3 2 - 2 3 Multiplying the numerator and denominator by 3 2 + 2 3 . 6 3 2 - 2 3 = 6 3 2 - 2 3 × 3 2 + 2 3 3 2 + 2 3 = 6 3 2 + 2 3 3 2 2 - 2 3 2 = 18 2 + 12 3 18 - 12 = 18 2 + 12 3 6 = 6 3 2 + 2 3 6 So, 6 3 2 - 2 3 = 3 2 + 2 3 Therefore, the value of a = 3 , b = - 2 .

If a and b are rational numbers and 2 + 2 3 2 2 + 3 = a + b 6 , find the values of a and b .

Given, 2 + 2 3 2 2 + 3 Multiplying the numerator and denominator by 2 2 - 3 . 2 + 2 3 2 2 + 3 = 2 + 2 3 2 2 + 3 × 2 2 - 3 2 2 - 3 = 2 + 2 3 2 2 - 3 2 2 2 - 3 2 = 2 2 2 - 3 + 2 3 2 2 - 3 8 - 3 = 4 - 6 + 4 6 - 6 8 - 3 = - 2 + 3 6 5 = - 2 5 + 3 6 5 So, 2 + 2 3 2 2 + 3 = - 2 5 + 3 6 5 Therefore, the value of a = - 2 5 , b = 3 5 .

If x = 1 3 - 2 2 and y = 1 3 + 2 2 , then find the value of x + y + x y .

Given, x = 1 3 - 2 2 and y = 1 3 + 2 2 We have to find the value of x + y + x y . x = 1 3 - 2 2 Multiplying the numerator and denominator by 3 + 2 2 . x = 1 3 - 2 2 × 3 + 2 2 3 + 2 2 = 3 + 2 2 3 2 - 2 2 2 = 3 + 2 2 9 - 8 x = 3 + 2 2 y = 1 3 + 2 2 × 3 - 2 2 3 - 2 2 Multiplying the numerator and denominator by 3 - 2 2 . = 3 - 2 2 3 2 - 2 2 2 = 3 - 2 2 9 - 8 y = 3 - 2 2 Now, we find x + y + x y , we get ⇒ 3 + 2 2 + 3 - 2 2 + 3 + 2 2 3 - 2 2 ⇒ 3 + 2 2 + 3 - 2 2 + 9 - 6 2 + 6 2 - 8 ⇒ 6 + 1 = 7 Therefore, the value of x + y + x y is 7 .

If x = 5 - 2 6 , then find the value of x + 1 x .

Given, x = 5 - 2 6 We have to find the value of x + 1 x . x = 5 - 2 6 = 3 + 2 - 2 2 3 = 3 - 2 2 ∵ x = 3 - 2 1 x = 1 3 - 2 × 3 + 2 3 + 2 = 3 + 2 3 2 - 2 2 = 3 + 2 3 - 2 ∵ 1 x = 3 + 2 Now, we find x + 1 x , we get x + 1 x = 3 - 2 + 3 + 2 = 2 3 Therefore, the value of x + 1 x is 2 3 .

Simplify: 8 - 4 3 + 2 - 2 1 2

Given, 8 - 4 3 + 2 - 2 1 2 We need to simplify. 8 - 4 3 + 2 - 2 1 2 ⇒ 2 3 - 4 3 + 2 - 2 1 2 ⇒ 2 3 × - 4 3 + 2 - 2 1 2 ⇒ 2 - 4 + 2 - 2 1 2 ⇒ 1 2 4 + 1 2 2 1 2 ⇒ 1 2 4 × 1 2 + 1 2 2 × 1 2 ⇒ 1 2 2 + 1 2 1 = 1 4 + 1 2 = 3 4 Hence. the required answer is 3 4 .