Number System

Question:

Prove that: $\frac{{a}^{-1}}{{a}^{-1}+{b}^{-1}}+\frac{{a}^{-1}}{{a}^{-1}-{b}^{-1}}=\frac{{\mathrm{2b}}^{2}}{{b}^{2}-{a}^{2}}$

(i) We have,

$\frac{{a}^{-1}}{{a}^{-1}+{b}^{-1}}+\frac{{a}^{-1}}{{a}^{-1}-{b}^{-1}}$

= $\frac{1/a}{1/a+1/b}+\frac{1/a}{1/a-1/b}$ = $\frac{1/a}{b+a/ab}+\frac{1/a}{b-a/ab}$

$=\frac{1}{a}$ $.\frac{ab}{b+a}$ $+\frac{1}{a}$ $.\frac{ab}{b-a}$ $=\frac{b}{b + a}$ $+\frac{b}{b - a}$

= $\frac{b\left(b - a\right)+b\left(b + a\right)}{\left(b + a\right)\left(b - a\right)}$ = $\frac{b2- ba + b2+ab}{b2-a2}$ = $\frac{2b2}{b2-a2}$

Number System

Q 1.

If $\frac{{9}^{n}×{3}^{2}×{\left({3}^{-n/2}\right)}^{-2}×{\left(27\right)}^{n}}{33m×23}=\frac{1}{27}$ , Prove that m - n = 1.

Q 2.

if x,y,z are positive real numbers show that :

$\sqrt{{x}^{-1}y}.\sqrt{{x}^{-1}z}.\sqrt{{z}^{-1}x}$

Q 3.

Prove that: $\frac{{a}^{-1}}{{a}^{-1}+{b}^{-1}}+\frac{{a}^{-1}}{{a}^{-1}-{b}^{-1}}=\frac{{\mathrm{2b}}^{2}}{{b}^{2}-{a}^{2}}$

Q 4.

If a = 2 and b= 3, then find the values of each of the following :

(i) aa + bb (ii) ab + ba (iii) ab

(iv) (a/b)a (v) (1/a +1/b)a

Q 5.

Show that :

$\frac{{x}^{a\left(b-c\right)}}{{x}^{b\left(a-c\right)}}÷{\left(\frac{{x}^{b}}{{x}^{a}}\right)}^{c}= 1\left(ii\right)\frac{{\left({x}^{a+b}\right)}^{2}{\left({x}^{b+c}\right)}^{2}{\left({x}^{c+a}\right)}^{2}}{\left(xaxbxc\right)4}= 1$

Q 6.

Find the value of x, if ${5}^{x-3}×{3}^{2x-8}=8$

Q 7.

Evalute each of the following:

(i) 52 x 54 (ii) 58 ÷ 53 (iii) (32)3 (iv) (11/12)3 (v) (3/4)-3

Q 8.

Evalute each of the following:

(i) (2/11)4 x (11/3)2 x (3/2)3 (ii) (1/2)5 x (-2/3)4 x (3/5)-1

(iii) 255 x 260 - 297 x 218 (iv) (2/3)2 x (2/5)-3 x (3/5)2

Simplify :

(i) (ii)

Q 10.

Prove that √3 - √2 is an irrational number.

Q 11.

Simplify each of the following :

(i) (625)-1/4 (ii) (256/81)5/4 (iii) (243/32)-4/5 (iv) 5√(32)-3

Simplify :

Q 13.

Express 0.999999... in the form p/q, where p and q are integers and q â‰ 0

Q 14.

Find a rational number between 2 and 1.

Q 15.

Give examples of two irrational numbers, the product of which is:

(i) a rational number (ii) an irrational number.

Q 16.

Find two rational numbers between 0.23332333233332… and 0.2525525552555-52…

Q 17.

Identify √45 as rational number or irrational number.

Q 18.

Write three numbers whose decimal expansions are non-terminating non recurring.

Q 19.

Represent 5/3 and -5/3 on the number line.

Q 20.

Express 0.001as a fraction in the simplest form.

Q 21.

Assuming that x, y are positive real numbers, simplify each of the following:

(i) $\sqrt{{x}^{-2}{y}^{3}}$ (ii) ${\left({x}^{-2/3}{y}^{-1/2}\right)}^{2}$ (iii) ${\left(\sqrt{{x}^{-3}}\right)}^{5}$ (iv) ${\left(\sqrt{x}\right)}^{-2/3}\sqrt{{y}^{4}}÷\sqrt{{y}^{4}}$

(v) $3\sqrt{{\mathrm{xy}}^{2}÷{x}^{2}y}$ (vi) $4\sqrt{3\sqrt{{x}^{2}}}$

Q 22.

Find three different irrational numbers between the rational numbers 5/7 and 9/11.

Q 23.

Insert 100 rational numbers between -3/13 and 9/13.

Q 24.

Find two irrational numbers lying between √2 and √3.

Q 25.

Find a rational number between - 2 and 6.

Q 26.

Express each of the following decimals in the form p/q :
(i)Â 0.4Â Â (ii)Â 0.2 Â Â (iii)Â 0.3
(iv)0.4 Â Â (v)Â 0.5Â Â (vi)0.6

Q 27.

Represent 8/5 and -8/5 on the the number line.

Q 28.

Insert 10 rational numbers betwveen -3/11 and 8/11.

Q 29.

Find one irrational number between the number a and b given below

a = 0.1111... = 0.1 and b = 0.1101

Q 30.

Evaluate each of the following removing radical signs and negative indices wherever they occur:

(i) (64)1/3 (ii) (125)-1/3[NCERT] (iii) (27) - 2/3 (iv) (64/25)-3/2

Q 31.

Find three rational numbers between -2 and 5.

Q 32.

Insert a rational and irrational number between 2 and 3.

Q 33.

Prove that √n is not a rational number, if n is not a perfect square.

Q 34.

Express each of the following decimals in the form p/q.
(i)Â 0.35Â Â Â (ii)  0.0.585

Q 35.

Show that 0.2353535... = 0.235 can be expressed in the form p/q, where p and q are integers and q â‰ 0.

Q 36.

Find two irrational numbers between 0.12 and 0.13

Q 37.

Examine whether the following numbers are rational or irrational:

(i) (√2 + 2)2 (ii) (5 + √5)(5 - √5) (iii) 6/ 2√3

Q 38.

Express each of the following numbers in the form p/q.
(i)Â 0.15 Â Â Â (ii)Â 0.675Â (iii)Â 0.00026

Q 39.

Simplify each of the following, removing radical signs and negative indices wherever they occur:

(i) (√4)-3 (ii) (√5)-3 (√2)-3 (iii) 1/ 3√4-5

(iv) (25)-1/3 x 3√16 (v) (3√8)-1/2 (vi) (√4)-7 (√2)-5

Q 40.

Express each of the following mixed recurring decimals in the form p/q. (i)4.32(ii)15.712

Q 41.

Find two irrational numbers between 2 and 2.5.

Q 42.

Convert the following decimal number in the form p/q.
(i) 5.2 (ii) 2343

Q 43.

Show that 1.272727 = 1.27 can be expressed in the form p/q , where p and q are integers and q ≠0.

Q 44.

Express each of the following numbers in the form p/q.
(i) 15.75 0 (ii) 8.0025Â (iii) -24.6875

Q 45.

Express the following decimals in the form p/q.
(i)0.32 (ii) 0.123 (iii) 0.00352