Mathematics

Number System

Question:

Prove that: a -1 a -1 + b -1 + a -1 a -1 b -1 = 2b 2 b 2 a 2

Answer:

(i) We have,

a -1 a -1 + b -1 + a -1 a -1 b -1

= 1/a 1/a + 1/b + 1/a 1/a 1/b = 1/a b + a / ab + 1/a b - a / ab

= 1 a . ab b+a + 1 a . ab b-a = b b + a + b b - a

= b(b - a)+b(b + a) (b + a)(b - a) = b2- ba + b2+ab b2- a2 = 2b2 b2- a2
previuos
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Number System

Q 1.

If 9 n × 3 2 × 3 -n/2 -2 × 27 n 33m× 23 = 1 27 , Prove that m - n = 1.

Q 2.

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

x -1 y . x -1 z . z -1 x

Q 3.

Prove that: a -1 a -1 + b -1 + a -1 a -1 b -1 = 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 :

x a(b-c) x b(a-c) ÷ x b x a c = 1 (ii) x a+b 2 x b+c 2 x c+a 2 (xaxbxc)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

Q 9.

Simplify :

(i) (25) 3/2 X (243) 3/5 (16) 5/4 X (8) 4/3 (ii) 16 X 2 n+1 - 4 X 2 n 16 X 2 n+2 - 2 X 2 n+2

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

Q 12.

Simplify : 81 16 -3/4 × 25 9 -3/2 ÷ 5 2 -3

Q 13.

Express 0.999999... in the form p/q, where p and q are integers and q ≠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) x -2 y 3 (ii) x -2/3 y -1/2 2 (iii) x -3 5 (iv) x -2/3 y 4 ÷ y 4

(v) 3 xy 2 ÷x 2y (vi) 4 3 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 ≠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