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R.D. Sharma solutions Mathematics for Class 9 by R D Sharma (2018-19 Session) chapter 1 Number Systems

Chapters

R.D. Sharma Mathematics Class 9 by R D Sharma (2018-19 Session)

Mathematics for Class 9 by R D Sharma (2018-19 Session)

Chapter 1 - Number Systems

Pages 0 - 3

Is zero a rational number? Can you write it in the form p/q, where p and q are integersand q ≠ 0?

Simplify of the following:

`root(3)4  xx root(3)16`

Simplify the following

`3(a^4b^3)^10xx5(a^2b^2)^3`

Simplify the following

`(2x^-2y^3)^3`

Simplify of the following:

`root(4)1250/root(4)2`

Simplify the following

`((4xx10^7)(6xx10^-5))/(8xx10^4)`

Simplify the following

`(4ab^2(-5ab^3))/(10a^2b^2)`

Simplify the following

`((x^2y^2)/(a^2b^3))^n`

Simplify the following

`(a^(3n-9))^6/(a^(2n-4))`

Find five rational numbers between 1 and 2.

Simplify the following expressions:

`(4 + sqrt7)(3 + sqrt2)`

If a = 3 and b = -2, find the values of :

aa + bb

 

Simplify the following expressions:

`(3 + sqrt3)(5 - sqrt2)`

If a = 3 and b = -2, find the values of :

ab + ba

If a = 3 and b = -2, find the values of :

(a + b)ab

 

Simplify the following expressions:

`(sqrt5 - 2)(sqrt3 - sqrt5)`

Find six rational numbers between 3 and 4.

Simplify the following expressions:

`(11 + sqrt11)(11 - sqrt11)`

Prove that:

`(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)=1`

Prove that:

`(x^a/x^b)^cxx(x^b/x^c)^axx(x^c/x^a)^b=1`

Simplify the following expressions:

`(5 + sqrt7)(5 - sqrt7)`

Simplify the following expressions:

`(sqrt8 - sqrt2)(sqrt8 + sqrt2)`

Simplify the following expressions:

`(3 + sqrt3)(3 - sqrt3)`

Simplify the following expressions:

`(sqrt5 - sqrt2)(sqrt5 + sqrt2)`

Find five rational numbers between 3/5 and 4/5.

Prove that:

`1/(1+x^(a-b))+1/(1+x^(b-a))=1`

Simplify the following expressions:

`(sqrt3 + sqrt7)^2`

Prove that:

`1/(1+x^(b-a)+x^(c-a))+1/(1+x^(a-b)+x^(c-b))+1/(1+x^(b-c)+x^(a-c))=1`

Simplify the following expressions:

`(sqrt5 - sqrt3)^2`

Simplify the following expressions:

`(2sqrt5 + 3sqrt2)^2`

Prove that:

`(a+b+c)/(a^-1b^-1+b^-1c^-1+c^-1a^-1)=abc`

State whether the following statement is true or false. Give reasons for your answers.

Every whole number is a natural number.

State whether the following statement is true or false. Give reasons for your answers.

Every integer is a rational number.

Prove that:

`(a^-1+b^-1)^-1=(ab)/(a+b)`

State whether the following statement is true or false. Give reasons for your answers.

Every rational number is an integer.

State whether the following statement is true or false. Give reasons for your answers.

Every natural number is a whole number.

State whether the following statement is true or false. Give reasons for your answers.

Every integer is a whole number.

State whether the following statement is true or false. Give reasons for your answers.

Every rational number is a whole number.

If abc = 1, show that `1/(1+a+b^-1)+1/(1+b+c^-1)+1/(1+c+a^-1)=1`

Simplify the following:

`(3^nxx9^(n+1))/(3^(n-1)xx9^(n-1))`

Simplify the following:

`(5xx25^(n+1)-25xx5^(2n))/(5xx5^(2n+3)-25^(n+1))`

Simplify the following:

`(5^(n+3)-6xx5^(n+1))/(9xx5^x-2^2xx5^n)`

Simplify the following:

`(6(8)^(n+1)+16(2)^(3n-2))/(10(2)^(3n+1)-7(8)^n)`

Solve the following equation for x:

`7^(2x+3)=1`

Solve the following equation for x:

`2^(x+1)=4^(x-3)`

Solve the following equation for x:

`2^(5x+3)=8^(x+3)`

Solve the following equation for x:

`4^(2x)=1/32`

Solve the following equation for x:

`4^(x-1)xx(0.5)^(3-2x)=(1/8)^x`

Solve the following equation for x:

`2^(3x-7)=256`

Solve the following equations for x:

`2^(2x)-2^(x+3)+2^4=0`

Solve the following equations for x:

`3^(2x+4)+1=2.3^(x+2)`

If 49392 = a4b2c3, find the values of a, b and c, where a, b and c are different positive primes.

If `1176=2^a3^b7^c,` find a, b and c.

Given `4725=3^a5^b7^c,` find

(i) the integral values of a, b and c

(ii) the value of `2^-a3^b7^c`

If `a=xy^(p-1), b=xy^(q-1)` and `c=xy^(r-1),` prove that `a^(q-r)b^(r-p)c^(p-q)=1`

Pages 0 - 15

Rationalise the denominator of each of the following

`3/sqrt5`

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

`(sqrt(x^-3))^5`

Express the following rational number as decimal:

`42/100`

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

`sqrt(x^3y^-2)`

Rationalise the denominator of each of the following

`3/(2sqrt5)`

Express the following rational number as decimal:

`327/500`

Express the following rational number as decimal:

`15/4`

Rationalise the denominator of each of the following 

`1/sqrt12`

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

`(x^((-2)/3)y^((-1)/2))^2`

Rationalise the denominator of the following

`sqrt2/sqrt5`

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

`(sqrtx)^((-2)/3)sqrt(y^4)divsqrt(xy^((-1)/2))`

Rationalise the denominator of each of the following

`(sqrt3 + 1)/sqrt2`

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

`root5(243x^10y^5z^10)`

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

`(x^-4/y^-10)^(5/4)`

Rationalise the denominator of the following

`(sqrt2 + sqrt5)/3`

Rationalise the denominator of the following 

`(3sqrt2)/sqrt5`

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

`(sqrt2/sqrt3)^5(6/7)^2`

Find the value to three places of decimals of the following. It is given that

`sqrt2 = 1.414`, `sqrt3 = 1.732`, `sqrt5 = 2.236` and `sqrt10 = 3.162`

`2/sqrt3`

Simplify:

`(16^(-1/5))^(5/2)`

Express the following rational number as decimal:

`2/3`

Express the following rational number as decimal:

`-4/9`

Find the value to three places of decimals of the following. It is given that

`sqrt2 = 1.414`, `sqrt3 = 1.732`, `sqrt5 = 2.236` and `sqrt10 = 3.162`

`3/sqrt10`

Simplify:

`root5((32)^-3)`

Express the following rational number as decimal:

`-2/15`

Find the value to three places of decimals of the following. It is given that

`sqrt2 = 1.414`, `sqrt3 = 1.732`, `sqrt5 = 2.236` and `sqrt10 = 3.162`

`(sqrt5 + 1)/sqrt2`

Simplify:

`root3((343)^-2)`

Find the value to three places of decimals of the following. It is given that

`sqrt2 = 1.414`, `sqrt3 = 1.732`, `sqrt5 = 2.236` and `sqrt10 = 3.162`

`(sqrt10 + sqrt15)/sqrt2`

`

Simplify:

`(0.001)^(1/3)`

Express the following rational number as decimal:

`-22/13`

Simplify:

`((25)^(3/2)xx(243)^(3/5))/((16)^(5/4)xx(8)^(4/3))`

Express the following rational number as decimal:

`437/999`

Find the value to three places of decimals of the following. It is given that

`sqrt2 = 1.414`, `sqrt3 = 1.732`, `sqrt5 = 2.236` and `sqrt10 = 3.162`

`(2 + sqrt3)/3`

Simplify:

`(sqrt2/5)^8div(sqrt2/5)^13`

Find the value to three places of decimals of the following. It is given that

`sqrt2 = 1.414`, `sqrt3 = 1.732`, `sqrt5 = 2.236` and `sqrt10 = 3.162`

`(sqrt2 - 1)/sqrt5`

Express the following rational number as decimal:

`33/26`

Simplify:

`((5^-1xx7^2)/(5^2xx7^-4))^(7/2)xx((5^-2xx7^3)/(5^3xx7^-5))^(-5/2)`

Look at several examples of rational numbers in the form p/q (q≠0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Express the following with rational denominator:

`1/(3 + sqrt2)`

Prove that:

`sqrt(3xx5^-3)divroot3(3^-1)sqrt5xxroot6(3xx5^6)=3/5`

Express of the following with rational denominator:

`1/(sqrt6 - sqrt5)`

Prove that:

`9^(3/2)-3xx5^0-(1/81)^(-1/2)=15`

Express the following with rational denominator:

`16/(sqrt41 - 5)`

Prove that:

`(1/4)^-2-3xx8^(2/3)xx4^0+(9/16)^(-1/2)=16/3`

Prove that:

`(2^(1/2)xx3^(1/3)xx4^(1/4))/(10^(-1/5)xx5^(3/5))div(3^(4/3)xx5^(-7/5))/(4^(-3/5)xx6)=10`

Express the following with rational denominator:

`30/(5sqrt3 - 3sqrt5)`

Prove that:

`sqrt(1/4)+(0.01)^(-1/2)-(27)^(2/3)=3/2`

Express the following with rational denominator:

`1/(2sqrt5 - sqrt3)`

Express the following with rational denominator:

`(sqrt3 + 1)/(2sqrt2 - sqrt3)`

Prove that:

`(2^n+2^(n-1))/(2^(n+1)-2^n)=3/2`

Prove that:

`(64/125)^(-2/3)+1/(256/625)^(1/4)+(sqrt25/root3 64)=65/16`

Express the following with rational denominator:

`(6 - 4sqrt2)/(6 + 4sqrt2)`

Express the following with rational denominator:

`(3sqrt2 + 1)/(2sqrt5 - 3)`

Prove that:

`(3^-3xx6^2xxsqrt98)/(5^2xxroot3(1/25)xx(15)^(-4/3)xx3^(1/3))=28sqrt2`

Express each one of the following with rational denominator:

`(b^2)/(sqrt(a^2 + b^2) + a)`

Prove that:

`((0.6)^0-(0.1)^-1)/((3/8)^-1(3/2)^3+((-1)/3)^-1)=(-3)/2`

Show that:

`1/(1+x^(a-b))+1/(1+x^(b-a))=1`

Rationales the denominator and simplify:

`(3 - sqrt2)/(3 + sqrt2)`

Show that:

`[{x^(a(a-b))/x^(a(a+b))}div{x^(b(b-a))/x^(b(b+a))}]^(a+b)=1`

Rationales the denominator and simplify:

`(5 + 2sqrt3)/(7 + 4sqrt3)`

Show that:

`(x^(1/(a-b)))^(1/(a-c))(x^(1/(b-c)))^(1/(b-a))(x^(1/(c-a)))^(1/(c-b))=1`

Rationales the denominator and simplify:

`(1 + sqrt2)/(3 - 2sqrt2)`

Show that:

`(x^(a^2+b^2)/x^(ab))^(a+b)(x^(b^2+c^2)/x^(bc))^(b+c)(x^(c^2+a^2)/x^(ac))^(a+c)=x^(2(a^3+b^3+c^3))`

Rationales the denominator and simplify:

`(4sqrt3 + 5sqrt2)/(sqrt48 + sqrt18)`

Show that:

`(x^(a-b))^(a+b)(x^(b-c))^(b+c)(x^(c-a))^(c+a)=1`

Show that:

`{(x^(a-a^-1))^(1/(a-1))}^(a/(a+1))=x`

Rationales the denominator and simplify:

`(2sqrt3 - sqrt5)/(2sqrt2 + 3sqrt3)`

Rationales the denominator and simplify:

`(2sqrt6 - sqrt5)/(3sqrt5 - 2sqrt6)`

Show that:

`(a^(x+1)/a^(y+1))^(x+y)(a^(y+2)/a^(z+2))^(y+z)(a^(z+3)/a^(x+3))^(z+x)=1`

Show that:

`(3^a/3^b)^(a+b)(3^b/3^c)^(b+c)(3^c/3^a)^(c+a)=1`

If 2x = 3y = 12z, show that `1/z=1/y+2/x`

Simplify:

`(5 + sqrt3)/(5 - sqrt3) + (5 - sqrt3)/(5 + sqrt3)`

Simplify

`1/(2 + sqrt3) + 2/(sqrt5 - sqrt3) + 1/(2 - sqrt5)`

Simplify

`2/(sqrt5 + sqrt3) + 1/(sqrt3 + sqrt2) + 3/(sqrt5 + sqrt2)`

If 2x = 3y = 6-z, show that `1/x+1/y+1/z=0`

In the following determine rational numbers a and b:

`(sqrt3 - 1)/(sqrt3 + 1) = a - bsqrt3`

In the following determine rational numbers a and b:

`(4 + sqrt2)/(2 + sqrt2) = n - sqrtb`

In the following determine rational numbers a and b:

`(3 + sqrt2)/(3 - sqrt2) = a + bsqrt2`

In the following determine rational numbers a and b:

`(5 + 3sqrt3)/(7 + 4sqrt3) = a + bsqrt3`

In the following determine rational numbers a and b:

`(sqrt11 - sqrt7)/(sqrt11 + sqrt7) = a - bsqrt77`

In the following determine rational numbers a and b:

`(4 + 3sqrt5)/(4 - 3sqrt5) = a + bsqrt5`

If ax = by = cz and b2 = ac, show that `y=(2zx)/(z+x)`

Find the value of `6/(sqrt5 - sqrt3)` it being given that `sqrt3 = 1.732` and  `sqrt5 = 2.236`

If 3x = 5y = (75)z, show that `z=(xy)/(2x+y)`

Find the values the following correct to three places of decimals, it being given that `sqrt2 = 1.4142`, `sqrt3 = 1.732`, `sqrt5 = 2.2360`, `sqrt6 = 2.4495` and `sqrt10 = 3.162`

`(3 - sqrt5)/(3 + 2sqrt5)`

Find the values the following correct to three places of decimals, it being given that `sqrt2 = 1.4142`, `sqrt3 = 1.732`, `sqrt5 = 2.2360`, `sqrt6 = 2.4495` and `sqrt10 = 3.162`

`(1 + sqrt2)/(3 - 2sqrt2)`

Simplify `(3sqrt2 - 2sqrt3)/(3sqrt2 + 2sqrt3) + sqrt12/(sqrt3 - sqrt2)`

If `27^x=9/3^x,` find x.

if `x = 2 +  sqrt3`,find the value of `x^2 + 1/x^2`

Find the value of x in the following:

`2^(5x)div2x=root5(2^20)`

Find the value of x in the following:

`(2^3)^4=(2^2)^x`

Find the value of x in the following:

`(3/5)^x(5/3)^(2x)=125/27`

Find the value of x in the following:

`5^(x-2)xx3^(2x-3)=135`

Find the value of x in the following:

`2^(x-7)xx5^(x-4)=1250`

Find the value of x in the following:

`(root3 4)^(2x+1/2)=1/32`

Find the value of x in the following:

`5^(2x+3)=1`

Find the value of x in the following:

`(13)^(sqrtx)=4^4-3^4-6`

Find the value of x in the following:

`(sqrt(3/5))^(x+1)=125/27`

If `x=2^(1/3)+2^(2/3),` Show that x3 - 6x = 6

if   `x= 3 + sqrt8`, find the value of `x^2 + 1/x^2`

Determine `(8x)^x,`If `9^(x+2)=240+9^x`

if `x =  (sqrt3 + 1)/2` find the value of `4x^2 +2x^2 - 8x + 7` 

If `3^(x+1)=9^(x-2),` find the value of `2^(1+x)`

If `3^(4x) = (81)^-1` and `10^(1/y)=0.0001,` find the value of 2^(-x+4y).

If `5^(3x)=125` and `10^y=0.001,` find x and y.

Solve the following equation:

`3^(x+1)=27xx3^4`

Solve the following equation:

`4^(2x)=(root3 16)^(-6/y)=(sqrt8)^2`

Solve the following equation:

`3^(x-1)xx5^(2y-3)=225`

Solve the following equation:

`8^(x+1)=16^(y+2)` and, `(1/2)^(3+x)=(1/4)^(3y)`

Solve the following equation:

`4^(x-1)xx(0.5)^(3-2x)=(1/8)^x`

Solve the following equation:

`sqrt(a/b)=(b/a)^(1-2x),` where a and b are distinct primes.

If a and b are distinct primes such that `root3 (a^6b^-4)=a^xb^(2y),` find x and y.

If a and b are different positive primes such that

`((a^-1b^2)/(a^2b^-4))^7div((a^3b^-5)/(a^-2b^3))=a^xb^y,` find x and y.

If a and b are different positive primes such that

`(a+b)^-1(a^-1+b^-1)=a^xb^y,` find x + y + 2.

If `2^x xx3^yxx5^z=2160,` find x, y and z. Hence, compute the value of `3^x xx2^-yxx5^-z.`

If 1176 = `2^axx3^bxx7^c,` find the values of a, b and c. Hence, compute the value of `2^axx3^bxx7^-c` as a fraction.

Simplify:

`(x^(a+b)/x^c)^(a-b)(x^(b+c)/x^a)^(b-c)(x^(c+a)/x^b)^(c-a)`

Simplify:

`root(lm)(x^l/x^m)xxroot(mn)(x^m/x^n)xxroot(nl)(x^n/x^l)`

Show that:

`((a+1/b)^mxx(a-1/b)^n)/((b+1/a)^mxx(b-1/a)^n)=(a/b)^(m+n)`

If `a=x^(m+n)y^l, b=x^(n+l)y^m` and `c=x^(l+m)y^n,` Prove that `a^(m-n)b^(n-l)c^(l-m)=1`

If `x = a^(m+n),` `y=a^(n+l)` and `z=a^(l+m),` prove that `x^my^nz^l=x^ny^lz^m`

Page 0

Express the following decimal in the form `p/q` : 0.39

Express the following decimal in the form `p/q` : 0.750

Express the following decimal in the form `p/q` : 2.15

Express the following decimal in the form `p/q`:

7.010

Express the following decimal in the form `p/q`: 9.90

Express the following decimal in the form `p/q`: 1.0001

Express the following decimal in the form `p/q`: `0.bar4`

Express the following decimal in the form `p/q`: `0.bar37`

Express the following decimal in the form `p/q`: `0.bar54`

Express the following decimal in the form `p/q`: `0.bar621`

Express the following decimal in the form `p/q`: `125.bar3`

Express the following decimal in the form `p/q`: `4.bar7`

Express the following decimal in the form `p/q`: `0.4bar7`

Page 0

Define an irrational number ?

Explain, how irrational numbers differ from rational numbers ?

Examine, whether the following number are rational or irrational:

`sqrt7`

Examine, whether the following number are rational or irrational:

`sqrt4`

Examine, whether the following number are rational or irrational:

`2+sqrt3`

Examine, whether the following number are rational or irrational:

`sqrt3+sqrt2`

Examine, whether the following number are rational or irrational:

`sqrt3+sqrt5`

Examine, whether the following number are rational or irrational:

`(sqrt2-2)^2`

Examine, whether the following number are rational or irrational:

`(2-sqrt2)(2+sqrt2)`

Examine, whether the following number are rational or irrational:

`(sqrt2+sqrt3)^2`

Examine, whether the following number are rational or irrational:

`sqrt5-2`

Classify the following number as rational or irrational :-

`sqrt23`

Classify the following number as rational or irrational :-

`sqrt225`

Classify the following number as rational or irrational :-

0.3796

Classify the following number as rational or irrational :-

7.478478...

Classify the following number as rational or irrational :-

1.1010010001...

Identify the following as rational or irrational number. Give the decimal representation of rational number:

`sqrt4`

Identify the following as rational or irrational number. Give the decimal representation of rational number:

`3sqrt18`

Identify the following as rational or irrational number. Give the decimal representation of rational number:

`sqrt1.44`

Identify the following as rational or irrational number. Give the decimal representation of rational number:

`sqrt(9/27)`

Identify the following as rational or irrational number. Give the decimal representation of rational number:

`-sqrt64`

Identify the following as rational or irrational number. Give the decimal representation of rational number:

`sqrt100`

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

x2 = 5

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

y2 = 9

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

z2 = 0.04

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

`u^2=17/4`

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

v2 = 3

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

w2 = 27

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

t2 = 0.4

Give two rational numbers lying between 0.232332333233332... and 0.212112111211112.

Give two rational numbers lying between 0.515115111511115... and 0.535335333533335...

Find one irrational number between 0.2101 and 0.222... = `0.bar2`

Find a rational number and also an irrational number lying between the numbers 0.3030030003... and 0.3010010001...

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

Give an example of two irrational numbers whose:

difference is a rational number.

Give an example of two irrational numbers whose:

difference is an irrational number.

Give an example of two irrational numbers whose:

sum is a rational number.

Give an example of two irrational numbers whose:

sum is an irrational number.

Give an example of two irrational numbers whose:

product is an rational number.

Give an example of two irrational numbers whose:

product is an irrational number.

Give an example of two irrational numbers whose:

quotient is a rational number.

Give an example of two irrational numbers whose:

quotient is an irrational number.

Find two irrational numbers between 0.5 and 0.55.

Find two irrational numbers lying between 0.1 and 0.12.

Prove that `sqrt3+sqrt5` is an irrational number.

Page 0

Complete the following sentence:

Every point on the number line corresponds to a _________ number which many be either _______ or ________.

Complete the following sentence:

The decimal form of an irrational number is neither ________ nor _________

Complete the following sentence:

The decimal representation of a rational number is either ______ or _________.

Complete the following sentence:

Every real number is either ______ number or _______ number.

Find whether the following statement is true or false.

Every real number is either rational or irrational.

Find whether the following statement is true or false.

π is an irrational number.

Find whether the following statement is true or false.

Irrational numbers cannot be represented by points on the number line.

Represent `sqrt6,` `sqrt7,` `sqrt8` on the number line.

Represent `sqrt3.5,` `sqrt9.4,` `sqrt10.5` on the real number line.

Page 0

Visualise 2.665 on the number line, using successive magnification.

Visualise the representation of `5.3bar7` on the number line upto 5 decimal places, that is upto 5.37777.

R.D. Sharma Mathematics Class 9 by R D Sharma (2018-19 Session)

Mathematics for Class 9 by R D Sharma (2018-19 Session)
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