# Class 10

Explore popular questions from Real Numbers for Class 10. This collection covers Real Numbers previous year Class 10 questions hand picked by experienced teachers.

## Mathematics

Real Numbers

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Q 1. Assertion: For any integers {tex} a {/tex} and {tex} b {/tex} with {tex} b \neq 0 {/tex} there exist unique integers {tex} q {/tex} and {tex} r , {/tex} such that {tex} a = b q + r , {/tex} {tex} 0 \leq r < | b | {/tex}
Reason: An integer a which is not exactly divisible by {tex}3{/tex} can be written in one of the forms {tex} a = 3 n + 1 {/tex} or a {tex} = 3 n + 2 {/tex} where {tex} n {/tex} is an integer

A

Both Assertion and Reason are true and Reason is the correct explanation of 'Assertion'

Both Assertion and Reason are true and Reason is not the correct explanation of 'Assertion'

C

Assertion is true but Reason is false

D

Assertion is false but Reason is true

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Q 2. Assertion: {tex} 9 / 4608 {/tex} is a rational number.
Reason: All terminating or non terminating recurring numbers are rational numbers.

A

Both Assertion and Reason are true and Reason is the correct explanation of 'Assertion'

B

Both Assertion and Reason are true and Reason is not the correct explanation of 'Assertion'

Assertion is true but Reason is false

D

Assertion is false but Reason is true

##### Explanation

Any real number that cannot be written in fraction form is an irrational number. These numbers include the non-terminating, non-repeating decimals (pi, 0.45445544455544445555..., 2, etc.). Any square root that is not a perfect root is an irrational number.

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Q 3. If {tex} x , y {/tex} and {tex} z {/tex} are real numbers such that {tex} x < y ( x , y > 0 ) {/tex} and {tex} z < 0 , {/tex} then the statement which is true is

A

{tex} x z < y z {/tex}

B

{tex} \left( \frac { x } { z } \right) < \left( \frac { y } { z } \right) {/tex}

C

{tex} \left( \frac { z } { x } \right) > \left( \frac { z } { y } \right) {/tex}

{tex} x y > y z {/tex}

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Q 4. Which of the following is a pair of coprimes?

A

{tex} ( 14,35 ) {/tex}

{tex} ( 18,25 ) {/tex}

C

{tex} ( 31,93 ) {/tex}

D

{tex} ( 32,62 ) {/tex}

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Q 5. {tex} \mathrm { HCF } {/tex} of {tex} 2 ^ { 3 } \times 3 ^ { 2 } \times 5,2 ^ { 3 } \times 3 ^ { 3 } \times 5 ^ { 2 } {/tex} and {tex} 2 ^ { 2 } \times 3 \times 5 ^ { 3 } \times 7 {/tex} is

A

{tex} 30{/tex}

B

{tex} 48{/tex}

{tex} 60{/tex}

D

{tex} 105{/tex}

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Q 6. The product of two digit number is {tex}2160{/tex} and their GCM is {tex} 12 . {/tex} The numbers are

A

{tex} 72,30 {/tex}

{tex} 36,60 {/tex}

C

{tex} 96,25 {/tex}

D

None

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Q 7. LCM of two numbers is 14 times their HCF. The sum of LCM and HCF is 600. If one number is 280, then the other is

A

40

B

60

80

D

100

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Q 8. The ratio of two numbers is {tex}3: 4{/tex} and their HCF is {tex} 4 . {/tex} Then LCM is

A

{tex}12{/tex}

B

{tex}16{/tex}

C

{tex}24{/tex}

{tex}48{/tex}

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Q 9. LCM of {tex}2{/tex} prime numbers {tex} x {/tex} and {tex} y ( x > y ) {/tex} is {tex} 161 . {/tex} The value of {tex} 3 y - x {/tex} is

{tex} - 2 {/tex}

B

{tex} - 1 {/tex}

C

{tex} 1{/tex}

D

{tex} 2{/tex}

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Q 10. The HCF of {tex} \left( 4 a ^ { 2 } b ^ { 3 } - 9 b \right) {/tex} and {tex} \left( 2 a ^ { 2 } b ^ { 2 } - a b - 3 \right) {/tex} is

A

{tex} ( 2 a - 3 ) {/tex}

{tex} ( 2 a b - 3 ) {/tex}

C

{tex} ( 2 b - 3 a ) {/tex}

D

None of these

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Q 11. What is the greatest number which shall divide 305 and 629 and leave a remainder 8 in each case?

A

24

B

36

27

D

35

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Q 12. Given {tex} \mathrm x {/tex} and {tex} \mathrm n {/tex} are integers {tex} \mathrm{\left( 15 n ^ { 2 } + 6 n ^ { 2 } + 5 n + x \right) / n }{/tex} is not an integer for what condition?

A

{tex} \mathrm { n } {/tex} is positive

B

{tex} \mathrm x {/tex} is divisible by {tex} \mathrm { n } {/tex}

{tex} \mathrm x {/tex} is not divisible by {tex} \mathrm { n } {/tex}

D

both {tex}\mathrm (A){/tex} & {tex} ( \mathrm { C } ) {/tex}

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Q 13. The set of even integers {tex} \mathrm { E } = \{ \ldots \ldots - 4 , - 2,0,2,4,6 \} {/tex} is closed under operation

A

B

Subtraction

C

Multiplication

All of the above

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Q 14. If {tex} \mathrm { n } {/tex} is a natural number, then {tex} \sqrt { \mathrm { n } } {/tex} can be

A

a natural number

B

always an rational number

C

an irrational number

Both (A) and (C)

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Q 15. a and b are prime numbers which of the following is/are true?

A

{tex} \mathrm { a } ^ { 2 } {/tex} has 3 positive integral factors

B

{tex} \mathrm { ab } {/tex} has 4 prositive integral factors

C

{tex} \mathrm { a } ^ { 3 } {/tex} has positive integral factors

All of the above

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Q 16. Which of the following statements for natural numbers a, b and c is/are true

A

If a is divisible by b and b is divisible by c, then a must be divisible by c.

B

If a is a factor of both b and c , then a must be a factor of b+c

C

If a is a factor of both b and c then a must be a factor of bc

All of the above

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Q 17. Which of the following rational numbers have terminating decimal expansion

A

{tex} 64 / 455 {/tex}

B

{tex} 29 / 343 {/tex}

{tex} 13 / 325 {/tex}

D

Both (B) and (C)

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Q 18. Which one of the following is correct?
The number {tex} \sqrt { 14 + 6 \sqrt { 5 } } + \sqrt { 14 - 6 \sqrt { 5 } } {/tex}

is a rational number

B

is not a rational number

C

simplifies to {tex} { 5 } {/tex}

D

simplifies to {tex} { 9 } {/tex}

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Q 19. For some integer {tex} m {/tex}, every even integer is of the form :

A

{tex} m {/tex}

B

{tex} m + 1 {/tex}

{tex} 2 m {/tex}

D

{tex} 2 m + 1 {/tex}

##### Explanation

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Q 20. For some integer {tex} q , {/tex} every odd integer is of the form:

A

{tex}q{/tex}

B

{tex} q + 1 {/tex}

C

{tex} 2 q {/tex}

{tex} 2 q + 1 {/tex}

##### Explanation

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Q 21. {tex} n ^ { 2 } - 1 {/tex} is divisible by {tex} 8 , {/tex} if {tex} n {/tex} is :

A

an integer

B

a natural number

an odd integer

D

an even integer

##### Explanation

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Q 22. The largest number which divides 70 and 125 , leaving remainders 5 and 8 , respectively, is:

13

B

65

C

875

D

1,750

##### Explanation

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Q 23. If two positive integers {tex} a {/tex} and {tex} b {/tex} are written as {tex} a = x ^ { 3 } y ^ { 2 } {/tex} and {tex} b = x y ^ { 3 } ; x , y {/tex} are prime numbers, then {tex} H C F ( a , b ) {/tex} is :

A

{tex} x y {/tex}

{tex} x y ^ { 2 } {/tex}

C

{tex} x ^ { 3 } y ^ { 3 } {/tex}

D

{tex} x ^ { 2 } y ^ { 2 } {/tex}

##### Explanation

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Q 24. The product of a non-zero rational and an irrational number is:

always irrational

B

always rational

C

rational or irrational

D

one

##### Explanation

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Q 25. The decimal expansion of the rational number {tex} \frac { 14587 } { 1250 } {/tex} will terminate after:

A

one decimal place

B

two decimal places

C

three decimal places

four decimal places