class 10 Real Numbers Guess Questions

1) State whether 6/15 will have a terminating decimal expansion or a non-terminating repeating decimal.

2) State whether 35/50 will have a terminating decimal expansion or a non-terminating repeatingdecimal.

3) Find the LCM and HCF of 192 and 8 and verify that LCM × HCF = product of the two numbers.

4) Use Euclid’s algorithm to find the HCF of 4052 and 12576.

5) Show that any positive odd integer is of the form of 4q + 1 or 4q + 3, where q is some integer.

6) Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

7) Prove that 3√2 5 is irrational.

8) Prove that 1/√2 is irrational. (3 marks)

9) In a school there are tow sections- section A and Section B of class X. There are 32 students in section A and 36 students in section B. Determine the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B.

10) Express 3825 as a product of its prime factors.

11) Find the LCM and HCF of 8, 9 and 25 by the prime factorization method.

12) Find the HCF and LCM of 6, 72 and 120, using the prime factorization method.

13) State whether 29/343 will have a terminating decimal expansion or a non-terminating repeating decimal.

14) State whether 23/ 23 52 will have a terminating decimal expansion or a non-terminating repeating decimal

15) Find the LCM and HCF of 336 and 54 and verify that LCM × HCF = product of the two numbers

CHAPTER: REAL NUMBER: MERIT GAIN SERIES CBSE CLASS X MATHEMATICS

1. Q. Show that there is no positive integer n so that √ (n-1 )+ √(n+1) is rational

2. Q. Prove that one of every three consecutive positive integers is divisible by 3 ?

3. Q. Show that m3-m is divisible by 6 for each natural number m.

4. Q. What is the product of a non-zero rational and an irrational number?

5. Q. Find the least number that is divisible by all the numbers from 1 to 10 ?

6. Q. Prove that n2-n is divisible by 2 for every positive integer n?
7. Q. Show that 

 (a) The sum, the difference, and the product of two even numbers is always even.

(b) The sum and the difference of two odd numbers is always even, whereas their product is always odd.
8. Q. Prove that the expression y ( y + 1) always represents an even number, where y is any positive integer.
9.Q. Prove that every positive integer is of the form 3 p , 3 p + 1, or 3 p + 2, where p is any integer?
10. Q. if 'a ' and 'b ' are odd positive integers,a2+b2 is even Prove that it is not divisible by 4?
11.Q. Show that the expressions given below are composite numbers.

12. Q. Find the largest number which divides 75 and 130, leaving remainders 3 and 5, respectively?
(a) 3 × 5 × 7 × 23 + 2 × 7 × 11 × 13 (b) 29 × 35 + 14 (c) 3power 4 + 6power 3

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CHAPTER: REAL NUMBER: MERIT GAIN SERIES CBSE CLASS X MATHEMATICS

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CBSE X Maths Sample paper for chapter Real number with ...

X Maths Sample paper for chapter Real number with ...:
1. The values of the remainder r, when a positive integer a is divided by 3 are 0 and 1 only. Justify your answer

No. According to Euclid’s division lemma,

a = 3q + r, where 0 ≤ r < 3 and r is an integer. Therefore, the values of r can be 0, 1 or 2.

2. Can the number 6n, n being a natural number, end with the digit 5? Give reasons.

: No, because 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ, so the only primes in the factorization of 6ⁿ are 2 and 3, and not 5. Hence, it cannot end with the digit 5.

3. Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer.

No, because an integer can be written in the form 4q, 4q+1, 4q+2, 4q+3......................................

CBSE Class X Self Evaluation Tests For Maths : Real Numbers

1. The values of the remainder r, when a positive integer a is divided by 3 are 0 and 1 only. Justify your answer

No. According to Euclid’s division lemma,

a = 3q + r, where 0 ≤ r < 3 and r is an integer. Therefore, the values of r can be 0, 1 or 2.

2. Can the number 6n, n being a natural number, end with the digit 5? Give reasons.

 No, because 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ, so the only primes in the factorization of 6ⁿ are 2 and 3, and not 5. Hence, it cannot end with the digit 5.

3. Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer.

No, because an integer can be written in the form 4q, 4q+1, 4q+2, 4q+3.

4. “The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.

True, because n (n+1) will always be even, as one out of n or (n+1) must be even

5. “The product of three consecutive positive integers is divisible by 6”. Is this statement true or false”? Justify your answer.

True, because n (n+1) (n+2) will always be divisible by 6, as at least one of the factors will be divisible by 2 and at least one of the factors will be divisible by 3.

6. Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer.

No. Since any positive integer can be written as 3q, 3q+1, 3q+2,

therefore, square will be 9q2 = 3m, 9q2 + 6q + 1 = 3 (3q2 + 2q) + 1 = 3m + 1, 9q2 + 12q + 3 + 1 = 3m + 1.

7. A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

No. (3q + 1)2 = 9q2 + 6q + 1 = 3 (3q2 + 2q) = 3m + 1.

8. The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer.

HCF = 75, as HCF is the highest common factor

9. Explain why 3 × 5 × 7 + 7 is a composite number.

3×5×7+7 = 7 (3×5 + 1) = 7 (16), which has more than two factors

10. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.

No, because HCF (18) does not divide LCM (380).

11. Without actually performing the long division, find if 987/10500 will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.

Terminating decimal expansion, because 987/ 10500 = 47/ 500  and 500 =5³ 2²

12. A rational number in its decimal expansion is 327/7081. What can you say about the prime factors of q, when this number is expressed in the form p/q ? Give reasons.

Since 327.7081 is a terminating decimal number, so q must be of the form 2^m.5ⁿ;m, n are natural numbers

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NCERT CBSE 10th Class Maths Test paper Real Numbers

1. Express 140 as a product of its prime factors

2. Find the LCM and HCF of 12, 15 and 21 by the prime factorization method.

3. Find the LCM and HCF of 6 and 20 by the prime factorization method.

4. State whether13/3125 will have a terminating decimal expansion or a non-terminating repeating decimal.

5. State whether 17/8 will have a terminating decimal expansion or a non-terminating repeating decimal.

6. Find the LCM and HCF of 26 and 91 and verify that LCM × HCF = product of the two numbers.

7. Use Euclid’s division algorithm to find the HCF of 135 and 225

8. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m

9. Prove that √3 is irrational.

10. Show that 5 – √3 is irrational

11. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

CBSE MATHS X Number system Questions Bank

1. Express 140 as a product of its prime factors

2. Find the LCM and HCF of 12, 15 and 21 by the prime factorization method.

3. Find the LCM and HCF of 6 and 20 by the prime factorization method.

4. State whether13/3125 will have a terminating decimal expansion or a non-terminating repeating decimal.

5. State whether 17/8 will have a terminating decimal expansion or a non-terminating repeating decimal.

6. Find the LCM and HCF of 26 and 91 and verify that LCM × HCF = product of the two numbers.

7. Use Euclid’s division algorithm to find the HCF of 135 and 225

8. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m

9. Prove that √3 is irrational.

10. Show that 5 – √3 is irrational

11. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

12. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

13. Express 156 as a product of its prime factors.

14. Find the LCM and HCF of 17, 23 and 29 by the prime factorization method.

15. Find the HCF and LCM of 12, 36 and 160, using the prime factorization method.

16. State whether 6/15 will have a terminating decimal expansion or a non-terminating repeating decimal.

17. State whether35/50 will have a terminating decimal expansion or a non-terminating repeating 18. decimal.

19. Find the LCM and HCF of 192 and 8 and verify that LCM × HCF = product of the two numbers.

20. Use Euclid’s algorithm to find the HCF of 4052 and 12576.

Prove that ( √n -1 + √n +1 ) is irrational, for every nÎN

21. Show that any positive odd integer is of the form of 4q + 1 or 4q + 3, where q is some integer.

Real Numbers examples and guess paper for class10

Sample Question 1: Using Euclid’s division algorithm, find which of the following
pairs of numbers are co-prime:
(i) 231, 396 (ii) 847, 2160

Solution : Let us find the HCF of each pair of numbers.
(i) 396 = 231 × 1 + 165
231 = 165 × 1 + 66
165 = 66 × 2 + 33
66 = 33 × 2 + 0
Therefore, HCF = 33. Hence, numbers are not co-prime.
(ii) 2160 = 847 × 2 + 466
847 = 466 × 1 + 381
466 = 381 × 1 + 85
381 = 85 × 4 + 41
85 = 41 × 2 + 3
41 = 3 × 13 + 2
3 = 2 × 1 + 1
2 = 1 × 2 + 0
Therefore, the HCF = 1. Hence, the numbers are co-prime.

Sample Question 2: Show that the square of an odd positive integer is of the form
8m + 1, for some whole number m.

Solution: Any positive odd integer is of the form 2q + 1,
where q is a whole number.
Therefore, (2q + 1)2 = 4q2 + 4q + 1
                                 = 4q (q + 1) + 1,----- (1)
q (q + 1) is either 0 or even.
So, it is 2m, where m is a whole number.
Therefore, (2q + 1)2 = 4.2 m + 1 = 8 m + 1. [From (1)]

Sample Question 3: Prove that √2 + √3 is irrational.


Solution : Let us suppose that √2 + √3 is rational. Let 2 + √3 = a , where a is
rational.

Therefore, √2 = a− √3
Squaring on both sides, we get
2 = a²  + 3 – 2a√3
2a√3 = a²  + 3 -2
2a√3 = a²  + 1
Therefore,
√3 = (a² + 1)/2a

which is a contradiction as the right hand side is a rational number while 3 is irrational. Hence,√2 + √3 is irrational.


Sample Question 4: 
Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q.

Solution : 

We know that any positive integer can be of the form 6m, 6m + 1, 6m + 2, 6m + 3, 6m + 4 or 6m + 5, for some integer m.

Thus, an odd positive integer can be of the form 6m + 1, 6m + 3, or 6m + 5
Thus we have:

(6 m +1)2 = 36 m2 + 12 m + 1 = 6 (6 m2 + 2 m) + 1 = 6 q + 1, q is an integer

(6 m + 3)2 = 36 m2 + 36 m + 9 = 6 (6 m2 + 6 m + 1) + 3 = 6 q + 3, q is an integer

(6 m + 5)2 = 36 m2 + 60 m + 25 = 6 (6 m2 + 10 m + 4) + 1 = 6 q + 1, q is an integer.

Thus, the square of an odd positive integer can be of the form 6q + 1 or 6q + 3.

                                             Now Solve these Questions

1. Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.

2. Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.

3. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.

4. Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.

5. Show that the square of any odd integer is of the form 4q + 1, for some integer q.

6. If n is an odd integer, then show that n2 – 1 is divisible by 8.

7. Prove that if x and y are both odd positive integers, then x2 + y2 is even but not divisible by 4.

8. Use Euclid’s division algorithm to find the HCF of 441, 567, 693.

9. Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.

10. Prove that √ 3+ √ 5 is irrational.

11. Show that 12n cannot end with the digit 0 or 5 for any natural number n.

12. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

13. Show that the cube of a positive integer of the form 6q + r, q is an integer and

r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.

14. Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is

any positive integer.

15. Prove that one of any three consecutive positive integers must be divisible by 3.

16. For any positive integer n, prove that n3 – n is divisible by 6.

17. Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible

by 5, where n is any positive integer.

[Hint: Any positive integer can be written in the form 5q, 5q+1, 5q+2, 5q+3,5q+4].

Real number Test paper

Real number Test paper  

1. Write whether every positive integer can be of the form 4q + 2, where q is an

integer. Justify your answer.

2. “The product of two consecutive positive integers is divisible by 2”. Is this statement

true or false? Give reasons.

3. “The product of three consecutive positive integers is divisible by 6”. Is this statement

true or false”? Justify your answer.

4. Write whether the square of any positive integer can be of the form 3m + 2, where

m is a natural number. Justify your answer.

5. A positive integer is of the form 3q + 1, q being a natural number. Can you write its

square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify

your answer.

6. The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is

HCF (525, 3000)? Justify your answer.

7. Explain why 3 × 5 × 7 + 7 is a composite number.

8. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.

9. Without actually performing the long division, find if 987/10500

will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.

10. A rational number in its decimal expansion is 327.7081. What can you say about

the prime factors of q, when this number is expressed in the form p/q ? Give reasons.

WHAT IS IRRATIONAL NUMBERS?

Math Adda

The invention of irrational numbers

We have seen  that when two squares are in the same ratio as two square numbers, then their sides will have the same ratio as the square roots.  Thus if CD were 1 meter, then we would like to say that AB is "Square root of 2" meters -- but there is no such rational number. For if there were, then again, AB, CD would have a common measure, which they do not.

If we insist, however, that there be a number to indicate the ratio of AB to CD, then we keep the name "Square root of 2," and we call it an irrational number.  Its numeral is .  That is,

 ×  = 2.

 is that number which when multiplied by itself -- when squared -- is 2.  "Square root of 2" is its name.  It is not any whole number, any fraction, any mixed number or any decimal -- it is not any rational number.  It is a pure creation.

Now,

7 5

is close to , because

7 5

×

7 5

=

49 25

-- which is almost 2.  But to prove that there is no rational number whose

square is 2, suppose there were.  Then we could express it as a fraction

m n

in lowest terms.  That is, suppose

m n

×

m n

=

m × m n × n

=  2.

But that is impossible.  Since

m n

is in lowest terms, then m and n

have no common divisors except 1.  Therefore, m × m and n × n also have no common divisors.  It will be impossible to divide n × n intom × m and get 2

There is no rational number whose square is 2.

Naturally, we wonder, "How much is ?"  We can only answer

approximately.  As a fraction, we saw it is almost

7 5

.   As a decimal, it is

approximately 1.414.  How would anyone know that?  Again, because 1.414 squared is almost 2.

1.414 × 1.414 = 1.999396.

We could come closer to  by approximating it to more decimal places.  But no decimal squared will ever be exactly 2.   is irrational.

It was argued for many centuries whether  "really" is a number. Mathematicians thought of  as a convenient symbol, but as for its being a number, that was something else.  Because again, if we ask, How much is it?. . . We cannot say.

In the following Topic, we will investigate in what sense irrational numbers "exist."  And we will return to our original inquiry:

If AB, CD are lengths, will there always be a number n -- rational or irrational -- such that, proportionally,

AB is to CD  as  1 is to n?

In any event,  is not the only irrational number.  Let us illustrate that by asking:

The square roots of which natural numbers are rational?

Answer.   Only the square roots of square numbers.  Thus,

 = 1  is rational.

,  are irrational.

 = 2  is rational.

,  ,  ,   are irrational.

 = 3  is rational.

And so on.

Problem 1.   Say the name of each number.

a)	Square root of 3.		b)	Square root of 5.
c)	Two. It's rational.		d)	Square root of 3/5.
e)	Three-fifths.		f)	One.

Problem 2.   Which of the following are rational numbers and which are irrational?

a)	Irrational.		b)	Rational.
c)	Rational.		d)	Irrational.

A common measure with 1

We have seen that every number has a ratio to 1.  A rational number is to 1 in the same ratio as two natural numbers.  That is, every rational number has a common measure with 1 (Topic 8).  We can say, then, that an irrational number is a number that has no common measure with 1.

1 and  are incommensurable.

Problem 3.   What number is a common measure of each pair?

a)   1 and 5   1

b)   1 and

5 8

1 8

c)   1 and 2.617    .001

d)   1 and     None.

Problem 4.   Why have irrational numbers been invented?

To express the ratios of incommensurable magnitudes; in particular, the ratio of a magnitude incommensurable with 1.

Problem 5.   The squares on the sides of triangle ABC are in the ratio 1 : 4 : 3.  Express the ratio of each pair of sides as a ratio of numbers, whether rational or irrational.

a)  AB : BC = 1 : 2         b)  BC : CA = 2 :          c)  CA : AB =  : 1

d)   is approximately 1

5 7

, or

12  7

.  How could you know that?

12  7

·

12  7

=

144  49

-- which is almost 3.

e)   Use that approximation to approximate the ratio of BC to CA as a
e)   ratio of natural numbers.

BC : CA  =  2 :

12  7

=  14 : 12 = 7 : 6

Problem 6.   One square is four fifths of another.

a)  Are the sides commensurable?   No. 4 and 5 are not both square
a) numbers.

b)  Express the ratio of the sides as a ratio of numbers.

2 : 

c)  Are the squares commensurable?

Yes. They are in the same ratio as natural numbers.

WHAT IS THE NATURAL NUMBERS?

Math Adda                           WHAT IS THE NATURAL NUMBERS?

MATHEMATICS IS NOT ONLY about numbers, it is about things that are not numbers; for example, length;  the distance from here to there,

A--------------------------B

C--------I------------------I-------------------D

from A to B.  Now a length is not a number,  yet we describe lengths by saying that they are like numbers.  For if CD is made up of three segments equal to AB, then we say,

AB is to CD  as  1 is to 3.

And if AB happens to be 1 centimeter, we would say that CD is 3 centimeters.

That is, we can say that CD is "three times" AB, and that is called theratio -- the relationship -- of CD to AB.  The eventual question will be:
                  1
A--------------------------B
                                 n
C---------------------------------------------D

If AB, CD are any lengths, will we always be able to express their ratio in words.  Will there always be a number n such that, proportionally,

AB is to CD  as  1 is to n?

That number will be called a real number, which is a number required to name the length of a line, relative to the line we have called 1.  We will see that there will be problems.  At the root of the problem is the difference between arithmetic and geometry.

The natural numbers

Arithmetic begins with counting, and a unit is whatever we would call one.  One apple, one orange, one person.  We count units -- which must have the same name. One apple, two apples, three apples. Therefore we must recognize whether things are the same or different. ("Here's one; here's one; these two are the same.  This is not one. Those two are different.")

A natural number is a collection of these indivisible ones.  5 people,
10 chairs, 32 names.  You cannot take half of any one.

The natural numbers have their everyday names and symbols:  1, 2,
3, 4, and so on.  Those symbols, however, are called numerals, and they represent the numbers.  They stand in for them.  Throughout history there have been many ways of representing numbers.  The student is surely familiar with the Roman numerals:  I, V, X, and so on.

The natural number is the actual collection of units, /////, whether strokes, apples, letters, or the idea of units.  For there is no '5' apart from five units, even though we do not say the word units.  The natural numbers are truly natural.  We find them in nature.

Nevertheless, it is conventional to refer to the numerals -- 1, 2, 3, 4, and so on -- as "numbers," which they are not.

Numerals are physical things. We write them on paper, and they obey rules of computation according to their physical appearance on the page. Mathematics at every level is concerned with physical form.

We are about to see that we can always name the ratio, the relationship, of any two natural numbers.

Cardinal and ordinal

The natural numbers have two forms, cardinal and ordinal.  The cardinal forms are

One, two, three, four,

and so on.  They answer the question How much? or How many?

The ordinal forms are

First, second, third, fourth,

and so on.  They answer the question Which one?

We will see that the ordinal numbers express division into equal parts. They will answer the question, Which part?

Parts of natural numbers

We say that a smaller number is a part of a larger number  if the smaller is contained in the larger an exact number of times. (That is called analiquot part.)  Equivalently, the larger number is a multiple of the smaller.

Consider these first few multiples of 5:

5, 10, 15, 20, 25, 30.

5 is the first multiple of 5.  10 is the second;  15, the third; and so on.

5 is a part of each multiple except itself.  It is a part of 10, of 15, of 20, and so on.

Now, since 15 is the third multiple of 5, we say that 5 is the third partof 15.  We use that same ordinal number to name the part.

The ordinal number names which part of fifteen 5 is.

5 is the fourth part of 20; it is the fifth part of 25; the sixth part of 30.  And so on.

5 is which part of 10?  We do not say the second part. We say half.  5 is half of 10.

It is extremely important to understand that we are not speaking here of proper fractions -- numbers that are less than 1 and that we need for measuring.  We are explaining how the ordinal numbers -- third, fourth, fifth, and so on -- name the parts of the cardinal numbers.  The names of the parts in fact are prior to the names of the fractions, as we will see. (Why do we call the number we write as 1 over 3 "one-third"? Because 1 is the third part of 3)

Note that 5 is not a part of itself.  There is no such thing as the first part.

So, with the exception of the name half, we name each part with ordinal number.  The ordinal number names which part.

What is a real number?

Math Adda                                      

Real numbers

What is a real number?

A real number is distinguished from an imaginary number.
It is a number we require for measuring rather than counting. Hence a real number is any rational or irrational number that we might name. They are the numbers we expect to find on the number line.

(An actual measurement can result only in a rational number.
 An irrational number can result only from a theoretical  calculation; examples are the Pythagorean theorem, and solving
 an equation such as x³ = 5.
 Any serious theory of measurement must address the question:  Which irrational numbers are theoretically possible? Which ones  could be actually predictive of a measurement?)

Problem 1.   We have categorized numbers as real, rational, irrational, andinteger.  Name all the categories to which each of the following belongs.

3  Real, rational, integer.		−3   Real, rational, integer.
−½   Real, rational.		Real, irrational.
5¾   Real, rational.		− 11/2   Real, rational.
1.732   Real, rational.		6.920920920. . .   Real, rational.
6.9205729744. . .   Real. And let us assume that it is irrational, that is, that the digits do not repeat. Moreover, we must assume that there is an effective procedure for computing each next digit. For if there were not, then we would not know which number we are computing. And that symbol would not refer to any "number"!
6.9205729744    Real, rational. Every exact decimal is rational.

7.  What is a real variable?

A variable is a symbol that takes on values. A value is a number.
A real variable takes on values that are real numbers.

Calculus is the study of functions of a real variable.

Problem 2.   Let x be a real variable, and let 3 < x < 4.  Name five values that x might have.