CBSE Class 10 Maths HOTs Number Systems

Numbers are intellectual witnesses that belong only to mankind.

1. If the HCF of 657 and 963 is expressible in the form of 657x + 963x - 15 find x.

   (Ans:x=22)

Ans: Using Euclid’s Division Lemma

a= bq+r , o  ≤ r < b

963=657×1+306

657=306×2+45

306=45×6+36

45=36×1+9

36=9×4+0

∴ HCF (657, 963) = 9

now 9 = 657x + 963× (-15)

657x=9+963×15

=9+14445

657x=14454

x=14454/657

 

 

 

6= 18×3 +48×(-1)

=18×3 +48×(-1) + 18×48-18×48

=18(3+48)+48(-1-18)

=18×51+48×(-19)

6=18x+48y

 

 

 

 

Hence, x and y are not unique.

 

3.  Prove that one of every three consecutive integers is divisible by 3. 

Ans:

n,n+1,n+2 be three consecutive positive integers

We know that n is of the form 3q, 3q +1, 3q + 2

So we have the following cases

 

Case – I when n = 3q

 

In the this case, n is divisible by 3 but n + 1 and n + 2 are not divisible by 3

 

Case -  II   When n = 3q + 1

Sub n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not divisible by 3

 

Case – III When n = 3q +2

Sub n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not divisible by 3

 

Hence one of n, n + 1 and n + 2 is divisible by 3

 

4.  Find the largest possible positive integer that will divide 398, 436, and 542 leaving remainder 7, 11, 15 respectively.

(Ans: 17) 

Ans: The required number is the HCF of the numbers 

Find the HCF of 391, 425 and 527 by Euclid’s algorithm

∴ HCF (425, 391) = 17

 

Now we have to find the HCF of 17 and 527

527 = 17 x 31 +0  

∴ HCF (17,527) = 17

∴ HCF (391, 425 and 527) = 17

 

5.  Find the least number that is  divisible by all  numbers between 1  and 10  (both inclusive). 

Ans:  The required number is the LCM of 1,2,3,4,5,6,7,8,9,10

∴ LCM = 2  ×  2  ×  3  ×  2  ×  3  ×  5  ×  7 = 2520

 

6.  Show that 571 is a prime number.

Ans: Let x=571==√x=√571 

Now 571 lies between the perfect squares of  (23)2 and (24)2

Prime numbers less than 24 are 2,3,5,7,11,13,17,19,23

Since  571 is not divisible by any of the above numbers

571 is a prime number

 

7.  If d is the HCF of 30, 72, find the value of x & y satisfying d = 30x + 72y.

(Ans:5, -2 (Not unique) 

Ans: Using Euclid’s algorithm, the HCF (30, 72)

72 = 30 ×   2 + 12

30 = 12 ×   2 + 6

12 = 6 ×   2 + 0

 

HCF (30,72) = 6

6=30-12×2

6=30-(72-30×2)2

6=30-2×72+30×4

6=30×5+72×-2

∴ x = 5, y = -2

Also 6 = 30 × 5 + 72 (-2) + 30 ×   72 – 30 ×   72 

Solve it, to get 

x = 77, y = -32

 

Hence, x and y are not unique 

 

8.  Show that the product of 3 consecutive positive integers is divisible by 6. 

Ans: Proceed as in question sum no. 3

 

9.  Show that for odd positive integer to be a perfect square, it should be of the form 8k +1. Let a=2m+1

Ans: Squaring both sides we get a2 = 4m (m +1) + 1

∴ product of two consecutive numbers is always even

m(m+1)=2k

a2=4(2k)+1 a2  = 8 k + 1

Hence proved

 

10. Find the greatest number of 6 digits exactly divisible by 24, 15 and 36. (Ans:999720) 

Ans: LCM of 24, 15, 36

LCM = 3 ×  2 ×  2 ×  2 ×  3 ×  5 = 360

 

Now, the greatest six digit number is 999999

Divide 999999 by 360

∴ Q = 2777 , R = 279

∴ the required number = 999999 – 279 = 999720 

 

11. If a and b are positive integers. Show that √2 always lies between

a/ b  anda − 2b / a + b 

 

Please refer to link below to download pdf file of CBSE Class 10 Number Systems HOTs

HOTS
REAL NUMBERS

2 MARKS QUESTIONS
1. Find the largest positive integer that will divide 122,150 and 115 leaving remainder 5, 7 and 11 respectively.
Solution 1: 122-5 = 117 is exactly divisible by the required number
150-7 = 143,
115-11 = 104
So, required number is the HCF of 117,143,104        1 Mark
117 = 3x3x13
143 = 11x13
104 = 2x2x2x13
\HCF ( 117,143,104 ) = 13
Hence, required number is 13.      1 Mark

2. Use Euclid’s division algorithm to find the largest number which divides 957 and 1280 leaving remainder 5 in each case.
Solution 2: 957-5=952 and 1280-5= 1275,are completely divisible by required number.
Now find the HCF by Euclid division lemma,
1275 > 952 by apply division lemma
1275 = 952 x 1 + 323 ( since R≠ 0 )
952 = 323 x 2 + 306 ( since R≠ 0 )      1 Mark
323 = 306 x 1 + 17 ( since R ≠ 0)
306 = 17 x 18 + 0 here R = 0
Divisor in the last step is 17
\ HCF of 1275 and 952 is 17.

Hence required number is 17.       1 Mark

3 MARKS QUESTIONS

1 In a school there are two sections of class 10th.there are 40 students in 1st section and 48 students in second section. Determine the minimum number of books required for their class library so that they can be distributed equally among students of both sections.
Solution 1: Required number of books = LCM ( 40,48 )
40 = 2x2x2x5        1 Mark
48 = 2x2x2x2x3     1 Mark
LCM (40,48) = 2x2x2x2x3x5 = 240
Hence, required number of books are 240       1 Marks

2. In a morning walk Nirmaljeet, Puneet, Rajiv step off together, their steps measuring 240 cm , 90 cm, 120cm respectively. What is the minimum distance each should walk so that one can cover the distance in complete steps?
Solution 2: 240 = 2x2x2x2x3x5     1 MARK
90 = 2x3x3x5
120 = 2x2x2x3x5 1 Mark
LCM = 2x2x2x2x3x3x5 = 720
Hence required distance 720 cm. 1 Mark

3. The sets of mathematics, physics and physical education books have to be stacked in such a way that all the books are stored topic wise. The number of mathematics , physics and physical education books are 14 , 18 and 22. Determine the number of stacks of each books provided books are of the same thickness.
Solution 3: Firstly , to arrange the books as according to condition,
Find HCF of 14, 18 and 22.
14 = 2x7

18 = 2x3x3
22 = 2x11     1 Mark
HCF = 2
So ,there are only 2 books in each stack.    1 Mark
Number of stack of Mathematics books =2/14= 7
Number of stack of Physics books =2/18= 9
Number of stack of Physical Education books =2/22 = 11. 1 Mark

More Questions

1 mark questions :-

Q. 1 State whether 6 / 300 has terminating or non-terminating repeating (recurring) decimal expansion.

Ans1. Terminating

Q. 2 If LCM (52, 182) = 364, write HCF (52, 182).

Ans2. 26

Q. 3 Write two rational numbers between 1/2 and 2/ 3 .

Ans3. 1 / 2< p/q <2 /3 but  < < but q #0

Q. 4 If a and b are two prime numbers, write their LCM.

Ans4. a × b

Q. 5 State whether 3×7×17×19+17 is a prime number or composite number.

Ans5. Coprime number

Q. 6 If HCF (24, 60) = 12., write LCM (24, 60).

Ans6. 120

Q. 7 State Fundamental Theorem of the Arithmetic.

Q. 8 State whether 123 / 2³ ×3×5² has terminating or non-terminating recurring decimal expansion.

Ans8. Terminating

Q. 9 Write HCF of 11 and 17.

Ans.9. 1

Q. 10 Write two irrational numbers between 1 and 2.

Ans10. Non-terminating recurring decimal

2 marks questions ( Question under HOTS) :-

Q. 11 Using Euclid's division algorithm, find HCF of 75 and 160.

Ans.11. 5

Q. 12 Decimal Expansion of two real numbers is given as (i) 0.20 200 2000 . . . . . . (ii) 3.333 . . . . . State whether they are rational or irrational numbers.

Ans.12. (i) Irrational (ii) rational

Q. 13 An army group of 308 members is to march behind an army band of 24 members in a parade.The two groups are to march in the same number of columns. What is the maximum number of column in which they can march?

Ans.13. 4

Q. 14 Using Euclid's division algorithm, find HCF of 135 and 225.

Ans.14. 45

Q. 15 Find HCF of 105, 120 and 150.

Ans.15. HCF = 15

Q. 16 Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.

Ans.16. 16

Q. 17 Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7, respectively.

Ans.17. 138

Q. 18  Two brands of chocolates are available in pack of 24 and 15 respectively. If I need to buy an equal number of chocolates of both kinds, what is the least number of boxes of each kind I would need to buy?

Ans.18. 5 boxes of first kind and 8 of second kind

Q. 19 Find HCF and LCM of 96 and 240.

Ans.19. HCF = 48, LCM = 480

Q. 20 Write two irrational numbers whose sum is rational.

Ans.20. (2 + √3) and (2 – √3) such other real numbers also

3 marks questions ( question under HOTS) :-

Q. 21 Show that 2 – √3 is an irrational number.

Q. 22 Show that √3 is an irrational number.

Q. 23 Check whether 4n can end with the digit 0 for any natural number n.

Q. 24 Show that 3 √5 is an irrational number.

Q. 25  Show that √2 + √3 is an irrational number.

Q. 26 Show that any positive odd integer is of the form 4q+1 or 4q + 3, where q is some positive integer.

Q. 27 The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm, respectively. Determine the longest rod which can measure the three dimensions of the room exactly.

Ans.27. 75 cm

Q. 28 Show that 3+ √5 is an irrational number.

Q. 29 Find the largest number that will divide 398, 436 and 540 leaving remainders 7, 11 and 13 respectively.

Ans.29. 17

Q. 30 Show that 7 √2–3 is an irrational number.

ADDITIONAL QUESTION

1. Given that H C F (2530, 4400) =110 and L C M (2530,4400)= 253 ×k, find the value of k.

2. If 0.2316 is expressed in the form of p/ 2^m 5ⁿ for the smallest value of the whole number n and m. Write the values of n, m and p.

3. Show that one and only one out of n, n+2 or n + 4 is divisible by 3, where n is any positive integer.

4. Prove that √3 + √5 is an irrational number.

5. Use Euchild’s Division Algorithm to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

6. Write the HCF of the smallest composite number and the smallest prime number.

7. Show that the square of any positive odd integer is of the form 8m+1 for some integer m.

Please refer to link below for CBSE Class 10 Mathematics HOTs Real Numbers Set A

Q1. What is the maximum no. of factors of a prime number?

Q2. Given HCF of (16, 100) = 4. find L.C.M of (16, 100).

Q3. Write a rational no. between √2 and √3 .

Q4. Write if 343/28 is a terminating or non-terminating repeating decimal without doing actual division.

Q5. Tell whether the prime factorization of 15 is 1X 3 X 5 or not.

Q6. If x and y are two irrational numbers then tell whether x + y is always irrational or not.

Q7. What is the L.C.M of x and y if y is a multiple of x?

Q8. Write the sum of exponents of prime factors of 98.

Q9. State if (√2 - √3)( √2 + √3) is rational or irrational.

Q10. Express 0.03 as a rational number in the form of p/q.