CBSE Class 10 Mathematics Real Numbers Worksheet Set D

Q.- Insert a rational and an irrational number between 2 and 3.

More question-

1. If 7 x 5 x 3 x 2 + 3 is composite number? Justify your answer

2. Show that any positive odd integer is of the form 4q + 1 or 4q + 3 where q is a positive integer

3. Show that 8n cannot end with the digit zero for any natural number n

4. Prove that 3√2 is irrational 5

5. Prove that √2 + √5 is irrational

6. Prove that 5 – 2 √3 is an irrational number

7. Prove that √2 is irrational

8. Use Euclid’s Division Algorithms to find the H.C.F of a) 135 and 225 (45)

b) 4052 and 12576 (4)

c) 270, 405 and 315 (45)

9. Using Euclid’s division algorithm, check whether the pair of numbers 50 and 20 are co-prime or not.

10. Find the HCF and LCM of 26 and 91 and verify that LCM X HCF = Product of two numbers (13,182)

11. Explain why 29 is a terminating decimal expansion 23 x 53

12. 163 will have a terminating decimal expansion. State true or false .Justify your answer. 150

13. Find HCF of 96 and 404 by prime factorization method. Hence, find their LCM. (4, 9696)

14. Using prime factorization method find the HCF and LCM of 72, 126 and 168 (6, 504)

15. If HCF (6, a) = 2 and LCM (6, a) = 60 then find a (20)

16. given that LCM (77, 99) = 693, find the HCF (77, 99) (11)

17. Find the greatest number which exactly divides 280 and 1245 leaving remainder 4 and 3 (138)

18. The LCM of two numbers is 64699, their HCF is 97 and one of the numbers is 2231. Find the other (2813)

19. Two numbers are in the ratio 15: 11. If their HCF is 13 and LCM is 2145 then find the numbers (195,143)

20. Express 0.363636………… in the form a/b (4/11)

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

22. Write whether 2√45 + 3√20 on simplification give a rational or an irrational number 2√5 (6)

23. State whether 10.064 is rational or not. If rational, express in p/q form

24. Write a rational number between √2 and √3

25. State the fundamental theorem of arithmetic

26. The decimal expansion of the rational number 74 will terminate after ………. Places

23 . 54

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

Sol. If a and b are two positive integers such that a is greater than b; then according to Euclid’s division algorithm; we have

a = bq + r; where q and r are positive integers and 0 ≤ r < b.

Let b = 6, then

a = bq + r

=> a = 6q + r; where 0 ≤ r < 6.

When r = 0

=> a = 6q + 0 = 6q;

which is even integer 

Q.- 

 More question-

Q01 :} Find the smallest number which when divided by 30, 40 and 60 leaves the remainder 7 in each case.

Q02 :} The dimensions of a room are 6 m 75 cm, 4 m 50 cm and 2 m 25 cm. Find the length of the largest measuring rod which can measure the dimensions in exact number of times.

Q03 :} The HCF of 2 numbers is 75 and their LCM is 1500. If one of the numbers is 300, find the other.

Q04 :} Prove that √6+√5  is irrational.

Q05 :} Can 72 and 20 respectively be the LCM and HCF of two numbers. Write down the reason.

Q06 :} If a and b are two prime numbers, write their HCF and LCM.

Q07 :} If p and q are two coprime numbers, write their HCF and LCM.

Q08 :} Without actual division, state whether the decimal form of 539/5 ³x2 ²x7 ²  is terminating OR recurring.

Q09 :} Find the HCF and LCM of 350 and 400 and verify that HCFxLCM=Product of the numbers.

Q10 :} Simplify: 2√45 + 3√20 + 10√125/2√5

Q11 :} Write down 5 irrational numbers in radical form which are lying between 4 and 5.

Q12 :} Write down 2 rational numbers lying between and .

Q13 :} Complete the missing entries in the following factor tree.

Q.- Find the prime factors of :

(i) 540   (ii) 21252    (iii) 8232

∴ 8232 = 2 × 2 × 2 × 3 × 7 × 7 × 7

= 2³ × 3 × 7³.

Q.- Find the missing numbers a, b and c in the following factorisation:

More question-

Q1. What is the H.C.F of the smallest composite number and the smallest prime number?

Q2. If 'p' is a prime number then what is the L.C.M of p , p2 , p3 ?

Q3. Two positive integers 'p' and 'q' can be expressed as p=ab2 and q=a2b , a and b are prime numbers . What is the L.C.M of ' p' and 'q' ?

Q4. Show that n2 -1 is divisible by 8 , if 'n' is an odd positive integer ?

Q5. Prove that n2 - n is divisible by 2 for every positive integer ' n' ?

Q6. Show that one and only one out of n , n+2 or n+4 is divisible by 3 , where n is any positive integer ?

Q7. Prove that one of every three consecutive positive integers is divisible by 3 ?

Q8. Find the H.C.F of 65 and 117 and express it in the form 65m+117n ?

Q9. If the H.C.F of 210 and 55 is expressible in the form of 210*5 + 55y , find 'y' ?

Q10. Find the largest positive integer that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively .

Q11. Find the greatest number of six digits exactly divisible by 24 , 15 and 36 ?

Q12. Three sets of English , Hindi and Mathematics books have to be stacked in such a way that all the books are stored topic wise and the height of each stack is the same . The number of English books is 96 , the number of Hindi books is 240 and the number of Mathematics books is 336 . Assuming that the books are of same thickness , determine the number of stacks of English , Hindi and Mathematics books ?

Q13. Two brand of chocolates are available in packs 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?

Q14. Prove that √2 + √5 is irrational .

Q15. Using Euclid's Division Algorithm , find whether the pair of numbers 847 and 2160 are co-prime or not .

ANSWERS

1) 2 2) P3 3) a2b2 10) 17 11) 999720 12) 2,5,7 13) 5,8

Very Short Answer type Questions

Question. What is the greatest possible speed at which a man can walk 52 km and 91 km in an exact number of minutes?
Answer : 13m / min

Question. The values of x and y in the given figure are:
Answer : x = 21 and y = 84

Question. Find the prime factorization of 1152
Answer : 1152= 2⁷ x 3²

Question. Find the sum of exponents of prime factors in the prime factorization of 216?
Answer : 15

Question. Show that the product of two numbers 60 and 84 is equal to the product of their HCF and LCM
Answer : LCM × HCF =420×12=5040 Also, 60×84=5040

Question. P and Q are two positive integers such that P =p³ q and Q = (pq)² , where p and q are prime numbers. What is LCM (P, Q)?
Answer : p³ × q²

Question. If p and q are two coprime numbers, then find the HCF and LCM of p and q.
Answer : HCF = 1 and LCM = pq

Question. If a=2³×3, b=2×3×5, c=3n×5 and LCM [a,b,c] = 2³×3²×5 then, n=?
Answer : 2

Q 1. The department of electricity sells the electric energy to the consumer inunits.

1 unit = 1 kilowatt.hour

Means electric energy consumed by 1000 watt electric appliance when it operates for one hour.

Household uses of electric appliances are given below of a family.

Note : 1000 Watt = 1 kilowatt

Now calculate the electricity bill of the households for the month of June. If 1 unit = Rs. 4.80.

Q 2. On a morning walk, three person step off together and their steps measure 40cm, 42 cm and 45 cm respectively. What is the minimum distance each shouldwalk so that each can cover the same distance in complete step?

Q 3. Two alarm clocks ring their alarms at regular intervals of 50 seconds and 48seconds respectively. If they first beep together at 12 noon at what time willthey beep together again.

1. Show that x2 – 3 is a factor of 2x4 + 3x3 - 2x2 - 9x – 12

2. Divide (6 + 19x + x2 – 6x3) by (2 + 5x – 3x2) and verify the division algorithm

3. Find other zeroes of the polynomial p(x) = 2x4 + 7x3 – 19x2 – 14x + 30 if two of its zeroes are √2 and - √2 (3/2, -5)

4. Find all the zeroes of 2x4 - 9x3 + 5x2 +3x – 1, if two of its zeroes are 2 + √3 and 2 - √3 (1, -1/2)

5. Find all the zeroes of polynomial 4x4 – 20x3 + 23x2 + 5x – 6 if two of its zeroes are 2 and 3 (1/2, -1/2)

6. When a polynomial f(x) is divided by x2 – 5 the quotient is x2 – 2x – 3 and remainder is zero. Find the polynomial and all its zeroes (3, -1, √5, -√5)

7. If the polynomial f(x) = x4 - 6x3 + 16x2 - 25x + 10, is divided by another polynomial x2 - 2x + k the remainder Comes out to be x + a,

Find k and a (k = 5, a = -5)

8. On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and -2x + 4, respectively. Find g(x) (x2 – x + 1)

9. If the polynomial 6x4 + 8x3 – 5x2 + ax + b is exactly divisible by the polynomial 2x2 – 5, then find the values of a and b (-20, -25)

10. What must be subtracted from 2x4 – 11x3 + 29 x2 – 40x + 29, so that the resulting polynomial is exactly divisible By x2- 3x + 4 (-2x + 5)

11. Find the polynomial, whose zeroes are 2 + √3 and 2 - √3 (x2 – 4x +) 12.Form a quadratic polynomial, one of whose zero is 2 + √5 and the sum of zeroes is 4 (x2 – 4x – 1)

13. Find a quadratic polynomial whose sum and product of the zeroes are 21/8 and 5/16 (16x2 - 42x +5)

14. Write a quadratic polynomial, the sum and product of whose zeroes are 3 and -2 (x2 – 3x – 2)

15. Find the zeroes of the polynomial and verify the relationship between the zeroes and the coefficient

a) 4x2 - 7 b) √3x2 – 8x + 4√3

16. If one root of the polynomial 5x3 + 13x + k is reciprocal of the other, then find the value of k? (k = 5)

17. If one zero of the polynomial (a2 + 9) x2 + 13x + 6a is reciprocal of the other. Find the value of a (3)

18. If α and β are the zeroes of the polynomial f(x) = 6x2 + x -2, find the value of 1 + 1 - α β (5/6) α β

19. If α and β are the zeroes of the polynomial f(x) = x2 – 8x + k such that α2 + β2 = 40, find k (12)

20. If α, β are the zeroes of a polynomial, such that α + β = 6 and α β = 4, then writes the polynomial

21. If the product of zeroes of the polynomial ax2 – 6x – 6 is 4, find the value of a (-3/2)

22.If α, β are the zeroes of quadratic polynomial 2x2 + 5x + k, find the value of k such that (α + β)2 – α β = 24 (- 71/2)

23. If α and β are zeroes of x2 + 5x + 5, find the value of α-1 + β-1 (-1)

24. α, β are the zeroes of the quadratic polynomial x2 – (k+6)x +2 (2k – 1). Find the value of k if α + β = ½ α β (7)

25. if α, β are the zeroes of the quadratic polynomial x2 – 7x + 10, find the value of α3 + β3 (133)

26. m, n are zeroes of ax2 – 12x + c. Find the value of a and c if m + n = m n = 3 (12)

27. Find the sum and the product of the zeroes of cubic polynomial 2x3 - 5x2 – 14x + 8 (5/2, -7, -4)

28. Find the sum and product of the zeroes of quadratic polynomial x2 – 3

29. If 1 is a zero of polynomial ax2 – 3(a-1) - 1, then find the value of a (1)

1 Mark

1.Determine .875 is terminating or non-terminating.

2.H.C.F of 3638 and 3587 is
(A) 13
(B) 17
(C) 19
(D) 23

3.Why is7x11x13+7 a composite integer.

4.Suppose you have 108 green marbles and 144 red marbles. You decide to separate them into packages of equal number of marbles. Find the maximum possible number of marbles in each package.
(A) 4
(B) 36
(C) 9
(D) 12

5.H.C.F of 3638 and 3587 is
(A) 13
(B) 17
(C) 19
(D) 23

6.Determine the prime factorization of the number 556920.

7.Find the HCF of 96 and 404 by prime factorization method. Hence, find the LCM
(A) 1000
(B) 9600
(C) 9640
(D) 9696

8.Determine the prime factorization of the number 556920.

9.H.C.F of two integers 26, 91 is 13 what will be its L.C.M.?

10.H.C.F of 3638 and 3587 is
(A) 13
(B) 17
(C) 19
(D) 23

11.Express 140 in its prime factor.

12.Without actual division, state whether the is terminating or non terminating rational numbers.

13.Why is7x11x13+7 a composite integer.

14.Explain why 7 x 11 x 13 + 13 and 7 x 6 x 5 x 4 x 3 x 1 + 5 are composite numbers
(A) Product of prime factor
(B) Composite None
(C) Both of these
(D) None of these

15.Find out HCF of 867 and 255 by using Euclid Division Algorithm
(A) 51
(B) 45
(C) 50
(D) 55

16.The length, breadth and height of a room are 8 m 25cm, 6m 75cm and 7m 50cm respectively. Determine the longest tape, which can measure the three dimensions of the room exactly.
(A) 75 cm
(B) 150 cm
(C) 90 cm
(D) 180 cm

17.If the HCF of 210 and 55 is expressible in the form 210 x 5 + 55y, find y
(A) 19
(B) 15
(C) -19
(D) -21

18.Why is7x11x13+7 a composite integer.

19.Determine the prime factorization of the number 556920.

20.Find the HCF of 96 and 404 by prime factorization method. Hence, find the LCM
(A) 1000
(B) 9600
(C) 9640
(D) 9696

21.If the HCF of 210 and 55 is expressible in the form 210 x 5 + 55y, find y
(A) 19
(B) 15
(C) -19
(D) -21

22.H.C.F of two integers 26, 91 is 13 what will be its L.C.M.?

23.Determine .875 is terminating or non-terminating.

24.Express 140 in its prime factor.

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

26.Why is7x11x13+7 a composite integer.

27.Find the greatest possible rate at which a man should walk to cover a distance of 70 km and 245 km in exact number of days?
(A) 55
(B) 60
(C) 35
(D) 45

28.Why is7x11x13+7 a composite integer.

29.Why is7x11x13+7 a composite integer.

30.Why is7x11x13+7 a composite integer.

31.Why is7x11x13+7 a composite integer.

32.Why is7x11x13+7 a composite integer.

33.Why is7x11x13+7 a composite integer.

34.Without actual division, state whether the 13/2⁰x5⁷ is terminating or non terminating rational numbers.

35.is a
(A) Terminating decimal
(B) Non-terminating decimal
(C) Cannot be determined
(D) None of these

36.Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of container which can measure the petrol of either tanker in exact number of times.
(A) 135
(B) 160
(C) 170
(D) 210

37.Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.
(A) 48
(B) 64
(C) 20
(D) 16

38.Find out HCF of 38,220 and 196 by using Euclid Division Algorithm
(A) 192
(B) 190
(C) 196
(D) 198

39.Which of the following is non terminating repeating decimals?
(A) 13/3125
(B) 17/8
(C) 64/455
(D) 129/2²5⁷7⁵

40.Find the greatest number of 6 digits exactly divisible by 24, 15 and 36
(A) 999999
(B) 999789
(C) 999000
(D) 999720

41.Find the HCF and LCM of 90 and 144 by the prime factorization method
(A) 15, 20
(B) 15, 720
(C) 18, 720
(D) None of these

42.2525 is
(A) a composite number
(B) a natural number
(C) an irrational number
(D) both (1) and (2)

43.If the sum of two numbers is 75 and the H.C.F. and L.C.M. of these numbers are 5 and 240 respectively, then the sum of the reciprocals of the numbers is equal to:
(A) 1/8
(B) 1/16
(C) 1/4
(D) 1/20

44.Three bells chime at an interval of 18, 24 and 32 minutes respectively. At a certain time they begin to chime together.What length of time will elapse before they chime together again?
(A) 2 hours 24 minutes
(B) 4 hours 48 minutes
(C) 1 hour 36 minutes
(D) 5 hours

45.Find the HCF of 65 and 117 and express it in the form 65m + 117n
(A) m = -2 , n = -1
(B) m = 2 , n = -1
(C) m = 3 , n = -1
(D) m = 2 , n = 1

46.Given H.C.F (306, 657) = 9, find L.C.M. (306, 637)
(A) 22222
(B) 22328
(C) 22302
(D) 22338

47.There is a circular path around a sports field. Priya takes 18 minutes to drive one round of the field, while Ravish takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point
(A) 30
(B) 36
(C) 40
(D) 26

48.Three men start together to travel the same way around a circular track of 11 kms. Their speeds are 4, 5(1/2), and 8 kms per hour respectively.When will they meet at the starting point?

49.Find the H.C.F and L.C.M. of 25152 and 12156 by using the fundamental theorem of Arithmetic
(A) 24457576
(B) 25478976
(C) 25478679
(D) 24456567

50.Find the largest number that will divide 2053 and 967 and leaves a remainder of 5 and 7 respectively.
(A) 128
(B) 54
(C) 256
(D) 64

51.The L.C.M. of two numbers is 45 times their H.C.F. If one of the numbers is 125 and the sum of H.C.F. and L.C.M. is 1150, the other number is:

52.A man was engaged for a certain number of days for Rs. 404.30 but because of being absent for some days he was paid only Rs. 279.90. His daily wages cannot exceed by:
(A) Rs. 29.10 p
(B) Rs. 31.30 p
(C) Rs. 31.10 p
(D) Rs. 31.41 p

53.The areas of three fields are 165m2 , 195m2 and 285m2respectively. From these flowers beds of equal size are to be made. If the breadth of each bed be 3 metres, what will be the maximum length of each bed.

54. Use Euclid's division algorithm to find the HCF of 210 and 55.

55.The length, breadth and height of a room are 8m25cm, 6m75cm and 4m50cmrespectively.
Determine the longest rod which can measure the three dimensions of the room exactly
(A) 65cm
(B) 77cm
(C) 75cm
(D) 80cm

56. In a seminar, the number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively. Find the maximum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.
(A) 17
(B) 21
(C) 27
(D) 19

57.In a school there are two sections - section A and section B of classX. 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
(A) 300
(B) 296
(C) 288
(D) 278

58.Find the HCF of 96 and 404 by prime factorization method. Hence, find there LCM
(A) 9595
(B) 9696
(C) 9292
(D) 9393

3 Marks

59.Express 32760 as the product of its prime factors.

60.Use Euclid's algorithm to find the HCF of 4052 and 12576.

61.Show that 3√2 is irrational.

62.is irrational.

63.Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

64.Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

65.Find the LCM and HCF of 6 and 20 by the prime factorisation method.

66.A sweetseller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray.What is the maximum number of barfis that can be placed in each stack for this purpose?

67.Show that 5 - √3 is irrational.

SECTION A:

1. If 241/4000  = 241/2^m5ⁿ, find the values of m and n where m and n are non-negative integers. Hence write its decimal expansion without actual division. 
m=5, n=3
0.06025

2. Express the number 0.3178 in the form of a rational number a/b.
635/1998

3. Can two numbers have 14 as their HCF and 325 as their LCM? Give reason.
No

SECTION B:

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

5. Find the least number that is divisible by all the numbers from 1 to 10.
2520

6. If the HCF of 65 and 117 is expressible in the form 65m – 117. Find the value of m.
2

7. What is the greatest possible speed at which a man can walk 52 km and 91 km in an exact number of minutes?
13m/min

8. Find the smallest natural number by which 1200 should be multiplied so that the square root of the product is a rational number. (CBSE 2015)
3

SECTION C:

8. On Darsait signal, three consecutive traffic lights change after 36, 42 and 72 seconds.
If the lights are first switched on at 9.00 am, at what time will they change simultaneously?
9:08:24

9. Using Euclid’s division algorithm, find whether the pair of numbers are co primes or not. 
Coprimes

10. Prove that p + q is irrational, where p and q are primes.

11. If n is any prime number and a2 is divisible by n, then n will also divide a. Justify.

SECTION D:

12. For any positive integer n, prove that n3 - n is divisible by 6.

13. Is the square of every non-square number always irrational? Find the smallest natural number which divides 2205 to make its square root a rational number.
Yes, 5

14. The floor of Manu’s drawing room is 306 inches long and 136 inches wide. He wishes to tile the floor with identical square tiles. Find the minimum number of tiles that he can use.
36

15. What is the sum of the digits of the smallest number, which leaves remainder 2 upon being divided by 10, 15 and 25?
8

1. Show that only one out of n, n + 4, n + 8, n + 12, n + 16 is divisible by 5, when n is a positive integer.

2. Use Euclid’s division algorithm to find the HCF of: (a) 135 & 225 (b) 196 & 38220 (c) 867 & 255.

3. Find the HCF of the following pairs of integers by the prime factorization method.
(a) 963 & 657 (b) 506 & 1155 (c) 1288 & 575

4. Find the greatest number which divides 285 and 1245 leaving remainders 9 & 7 respectively.

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

6. Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively.

7. A rectangular courtyard is 18m 72cm long and 13m 20cm broad. It is to be paved with square tiles of the same size. Find the least possible number of such square tiles required.

8. HCF of two numbers is 145 and their LCM is 2175. If one number is 725, find the other.

9. Prove that 3 + 2√5 is irrational.

10. Prove that √5 + √3 is irrational.

11. What can you say about prime factors of denominators of following real numbers?
(i):34.12345 (ii): 25. 567 (iii): 2.5055055505555….