X Maths:Real Number : Edugain series practice paper

1) Show that 3√ 2 is irrational.

2) Prove that 3 + 2 √5 is irrational.

3) A sweet seller 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?

4) Use Euclid’s division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

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

6) 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? Sol. Hints: Find the HCF of 616 and 32

7) 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. [Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]


8) Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

9) Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero.

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

11) Find the HCF of 96 and 404 by the prime factorization method. Hence, find their LCM.

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

13) Find the value of y if the HCF of 210 and 55 is expressible in the form 210 x 5 + 55y

14) Prove that no number of the type 4K + 2 can be a perfect square.

15) Express each number as a product of its prime factors:

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

You may also see
CBSE :10th Real Numbers Extra score Test paper

X Maths:Real Number Practice paper

1. Use Euclid’s division algorithm to find the HCF of 867 and 255

2. Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

3. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9lm + 1 or 9m + 8.

4. Prove that 7 √5 is irrational.

5. Prove that √5 is irrational.

6. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi 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?

7. Express 5005 as a product of its prime factors.

8. Find the LCM and HCF of 24, 36 and 72 by the prime factorization method.

9. Find the LCM and HCF of 96 and 404 by the prime factorization method

10. State whether 64/455 will have a terminating decimal expansion or a non-terminating repeating decimal

11. State whether15/ 1600 will have a terminating decimal expansion or a non-terminating repeating decimal.

12. Find the LCM and HCF of 510 and 92 and verify that LCM × HCF = product of the two numbers.

13. Use Euclid’s division algorithm to find the HCF of 196 and 38220 (3 marks)

14. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m,9m + 1 or 9m + 8

15. Show that every positive odd integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer

10th Real Number FA test paper

Chapter: 01- Real numbers           Test paper – 01                     FM: 30               Time: 45 min

1 mark questions

1. 3.24636363... is:

(a) a terminating decimal number                          (b) a non-terminating repeating decimal  number

(c) a rational number                                           (d) both  (b) and (c)

2. For some integer q, every odd integer is of the form :

(a) 2q                (b) 2q + 1                      (c) q                                         (d) q + 1

3. If the HCF of 85 and 153 is expressible in the form 85m – 153, then the value of m is :

(a) 1                  (b) 4                             (c) 3                                         (d) 2

4. If two integers a and b are written as a = x³y² and b = xy⁴; x, y are prime numbers, then H.C.F. (a, b) is :

(a) x³y³              (b) x²y²                          (c) xy                                        (d) xy²

5.  If least prime factor of a is 3 and least prime factor of b is 7, the least prime factor of (a + b) is:

 (a) 2                 (b) 3                            (c) 5                                         (d) 11

2 marks questions

6. Show that every positive even integer is of the from 2m, and that every positive odd integer is of the form 2m + 1, where m is some integer.

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

8. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

9. Show that any positive integer is of the form 3q or 3q + 1 or 3q + 2 for some integer q.

10. Show that 5-√3 is irrational.

3 marks questions

11. Check whether 6ⁿ can end with the digit 0, for any natural number n.

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

13. Prove that n²−n is divisible by 2 for every positive integer n.

14. Use Euclid division lemma to show that cube of any positive integer is either of the form 9m, 9m + 1, or 9m + 8

                                                                                  OR,

 If d is the HCF of 45 and 27, find x & y satisfying d=27x +45y.                               (Ans d=9, x=2, y= -1 )

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

                                                                                   OR,   

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

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