REAL NUMBERS CLASS 10 EXTRA QUESTIONS AND ANSWERS

 HIGHER ORDER THINKING SKILLS (HOTS) QUESTIONS

Real Numbers

1.  Find the HCF of 52 and 117 and express it in form 52x + 117y.

2.  Prove that x2 – x is divisible by 2 for all positive integer x.

3.  If m and n are odd positive integers, then m2 + n2 is even, but not divisible by 4. Justify.

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

5.  If remainder of  is a natural number, then find it.

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

7.  Find the missing numbers in prime factors tree.

8.  Find the greatest number of 5 digits exactly divisible by 12, 15 and 36.

9.  Find the smallest number which when increased by 20 is exactly divisible by 90 and 144.

10.  Find the smallest number which leaves remainder 8 and 12 when divided by 28 and 32 respectively.

11.  Floor of a room is to be fitted with square marble tiles of the largest possible size. The size of the room is 10 m × 7 m. What should be the size of tiles required that has to be cut and how many such tiles are required?

12.  If the HCF of 408 and 1032 is expressible in the form 1032p – 408 × 5 find p.

MULTIPLE CHOICE QUESTIONS
REAL NUMBERS

1.  The decimal expansion of number  has:

      (a)   a terminating decimal

(b)   non-terminating but repeating

      (c)   non-terminating non repeating

(d)   terminating after two places of decimal

2.  The values of x and y in the given figure are:
        

      (a)   x = 10; y = 14

(b)   x = 21; y = 84

(c)   x = 21; y = 25

(d)   x = 10; y = 40

3.  For any positive integer a and 3, there exist unique integers q and r such that a = 3q + r, where r must satisfy :

      (a)   0 ≤ r < 3

(b)   1 < r < 3

(c)   0 < r < 3

(d)   0 < r ≤ 3

4.   is:

      (a)   a rational number

(b)   an irrational number

      (c)   a prime number

(d)   an even number

5.  L.C.M. of 23 × 32 and 22 × 33 is :

      (a)   23

(b)   33

(c)   23 × 33

(d)   22 × 32

6.  The HCF and LCM of two numbers are 33 and 264 respectively. When the first number is completely divided by 2 the quotient is 33. The other number is:

      (a)   66

(b)   130

(c)   132

(d)   196

7.  What will be the least possible number of the planks, if three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length?

      (a)   5

(b)   6

(c)   7

(d)   none of these

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

      (a)   17 m/min

(b)   7 m/min

(c)   13 m/min

(d)   26 m/min

9.  If A = 2n + 13, B = n + 7, where n is a natural number then HCF of A and B is:

      (a)   2

(b)   1

(c)   3

(d)   4

10.  Pairs of natural numbers whose least common multiple is 78 and the greatest common divisor is 13 are:

      (a)   58 and 13 or 16 and 29

(b)   68 and 23 or 36 and 49

      (c)   18 and 73 or 56 and 93

(d)   78 and 13 or 26 and 39

11.  Two natural numbers whose sum is 85 and the least common multiple is 102 are:

      (a)   30 and 55

(b)   17 and 68

(c)   35 and 55

(d)   51 and 34

12.  4 Bells toll together at 9.00 am. They toll after 7, 8, 11 and 12 seconds respectively. How many times will they toll together again in the next 3 hours?

      (a)   3

(b)   4

(c)   5

(d)   6

13.  A forester wants to plant 66 apple trees, 88 banana trees and 110 mango trees in equal rows (in terms of number of trees). Also he wants to make distinct rows of trees (i.e., only one type of trees in one row). The number of minimum rows required are

      (a)   2

(b)   3

(c)   10

(d)   12

14.  A number 10x + y is multiplied by another number 10a + b and the result comes as 100p + 10q +r, where r = 2y, q = 2(x + y) and p = 2x; x, y < 5, q ≠ 0. The value of 10a + b may be:

      (a)   11

(b)   13

(c)   31

(d)   22

15.  If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is

      (a)   4

(b)   2

(c)   1

(d)   3

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

      (a)   13

(b)   65

(c)   875

(d)   1750

17.  If two positive integers a and b are written as a = x3y2 and b = xy3; x, y are prime numbers, then HCF (a, b) is

      (a)   xy

(b)   xy2

(c)   x3y3

(d)   x2y2

18.  If two positive integers p and q can be expressed as p = ab2 and q = a3b; a, b being prime numbers, then LCM (p, q) is

      (a)   ab

(b)   a2b2

(c)   a3b2

(d)   a3b3

19.  The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is

      (a)   10

(b)   100

(c)   504

(d)   2520

20.  The decimal expansion of the rational number  will terminate after

      (a)   one decimal place

(b)   two decimal places

      (c)   three decimal places

(d)   four decimal places

1. (a)	2. (b)	3. (a)	4. (b)	5. (c)
6. (c)	7. (d)	8. (c)	9. (b)	10 (d)
11. (d)	12. (c)	13. (d)	14. (d)	15. (b)
16. (a)	17. (b)	18. (c)	19. (d)	20. (d)

Sample Paper
REAL NUMBERS

1.  Has the rational number  a terminating or a non-terminating decimal representation?

2.  Prove that  is an irrational number.

3.  (i)   Find the LCM and HCF of 120 and 144 by using Fundamental Theorem of Arithmetic.
      (ii)    find the values of m and n where m and n are non-negative integers.
      Hence write its decimal expansion without actual division.

4.  Prove that  is irrational.

5.  Express the number  in the form of rational number 

6.  The LCM of 2 numbers is 14 times their HCF. The sum of LCM and HCF is 600. If one number is 280, then find the other number.

7.  The decimal expansion of π is:

      (a)   terminating

(b)   non-terminating and non-recurring

      (c)   non-terminating and recurring

(d)   doesn’t exist

8.  Which of the following rational numbers have a terminating decimal expansion?

      (a)   

(b)   

(c)   

(d)   

9.  If the HCF of 55 and 99 is expressible in the form 55m � 99, then the value of m is:

      (a)   4

(b)   2

(c)   1

(d)   3

10.  Given that LCM (91, 26) = 182, then HCF (91, 26) is:

      (a)   13

(b)   26

(c)   7

(d)   9

Class X Math
Test for Real Numbers

Max Marks : 35

Max Time : 45 mins

1.  The LCM and HCF of two numbers are 240 and 12 respectively. If one of the numbers is 60, then find the other number.            3 marks

2.  Without actually performing the long-division, state  will have a terminating or non-terminating repeating decimal expansion.            3 marks

3.  Write whether the rational number  will have a terminating decimal expansion or a non-terminating repeating decimal expansion.            3 marks

4.  The HCF and LCM of two numbers are 9 and 360 respectively. If one number is 45, write the other number.            3 marks

5.  What is the HCF of 52 and 130?            3 marks

6.  Express  as a rational number in the simplest form.            4 marks

7.  Prove that  is an irrational number.            4 marks

8.  Show that  is an irrational number.            4 marks

9.  Show that one and only one of n, n + 2 and n + 4 is divisible by 3.            4 marks

10.  Using Euclid’s division algorithm, find the HCF of 56, 96 and 404.            4 marks