Competency Based Questions for Class 10 Maths Chapter 1 Real Numbers

Competency Based Questions are new type of questions asked in CBSE Board exam for class 10. Practising the following Competency Based Questions will help the students in facing Board Questions.

Hint: Apply Euclid Division Algorithm in order to obtain HCF of 2 given numbers in the context of the given problem.

Question.1. A worker needs to pack 350 kg of rice and 150 kg of wheat in bags such that each bag weighs the same.
Each bag should either contain rice or wheat. Which option shows the correct steps to find the greatest amount of rice/wheat the worker can pack in each bag?

(a)
Step 1: 350 = 2(150) + 50
Step 2: 150 = 3(50) + 0
Step 3: Greatest amount = 50 kg
(b)
Step 1: 350 = 2(150) + 50
Step 2: 150 = 2(50) + 0
Step 3: Greatest amount = 50 kg
(c)
Step 1: 350 = 2(150) + 50
Step 2: 150 = 3(50) + 0
Step 3: Greatest amount = 150 kg
(d)
Step 1: 350 = 2(150) + 50
Step 2: 150 = 2(50) + 0
Step 3: Greatest amount = 150 kg

Answer. (a)
Step 1: 350 = 2(150) + 50
Step 2: 150 = 3(50) + 0
Step 3: Greatest amount = 50 kg

Question.2. Pranay wants to stack few one-rupee coins and some five-rupee coins in such a way that:

a. Each stack has the same number of coins.
b. There is least number of stacks.
c. Each stack either has one rupee or five-rupee coins.
d. No coins are left over after creating stacks.

His first step to find the number of coins that should be in each stack is 195=1(180)+15. Given that he has more of five-rupee coins than one-rupee coins, how many one-rupee coins stack can he make?

(a) 12
(b) 13
(c) 15
(d) 25

Answer. (a) 12

Hint: Apply Euclid Division Algorithm in order to prove results of positive integers in the form of ax+b where a and b are constants.

Question.3. Given that p is a non-negative integer, which of these gives positive integers that are multiple of 5?

(a) 10p and 10p+2 .
(b) 10p and 10p+3 .
(c) 10p and 10p+4 .
(d) 10p and 10p+5 .

Answer. (d) 10p and 10p+5 .

Question.4. In the equation below, a, b, q, r are integers, 0 \leq r <b , a is a multiple of 3 and b=9.
a=bq+r
Which of the following forms represent a?

(a) Only 9q and 9q+3 , as only these forms when divided by 3 gives r=0 .
(b) Only 9q+1 and 9q+4 , as only these forms when divided by 3 gives r=1 .
(c) Only 9q , 9q+3 , and 9q+6 , as only these forms when divided by 3 gives r=0 .
(d) Only 9q+1 , 9q+4 , and 9q+7 , as only these forms when divided by 3 gives r=1 .

Answer. (c) Only 9q , 9q+3 , and 9q+6 , as only these forms when divided by 3 gives r=0 .

Hint: Use the Fundamental Theorem of Arithmetic in order to calculate HCF and LCM of the given numbers in the context of the given problem.

Question.5. Rahul has 40 cm long red and 84 cm long blue ribbon. He cuts each ribbon into pieces such that all pieces are of equal length. What is the length of each piece?

(a) 4 cm as it is the LCM of 40 and 84
(b) 4 cm as it is the HCF of 40 and 84
(c) 8 cm as it is the LCM of 40 and 84
(d) 8 cm as it is the HCF of 40 and 84

Answer. (b) 4 cm as it is the HCF of 40 and 84

Question.6. Three bulbs red, green and yellow flash at intervals of 80 seconds, 90 seconds and 110 seconds. All three flash together at 8:00 am. At what time will the three bulbs flash altogether again?

(a) 9:00 am
(b) 9:12 am
(c) 10:00 am
(d) 10:12 am

Answer. (d) 10:12 am

Hint: Recall the properties of irrational number in order to prove that whether the sum / difference / product / quotient of 2 numbers is irrational or not.

Question.7. Which of the following is an irrational number?

(a) \frac{\sqrt{2}}{\sqrt{8}}
(b) \frac{\sqrt{3}}{3 \sqrt{5}}
(c) \frac{\sqrt{5}}{\sqrt{20}}
(d) \frac{\sqrt{63}}{\sqrt{7}}

Answer. (b) \frac{\sqrt{3}}{3 \sqrt{5}}

Question.8. A teacher creates the question “Which of the following could be the sum of two rational numbers?”.
She now needs to create three incorrect choices and one correct answer. Which option shows the choices that the teacher should create?

(a) First choice: \pi ; Second choice: 20+16; Third choice: 50-1; Correct Answer: 49
(b) First choice: 227; Second choice: 25+16; Third choice: 64; Correct Answer: 5
(c) First choice: 125; Second choice: 36+42; Third choice: 81; Correct Answer: 169
(d) None of them

Answer. (a) First choice: \pi ; Second choice: 20+16; Third choice: 50-1; Correct Answer: 49

Hint: Apply theorems of irrational number in order to prove whether a given number is irrational or not.

Question.9. Which of the following is NOT an irrational number?

(a) 2 \times \frac{1}{\sqrt{2}}
(b) \sqrt{2} \times \frac{1}{{2}}
(c) \sqrt{2} \times \frac{1}{{\sqrt{3}}}
(d) \sqrt{2} \times \frac{1}{\sqrt{2}}

Answer. (d) \sqrt{2} \times \frac{1}{\sqrt{2}}

Question.10. Is 9+\sqrt{2} an irrational number?

(a) No, because if 9+\sqrt{2}=\frac{a}{b} where a and b are integers and b \neq 0 , then \sqrt{2}=\frac{9b-a}{b} , but \sqrt{2} is an irrational number. So, 9+\sqrt{2} \neq \frac{a}{b} .
(b) No, because if 9+\sqrt{2}=\frac{a}{b} , where a and b are integers and b \neq 0 , then \sqrt{2}=\frac{9b+a}{b} , but \sqrt{2} is an irrational number. So, 9+\sqrt{2} \neq \frac{a}{b} .
(c) Yes, because if 9+\sqrt{2}=\frac{a}{b} , where a and b are integers and b \neq 0 , then \sqrt{2}=\frac{9b-a}{b} , but \sqrt{2} is an irrational number. So, 9+\sqrt{2} \neq \frac{a}{b} .
(d) Yes, because if 9+\sqrt{2}=\frac{a}{b} , where a and b are integers and b \neq 0 , then \sqrt{2}=\frac{9b+a}{b} , but \sqrt{2} is an irrational number. So, 9+\sqrt{2} \neq \frac{a}{b} .

Answer. (b) No, because if 9+\sqrt{2}=\frac{a}{b} , where a and b are integers and b \neq 0 , then \sqrt{2}=\frac{9b+a}{b} , but \sqrt{2} is an irrational number. So, 9+\sqrt{2} \neq \frac{a}{b} .

Hint: Apply theorems of rational numbers in order to find out about the nature of their decimal representation and their factors.

Question.11. Which of the following is equivalent to a decimal that terminates?

(a) \frac{1}{5^{2}2^{2}}
(b) \frac{1}{2^{2}3}
(c) \frac{1}{5^{2}7}
(d) \frac{1}{5^{2}11^{2}}

Answer. (a) \frac{1}{5^{2}2^{2}}

Question.12. The fractions \frac{3}{a} and \frac{7}{b} are equivalent to decimals that terminate. Which best describes the product of a and b?

(a) It is a prime number.
(b) It cannot be an odd number.
(c) It is of the form 21 k , where k could be multiples of 2 or 5.
(d) It is of the form 21 k , where k could be multiples of 7 or 9.

Answer. (c) It is of the form 21 k , where k could be multiples of 2 or 5.

More Practice Questions

Question.1. In my colony, number of males and females living are 32 and 36 respectively. What is the minimum number of sweets to be purchased on 77th Independence Day 2023 so that everyone will get equal number of sweets?

(a) 272
(b) 288
(c) 274
(d) 144

Answer. (b) 288

Question.2. In the following factorization of x , which relationship is true?

(a) y = x
(b) y > x
(c) z < y
(d) z > y

Answer. (d) z > y

Question.3. If x is a prime number and a be any positive integer such that a^{2} is a multiple of x , the which of the following statement is true?

(a) x can divide a^{2}
(b) x may or may not divide a
(c) x divides a
(d) Both (a) and (b) are correct

Answer. (d) Both (a) and (b) are correct

Question.4. If a and b are any two integers and b = 3, then a can be expressed as

(a) 3q, 3q+1, 3q+2
(b) 3q
(c) none of the above
(d) 3q+ 1
* Here q is some integer

Answer. (a) 3q, 3q+1, 3q+2

Question.5. In a Block, the villagers from three Panchayats with strength 280, 504 and 672 respectively want to go on a picnic. Buses are to be hired to take these villagers to the picnic. What is the maximum number of villagers who can sit in a bus if each bus takes equal number of villagers?

(a) 72
(b) 56
(c) 77
(d) 85

Answer. (b) 56

Question.6. If X, Y and Z are three are positive integers such that X = m^{3}n^{2}p , Y = mn^{3}p^{2} and Z = m^{2}n^{2}p^{2} , the find the HCF(X,Y,Z).

(a) mnp
(b) m^{2}np
(c) mn^{2}p
(d) mnp^{2}

Answer. (c) mn^{2}p

Question.7. The HCF of y and 153 is 51, where y is a natural number. Which of these can be true for some values of y?

(i) y is a multiple of 51
(ii) y is a multiple of 153
(iii) y is a an even number
(iv) y is a an odd number

(a) only (ii) and (iii)
(b) only (i), (ii) and (iii)
(c) only (i), (iii) and (iv)
(d) all (i), (ii),(iii) and (iv)

Answer. (c) only (i), (iii) and (iv)

Question.8. The LCM of the two numbers is 20^{2} \times 3. Which of the following cannot be their HCF? Justify your reason.

(a) 200
(b) 400
(c) 600
(d) 500

Answer. (d) 500

Question.9. If 3 is the least prime factor of number a and 7 is the least prime factor of number b, then the least prime factor of a + b is

(a) 5
(b) 2
(c) 3
(d) 10

Answer. (b) 2

Question.10. If HCF (26, 169) = 13, then LCM (26, 169) =

(a) 26
(b) 52
(c) 338
(d) 13

Answer. (c) 338

Question.11. If two positive integers m and n are expressible in the form m = pq^{3} and n = p^{3}q^{2} where p, q are prime numbers, then HCF (m, n) =

(a) pq
(b) pq^{2}
(c) p^{3}q^{3}
(d) p^{2}q^{3}

Answer. (b) pq^{2}

Question.12. If n = 23 × 34 × 54 × 7, then the number of consecutive zeros in n, where n is a natural number, is

(a) 2
(b) 3
(c) 4
(d) 7

Answer. (b) 3

Question.13. To enhance the reading skills of grade X students, the school nominates you and two of your friends to set up a class library. There are two sections- section A and section B of grade X. There are 32 students in section A and 36 students in section B.
What is the minimum number of books you will acquire for the class library, so that they can be distributed equally among students of Section A or Section B?

(a) 144
(b) 128
(c) 288
(d) 272

Answer. (c) 288

Question.14. To enhance the reading skills of grade X students, the school nominates you and two of your friends to set up a class library. There are two sections- section A and section B of grade X. There are 32 students in section A and 36 students in section B.
If the product of two positive integers is equal to the product of their HCF and LCM is true then, the HCF (32, 36) is

(a) 2
(b) 4
(c) 6
(d) 8

Answer. (b) 4

Question.15. Three farmers have 490 kg, 588 kg and 882 kg of wheat respectively. Find the maximum capacity of a bag so that the wheat can be packed in exact number of bags.

(a) 98 kg
(b) 290 kg
(c) 200 kg
(d) 350 kg

Answer. (a) 98 kg

Question.16. A seminar is being conducted by an Educational Organization, where the participants will be educators of different subjects. The number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively. In each room the same number of participants are to be seated and all of them being in the same subject, hence maximum number participants that can accommodated in each room are

(a) 14
(b) 12
(c) 16
(d) 18

Answer. (b) 12

Question.17. A seminar is being conducted by an Educational Organization, where the participants will be educators of different subjects. The number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively. What is the minimum number of rooms required during the event?

(a) 11
(b) 31
(c) 41
(d) 21

Answer. (d) 21

Question.18. A seminar is being conducted by an Educational Organization, where the participants will be educators of different subjects. The number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively. The LCM of 60, 84 and 108 is

(a) 3780
(b) 3680
(c) 4780
(d) 4680

Answer. (a) 3780

Question.19. A Mathematics Exhibition is being conducted in your School and one of your friends is making a model of a factor tree. He has some difficulty and asks for your help in completing a quiz for the audience. Observe the following factor tree and answer the following:
What will be the value of x ?

(a) 15005
(b) 13915
(c) 56920
(d) 17429

Answer. (b) 13915

Question.20. If n is a natural number, then 9^{2n}-4^{2n} is always divisible by

(a) 5
(b) 13
(c) Both 5 and 13
(d) None of these

Answer. (c) Both 5 and 13

Question.21. Ishani was teaching counting from 1 to 10 to her brother who is a Kindergarten student. Suddenly one question came to her mind. Are there any numbers which are divisible by all the numbers from 1 to 10? She found that least number that is divisible by all the numbers from 1 to 10 which is

(a) 10
(b) 100
(c) 504
(d) 2520

Answer. (d) 2520

Question.22. If 3 is the least prime factor of p, and 7 is the least prime factor of q, then the least prime factor of (p+q) is:

(a) 11
(b) 2
(c) 5
(d) 10

Answer. (b) 2

Question.23. n^{2}-1 is divisible by 8 if n is

(a) an integer
(b) a natural number
(c) an odd integer
(d) an even integer

Answer. (c) an odd integer

Question.24. L.C.M. of two co-prime numbers is always

(a) product of numbers
(b) sum of numbers
(c) difference of numbers
(d) none

Answer. (a) product of numbers

Question.25. The smallest irrational number which should be added to 4 − \sqrt{5} to get a rational number is:

(a) \sqrt{5} – 5
(b) – \sqrt{5}
(c) -4 + \sqrt{5}
(d) \sqrt{5}

Answer. (d) \sqrt{5}