Class 10 Mathematics Ch 1 MCQ with Answers: Real Numbers

Contents hide

1 Direct Questions

2 Assertion-Reason

3 Match the Following

4 Statement-Based Questions

5 Arrange in the Correct Order

6 Fill in the Blanks

7 Odd One Out

8 Passage-Based Questions

9 True / False

20 multiple-choice (MCQ / objective) questions from this chapter, in the common exam-style shapes. Every question comes with an explanation along with the answer, so you remember why the option is correct, not just which one it is.

1 Direct Questions

Direct Questions

1 markThe LCM of 26 and 91 is:
(A) 91(B) 117(C) 169(D) 182
1 mark5005 written as a product of primes is:
(A) $5\times7\times13\times13$ (B) $5\times7\times11\times13$ (C) $5^2\times7\times11$ (D) $5\times7\times11\times17$
1 markThe HCF of two numbers is 6 and their LCM is 3024. If one number is 336, the other number is:
(A) 27(B) 36(C) 54(D) 48
1 markThe LCM of 12, 15 and 21 is:
(A) 630(B) 420(C) 840(D) 1260
1 markWhich of the following numbers ends with the digit 0 for every natural number $n$ ?
(A) $4^n$ (B) $6^n$ (C) $15^n$ (D) $20^n$
1 markIf $p$ is a prime number and $p$ divides $a^2$ , where $a$ is a positive integer, then:
(A) $p$ divides $a$ (B) $a$ divides $p$ (C) $p=a$ (D) $p$ divides $a+1$
1 markIn the proof that $\sqrt{2}$ is irrational, we begin by assuming $\sqrt{2}=\dfrac{a}{b}$ , where $a$ and $b$ are:
(A) coprime integers with $b\ne0$ (B) both even integers(C) both prime numbers(D) both odd integers
1 markA number that cannot be written in the form $\dfrac{p}{q}$ , where $p$ and $q$ are integers and $q\ne0$ , is called:
(A) a rational number(B) a prime number(C) a composite number(D) an irrational number
1 markThree bells ring at intervals of 6, 8 and 12 minutes. They ring together at 9:00 a.m. When will they next ring together?
(A) 9:12 a.m.(B) 9:18 a.m.(C) 9:24 a.m.(D) 9:48 a.m.
1 markIf $a=2^3\times3$ and $b=2\times3\times5$ , then $\text{LCM}(a,b)$ is:
(A) 6(B) 30(C) 120(D) 60

2 Assertion-Reason

Assertion-Reason

1 markAssertion (A): $\text{HCF}(6,20)\times\text{LCM}(6,20)=120$ .
Reason (R): For any two positive integers $a$ and $b$ , $\text{HCF}(a,b)\times\text{LCM}(a,b)=a\times b$ .
(A) Both A and R are true, and R is the correct explanation of A(B) Both A and R are true, but R is not the correct explanation of A(C) A is true, but R is false(D) A is false, but R is true
1 markAssertion (A): The HCF of 8 and 9 is 1.
Reason (R): 8 and 9 are both prime numbers.
(A) Both A and R are true, and R is the correct explanation of A(B) Both A and R are true, but R is not the correct explanation of A(C) A is true, but R is false(D) A is false, but R is true

3 Match the Following

Match the Following

1 markMatch Column A with Column B.
Column A: (i) HCF(26, 91) (ii) LCM(26, 91) (iii) HCF(510, 92) (iv) LCM(12, 15, 21)
Column B: (a) 420 (b) 2 (c) 182 (d) 13
(A) i-a, ii-b, iii-c, iv-d(B) i-d, ii-c, iii-b, iv-a(C) i-c, ii-d, iii-a, iv-b(D) i-b, ii-a, iii-d, iv-c

4 Statement-Based Questions

Statement-Based Questions

1 markStatement I: $\text{HCF}(6,72,120)\times\text{LCM}(6,72,120)=6\times72\times120$ .
Statement II: $\text{HCF}(6,72,120)=6$ and $\text{LCM}(6,72,120)=360$ .
(A) Both I and II are true(B) I is true, II is false(C) I is false, II is true(D) Both I and II are false

5 Arrange in the Correct Order

Arrange in the Correct Order

1 markArrange the steps of the reasoning that $4^n$ never ends with the digit 0 in the correct order:
(I) The prime 5 would have to appear in the prime factorisation of $4^n$
(II) If $4^n$ ended with 0, it would be divisible by 5
(III) But $4^n=2^{2n}$ , so its only prime factor is 2
(IV) By the uniqueness of prime factorisation, 5 can never appear, so $4^n$ never ends with 0
(A) II, I, III, IV(B) I, II, III, IV(C) II, III, I, IV(D) III, II, I, IV

6 Fill in the Blanks

Fill in the Blanks

1 markThe Fundamental Theorem of Arithmetic says that the prime factorisation of a composite number is unique, apart from the ______ in which the prime factors occur.
(A) sum(B) order(C) size(D) number

7 Odd One Out

Odd One Out

1 markThree of these numbers are irrational. Which one is NOT?
(A) $\sqrt{2}$ (B) $\sqrt{5}$ (C) $\pi$ (D) $\sqrt{49}$

8 Passage-Based Questions

Passage-Based Questions

Passage

On a school’s circular running track, Aman completes one round in 16 minutes and Bhavna completes one round in 20 minutes. Both start together from the starting point at 7:00 a.m. and run in the same direction at steady speeds.

1 markAfter how many minutes will Aman and Bhavna next meet at the starting point?
(A) 40 minutes(B) 80 minutes(C) 160 minutes(D) 320 minutes
1 markAt what time will they next meet at the starting point?
(A) 8:00 a.m.(B) 8:10 a.m.(C) 7:40 a.m.(D) 8:20 a.m.

9 True / False

True / False

1 markConsider the following statements:
(i) Every composite number can be written as a product of primes.
(ii) 1 is a composite number.
(iii) The prime factorisation of 32760 is unique apart from the order of its factors.
Which of the above statements are true?
(A) (i) and (ii) only(B) (ii) and (iii) only(C) (i) and (iii) only(D) All three

Answer Key and Explanations

Direct Questions

(D) $26=2\times13$ , $91=7\times13$ . LCM $=2\times7\times13=182$ .
(B) $5005=5\times1001=5\times7\times143=5\times7\times11\times13$ .
(C) HCF $\times$ LCM $=$ product of the two numbers, so the other number $=\dfrac{6\times3024}{336}=\dfrac{18144}{336}=54$ .
(B) $12=2^2\times3$ , $15=3\times5$ , $21=3\times7$ . Greatest powers: $2^2,3,5,7$ . LCM $=4\times3\times5\times7=420$ .
(D) A number ends in 0 only if its prime factorisation has both 2 and 5. $20^n=2^{2n}\times5^n$ always has both. $4^n=2^{2n}$ and $6^n=2^n\times3^n$ have no 5; $15^n=3^n\times5^n$ has no 2.
(A) This is Theorem 1.2: if a prime $p$ divides $a^2$ , then $p$ divides $a$ .
(A) We take the fraction in lowest terms, so $a$ and $b$ are coprime (no common factor other than 1), and $b\ne0$ .
(D) That is the definition of an irrational number.
(C) $6=2\times3$ , $8=2^3$ , $12=2^2\times3$ . LCM $=2^3\times3=24$ minutes, so they ring together again at 9:24 a.m.
(C) Take the greatest power of each prime: $2^3$ , $3^1$ , $5^1$ . LCM $=8\times3\times5=120$ . (HCF would be $2\times3=6$ .)

Assertion-Reason

(A) HCF $=2$ , LCM $=60$ , and $2\times60=120=6\times20$ , so A is true. R is a general result: applied to $a=6$ , $b=20$ it gives $6\times20=120$ , which is exactly A. So R explains A.
(C) A is true: $8=2^3$ and $9=3^2$ have no common prime, so HCF $=1$ . Reason (R) is false: 8 and 9 are composite numbers, not primes.

Match the Following

(B) HCF(26, 91) $=13$ (d); LCM(26, 91) $=182$ (c); HCF(510, 92) $=2$ (b); LCM(12, 15, 21) $=420$ (a).

Statement-Based Questions

(C) Statement I is false: $6\times360=2160\ne51840$ . Statement II is true (Example 4 of the book).

Arrange in the Correct Order

(A) Start from what ending in 0 would mean (II), turn it into a statement about prime factors (I), compare with the actual factorisation $2^{2n}$ (III), and conclude using uniqueness (IV). This is Example 1 of the book.

Fill in the Blanks

(B) The factorisation is unique apart from the order: $2\times3\times5\times7$ and $3\times5\times7\times2$ count as the same.

Odd One Out

(D) $\sqrt{49}=7$ is rational. $\sqrt{2}$ , $\sqrt{5}$ and $\pi$ are irrational.

Passage-Based Questions

(B) They meet at the start when both have completed whole rounds, i.e. after LCM(16, 20). $16=2^4$ , $20=2^2\times5$ , so LCM $=2^4\times5=80$ minutes.
(D) 80 minutes $=1$ hour 20 minutes after 7:00 a.m., that is 8:20 a.m.

True / False

(C) (i) is true (Fundamental Theorem of Arithmetic). (ii) is false: 1 is neither prime nor composite. (iii) is true, by the uniqueness part of the theorem. So (i) and (iii) only.

More for this chapter

Notes Important Questions NCERT Solutions Short Notes