If n is a natural number, then $9^{2 n }-4^{2 n }$ is always divisible by:
- A$5$
- B$3$
- ✓both $5$ and $13$
- DNone of these.
Answer: C.
View full solution →Question types
Real Numbers question types
341 questions across 9 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.
341
Questions
9
Question groups
5
Question types
01
M.C.Q (1 Marks)
117 Q→02Assertion (A) & Reason (B) MCQ
18 Q→03True False[1 Marks ]
15 Q→04Fill In The Blanks[1 Marks ]
16 Q→051 Marks Question
39 Q→062 Marks Questions
58 Q→073 Marks Question
50 Q→085 Marks Questions
19 Q→09Case study (4 Marks)
9 Q→Sample Questions
Real Numbers questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The exponent of $2$ in the prime factorisation of $144$, is:
- ✓$4$
- B$5$
- C$6$
- D$3$
Answer: A.
View full solution →For some integer q, every odd integer is of the form:
- Aq
- Bq + 1
- C2q
- ✓2q + 1
Answer: D.
View full solution →The smallest rational number by which $\frac{1}{3}$ should be multiplied so that its decimal expansion terminates after one place of decimal, is:
- ✓$\frac{3}{10}$
- B$\frac{1}{10}$
- C$3$
- D$\frac{3}{100}$
Answer: A.
View full solution →If $n =2^3 \times 3^4 \times 5^4 \times 7$, then the number of consecutive zeroes in $n$ , where $n$ is a natural number, is:|
- A$2$
- ✓$3$
- C$4$
- D$7$
Answer: B.
View full solution →Statement-1 (A): $ 2+\sqrt{2}$ is an irrational number.
Statement-2 (R): The sum of a non-zero rational number and an irrational number is always an irrational number.
Statement-2 (R): The sum of a non-zero rational number and an irrational number is always an irrational number.
- ✓Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- CStatement-1 is true, Statement-2 is false.
- DStatement- 1 is false, Statement- 2 is true.
Answer: A.
View full solution →Statement-1 $(A)$ : The number $5^n$ cannot end with the digit 0 , where $n$ is a natural number.
Statement-2 $(R)$: Prime factorisation of 5 has only two factors 1 and 5.
Statement-2 $(R)$: Prime factorisation of 5 has only two factors 1 and 5.
- ✓Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- CStatement-1 is true, Statement-2 is false.
- DStatement- 1 is false, Statement-2 is true.
Answer: A.
View full solution →
- AStatement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- CStatement-1 is true, Statement-2 is false.
- ✓Statement- 1 is false, Statement- 2 is true.
Answer: D.
View full solution →Statement-1 $(A)$ : The product of $(5+\sqrt{3})$ and $(2-\sqrt{3})$ is an irrational number.
Statement-2 $(R)$: The product of two irrational numbers is an irrational number.
Statement-2 $(R)$: The product of two irrational numbers is an irrational number.
- AStatement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- ✓Statement-1 is true, Statement-2 is false.
- DStatement- 1 is false, Statement-2 is true.
Answer: C.
View full solution →Statement-1 $(A)$: If 11 divides 627264, then 11 divides 792.
Statement-2 $(R)$ : Let $p$ be a prime number and a be a positive integer, if $p$ divides $a^2$, then pdivides $a$.
Statement-2 $(R)$ : Let $p$ be a prime number and a be a positive integer, if $p$ divides $a^2$, then pdivides $a$.
- ✓Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- CStatement-1 is true, Statement-2 is false.
- DStatement- 1 is false, Statement-2 is true.
Answer: A.
View full solution →The product of two irrational numbers is an irrational number (True/ False).
Two numbers have 12 as their HCF and 350 as their LCM (True/ False).
Every even integer is of the form 2m, where m is an integer (True/ False).
Every odd integer is of the form 2m - 1, where m is an integer (True/ False).
The product of any three consecutive natural number is divisible by 6 (True/ False).
If the least prime factors of two positive integers a and b are 5 and 13 respectively, then the least prime factor of a + b, is __________.
The LCM of the smallest prime number and the smallest composite number is __________.
The LCM of the smallest prime number and the smallest odd composite number is __________.
The LCM of 2x, 5x and 7x is __________, where x is a positive integer.
Two numbers are in the ratio 3: 4 and their LCM is 120. The sum of the numbers is __________.
If a and b are relatively prime numbers, then what is their LCM?
Write the condition to be satisfied by q so that a rational number $\frac{\text{p}}{\text{q}}$ has a terminating decimal expansion.
View full solution →What is a lemma?
Given that HCF (306. 657) = 9, find LCM (306, 657).
Determine the prime factorisation of the following positive integer:
20570
20570
Check whether$ 6^n$ can end with the digit $0$ for any natural number n.
View full solution →Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non terminating repeating decimal expansion.
$\frac{987}{10500}$
View full solution →$\frac{987}{10500}$
Find the LCM and HCF of the following integer by applying the prime factorisation method.
17, 23 and 29
17, 23 and 29
Define HCF of two positive integers and find the HCF of the following pairs of numbers:
475 and 495
475 and 495
What is a composite number?
During a sale, colour pencils were being sold in packs of 24 each and crayons in packs of 32 each. If you want full packs of both and the same number of pencils and crayons, how many of each would you need to buy?
The length, breadth and height of a room are 8m 25cm, 6m 75cm and 4m 50cm, respectively. Determine the longest rod which can measure the three dimensions of the room exactly.
Prove that the square of any positive integer is of the form $5q, 5q + 1, 5q + 4$ for some integer q.
View full solution →Find the HCF of the following pairs of integers and express it as a linear combination of them.
963 and 657
963 and 657
Show that the following numbers are irrational.
$3-\sqrt{5}$
View full solution →$3-\sqrt{5}$
Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.
Determine the number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21.
In a morning walk three persons step off together, their steps measure 80cm, 85cm and 90cm respectively. What is the minimum distance each should walk so that he can cover the distance in complete steps?
Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3 respectively.
Prove that if a positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely.
Teaching Mathematics through activities is a powerful approach that enhances students' understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got 173250.
Now, Mukta asked some questions as given below to the students:
(i) What is the least prime number used by students?
(ii) (a) How many students are in the class?
OR
(b) What is the highest prime number used by students?
(iii) Which prime number has been used maximum times?
Now, Mukta asked some questions as given below to the students:
(i) What is the least prime number used by students?
(ii) (a) How many students are in the class?
OR
(b) What is the highest prime number used by students?
(iii) Which prime number has been used maximum times?
represents the golden ratio spiral. The sequence $0,1,1,2,3,5,8,13$ is called a Fibonacci sequence in which each term (except the first two terms) is obtained by adding the previous two terms of the sequence.
(i) The next two numbers in the sequence are
(a) 223 and 367 $\qquad$ (b) 233 and 367 $\qquad$ (c) 223 and 377 $\qquad$ (d) 233 and 377
(ii) A rational number has terminating decimal expansion, if the denominator is of the form
(a) $2^m \times 5^n$ $\qquad$ (b) $3^m \times 5^n$ $\qquad$ (c) $2^m \times 3^n$ $\qquad$ (d) $3^m \times 4^n$
where $m, n$ are non-negative integers.
(iii) Which of the following rational numbers (formed by taking the ratio of a term to its previous term) has terminating decimal expansion?
(a) $\frac{5}{3}$ $\qquad$ (b) $\frac{21}{13}$ $\qquad$ (c) $\frac{13}{8}$ $\qquad$ (d) $\frac{34}{21}$
(iv) HCF of two consecutive numbers in the given sequence (except the first 3 terms) is
(a) 1 $\qquad$ (b) 0 $\qquad$ (c) 2 $\qquad$ (d) 3
View full solution →(i) The next two numbers in the sequence are
(a) 223 and 367 $\qquad$ (b) 233 and 367 $\qquad$ (c) 223 and 377 $\qquad$ (d) 233 and 377
(ii) A rational number has terminating decimal expansion, if the denominator is of the form
(a) $2^m \times 5^n$ $\qquad$ (b) $3^m \times 5^n$ $\qquad$ (c) $2^m \times 3^n$ $\qquad$ (d) $3^m \times 4^n$
where $m, n$ are non-negative integers.
(iii) Which of the following rational numbers (formed by taking the ratio of a term to its previous term) has terminating decimal expansion?
(a) $\frac{5}{3}$ $\qquad$ (b) $\frac{21}{13}$ $\qquad$ (c) $\frac{13}{8}$ $\qquad$ (d) $\frac{34}{21}$
(iv) HCF of two consecutive numbers in the given sequence (except the first 3 terms) is
(a) 1 $\qquad$ (b) 0 $\qquad$ (c) 2 $\qquad$ (d) 3
A seminar is being conducted by an educational organisation, where the participants will be educators of different subjects. The numbers of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively.
(i) In each room the same number of participants are to be seated and all of them being in the same subject, hence the maximum number of participants that can be accomodated in each room is
(a) 14 $\qquad$ (b) 12$\qquad$ (c) 16 $\qquad$ (d) 18
(ii) The minimum number of rooms required during the event, is
(a) 11 $\qquad$ (b) 9 $\qquad$ (c) 7 $\qquad$ (d) 21
(iii) The LCM of 60,84 and 108 is
(a) 3780 $\qquad$ (b) 3680 $\qquad$ (c) 4780 $\qquad$ (d) 4680
(iv) The product of HCF and LCM of 60,84 and 108 is
(a) 55360 $\qquad$ (b) 35360 $\qquad$ (c) 45500 $\qquad$ (d) 45360
View full solution →(i) In each room the same number of participants are to be seated and all of them being in the same subject, hence the maximum number of participants that can be accomodated in each room is
(a) 14 $\qquad$ (b) 12$\qquad$ (c) 16 $\qquad$ (d) 18
(ii) The minimum number of rooms required during the event, is
(a) 11 $\qquad$ (b) 9 $\qquad$ (c) 7 $\qquad$ (d) 21
(iii) The LCM of 60,84 and 108 is
(a) 3780 $\qquad$ (b) 3680 $\qquad$ (c) 4780 $\qquad$ (d) 4680
(iv) The product of HCF and LCM of 60,84 and 108 is
(a) 55360 $\qquad$ (b) 35360 $\qquad$ (c) 45500 $\qquad$ (d) 45360
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.
(i) 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 $\qquad$ (b) 128 $\qquad$ (c) 288 $\qquad$ (d) 272
(ii) 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 $\qquad$ (b) 4 $\qquad$ (c) 6 $\qquad$ (d) 8
(iii) 36 can be expressed as a product of its primes as
(a) $2^2 \times 3^2$ $\qquad$ (b) $2^1 \times 3^3$ $\qquad$ (c) $2^3 \times 3^1$ $\qquad$ (d) $2^0 \times 3^0$
(iv) $7 \times 11 \times 13 \times 15+15$ is a
(a) Prime number
(b) Composite number
(c) Neither prime nor composite
(d) None of the above
View full solution →(i) 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 $\qquad$ (b) 128 $\qquad$ (c) 288 $\qquad$ (d) 272
(ii) 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 $\qquad$ (b) 4 $\qquad$ (c) 6 $\qquad$ (d) 8
(iii) 36 can be expressed as a product of its primes as
(a) $2^2 \times 3^2$ $\qquad$ (b) $2^1 \times 3^3$ $\qquad$ (c) $2^3 \times 3^1$ $\qquad$ (d) $2^0 \times 3^0$
(iv) $7 \times 11 \times 13 \times 15+15$ is a
(a) Prime number
(b) Composite number
(c) Neither prime nor composite
(d) None of the above
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