Sample QuestionsReal Numbers questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The common difference of an $AP,$ whose $\mathrm{n}^{\text {th }}$ term is $a_n=(3 n+7)$, is :
Answer: A.
View full solution →The $\text{HCF}$ of $135$ and $225$ is :
Answer: C.
View full solution →The sum of exponents of prime factors in the prime-factorisation of $196$ is :
Answer: B.
View full solution →Euclid’s division Lemma states that for two positive integers $a$ and $b,$ there exists unique integer $q$ and $r$ satisfying $a = bq + r,$ and.
- A
$0 < r < b$
- B
$0 < r ≤ b$
- ✓
$0 ≤ r < b$
- D
$0 ≤ r ≤ b$
Answer: C.
View full solution →The $\text{HCF}$ and the $\text{LCM}$ of $12, 21, 15$ respectively are :
- A
$3, 140$
- B
$12, 420$
- ✓
$3, 420$
- D
$420, 3$
Answer: C.
View full solution →Directions : In the following questions, the Assertions $(A)$ and Reason $(s) \ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion : Euclid division lemma States that $a = bq + r$ where $0 < = r < b.$
Reason : Dividend $=$ divisor $\times $ quotient $+$ remainder is called Euclid multiplication lemma.
- A
Both assertion $(A)$ and reason $(R)$ are true and reason $(R)$ is the correct explanation of assertion $(A)$.
- B
Both assertion $(A)$ and reason $(R)$ are true but reason $(R)$ is not the correct explanation of assertion $(A)$.
- ✓
Assertion $(A)$ is true but reason $(R)$ is false.
- D
Assertion $(A)$ is false but reason $(R)$ is true.
Answer: C.
View full solution →Directions : In the following questions, the Assertions $(A)$ and Reason $(s) \ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion : $\frac{3}{5}$ has terminating decimal representaion.
Reason : The prime factor of $5$ is $ 5$.
- ✓
Both assertion $(A)$ and reason $(R)$ are true and reason $(R)$ is the correct explanation of assertion $(A)$.
- B
Both assertion $(A)$ and reason $(R)$ are true but reason $(R)$ is not the correct explanation of assertion $(A)$.
- C
Assertion $(A)$ is true but reason $(R)$ is false.
- D
Assertion $(A)$ is false but reason $(R)$ is true.
Answer: A.
View full solution →Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion : When positive integer a is divided by $3$ the value of remainder can be $0, 1$ or $2.$
Reason : According to Euclid’s division lemma $a = bq + r $ where $0 \leq r < b r$ is integer.
- ✓
Both assertion $(A)$ and reason $(R)$ are true and reason $(R)$ is the correct explanation of assertion $(A)$.
- B
Both assertion $(A)$ and reason $(R)$ are true but reason $(R)$ is not the correct explanation of assertion $(A)$.
- C
Assertion $(A)$ is true but reason $(R)$ is false.
- D
Assertion $(A)$ is false but reason $(R)$ is true.
Answer: A.
View full solution →Directions : In the following questions, the Assertions $(A)$ and Reason $(s) \ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion : The decimal expansion of rational no. $\frac{33}{2^{2}}5$ will terminate after two decimal place.
Reason : The termination of any rational no. depends upon the power of $2$ in prime factorization of denominator.
- ✓
Both assertion $(A)$ and reason $(R)$ are true and reason $(R)$ is the correct explanation of assertion $(A)$.
- B
Both assertion $(A)$ and reason $(R)$ are true but reason $(R)$ is not the correct explanation of assertion $(A)$.
- C
Assertion $(A)$ is true but reason $(R)$ is false.
- D
Assertion $(A)$ is false but reason $(R)$ is true.
Answer: A.
View full solution →Directions : In the following questions, the Assertions $(A)$ and Reason $(s) \ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion : $ 3+2\sqrt{5}$ is a rational number.
Reason : Sum of rational and irrational number is always irraational.
- ✓
Both assertion $(A)$ and reason $(R)$ are true and reason $(R)$ is the correct explanation of assertion $(A)$.
- B
Both assertion $(A)$ and reason $(R)$ are true but reason $(R)$ is not the correct explanation of assertion $(A)$.
- C
Assertion $(A)$ is true but reason $(R)$ is false.
- D
Assertion $(A)$ is false but reason $(R)$ 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).
$\Big(\frac{2+\sqrt{5}}{3}\Big)$is _________ number.
View full solution →Prove that $\sqrt{5}$ is an irrational number.
View full solution →Prove that $\sqrt{5}$ is an irrational number.
View full solution →Express 5005 as a product of its prime factors.
Express 3825 as a product of its prime factors.
Express 140 as a product of its prime factors.
Check whether $6^n$ can end with the digit 0 for any natural number $n$.
View full solution →Given that HCF (306, 657) = 9, find LCM (306, 657).
Find the LCM and HCF of 12, 15 and 21 integers by applying the prime factorisation method.
Find the LCM and HCF of 8, 9 and 25 integers by applying the prime factorisation method.
Find the LCM and HCF of 17, 23 and 29 integers by applying the prime factorisation method.
Find the LCM and HCF of 336 and 54 pairs of integers and verify that LCM $\times$ HCF = product of the two numbers.
View full solution →Prove that $6+\sqrt { 2 }$ is irrational.
View full solution →Prove that $7 \sqrt { 5 }$ is irrational.
View full solution →Prove that $\frac{1}{{\sqrt 2 }}$ is irrational.
View full solution →Prove that $3+2\sqrt { 5 }$ is irrational.
View full solution →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?
Explain why 7 $\times$ 11 $\times$ 13 + 13 and 7 $\times$ 6 $\times$ 5 $\times$ 4 $\times$ 3 $\times$ 2 $\times$ 1 + 5 are composite numbers.
View full solution →Prove that $\sqrt 5 $ is irrational.
View full solution →Decimal form of rational numbers can be classified into two types.Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form $\frac{\text{p}}{\text{q}}$, where p and q are co-prime and the prime factorisation of q is of the form $2^\text{n}\times5^\text{m},$ where n, rn are non-negative integers and vice-versa.
Let $\text{x}=\frac{\text{p}}{\text{q}}$ be a rational number, such that the prime factorisation of q is not of the form $2^\text{n}\times5^\text{m},$ where n and m are non-negative integers. Then x has a non-terminating repeating decimal expansion.
- $\frac{441}{(2^{2}\times5^7\times7^2)}$ is which decimal?
- $\frac{251}{(2^5\times\text{5}^3)}$ is which decimal?
- does $\frac{15}{1600}$ have a terminating decimal expansion?
Or
$\frac{23}{(2^{5}\times5^3)}=$
View full solution →HCF and LCM are widely used in number system especially in real numbers in finding relationship between different numbers and their general forms. Also, product of two positive integers is equal to the product of their HCF and LCM.
Based on the above information answer the following questions.
- A boy with collection of marbles realizes that if he makes a group of 5 or 6 marbles, there are always two marbles left, then p will be odd, even, prime or not prime?
- Find the least positive integer which on adding 1 is exactly divisible by 126 and 600.
- Find the largest possible positive integer that will divide 398, 436 and 542 leaving remainder 7, 11, 15 respectively.
Or
If A, B and Care three rational numbers such that 85C - 340A = 109, 425A + 85B = 146, then the sum of A, B and C is divisible by?
ln a classroom activity on real numbers, the students have to pick a number card from a pile and frame question on it if it is not a rational number for the rest of the class. The number cards picked up by first 5 students and their questions on the numbers for the rest of the class are as shown below. Answer them.
- Ananya picked up $\sqrt{15}-\sqrt{10}$ and her question was $\sqrt{15}-\sqrt{10}$ is ______ number.
- Suraj picked up $\sqrt{8}$ and his question was which type of number it was?
- Preethi picked up $\sqrt{6}$ and her question was what will be irrational number of $\sqrt{6}$?
Or
Preethi picked up $\sqrt{9}$ and her question was what will be irrational number of $\sqrt{9}$?
View full solution →Srikanth has made a project on real numbers, where he finely explained the applicability of exponential laws and divisibility conditions on real numbers. He also included some assessment questions at the end of his project as listed below. Answer them.
1. The statement 'One of every three consecutive positive integers is divisible by 3 ' is?
2. For what value of $n, 4^n$ ends in 0 ?
3. If $a$ is a positive rational number and $n$ is a positive integer greater than $I$, then for what value of $n, 4^n$ is a rational number?
Or
If n is any odd integer, then $n ^2-1$ is divisible by?
View full solution →Real numbers are extremely useful in everyday life. That is probably one of the main reasons we all learn how to count and add and subtract from a very young age. Real numbers help us to count and to measure out quantities of different items in various fields like retail, buying, catering, publishing etc. Every normal person uses real numbers in his daily life. After knowing the importance of real numbers, try and improve your knowledge about them by answering the following questions on real life based situations.
- Two tankers contain 768 litres and 420 litres of fuel respectively. Find the maximum capacity of the container which can measure the fuel of either tanker exactly.
- Pens are sold in pack of 8 and notepads are sold in pack of 12. Find the least number of pack of each type that one should buy so that there are equal number of pens and notepads.
- Three people go for a morning walk together from the same place. Their steps measure 80cm, 85cm and 90cm respectively. What is the minimum distance travelled when they meet at first time after starting the walk assuming that their walking speed is same?
Or
ln a school Independence Day parade, a group of 594 students need to march behind a band of 189 members. The two groups have to march in the same number of columns. What is the maximum number of columns in which they can march?