CBSE Class 10 Mathematics Real Numbers VBQs Set 04

Assertion and Reason Questions

Question. Assertion (A): \( 2 \times 3 \times 5 \times 7 + 7 \) is a composite number.
Reason (R): Every composite number can be expressed as product of primes.
(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.
Answer: (a) Both A and R are true and R is the correct explanation of A.

Question. Assertion (A): For no value of \( n \), where \( n \) is a natural number, the number \( 6^n \) ends with the digit zero.
Reason (R): For a number to end with digit zero, its prime factors should have 2 and 5.
(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.
Answer: (a) Both A and R are true and R is the correct explanation of A.

Question. Assertion (A): If LCM of two numbers is 2475 and their product is 12375, then their HCF is 5.
Reason (R): HCF (a, b) \( \times \) 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.
Answer: (a) Both A and R are true and R is the correct explanation of A.

Case-based Questions

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.

Question. 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?
Answer: Minimum number of books \( = LCM(32, 36) = 288 \).

Question. Express 36 as a product of its primes.
Answer: \( 36 = 2^2 \times 3^2 \).

Question. If there are 36 students in section A and 44 students in section B, what is minimum number of books you will acquire for the class library so that they can be distributed equally among students of section A or B.
Answer: Minimum number of books \( = LCM(36, 44) = 396 \).

Question. Find HCF of 32, 36 and 44.
Answer: \( 32 = 2^5, 36 = 2^2 \times 3^2, 44 = 2^2 \times 11 \). HCF \( = 2^2 = 4 \).

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

Question. In each room the same number of participants are to be seated and all of them being in the same subject. Find the maximum number of participants that can be accommodated in each room.
Answer: Maximum number of participants \( = HCF(60, 84, 108) = 12 \).

Question. What is the minimum number of rooms required during the event?
Answer: Total participants \( = 60 + 84 + 108 = 252 \).
Number of rooms \( = 252 / 12 = 21 \).

Question. Find the LCM of 60, 84 and 108.
Answer: \( LCM = 3780 \).

Question. Find the product of HCF and LCM of 60, 84 and 108.
Answer: \( 12 \times 3780 = 45360 \).

Aditi plantations have two rectangular fields of the same width but different lengths. They are required to plant 168 trees in the smaller field and 462 trees in the larger field. In both fields, the trees will be planted in the same number of rows but in different number of columns.

Question. What is the maximum number of rows in which the trees can be planted in each of the fields?
Answer: Maximum rows \( = HCF(168, 462) = 42 \).

Question. If the trees are planted in the number of rows obtained in part (i), how many columns will each field have?
Answer: Columns in smaller field \( = 168 / 42 = 4 \).
Columns in larger field \( = 462 / 42 = 11 \).

Question. If total cost of planted trees in one column is Rs. 500, then find the cost to plant the trees in smaller field.
Answer: Cost for smaller field \( = 4 \times \text{Rs. } 500 = \text{Rs. } 2000 \).

Question. If the total cost of planted trees in one column is Rs. 500, find the cost to plant the trees in larger field.
Answer: Cost for larger field \( = 11 \times \text{Rs. } 500 = \text{Rs. } 5500 \).

Assess Yourself

Question. On MG road, three consecutive traffic lights change after 36, 42 and 72 seconds. If the lights are first switched on at 9.00 am, at what time will they change simultaneously?
(a) 9:08:04
(b) 9:08:24
(c) 9:08:44
(d) None of the options
Answer: (b) 9:08:24

Question. What is the smallest number which when increased by 6 becomes divisible by 36, 63 and 108?
(a) 750
(b) 752
(c) 754
(d) 756
Answer: (a) 750

Question. When \( 2^{256} \) is divided by 17 the remainder would be
(a) 1
(b) 16
(c) 14
(d) None of the options
Answer: (a) 1

Question. If N is the sum of first 13986 prime numbers, then N is always divisible by
(a) 6
(b) 4
(c) 8
(d) None of the options
Answer: (d) None of the options

Question. If \( (-1)^n + (-1)^{4n} = 0 \), then \( n \) is
(a) any positive
(b) any negative integer
(c) any odd natural number
(d) any even natural number
Answer: (c) any odd natural number

Question. The HCF of two numbers is 145 and their LCM is 2175. If one number is 725, find the other number.
Answer: \( \text{Other number} = \frac{145 \times 2175}{725} = 435 \).

Question. Find HCF and LCM of 270, 405 and 315 using Fundamental Theorem of Arithmetic method.
Answer: HCF = 45, LCM = 5670.

Question. Find HCF and LCM of 377, 435 and 667 using Fundamental Theorem of Arithmetic method.
Answer: HCF = 29, LCM = 130065.

Question. Show that \( 3 + 5\sqrt{2} \) is an irrational number.
Answer: Let \( 3 + 5\sqrt{2} = r \), where \( r \) is rational.
\( \implies 5\sqrt{2} = r - 3 \)
\( \implies \sqrt{2} = \frac{r - 3}{5} \).
Since \( r \) is rational, \( \frac{r - 3}{5} \) is rational. But \( \sqrt{2} \) is irrational. This is a contradiction. Hence \( 3 + 5\sqrt{2} \) is irrational.

Question. Prove that \( 2\sqrt{3} - 1 \) is an irrational number.
Answer: Similar to above, let \( 2\sqrt{3} - 1 = r \)
\( \implies 2\sqrt{3} = r + 1 \)
\( \implies \sqrt{3} = \frac{r + 1}{2} \).
LHS is irrational, RHS is rational. Contradiction. Hence irrational.

Question. Prove that \( \frac{2\sqrt{3}}{5} \) is irrational.
Answer: Let \( \frac{2\sqrt{3}}{5} = r \)
\( \implies \sqrt{3} = \frac{5r}{2} \).
LHS is irrational, RHS is rational. Contradiction. Hence irrational.

Question. Two positive numbers M and N are both divisible by 3, 5, 15, 25 and 75. What is the HCF of M and N?
Answer: HCF of M and N is M if M < N (or specifically, it is a multiple of 75).

Question. Find the number nearest to 100000 and greater than 100000 which is exactly divisible by each of 8, 15 and 21.
Answer: LCM(8, 15, 21) = 840. Dividing 100000 by 840 gives remainder 40.
Number \( = 100000 + (840 - 40) = 100800 \).

Question. Find the least number that is divisible by all the natural number from 5 to 15 (both inclusive).
Answer: LCM(5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15) = 360360.

Question. A garden has 48 guava trees, 60 pineapple trees and 96 mango trees. These have to be arranged in rows such that each row has same number of trees and all are of same type. Find the minimum numbers of such rows that can be formed?
Answer: Trees per row \( = HCF(48, 60, 96) = 12 \).
Total rows \( = (48/12) + (60/12) + (96/12) = 4 + 5 + 8 = 17 \).

Question. Four bells commence tolling together. They toll at intervals of \( 2, 2\frac{1}{4}, 4\frac{1}{2} \) and \( 2\frac{3}{4} \) seconds respectively. After what time will they toll together again?
Answer: LCM of 2, 9/4, 9/2, 11/4 \( = \frac{LCM(2, 9, 9, 11)}{HCF(1, 4, 2, 4)} = \frac{198}{2} = 99 \) seconds or 1 min 39 sec.

Question. Assertion (A): LCM of 13 and 61 is 793.
Reason (R): If \( a \) and \( b \) are prime, then 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.
Answer: (a) Both A and R are true and R is the correct explanation of A.

Question. Assertion (A): HCF of 23 and 37 is 23.
Reason (R): If \( a \) and \( b \) are prime, then HCF (a, b) = 1.
(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.
Answer: (d) A is false but R is true.

Question. Assertion (A): HCF (105, 225) = 15 and LCM (105, 225) = 225 \( \times \) k, then the value of \( k \) is 7.
Reason (R): For any two positive numbers \( a \) and \( b \), HCF (a, b) \( \times \) 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.
Answer: (a) Both A and R are true and R is the correct explanation of A.

Question. Assertion (A): \( \sqrt{3} \) is an irrational number.
Reason (R): If \( p \) is prime, then \( \sqrt{p} \) is an irrational number.
(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.
Answer: (a) Both A and R are true and R is the correct explanation of A.

Ravish runs a book shop at school of Math, Gurgaon. He received 486 Chemistry books, 216 Physics books and 702 Mathematics books of class XI. He wishes to arrange these books in minimum numbers of stacks such that each stack consists of the books on only one subject and the number of books in each stack is the same.

Question. Find the number of books in each stack.
Answer: Books per stack \( = HCF(486, 216, 702) = 54 \).

Question. Find the number of stacks of Mathematics books.
Answer: \( 702 / 54 = 13 \) stacks.

Question. Find the difference in number of stacks of Mathematics books and sum of stacks of Physics and Chemistry books.
Answer: Chemistry stacks \( = 486 / 54 = 9 \). Physics stacks \( = 216 / 54 = 4 \).
Difference \( = 13 - (9 + 4) = 0 \).

Question. If the thickness of each book of Physics is 2.5 cm, then find the height of each stack of Physics book.
Answer: Height \( = 54 \times 2.5 = 135 \text{ cm} \) or 1.35 m.