Answer :
Sure, let's go through each of the questions and provide detailed explanations for each.
### Question (i)
For what value of \( n, 4^n \) ends in 0?
Options:
(a) 10
(b) when \( n \) is even
(c) when \( n \) is odd
(d) no value of \( n \)
To answer this question, we need to understand the form of \( 4^n \). The number \( 4^n \) is \( (2^2)^n = 2^{2n} \). Since \( 2^{2n} \) is a power of 2, and powers of 2 end in either 2, 4, 8, or 6, but never in 0. Therefore, there is no value of \( n \) for which \( 4^n \) ends in 0.
Answer: (d) no value of \( n \)
### Question (ii)
If \( a \) is a positive rational number and \( n \) is a positive integer greater than 1, then for what value of \( n, a^n \) is a rational number?
Options:
(a) when \( n \) is any even integer
(b) when \( n \) is any odd integer
(c) for all \( n > 1 \)
(d) only when \( n = 0 \)
A positive rational number raised to any positive integer power is always rational. Therefore, for any \( n > 1 \), \( a^n \) will be rational.
Answer: (c) for all \( n > 1 \)
### Question (iii)
If \( x \) and \( y \) are two odd positive integers, then which of the following is true?
Options:
(a) \( x^2 + y^2 \) is even
(b) \( x^2 + y^2 \) is not divisible by 4
(c) \( x^2 + y^2 \) is odd
(d) both (a) and (b)
For two odd integers \( x \) and \( y \), their squares \( x^2 \) and \( y^2 \) will both be odd because the square of an odd number is always odd. The sum of two odd numbers is even, so \( x^2 + y^2 \) is even.
Moreover, if you look at the remainders when squaring odd numbers (they are always 1 modulo 4), the sum \( x^2 + y^2 \) will be \( 1 + 1 = 2 \) modulo 4, which means \( x^2 + y^2 \) is not divisible by 4.
Answer: (d) both (a) and (b)
### Question (iv)
The statement 'One of every three consecutive positive integers is divisible by 3' is
Options:
(a) always true
(b) always false
(c) sometimes true
(d) None of these
For any set of three consecutive integers \( n, n+1, n+2 \), one of these numbers must be divisible by 3 due to the properties of consecutive integers and the nature of division by 3.
Answer: (a) always true
### Question (v)
If \( n \) is any odd integer, then \( n^2 - 1 \) is divisible by
Options:
(a) 22
(b) 55
(c) 88
(d) 8
For any odd integer \( n \), \( n^2 - 1 \) can be factored as \( (n-1)(n+1) \). This expression consists of two consecutive even integers because \( n \) is odd, and one of those integers will be a multiple of 4 and the other will be an even number. Thus, their product will always be divisible by 8.
Answer: (d) 8
So, the detailed step-by-step answers are:
(i) (d) no value of \( n \)
(ii) (c) for all \( n > 1 \)
(iii) (d) both (a) and (b)
(iv) (a) always true
(v) (d) 8
### Question (i)
For what value of \( n, 4^n \) ends in 0?
Options:
(a) 10
(b) when \( n \) is even
(c) when \( n \) is odd
(d) no value of \( n \)
To answer this question, we need to understand the form of \( 4^n \). The number \( 4^n \) is \( (2^2)^n = 2^{2n} \). Since \( 2^{2n} \) is a power of 2, and powers of 2 end in either 2, 4, 8, or 6, but never in 0. Therefore, there is no value of \( n \) for which \( 4^n \) ends in 0.
Answer: (d) no value of \( n \)
### Question (ii)
If \( a \) is a positive rational number and \( n \) is a positive integer greater than 1, then for what value of \( n, a^n \) is a rational number?
Options:
(a) when \( n \) is any even integer
(b) when \( n \) is any odd integer
(c) for all \( n > 1 \)
(d) only when \( n = 0 \)
A positive rational number raised to any positive integer power is always rational. Therefore, for any \( n > 1 \), \( a^n \) will be rational.
Answer: (c) for all \( n > 1 \)
### Question (iii)
If \( x \) and \( y \) are two odd positive integers, then which of the following is true?
Options:
(a) \( x^2 + y^2 \) is even
(b) \( x^2 + y^2 \) is not divisible by 4
(c) \( x^2 + y^2 \) is odd
(d) both (a) and (b)
For two odd integers \( x \) and \( y \), their squares \( x^2 \) and \( y^2 \) will both be odd because the square of an odd number is always odd. The sum of two odd numbers is even, so \( x^2 + y^2 \) is even.
Moreover, if you look at the remainders when squaring odd numbers (they are always 1 modulo 4), the sum \( x^2 + y^2 \) will be \( 1 + 1 = 2 \) modulo 4, which means \( x^2 + y^2 \) is not divisible by 4.
Answer: (d) both (a) and (b)
### Question (iv)
The statement 'One of every three consecutive positive integers is divisible by 3' is
Options:
(a) always true
(b) always false
(c) sometimes true
(d) None of these
For any set of three consecutive integers \( n, n+1, n+2 \), one of these numbers must be divisible by 3 due to the properties of consecutive integers and the nature of division by 3.
Answer: (a) always true
### Question (v)
If \( n \) is any odd integer, then \( n^2 - 1 \) is divisible by
Options:
(a) 22
(b) 55
(c) 88
(d) 8
For any odd integer \( n \), \( n^2 - 1 \) can be factored as \( (n-1)(n+1) \). This expression consists of two consecutive even integers because \( n \) is odd, and one of those integers will be a multiple of 4 and the other will be an even number. Thus, their product will always be divisible by 8.
Answer: (d) 8
So, the detailed step-by-step answers are:
(i) (d) no value of \( n \)
(ii) (c) for all \( n > 1 \)
(iii) (d) both (a) and (b)
(iv) (a) always true
(v) (d) 8
