Principle of Mathematical Induction - Online Test
Q1. For each n N , is divisible by :
Answer : Option D
Explaination / Solution:
when n = 1 the value is 8 which is the multiple pf 8.
Q2. The statement P ( n ) : “ “ is true for :
Answer : Option D
Explaination / Solution:
When n = 1 we get 16>16, which is false. when n = 2 we get 25>32,which is false as well. As n = 3,4,5....the inequalty does not hold correct.
Q3.
The smallest positive integer ‘ n ‘ for which holds is :
Answer : Option B
Explaination / Solution:
When n = 1 ,we get ,which is not valid. When n takes values 2,3,4,5 the inequality is invalid . But when n = 6 we have ,which is valid.
Q4.
If is divisible by 9 for all n N , then the least positive integral value of k is :
Answer : Option C
Explaination / Solution:
When n = 1 we have 58. Given that the expression is divisible by 9 . Hence the least value of k will be 5,so that the value becomes 63 which is divisible by 9.
Q5. The smallest positive integer n , for which holds :
Answer : Option C
Explaination / Solution:
When n = 1 we get ,which is invalid. When n = 2 we get, which is valid.
Q6.
Let is a prime number . then :
Answer : Option C
Explaination / Solution:
Since when n = 41 we have , which is not a prime number.
Q7. Let that for all natural numbers n. also , if P ( m ) is true , m N , then we conclude that
Answer : Option D
Explaination / Solution:
This criteria is from the basic principle of mathematical induction.
Q8. Consider the statement P ( n ) : “ “ . Here for all natural numbers n . Does it mean
Answer : Option B
Explaination / Solution:
Since the statement is not true for n = 1 ,we cannot predict the validity for all natural numbers n.
Q9. The smallest positive integer ‘ n ‘ for which P ( n ) : holds is :
Answer : Option D
Explaination / Solution:
Since the values of n as 1,2,3gives inavlid inequations but when n = 4 we have 16<24, which is valid.
Q10.
If is divisible by x – k for all n belongs to natural numbers N , then the least positive integral value of k is :
Answer : Option A
Explaination / Solution:
since we have x - 1 as a factor of xn - 1n.
