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Subject  >  Math  >  Number Theory  >  Question and Answers 4

Number Theory

   
 
Question (1): Define an even number.

Answer:  A number which is divisible by 2 is called an even number.
Question (2): Define an odd number.

Answer:  A number when divided by 2 gives remainder 1 is called an odd number.
Question (3): What is a prime number?

Answer:  A number which has only two factors, namely 1 and then number itself is called a prime number.
Question (4): What is a composite number?

Answer:  A number which has more than two factors is called composite number.
Question (5): When a rational number can be expressed as terminating decimal?

Answer:  A rational number can be expressed as terminating decimal, when the factors of its denominator are in the form 2n x 5m
Question (6):
Answer: 
We know that the (r + 1)th term of (x + a)n is given by









Question (7):
Answer:  Expanding by binomial theorem, we get






Question (8):
Answer: 









Question (9):
Answer: 
To find the two middle terms:

Since n = 9, the two middle terms are



Question (10):
Answer: 




Question (11):
Answer: 




The greatest value of 'r' can be 8. Then, the greatest term in the expansion is the 9th term.




Question (12): Find the greatest term in the expansion of
(2a + b)14 when a = 4, b= 5.
Answer: 


Hence the greatest term is the 6th term.

Question (13): If A and B be the sum of the odd terms and the sum of the even terms separately in the expansion of (x + a)n, then

Answer: 


Again,




Question (14): In the expansion (x + y)n the coefficients of 4th and 13th terms are equal. Find n.
Answer: 






Question (15):
Answer: 
Here y = 4x, x = 1, n = 8






6th term is the greatest.



Question (16): Evaluate (0.998)8 correct to 6 decimal places.
Answer: 





Question (17):
Answer: 
Put x = -1








Question (18): Prove that

Answer: 



Question (19): Use the Binomial Theorem to (expand) evaluate.

Answer: 

Question (20): Write and simplify the term involving in the expansion
Answer: 






Question (21): Simplify the term independent of x in the expansion of
Answer: 





Question (22): Find the term independent of x in the expansion of

Answer: 


\There is no term containing xo i.e., independent of x.
Question (23):
Answer: 




Question (24):
Answer: 





Note:

Question (25): Find the greatest term in the expansion of (2x + 3y)28 when x = 9, y = 4.
Answer: 




The greatest value of 'r' can be 11.
Hence, the greatest term is the 12th term.

Question (26):
Answer: 






Question (27): The first three terms in the expansion of (x + y)n are 1, 56 and 1372 respectively. Find the values of x and y.
Answer: 



But x = 1



Question (28): Using binominal theorem, show that

Answer:  i)

which is divisible by 64.
ii)

which is divisible by 25.
Question (29):
Answer: 









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