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Numbers & Algebra - Numbers and Algebra MCQ

21:  

Which of the following is true?

A.

Sum of four consecutive even numbers is always divisible by 8.

B.

Sum of four consecutive odd numbers is always divisible by 8.

C.

Product of any n consecutive natural numbers may not be divisible by n!.

D.

Product of 4 consecutive odd numbers is always divisible by 15.

 
 

Option: B

Explanation :

(A) Four consecutive even numbers can be written as 2n, 2n + 2, 2n + 4 and 2n + 6,where n is any natural number.
Sum = 2n +(2n + 2) + (2n + 4) + (2n +6) = 8n + 12
= 4(2n + 3) not always divisible by 8.
Thus, (A) is not true.
(B) Four consecutive odd numbers can be written as 2n-1, 2n+ 1, 2n + 3, 2n + 5 where n is a natural number
Sum = (2n-1) + (2n + 1) + (2n + 3) + (2n + 5)
= 8n + 8 = 8(n + 1) divisible by 8
Thus, (B) is true
(C) In product of n consecutive natural numbers atleast one is divisible by n, atleast one by n - 1 ... till 1.
Thus product is atleast divisible by
n * (n - 1) * (n - 2) *1  =  n!.
Thus, (C) is not true.
(D) Take four consecutive odd numbers as 7 x 9 x 11 x 13 which is not divisible by 15.
Thus, (D) is not true.

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22:  

Sum of an infinite geometric series is 3 and sum of an infinite geometric series formed from squares of the terms of the original series is 6. First term of the first series will be

A.

B.

C.

D.

 
 

Option: D

Explanation :

Also,

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23:  

 Sum of all odd numbers up to 100 is

A.

5050

B.

2500

C.

2525

D.

5000

 
 

Option: B

Explanation :

Given numbers are 1, 3, 5.....99.
This is an A.P with a = 1, d = 2
Let it contain n terms
∴ 1 +(n-1)2 = 99
=> n = 50
Hence required sum = n/2 (first term +last term)
= 50/2(1 + 99) = 2500

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24:  

If x+y+z=0,then x2/yx + y2/zx + z2/xy equal

A.

0

B.

1

C.

2

D.

3

 
 

Option: D

Explanation :

x2/yx + y2/zx + z2/xy
             ...(since, x+y+z=0  'x3+y3+z3=3xyz)
(x3+y3+z3) / xyz = 3xyz / xyz = 3.

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Devanshu said: (4:20am on Friday 1st July 2016)
Let x=2 y =-1 z=-1 then x y z=0 but RHS is - 3?????Btw if x y z = 0 then x^3 y^3 z^3=3xyz???? I think it should be - 3xyz then ans satisfied

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25:  

Average age of a committee of seven trustees is the same as it was five years ago, a younger man having been substituted for one of them. How much younger was he then trustee whose place he took?

A.

32 years

B.

35 years

C.

33 years

D.

34 years

 
 

Option: B

Explanation :

Increase after 5 years should have been 7 x 5 = 35 years.
But since average remains same, diference in their age is 35 years.

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