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Mathematics forMathematics forcompetitivecompetitive

examinationexaminationDr. T.K. Jain.Dr. T.K. Jain.

AFTERSCHOOOLAFTERSCHOOOLM: 9414430763M: 9414430763

tkjainbkn@yahoo.co.intkjainbkn@yahoo.co.in

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Mention smallest prime

number? 2

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How many prime numbers

are there till 100? 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.

Count these: Total 25

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Is 1027 a prime number? Step 1. take square root of 1027 to nearestplace lets say 33.

Take all prime numbers till 33

Divide 1027 by each of these numbers and see what remainder do you get - if 1027 is notdivisible by any one of these it is prime number.

2,3,5,7,11,13,17,19,23,29,31 try with these

Since none of these words divide 1027, so it is aprime no.

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ow many num ers arethere till 1000, which are

divisible by 7 and 11? Steps : Divide 1000 by 7= 142 (ignore decimals)

Divide 1000 by 11=90 (ignore decimals) Divide 1000 by 77 =12(ignore decimals)

Add 142+90 and deduct 12

142+90-12 = 220 Ans.

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ic o t e o owing iscoprime:

(23:46),(49:210),(18:25) Find the two numbers which dont

have even one number common among

them. Obviously, 18 and 25 dont have any

common factor as the factors of 18

are 2*3*3, and factors of 25 are5*5 so this is the answer.

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How many our igitnumbers can you make with

1,2,3,4 (repeatition notallowed)?

There are four digits so you canmake 4! (factorials) numbers i.e.

4*3*2*1 = 24 Ans.

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ow many num ers can youmake with 1,2,3,4

(repeatition not allowed)? Four digit numbers: 24 (as per previous solution) Three digit numbers: For first digit, you have four choices, and for

second digit you have three choice and for thirddigit you have 2 choice so ans is : 4*3*2 = 24

For two digit numbers : For first digit you have 4 choice and for second

digit you have 3 choices so : 4*3 = 12

For one digit number you have 4 choice only. Total them up: 24+24+12+4 = 64 Ans.

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How many four digitnumbers can you make with1,2,3,4 (repetition allowed)?

There are four choices for you for

each digit, so the answer is4*4*4*4=256 Ans.

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How many four numbers canyou make with 1,2,3,4(repetition allowed)?

Four digit numbers : 256 Three digit numbers: 4*4*4 =64 Two digit numbers: 4*4=16 One digit numbers: 4 Total: 256+64+16+4= 340

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How many four digitnumbers can you make with

1,2,3,4 (repetition notallowed) which are exactly

divisible by 4? Any number will be divisible by four only if its last twodigits are divisible by 4, so find out all the possibilities for

last two digits. These are : 24,12,32 (3 only), for the firsttwo digits, you have remaining two choices. Thus totalpossibilities are:

3*2 =6 ans.

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How many numbers can you make

with 1,2,3,4 (repetition notallowed) which are exactly divisible

by 4?

Four digit numbers = 6 (previous solution) Three digit numbers = 6 (we have two choices for first digit

and three choices for last two digits, so 2*3=6) Two digit numbers = 3 One digit number =1

Total = 16

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In how many ways, can 4person sit on a round table?

The formulae is (n-1)!

So solution is (4-1)! = 3!

=3*2*1 =6 Ans.

I

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In ow many ways can wemake a necklace from 4

diamonds? Here the formulae is :1/2 (n-1)! =1/2(4-1)!

=1/2(3)! =1/2*(3*2*1)

=3 Ans.

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What is the average speed

of an airplane whichtraveled by 200 km perhour once and 300 Km. per

hour next time (samedistance)?

For all such questions (speed) use harmonic mean. Take harmonic mean of 200 and 300, you will get

240: formulae is ( inverse of (1/200+1/300) )Ans.

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