100 out of 100 my doing that notes

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T APPLIED MATHEMATICS
T
FORMULAE AND CONCEPTS CLASS 12




CBSE




Compiled by
Dr. R. Jayasankar M.Sc., M.Phil., Ph.D.
Head Of Mathematics Department
The Indian High School, Dubai

,This book presents the essential formulae and key concepts
required for students’ learning and reference. While every effort
has been made to ensure accuracy, any typographical or
conceptual errors may kindly be brought to my attention through
WhatsApp.
Your valuable suggestions for improvement are most welcome, as
they will help us serve the student community more effectively.

,1. a mod b is a remainder when a is divided by b.
2. a mod 10 unit’s digit of a
3. a mod 100 is last tow digit’s digit of a
4. a mod 1000 is last three digit’s digit of a
5. a+b ( mod m )= a ( mod m ) + b ( mod m )  mod m

6. a-b ( mod m )=  a ( mod m )-b ( mod m ) mod m

7. a  b ( mod m )= a ( mod m )  b ( mod m )  mod m

8. If a and b are positive integers, then
 a+b if a+b < m
9. a+m b ( mod m )= 
remainder when a+b is divided by m if a+b  m
 a-b if a-b < m
10. a-m b ( mod m )= 
remainder when a-b is divided by m if a-b  m
 ab if a  b < m
11. a m b ( mod m )= 
remainder when a-b is divided by m if a  b  m
12. Congruence modulo: Let a and b be any two positive integers and m is a fixed positive integer > 1
then we write a  b (mod m) which means a and b leaves same remainder when divided by m.
13. a  b (mod m)  a-b is divisible my m (or) m divides a-b (or) a-b is a multiple of m or m is a
factor of a-b.
14. a  0 (mod m)  m divides a
15. a  a (mod m)
16. a  b (mod m)  b  a (mod m)
17. a  b (mod m)  -a  -b (mod m)

18. a  b (mod m)  a n  b n (mod m)
19. a  b (mod m)  a + k  b+k (mod m) for some integer k.
20. a  b (mod m)  a - k  b-k (mod m) for some integer k.
21. a  b (mod m)  a  k  b  k (mod m) for some integer k.
22. a  b (mod m) and c  d (mod m) then the following are true
23. a +c  b+c (mod m)
24. a -c  b-c (mod m)
25. a  c  b  c (mod m)

, 1. Let q1 and q2 be quantities of type A and B respectively. Let c, d and m be the price of cheaper,


Quantity of cheaper q1 d − m
dearer and mean price then = =
Quantity of dearer q2 m − c


 a   c 
  qA +   qB
e  a+b  c+d 
2. Mixture of Mixtures =
f  b   d 
  qA +   qB
 a+b c+d 

a:b is the ratio of two ingredients of quantity A (qA)

c:d is the ratio of two ingredients of quantity B (qB)

e:f is the ratio of two ingredients in the mixture of A and B

3. If a container contains x Litres of liquid A from which y litres are taken out and replaced by y litres

of another liquid B and the process is repeated n-times, then the

n
 y
4. Quantity of A left is x 1 − 
 x

n
 y
5. Quantity of b left is x-x 1 − 
 x

n
 y
1 − 
6. Ratio of two quantities is  x
n
 y
1 − 1 − 
 x