Mathematics Dictionary

Factor theory : (x-a) is a factor of F(x) if F(a) = 0
Factorial n! = n*(n-1)*(n-2)*....*3*2*1
Fibonacci's sequence
- T(0) = 0 and T(1) = 1
- T(n) = T(n-1) + T(n-2) and n GT 1
- Sequence : 1, 1, 2, 3, 5, 8, 13, 21, ....
05 04 Focus of parabola

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Q07. G

08 02 Geometric series

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Q08. H

Hyperbolic functions

cosh(x) = ((e^x) + (e^(-x)))/2
sinh(x) = ((e^x) - (e^(-x)))/2
tanh(x) = sinh(x)/cosh(x)

Hyperbolic functions : Identities

cosh(x)^2 - sinh(x)^2 = 1
sinh(x+y) = sinh(x)*cosh(y) + cosh(x)*sinh(y)
cosh(x+y) = cosh(x)*cosh(y) + sinh(x)*sinh(y)
sinh(x-y) = sinh(x)*cosh(y) - cosh(x)*sinh(y)
cosh(x-y) = cosh(x)*cosh(y) - sinh(x)*sinh(y)
sinh(2*x) = 2*sinh(x)*cosh(y)
cosh(2*x) = cosh(x)^2 + sinh(x)^2

Hyperbolic functions : Compare with trigonometric function in PM section 14

Hypergeometric probability

H(x) = C(n,x)*C(N-n,n-x)/C(N,n)
H(x) = C(n,x)*C(N-n,s-x)/C(N,s)

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Q09. I

Induction method

14 00 Induction method
14 06 Induction : S(n) = 1/(1*3) + 1/(3*5) + .... = (1 - 1/(2*n+1))/2
14 07 Induction : S(n) = 1/(1*2*3) + 1/(2*3*4) + .... = n*(n+3)/(4*(n+1)*(n+2)
14 08 Induction : S(n) = 1^2 + 3^2 + 5^2 + ... = n*(4*n^2 - 1)/3
14 09 Induction : S(n) = 1*(2^2) + 2*(3^2) + ... = n*(n+1)*(n+2)*(3*n+5)/12
14 10 Induction : S(n) = (1 -4/1)*(1 - 4/9)*(1 - 4/25)*(1 - 4/49)*......
14 11 Induction : S(n) = (1 + 1)*(1 + 1/2)*(1 + 1/3)*.....*(1 + 1/n) = n + 1
14 12 Induction : S(n) = (1^2)/(1*3) + (2^2)/(3*5) + .... = (n*(n+1))/(2*(2*n+1))
14 13 Induction : S(n) = 2 + 4 + 6 + ..... + 2*n = (n + 1/2)^2. Is it true
14 14 Induction : S(n) = n^2 + n + 41. Is it always a prime number ?
14 17 Induction : Prove that 1! + 2! + 3! + 4! + ..... + n! = 3^(n-1)

15 00 Inequality

15 02 Solve x^2 - 6*x + 8 < 0
15 08 Solve (x-1)/(x+1) > 1
15 09 Solve (x-1)/(x+1) > 2

Inverse of y = e^x is y = ln(x)

e^(ln(x)) = x
ln(e^x) = x

Inverse of y = ln(x) is y = e^x

e^(ln(x)) = x
ln(e^x) = x

05 03 Inverse of y = a*x^2 + b*x + c is x = a*y^2 + b*y + c
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Q10. J

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Q11. K

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Q12. L

11 01 Linear equations
09 02 Logarithmic laws

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Q13. M

03 04 Magic number patterns
03 05 Matrix pattern : Find row number and column number of 100
- 1, 03, 06, 10, 15, 21, ...
- 2, 05, 09, 14, 20, .......
- 4, 08, 13, 19, ...........
- 7, 12, 18, ...............
03 10 Multiple of number whose digits are same.

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Q14. N

17 12 Normal distribution : N(z LT a)
03 11 Number : Amicable number pairs.
03 05 Number : Numbers arranged in matrix. 1st row sequence 1,3,6,10, ...
03 12 Number : Perfect numbers.
03 03 Number : Properties based on factors of number.
14 15 Number : Prove that ((n^3) + 3*(n^2) + n)/3 is an integer
14 16 Number : Prove that (2*(n^3) + 3*(n^2) + n) is divisible by 6
07 01 n! : definition and trailor zeros in n!

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Q15. O

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Q16. P

05 04 Parabola : Locus, Focus and directrix
08 05 Pascal triangle : Sequence and series
03 12 Perfect number : How to find the third perfect number ?
03 07 Perfect square numbers.
17 01 Permutation : P(n,r)
- 17 02 Take r from N symbols for arrangement without duplicate : P(n,r)
- 17 02 Take r from N symbols for arrangement with duplicate : n^r
08 13 Pi in series
21 00 Polynomial equation
- 21 11 Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + x + 1 = 0

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Q17. Q

11 04 Quaint equation
21 01 Quaint equation : Graphic solution
12 06 Quartic equation : (x-p)*(x-q)*(x-r)*(x-s) = t
11 03 Quartic equation : (x-5)*(x-7)*(x+4)*(x+6) = 504
21 01 Quartic function : y = a*x^4 + b*x^3 + ....
21 01 Quartic function : y = (x-p)*(x-q)*(x-r)*(x-s) - t
11 01 Quadratic equations
11 10 Quadratic equations : x^2 -5*x + 2*Sqr(x^2 - 5*x + 3) = 12
05 01 Quadratic functions : Defintion and application
05 02 Quadratic functions : With absolute operation
05 03 Quadratic functions : Inverse
05 04 Quadratic functions : Parabola

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Q18. R

16 01 Rational function : y = 1/(x + 1)
16 02 Rational function : y = 1/(x^2 - 1)
16 03 Rational function : y = 1/(x^3 -2*x - x + 2)
16 06 Rational function : y = x + 4/(x^2)
16 04 Rational function : y = (x^2 -2*x + 1)/(x)
16 05 Rational function : y = ((x - 1)^3)/(2*x)
03 01 Real number system
01 04 Remainder theory

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Q19. S

07 04 Sequence in Pascal triangle.
- T(1)
- T(n)
- T(n*(n+1)/2!)
- T(n*(n+1)*(n+2)/3!)
- T(n*(n+1)*(n+2)*(n+3)/4!)
08 01 Series : A.P.
08 02 Series : G.P.
08 13 Series : pi
08 03 Series : Sum[n^2] = n*(n+1)*(2*n+1)/6.
08 04 Series : Sum[n^3] = (n*(n+1)/2)^2.
07 08 Series : Sum[n^4] = ?
08 05 Series : Series and sequence in Pascal triangle.
- Sum[n*(n+1)/2!] = n*(n+1)*(n+2)/3!
- Sum[n*(n+1)*(n+2)/3!] = n*(n+1)*(n+2)*(n+3)/4!
- Sum[n*(n+1)*(n+2)*(n+3)/4!] = n*(n+1)*(n+2)*(n+3)*(n+4)/5!
08 05 Series : Series and sequence in Pascal triangle.
- Sum[C(n+1),2)] = C(n+2,3)
- Sum[C(n+2),3)] = C(n+3,4)
- Sum[C(n+3),4)] = C(n+4,5)
08 06 Series : Special
- S(n) = 1/(1*2) + 1/(2*3) + 1/(3*4) + ........ + 1/((n-1)*n) = ?.
- S(n) = 1/(1*2*3) + 1/(2*3*4) + ... + 1/((n*(n+1)*(n+2)) = ?
- S(n) = 1/3 + 1/15 + 1/35 + ...... + 1/((2*n-1)*(2*n+1)) = ?
08 08 Series : 1^3 + 3^3 + 5^3 + ..... (2*n-1)^3 = ?
08 09 Series : 1*(2^2) + 2*(3^2) + 3*(4^2) + ..... n*((n+1)^2) = ?
08 07 Series : S(n) = 1 - 2 + 3 - 4 + 5 - 6 + ....... - n if n is even
08 08 Series : 1^3 + 3^3 + 5^3 + .... = ?
08 09 Series : 1*(2^2) + 2*(3^2) + 3*(4^2) + .... = ?
08 10 Series : 1^3 + 2^3 + ... + n^3 GT (n^4)/4 GT 1^3 + 2^3 + ... +(n-1)^3
08 11 Series : (1-1/4)*(1-1/9)*(1-1/16)*....*(1-1/((n+1)^2 = (n+1)/2*n. n GT 1
08 12 Series : 1 - 2 + 3 - 4 + ...... + n and n is odd
11 10 Simultaneous equations
- x + y = 3 (line)
- x*y = 2 (Hyperbola)
11 10 Simultaneous equations
- x - y = 1
- x*y = 2 (Hyperbola)
11 10 Simultaneous equations
- x^2 + y^2 = 4 (Circle)
- x*y = 1 (Hyperbola)
16 06 Slant asymptote in y = x + 4/(x^2)
03 07 Square free numbers.
03 08 Square root : How to find Sqr(3) ?