Diploma Maths 3 Imp Notes

M3 Notes

·       

For Metal wire 100cm X & Y = I & b 2x+2y=100 Y=50-x Area =x.y A=x(50-x)

80 By 2 parts 

1.    Check condition

2.   Find eq^{n    |}   _{x+y=80 ; (product=x(80-x)}

3.   Diff. w.r.t x

4.   =0

5.   Find x & y

6.   Last answer

·       

For Metal Sheet L=16-2x B=10-2x H=x V= l *b * h Diff. w .r .t.x (v)

Maxima & Minima      ( Least/Less*120/x}

1.    Y=Eqⁿ

2.   Diff. w.r.t x

3.   =0

4.   Find X

5.   Again Diff. w.r.t x

6.   Max=negative & Min=Positive ……..Put X and find it

7.   Find Ymax & Ymin By Substituting x in Original Eqⁿ

·        Tangent Normal

1.    Y=Eqⁿ

2.   Find (x,y) co-ordinates from quadratic Eqn

3.   Diff. w.r.t x

4.   Dy/dx = m

5.   M’=   -1

       

6.   T → y-y1 = m(x-x1)

7.   N→ y-y1 = m’(x-x1)

·        Radius of Curvature

1.    Diff. w.r.t x

2.   Find

3.   Again Diff. w.r.t x

4.   Find

5.  R=     [²] ^3/2

                  

·       Integration – Third term ( Denominator = quadratic eqn)

1.   Check eqⁿ

2.  Find Third term (TT)  →      ( ) ²

3.  Add & Subtract it

4.  Make it in like (x ²  +/-  a² ) Form

5.  Use (x & a ) Formula

6.  Write I = Solved eqⁿ

·       Integration By Parts  -  ( L I A T E )

1.   Use LIATE and find  u & v

2.  Use

3.   =  u

4.  Then  solve that eqⁿ

·       Integration : Denominator = sin²/ cos²

1.   Divide   by cos²x

2.  Put tan x = t

3.  Diff. tan x  w.r.t. “t”

4.  Then solve

·       Integration : By Substitution

1.   Find the term

2.  Put tan x/sin x  = t

3.  Diff. tan x/sin x  w.r.t. “t”

4.  Find dt = ?

5.  Put t & dt = I

6.  Solve

·       Integrals of form

1.   Put sin 2x =  OR  sin x =

2.  Put  & t = tan x

3.  Put & solve

·    Partial Fraction  

1.   Find Eqⁿ : Put as A,B,C……

2.  Cancel Denominator

3.  Find  from eqⁿ

4.  Find A , B , C  values = ?

5.  Put A,B,C in eqⁿ and  solve

·       Evaluate Integration

1.   Solve derivative using formulas (carefully)

2.  Solve integration

3.  Put the limits

4.  Solve

·   Problems based on

1.   Write eqⁿ as no. 1

2.  Use the property

3.  Replace

4.  Solve  and named as no. 2

5.  Add results (1) & (2)

6.  Solve & put limits

·       Find area under the curves

1.   Draw diagram

2.  Required area =

3.  Solve & last ans = sq. units

·       Find area between two curves

1.   Euate both x & y

2.  Find when x = ……………

3.  Diagram

4.  Write y1 & y2

5.  Required area =  

·    Exact D.E {  }      

1.    Put m = dx & n = dy

2.   Diff w.r.t x & y partially

3.   If equal then the given eqⁿ is Exact D.E

4.   Last ans

·        Verify eqⁿ

1.    Write eqⁿ

2.   Diff. w.r.t. x

3.   Again Diff. w.r.t.x

4.   Find the eqⁿ

·        Probability

1.    P () = A

2.  Addition theorem = P(AB) = P(A) + P(B) – P(AB)

3.   P(A) + P(A)’ = 1

4.   P(A)’ = 1 - P(A)

5.   P(A) = 1 - P(A)’

6.  (AB)’ = AB’

7.  = Multiplication  |   = Addition

·        Mean = m = n.p

·    S.d =

·        Variance = npq

·     Coefficient of variation =

·        Poisson’s distribution

P(x = r) → e^-m  * m^r

                     r!

·    Z=

·    Mean =

·        Phase card = queen / king / joker

·        Sample space = nC ^r

·        Problem Statements :

1.   Solved = 1- (AB’C’ )

2.   Not solved = P(A)* P(B)*P(C)

·    E() =

       =

·        Variance = v(x)

·    V(x) = ²) - ²

·    ²) = ²)^{2 2})²

·        P.D.

P(x=t) → ⁿC_rP^rq^n-r

·       q= 1-p

ü n= no. of trails

ü r =selected trails

ü p = probability

ü q = failure