# 158 - Mathematics Questions Answers

help me understand lamis theorem

Joshi sir comment

It is the theorem for the equilibrium of a body under 3 forces

According to the theorem if three forces are acting on a body and body is in equilibrium then

P/sinA = Q/sinB = R/sinC Maths >> Calculus >> Differential Equations Engineering Exam
Differential form of ol conics whose axes coincide with the co-ordinate axes??
Joshi sir comment

let us consider the case of ellipse with x and y axes as their axes

eq. is x2/a2 + y2/b2= 1

on differentiating we get 2x/a2 + 2yy'/b2 = 0                                y'is first differential

so yy'/x = -b2/a2

You should remember that you should differentiate as many times as the number of constants.

for ex. in the case of parabola only first diffrentiation is sufficient.

now complete it for all conics

Maths >> Algebra >> Matrices and Determinant Engineering Exam

prove using property of determinant

1.  a+b+c      -c                 -b

-c               a+b+c           -a                   =  2(a+b)( b+c ) (c+a)

-b              -a                     a+b+c

Joshi sir comment

first use C1 = C1+C2+C3

then R1 = R1+R2 and R2 = R2+R3

after it solve mathematically

using the transformation x=r cosθ and y=r sinθ,find the singular solutionof the differential equation x+py=(x-y)(p2+1)½           where p=dy/dx

Joshi sir comment

x=rcosθ and y=rsinθ

so dx=-rsinθdθ  and dy = rcosθdθ

so p = -cotθ

so given eq. will become x-cotθy = (x-y)cosecθ

on putting values of x and y we get

0 = r(cosθ-sinθ)/sinθ

so tanθ = 1  so θ = nπ+π/4

so x = rcos(nπ+π/4) and y = rsin(nπ+π/4)

show that the family of parabolas y2=4a(x+a) is self orthogonal.

Joshi sir comment

y2= 4a(x+a)

so 2yy' = 4a so a = yy'/2

on putting a we get y2= 4yy'/2(x+yy'/2)

so y2 = yy' (2x+yy')  or y = 2xy' + yy'2    (1)

now on putting -1/y' in the place of y'

we get y2 = -y/y'[2x-y/y']

so -yy'2 = 2xy' - y (2)

similarity of (1) and (2) shows that the given curve is self orthogonal

rank of the number 3241?(digits cannot be repeated)

Joshi sir comment

total numbers = 4*3*2*1 = 24

on arranging these numbers in order we get 1234

so rank = [24/4]*2+[6/3]*1+[2/2]*1+1 = 16

Maths >> Algebra >> Matrices and Determinant Engineering Exam

## which is the 4 digit number whose second digit is thrice the first digit and 3'rd digit is sum of 1'st and 2'nd and last digit is twice the second digit.

Joshi sir comment

let the 4 digit number is 1000a+100b+10c+d

then according to the given condition b=3a, c=a+b, d=2b

so c=4a, d=6a and b=3a

on putting all the values we get number = 1000a+100(3a)+10(4a)+6a = 1346a

on putting a =1, number = 1346

Sir , i have solved   limx-->0    (sin-1x  - tan-1x )/x. plz have a look on others.

Joshi sir comment

limx-->0    (sin-1x  - tan-1x )/x3

use D L Hospital rule or expansion of sin-1x and tan-1x to solve it

no of solutions of  e-x^2/2 - x2 = 0 are

Joshi sir comment two intersecting points means two solutions

lim x»infinity( ((x+1)(x+2)(x+3)(x+4))^¼ - x )