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Let g_i (x) represent any odd function,

Let $f(x)=g_1(x).g_2(x)...g_{2n+1}(x)$

For an odd function,

f(-x)=-f(x)

$f(-x)=g_1(-x).g_2(-x)...g_{2n+1}(-x)$

$=(-1)(2n+1)[g_1(x).g_2(x)...g_{2n+1}(x)]$

$\Rightarrow$

$\Rightarrow\ it\ is\ an\ odd\ function$

i have heard that intermediate marks weightage fr eamcet is being removed...is it true?

Vapour pressure does not depend on the surface area of the liquid.

No doubt increase in surface area increases the total rate of evaporation, but at the same time it increases the rate at which particles condense...and thus the vapour pressure remains unaffected

The basic principle is that,

The no of ways of dividing n identical things into r ordered groups with 1,2,3,...k things in any group is same as the no of ways of expressing the natural number n as the sum of r integers using and considering the order of addition.

This is same as the no of solutions of the equation

x_1+x_2+...x_r = n ,

where x_i can take any value and this is equal to the coefficient of x^n in the expansion of

${(x^1+x^2+...x^k)}^r$

For example, consider a case where 3 identical balls are to be arranged into 2 groups, each group containing any number of things, $r(r \ge 1)$

the possible groups are group A with 1 ball, group B with 2 balls -(1,2)

group A with 2 balls, group A with 1 ball -(2,1)

consider the expansion of (x+x^2+x^3)^2

=(x+x^2+x^3)(x+x^2+x^3)

$=x^{1+1}+\boxed{x^{2+1}}+x^{3+1}+\boxed{x^{2+1}}+x^{2+2}+x^{2+3}+x^{3+1}+x^{3+2}+x^{3+3}$

the exponents which add to 3 are the second n 4th terms, $x^{2+1}$

which is same as the coefficient of x^3 in the expansion ...

once u have understood the basic idea you yourself will come to know when and where to apply it...

Statement 1:The differential equation of all the circles on a plane must be of order 3

Statement 2:There is only one circle passing through three non collinear points

Does statement 2 explain statement 1? how?

hey u said the correct ans is 89...?

From among 4 oranges you can select either 0 or 1 or 2 or 3 or 4 oranges, so this can be done in (4+1)=5 ways

lly from among 5 apples..u can select 1 or more in (5+1) ways

lly from among 6 mangoes, u can select 1 or more in (6+1) ways.

but this includes the selection in which no fruit is selected...so no of selections which can be made = (4+1)(5+1)(6+1)-1

so total no of selections = 209

yeah...sry...they are conducting shells

Find the electric potentials at A,B,C,D,E,F & G, if the inner shell is given a charge +q and the outer shell is given a charge +q'

(explain ur working)

How to find the effective capacitance of a capacitor having many parallel plates as shown ?

Take surface area of each plate = A

AB=CD=EF=2d

BC=DE=d

(edited: P and Q are connected to the battery)

u are welcome :)

@v89

ofc it can...infact to get the maximum number you SHOULD take that into consideration..

Let me explain my working..

suppose there are only two circles..they can intersect atmost at only 2 points..

Now take the case of three circles..the 1st can intersect the 2nd at points, 2nd can intersect the third at 2 points, and third can intersect the 1st at two

so total no of points of intersection = 6 = 3C2 * 2

now take the case of 4 circles...

1-->2 at 2 pts

1-->3 at 2 pts

1-->4 at 2 pts

2-->3 at 2 pts

2-->4 at 2 pts

3-->4 at 2 pts

so total no of points of intersection = 4C2 * 2

extending this to n circles...it will be nC2 * 2