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for the equality use:
tan(pi/2 -x)= cotx

for the second equality
let sum=S
then S+S will be the summation of {tanr(pi)/16}^2+ {cotr(pi)/16}^2
2S=summation of [{tanr(pi)/16+ cotr(pi)/16}^2 -2]
2S=summation of [{tanr(pi)/16+ cotr(pi)/16}^2] -14
the term inside the bracket reduces to {cosec2r(pi)/16}^2

Now use the formula
{cosecx}^2={cotx}^2+1

then, ithe cases in which r>3,
use cot(pi-x)=-cotx

I hope it helps

for 1st
let f(x)=x^(1/x)
find its extreme.
you will find that it is maximum at x=e
hence
e^(1/e)>pi^(1/pi)
raise both sides to the power e*pi and you will get your answer.

for second question, analyse the function x*(square root of x).Examine how it varies with x.

the factors are
1, 2, 2^2, 2^3, .......2^(m-1)
and p,2p,2^2*p,, where p is the prime.

sum all of them using gp and it completes your proof.

the basic funda is: volume of water displaced=rise if level of water.

volume of water displaced depends upon densityof the object(in fact, relative density between the object and the liquid).

If density is gerater than that of water, volume displaced= volume of liquid.

If density is less than that of water, volume displaced=V*a/b
where,
V=volume of object
a=density of object
b=density of water.

in the first case, volume displaced=V*c/b
{V is volume of boat, c is combined density of booat and stone)

in the second case, volume displaced=V*a/b + v
{v is volume of stones, d is their density)

now v< hence in second case, volume displaced=V*a/b

since a is definitely less than c, volume displaced in first case is greater.

is the answer 1.643*10^5m/s????

I think numerator is finite at x=4 and denominatr is zero at x=4

if any two diagonals intersect, then the no. of diagonals cannot be 70.

If there are 13 diagonals, and any two diagonals intersect and no 3 diagongonals are congruent, then the number of intersection points are 78{C(13,2)}.

If there are 12 diagonals, and any two diagonals intersect and no 3 diagongonals are congruent, then the number of intersection points are 66{C(12,2)}.

Hence if 70 intersection points are there, thre should be pairs of parallel diagonals.

(I made the mistake in the last post, the answer may be 13......Sorry for the mistake)

could someone give the slution please....

Let denominator be D
break the integrals into two integrals
pi/D + 4x^3/D
the second evalutes to zero( propertiy of an odd function)

break the first integral again in two parts, in first take limit -pi/3 to 0, in secont take limit 0 to pi/3

I hope it helps

after the initial impulse, the system will attain equilibrium.
Let velocity of disc at equilibrium is w and elongation is I/4(by equlibrium, I mean there is no furhter change in speed and disc will move in a circle)

from law of conservation of energy
m*v*v/2=m*w*w/2 + k*I*I/32

since the disc is in equilibrium
m*w*w/(5I/4)=k*I/4

I am not an expert, but i can sove it.
acceleration of the bar is gsin(alpha)-asgcos(alpha)
{I will use x instead of alpha,}
gsinx-gascosx=dv/dt=dv/dt*ds/ds=vdv/ds
vdv=(gsinx-asgcosx)ds
integrating both sides we hace
v*v/2=gsinx*s-agcosx*s*s/2
to find the total distance travelled, v=0
giving s=2tanx

use differentiation to find maximum vale of v{if we can find max. value of v*v, it will suffice}
hence v*v/2 is max. at gsinx-asgcosx=0

If there are n diagonals, none of them concurrent and none of them parallel to each other, the no. of intersection points is C(n,2).

So the no. of diagonals may be 9,

I think the answer is C(6,4)*C(4,2)/C(11,6)

THE ANSWER IS 29.

Let n be a number of the form6x+10y+15z
n+1=6(x+1)+10(y-2)+15(z+1)
n+2=6(x+2)+10(y-1)+15(z+0)
n+3=6(x-2)+10(y-0)+15(z+1)
n+4=6(x+4)+10(y-2)+15(z+0)
n+5=6(x+0)+10(y-1)+15(z+1)
n+6=6(x+1)+10(y-0)+15(z+0)
n+7=(n+6)+1
and so. on

hence, if x-2,y-2 are non-negative, we can express any number greater than it in the given form.
So the minimum number beyond which every number is in the given form is 32.

To find the greatest number which is not expressible in the given form, we start backwards from 31 and we find that the answer is 29
{31=6+10+15, 30=15+15}

ADcotB+ADcotC=a
substitute the value and get the answer