Messages posted by magiclko

sinx cosx dx/ [a²cos²x + b²sin²x]

=> sin2x dx/2 [a²cos²x + b²sin²x] .............(A)

Let [a²cos²x + b²sin²x] = t

Differentiating both the sides, we have

sin 2x [b²-a²] dx = dt

=> sin 2x dx = dt / [b²-a²]

Substituting in equation (A)

integral of dt /{2t [b²-a²] }

=> ln (t) / {2 [b²-a²]}

substitute the value of 't' and get the answer :)

jagdish has already solved the question...thank u for reducing the wrk :D

well done :)

question isnt complete.. :(

Karyotaxonomy is itself the most recent branch of Taxonomy !!!

mayank has solved the question correctly !

well done :)

[2*3.14]-[-3.14]

=6-(-4)

=10

probability needs alot of practice, and series logical thinking.

brush up ur permutations and combination, and practice probability problems, as much as possible.

solution to Q3:

The equation of the normal in terms of 'm' (slope) is given by,

y = mx – 2am – am³

here the parabola is y² = 4x ....i.e. a=1

therefore, the equation of the normal is y = mx – 2m – m³

now, all the 3 normals passes thru point (15,12) ...hence this point shud satisfy above equation.

substituting the value, we have 12=15m – 2m – m³

solving the equation for m, we have m = 1, 3, -4

thus the three normals are, y= x -3

y= 3x-33

y=-4x+72

The solution to this problem, is beyond the scope of a 12th standard student, and is irrevelant for JEE aspirants. still, if u want the solution, reply back... its highly complex and involves, some trigonometric integrals etc. ...
I'll give the starting, and then u can solve on ur own !!

Physical, chemical and mathematical sciences follow a specific set of rules and regulations. They are governed by a set of laws such as the law of gravity, the laws of thermodynamics, the law of conservation of mass and so on. However, biological sciences, in the strict sense, do not follow any rules or regulations. The reason for this is that living organisms show a rich diversity that has resulted from mutations, genetic re-combinations and evolution. Biology is thus, regarded as 'a science of exceptions

cot @ + cot # = 0

=> cot @ = -cot #

=> tan @ = -tan # = m (let) ....................(A)

now Let the point of intersection be P(h,k)

Thus, the equation of the tangents will be

(y-k) / (x-h) = m and, (y-k) / (x-h) = -m

=> y=mx + (k-mh) and y = -mx + (k+mh)

The distance of origin, which is the centre of the circle should be same for the two tangents, and equal to the radius 'a'

Thus (k-mh)/ [(1+m²)]^(1/2) = a = (k+mh)/ [(1+m²)]^(1/2)

=> k-mh = k+mh

=> h =0

The locus is x=0 ...i.e. the y-axis

Also, from the equation (A) ...we can infer, that the inclination angles are supplementary, which is possible only when the point P always lie on y-axis ...

the LHS is a sine function, hence, the range of LHS = [-1,1]

for a solution, the graph of RHS should intersect the graph of sine function of the LHS,

which is possible only if

for some real 'x', -1=< RHS <= 1

now for x>0 , RHS > 0

also for x = -sqrt(3), LHS=0 and RHS = 1.9

x = - 3 sqrt(3) /2 , LHS=1 and RHS = -8.5 (approx)

i.e. the polynomial in the RHS changes its sign for some x in the range, ( - 3 sqrt(3) /2 , sqrt(3) ) ...

Thus there exist only one solution to the equation

u want the entire chapter's solution

i dnt think, any such thing is available till now .... but u can create one of urs, by solving all the problems...

...so that ur friends can be benefited :)

gud luck

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weight loss

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poor appetite

cough

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abdominal pain

joint pain

Leishmaniasis or Kala Azar, is caused by protozoan parasites of the genus Leishmania. It appears in the form of localized ulcer or nodule. The infection is spread by the bite of female sand flies, which usually get infected by first biting an infected person and then inject parasites called mastigotes into the bloodstream or skin of a healthy person. The parasite migrates to the bone marrow, spleen, and lymph nodes.

relative growth rate of what???

no.. of handshakes between 'n' no. of people = ⁿC₂

if 'b' be the no. of boys, and 'g' be no. of girls , then

^bC₂ + ^gC₂= 27

b(b-1) + g(g-1)=54

the only integral solution is (4,7) or (7,4)

hence, b=7 and g= 4

S = 3 + 7x + 13x²+ 21x³ +......... (A)

Multiplying both sides by 'x'

Sx = 3x + 7x² + 13x³+........ (B)

Subtracting (B) from (A),

S(1-x) =3 + 4x + 6x² + 8x³........ (C)

again, multiplying both sides by 'x'

S(1-x)x = 3x + 4x² + 6x³ + 8x⁴........ (D)

Subtracting (D) from (C)

S(1-x)² = 3 + x + 2(x²+ x³+ x⁴+........)

= 3 +x + 2[x²/(1-x)]

S = { 3 +x + 2[x²/(1-x)] } / { (1-x)² }

substituting the value, x=1/2 , in the above equation, we have

S = 18

Let Image of Point P(a,b) wrt y=-x be point Q (h,k)

Then, h=-b and k=-a

i.e. Q(h,k)= Q(-b, -a)

Now, let Image of Point Q(h,k) wrt y=x be point R (m,n)

Then, m=k=-a and n=h=-b

i.e. R(m,n)= R( -a,-b)

Now we have three points, P(a,b) ; Q(-b,-a) ; R(-a, -b)

midpoint of PQ = {(a-b)/2 , -(a-b)/2 } ............which lies on the line y=-x

midpoint of QR = {(-a-b)/2 , (-a-b)/2 } ............which lies on the line y=x

and

midpoint of PR = (0,0) ...which will always be the origin, irrespective of the value of a and b

And one can crosscheck it using geometry also :)

for more info, u can chck this http://cseligman.com/text/history/ellipses.htm

simply because it continues indefinitely to infinity