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A : The radical centre of the circlesx^(2) + y^(2) + 4, x^(2) + y^(2) - 3x = 4 , x^(2) + y^(2) - 3x = 4, x^(2) + y^(2) - 4y = 4is (0,0). R : Radical centre of three circles is the point of concurrence of the radical axes of the circles taken in pairs .

A couple has 3 children and it is known that atleast one of them is a boy. Then the probability that the couple will have exactly two boys is

If the line r = a + lb is parallel to the plane r = c + ld + "me", then

A particle stays at rest as seen in frame. Then which of the following statement is/are TRUE ?

If a,b,c are distinct and the roots of (b-c)x^(2)+(c-a)x+(a-b)=0 are equal, then a,b and c are in

Integrate the following functions (2x)/(1+x^2)

Differentiate the following with respect to x: sin [log (e^(x))]

Ifx^(y) = e^(x-y) then (dy)/(dx) is equal to

Consider the functionf:(-oo, oo) -> (-oo ,oo) defined byf(x) =(x^2 - ax + 1)/(x^2+ax+1) ;0 lt a lt 2. Let g(x)=int_(0)^(e^(x))(f'(t))/(1+t^(2)) dt. Which of the following is true?

If the normal at the point P(x_1y_1),i=1.2,3,4 on the hyperbola xy=c^2 are concurrent at the point Q(h, k), then ((x_1+x_2+x_3+x_4)(y_1+y_2+y_3+y_4))/(x_1x_2x_3x_4) is:

If (1+x+2x^(2)+4x^(3))^(10)=a_(0)+a_(1)x+a_(2)x^(2)+….+a_(30)x^(30). Find the value of a_(0) +a_(1)+a_(2)+….+a_(30)

int(1)/((1+x)sqrt(x))dx=f(x)+c then f(x)=.....

Find the equation and length of the common chord of the two circles S -= x^2 + y^2 + 3x + 5y + 4 =0 and S^1 -= x^2 + y^2 + 5x + 3y + 4 = 0

Integration of a binomialdifferential introot(3)x^(7)sqrt(1+root(3)(x^(4)))dx.

The maximum areain squre units of an isosceles triangle inscribed in an ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 with its vertex at one end of the major axis is

Which of the following compouds on direct heating can not produce anhydrous form of it.

Let P be an odd prime number and T_(p) be the following set of 2xx2 matrices : The number of A in T_(p) such that the trace of a is not divisible by p but det (A) divisible by p is [Note : The trace of matrix is the sum of its diaginal entries].

No part of the hyperbola x^(2)/(a^(2))-y^(2)/b^(2)=1. Lies between which of the following

Let a,b gt 0 and vecalpha=(veci/a+(4hatj)/b+bhatk) and vecbeta = bhati+ahatj+1/bhatk, then the maximum value of 10/(5 + vecalpha.vecbeta) is

State with reason wheher following functins have inverse (i) f : {1,2,3,4} to {10} with f = {(1,10), (2,10), (3,10) ,(4,10)} (ii) g: {5,6,7,8} to {1,2,3,4} with g = {(5,4) , (6,3), (7,4), (8,2)} (iii) h : {2,3,4,5} to {7,9,11,13} with h = {(2,7), (3,9) , (4, 11), (5,13)}

Findthepolynomialequationwhose roots arethe negativesof therootsof theequation x^4 -6x +7x^2 -2x +1=0

Statement - I : The lines vec(r)=hat(i)+hat(j)-hat(k)+S(3hat(i)-hat(j)) and vec(r)=4hat(i) -

If 0 ltphilt (pi)/(2) ,x= sum_(n=0)^(oo)cos^(2n) phi,y sum _(n=0)^(oo)sin^(2n) phiandz= sum_(n=0) ^(oo) cos ^(2n) phisin^(2n) phi ,then

If cotA=sqrt(ac),cotB=sqrt((c )/(a)),cot C =sqrt((a^(3))/(c ))& c=a^(2)+a+1 then

Three normals are drawn from the point (c, 0) to the curve y^(2)=x. If one of the normals is X-axis, then the value of c for which the other two normals are perpendicualr to each other is

{:(" " Lt),(x rarr 0):} (int_(0)^(x^(2))sec^(2) t dt)/(x sin x) =

Threenormals are dran toa parabola y^(2)=4ax from a given point (x_(1),y_(1)) the algebraic sum of theordinates of theirfeet is

If f(x)=underset(-1)overset(x)(int)|t|dt,x ge-1, then :

(dy)/(dx) = (1 + x^(2))(1 + y^(2))

An urn contains 10 white balls and 5 black balls. Two players Q and R alternatively draw a ball with replacement from the urn. The player that draws a white ball first wins the game. If Q begins the game, find the probability of his winning the game.

The solution of the differential equation (x+y-1)/(x+y-2)(dy)/(dx)=(x+y+1)/(x+y+2), given that y = 1 when x = 1, is

Let T be set of all triangle in the Euclidean plane , and let a relation R on T be defined as aRb if a is congruent tob,AA "a",b inT .Then, R is ....

What are the foci of the hyperbola x^(2)/(36)-y^(2)/(16)=1

The radius of the auxiliary circle of the hyperbola x^(2)//25-y^(2)//9=1 is

Which of the following alkyl halides would be most likely to give a rearranged product under S_(N)1 conditions ?

Find the value ofroot(5)(32.16)correct to 4 decimals places

int_(0)^(pi//4) tan^(6) x dx=

If the directions cosines of a line are k, k and k then ..........

8 coins are tossed togather. Then ………. is the probability of an event that head H comes up at least 6 times.

Let OA, OB, OC be the co-terminal edges of a rectangular parallelopiped of volume V and let p be the vertex opposite to O. Then, [vec(AP) vec(BP) vec(CP)] is equal to

Find the area bounded by the curve y = sin x between x = 0 and x = 2pi.

Evaluation of definite integrals by subsitiution and properties of its : int_(0)^(2)[x^(2)]dx=....... where [.] denotes maximum integer function.

Evaluation of definite integrals by subsitiution and properties of its : int_(4)^(9)(dx)/(x-sqrtx)=..........

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Find the direction cosines of the line passing through the two points (-2, 4, -5) and (1, 2, 3).

Using differentials, find the approximate value of each of the up to 3 places of decimal. (81.5)^((1)/(4))

A circle touches x-axis at (2,0) and also the line y=x in first quadrant then its radius is

Evaluateint_(0)^(infty)e^(-x) sin^(n) x dx, if nis an even integer.

Find x, if [1" "x" "1][{:(1,3,2),(2,5,1),(15,3,2):}][{:(1),(2),(x):}]=O.