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Two events A and B are such that P(A) = (1)/(2), P(A|B) = (1)/(4) and P(B|A) = (1)/(2) Consider the following statements : (I) P(bar(A) | bar(B)) = (3)/(4) (II) A and B are mutually exclusive (III) P(A|B) + P(A|B) = 1. Then

Let f be a function defined on [a,b] such that f'(x)gt 0, for all x in (a, b). Then prove that f is an increasing function on (a, b).

An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.

Find the equation of circle passing through each of the following three points. (5,7),(8,1),(1,3)

Find the values of a and b such that the function defined by is continuous. f(x) = {(x^(2)+1",",""xle 2),(ax + b",", 2 lt x lt 10),(2x+1",", ""x ge 10):}

Find the unit vector in the direction of the vectorveca=hati+hatj+2hatk.

Sum of coefficients of expansion of B is 6561 . The difference of the coefficient of third to the second term in the expansion of A is equal to 117 . Ifn^(m) is divided by 7 , theremainder is

Sum of coefficients of expansion of B is 6561 . The difference of the coefficient of third to the second term in the expansion of A is equal to 117 . The ratio of the coefficient of second term from the beginning and the end in the expansion of B , is

A= (5/2+x/2)^n, B=(1+3x)^m Sum of coefficients of expansion of B is 6561 . The difference of the coefficient of third to the second term in the expansion of A is equal to 117 . The value of m is

Find the sum of all those integers n for which n^2+20n+15 is the square of an integer.

The point 'E' divides the segment PQ internally in the ratio 1 : 2 and R is any point not on the line PQ. If F is a point on QR such that QF : FR = 2 : 1 then show that EF is parallel to PR.

Prove that (1- sec 8 alpha )/(1- sec 4 alpha )=(tan 8 alpha )/( tan 2 alpha )

A tangent drawn to hyperbola(x^(2))/( a^(2)) -(y^(2))/( b^(2)) =1atP((pi )/( 6))forms a triangle of area 3a^(2)square units , with coordinates axes, then the square of its eccentricity is

The value of int_(0)^(2) | cos ""(pi)/( 2) t|dtis equal to

If a = I + j - k, b = I - j + k, c = I - j - k then a xx (b xx c) =

Find derivatives of the following functions.sin 5x + cos7x

P points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is

Solve sqrt(1-y^(2))dx = [ sin^(-1) (y-x) ] dy

Let a, b, c be district real numbers. If a, b, c are in G.P and a + b + C = bx, the x in

Which condition of Rolle's theorem is violated by the functionf(x) = sin x in [0,(3pi)/\4]

If r is a real number |r| lt 1 and if a= 5(1-r) , then

Find the equation of the common chord of the following pair of circles x^2+y^2-4x-4y+3=0 x^2+y^2-5x-6y+4=0

lim _(x to (pi)/(2))(sin x )/(cos ^(-1)[(1)/(4 ) (3 sin x- sin 3x)]),(where [.] denotes greatest integer function) is :

Show that sin 78^(@)-sin18^(@)+cos132^(@)=0

If (1,2) ,(4,3) are the limiting points of a coaxal system , then the equation of the circle in its conjugate system having minimum area is

int{1+2 tan x(tanx+sec x)}^((1)/(2))dx=....

Show that the line 2x - y+12=0 is a tangent to the parabola y^(2) = 16x. Find the point of contact also.

Solve the following differential equations. (1-x^(2))(dy)/(dx)+2xy=x sqrt(1-x^(2))

Find the area of the parallelogram whose diagonals are determined by the vectors 3hati+hatj-2hatk and hati-3hatj+4hatk.

8 different things are arranged along a circle. In how many ways can 3 objects be selected such that no two of the selected objects are consecutive.

Match the following lists:

If A={x:f(x)=0} and B={x:g(x)=0} then AcapB will be

Match the following lists:

Match the following lists:

Match the following lists:

If x=a+b, y=a omega+b omega^2,z=a omega^2+bomega show thatx^3+y^3+z^3=3(a^3+b^3)

If 0lt plt 2pi and sin^(-1)(sinp)lt (x^(2))/(y)+(3y^(2))/(x)+(1)/(81xy)forall x ,yin R^(+) then p in

Find the number of zeros at the end of 15!

Draw a coordinate plane and plot the following points : x=0

(1 + omega) (1 + omega^(2)) (1 + omega^(3)) (1 + omega^(4)) (1 + omega^(5)) (1 + omega^(6)) …. (1 + omega^(3n)) =

Miki has 7 jazz CDs for every 12 classical CDs in his collection. If he has 60 classical CDs, how many jazz CDs does he have?

If 4P(A) = 6 P(B) = 10 P(A cap B) = 1then P(B | A)= ……….

The value of ((1+sqrt3i)/(1-sqrt3i))^(64)+((1-sqrt3i)/(1+sqrt3i))^(64) is

P is a point denoting z in the argand diagram and if (z-i)/(z-1) is always purely imaginary, then locus of P is

1 + i^(2) + i^(4) + i^(6) + … +i^(2n) =

Let p(gt0) be the first of the n arthimaticmeans betweens between two numbers and q(gt0) the first of n harmonic means between the same numbers. Then prove that qnotin(p,((n+1)/(n-1))^(2)p)andpnotin(((n-1)/(n+1))^(2)q,q)

By solving linear programming problem maximize Z=9x+3y subject to 2x+3y le13 , 3x+y le5 and x,yge0 using graphical method we get x =2,y=k and z=28 find the value of k

The maximum number of equivalence relations on the set A = {1,2,3}are ..........

Find the unit vector in the direction of vector vec(PQ) , where P and Q are the points (1,2,3) and (4,5,6) , respectively.

If [] denotes the greatest integer function, then the integral int_(0)^(pi)[cosx]dx is equal to