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If f(x) =ax^(2) +b, b ne 0, x le 1 =bx^(2) +ax + c, x gt 1 , then f(x) is continous and differentiable at x=1, if

If L and L' are the ends of the latus rectum of the parabola x^(2)=6y find the equations of OL and OL' where 'O' is the origin.Also find the angle between them.

If the foot of the perpendicular drawn from the origin to a plane is (1,2,3), then a point on the plane is

Let f(x) = max (x, x^(3)) AA x in R. Calculate the area bounded by the curves y = f(x) and the x-axis between the ordinates x = - 1 and x = 1

The value of int_1^2(g(x) -f(x))dx is-

If the x-axis divides the area of the rigion bounded by the parabolas y= 4-x-^(2) and y^(2) -x^(2)-xin the ratio of a: b, then ab is equal to

If a,b,care distinctpositivenumbersthen the3expression( b+c-a) (c+a-b)(a+b-c)-abcis

A circle whose diameter is major aixs of ellipe (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 (agtbgt0) meets axis at point P. Ifthe orthocentre of DeltaPF_(1)F_(2)lies on ellipse where F_(1)and F_(2) are foci of ellipse , then find the eccenricity of theellipse

Let f (x) = {{:( x ^(2) - 3x + 2 "," , x lt 2 ), ( x ^(3) - 6x ^(2) + 9x + 2 "," , xge 2 ):}Then

"Two curves "C_(1)equiv[f(y)]^(2//3)+[f(x)]^(1//3)=0 and C_(2)equiv[f(y)]^(2//3)+[f(x)]^(2//3)=12," satisfying the relation "(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^(2)-y^(2)) The area bounded by C_(1) and C_(2) is

"Two curves "C_(1)equiv[f(y)]^(2//3)+[f(x)]^(1//3)=0 and C_(2)equiv[f(y)]^(2//3)+[f(x)]^(2//3)=12," satisfying the relation "(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^(2)-y^(2)) The area bounded by C_(1) and x+y+2=0 is

"Two curves "C_(1)equiv[f(y)]^(2//3)+[f(x)]^(1//3)=0 and C_(2)equiv[f(y)]^(2//3)+[f(x)]^(2//3)=12," satisfying the relation "(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^(2)-y^(2)) The area bounded by the curve C_(2) and |x|+|y|=sqrt(12) is

Determine the sign of the expression i) x^(2)+x+1 ii) -x^(2)+x-1 for x in R.

If the sum of the distance from a variable point P to the given points A(1,0) and B(0,1) is 2, then the locus of P is

If lim_(xto oo){(x^2+1)/(x+1)-(ax+b)}to oo, then

The vector equation of the line passing through (2,3,4) and parallel to Z-axis is

Ify = tan^(-1) ((sqrt(1+x^(2)) -sqrt(1-x^(2)))/(sqrt(1+x^(2)) + sqrt(1-x^(2)))) then dy/dx =

Thetransformedequationwithintegerco-effcientswhoserootsaremultipledby someconstantof thoseofx^4 -1/2 x^3 +3/4x^2 -5/4 x+(1)/(16 )=0 is

Prove that the following functions do not have maxima or minima h(x)=x^3+x^2+x+1

Evaluate the following integrals. int((1+logx)^n)/(x)dx

If a,b in R^(+) then find Lim_(nrarroo)sum_(k=1)^(n) ( n)/((k+an)(k+bn)) is equal to

int (x dx)/(1-x cotx)=.....

If D=|{:(,a^(2)+1,ab,ac),(,ba,b^(2)+1,bc),(,ca,cb,c^(2)+1):}| then D=

Obtain the differental equations of the family of parabolas having their focus at the origin and the axis along the x-axis.

The non-zero vectors a, b and c are related by a = 8b and c = - 7b. Then the angle between a and c is

If the letters of the 'AJANTA' are permutated in all possible ways and the words thus formed are arranged in dictionary order. Find the rank of 'JANATA'

If x+y=1, then sum_(r=6)^(n) r ^n C_r x^r . Y^(n-r)=

Equation of the plane containing the lines. r=i+2j-k+lambda(i+2j-k) and r=i+2j-k+mu(i+j-3k) is

Evaluate: int(dx)/(cos^(3)xsqrt(2sin2x))

If alpha+beta=gamma.then cos^(2)alpha+cos^(2)beta+cos^(2)gamma-2 cos alpha cos beta cos gamma=

sec(x+y)= xy

The sum (2^(1))/(4^(1) - 1) + (2^(2))/(4^(2) - 1) + (2^(4))/(4^(4) - 1) + (2^(8))/(4^(8) - 1) +... oo is equal to

Let vec(alpha)=hat(i)+hat(j)+hat(k),vec(beta)=hat(i)-hat(j)-hat(k)andvec(gamma)=-hat(i)+hat(j)-hat(k) be three vectors. A vector vec(delta), in the plane of vec(alpha)andvec(beta), whose projection on vec(gamma)" is "(1)/sqrt(3), is given by

If veca=hati+6hatj+3hatk, vecb=3hati+2hatj+hatk and vec c=(alpha+1)hati+(beta-1)hatj+hatkare linearlydependentvectors and |vec c|=sqrt(6), then the possible value(s) of (alpha+beta) can be :

A coin is tossed 2n times. During the toss any person can't get head and tail with equal numbers. Its probability is ........

Let a and b be natural numbers and let q and r be the quotient and remainder respectively when a^2+b^2 is divided by a + b . Determine the number q and r if q^2+ r = 2000.

int (1)/(2 + 3 cos x) dx =

If the curves x^2+py^2=1 and qx^2+y^2=1 are orthogonal to each other, then

Integrate the function is (xcos^-1x)/sqrt(1-x^2)

lim_(xto0) |x|

The value of Cos(tan^(-1)(3/4))is:

If the letters of the word 'QUESTION' are arranged at random. What is the probability that there are exactly two letters between Q and U.

Find the values of (dy)/(dx), if y = x^(tanx)+sqrt((x^(2)+1)/(2)).

If (1+x)^n = C_0 + C_1x+ C_2 x^2 + ….....+ C_n x^n,then C_0+2. C_1 +3. C_2 +….+(n+1) . C_n=

Resolve into Partial Fractions (vi) (x^(2)+13x+15)/((2x+3)(x+3)^(2))

Prove that .^(n)C_(0) + .^(n)C_(5) + .^(n)C_(10) + "….." = (2^(n))/(5) (1+2cos^(n)'(pi)/(5)cos'(npi)/(5)+2cos'(pi)/(5)cos'(2npi)/(5)).