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The value of sin(cot^-1x) is equal to

Let a=alphahat(i)+2hat(j)-3hat(k), b=hat(i)+2alphahat(j)-2hat(k) and c=2hat(i)-alphahat(j)+hat(k). Then the value of 6alpha, such that {(atimesb)times(btimesc)}times(ctimesa)=a, is

Let m,n be two positive real numbers and define f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx and g(m,n)=int_(0)^(1)x^(m-1)(1-m)^(n-1)dx. It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0. int_(0)^(1)x^(m)(log_(e).(1)/(x))dx=

Let m,n be two positive real numbers and define f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx and g(m,n)=int_(0)^(1)x^(m-1)(1-m)^(n-1)dx. It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0. int_(0)^(oo)(x^(m-1))/((1+x)^(m+n))dx=

Let m,n be two positive real numbers and define f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx and g(m,n)=int_(0)^(1)x^(m-1)(1-m)^(n-1)dx.It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0.int_(0)^(1)(x^(m-1)+x^(n-1))/((1+x)^(m+n))dx=

Differentiate the following w.r.t.x. cos^(-1) (sin x).

Find the probability of getting equal numbers, when two dice are rolled.

The statement P(n) = 9^(th)- 8^(n),whendividedby 8,alwaysleaves the remainder

lim_(xtosqrt3)[x]

Find all points of discontinuity of f(x) where f is defined by f(x) = {(x^3-3,if, x le 2),(x^3+1,if, x gt 2):}.

The velocity of a particle moving along positive X-axis varies as v=alphasqrtx where alpha is a constant.if particle is at x=0 at t=0 , what will be the average velocity of particle during the time it moves a distance S ?

Prove that the least perimeter of an isosceles triangle in which a circle of radius can be inscribed is 6rsqrt3.

Delta=|{:(p,q,r),(p+2a,q+2b,r+2c),(a,b,c):}| then

y=(x^(3)+(x^(6))/(2)+(x^(9))/(3)+……) then

Evaluate the integral underset(0)overset(pi//4)int log(1+tanx)dx

If the standard deviation of the binomial distribution (q+p)^(16)is 2, then mean of the distribution is….

Find the direction cosines of the line joining the two points P(-2, 4, -5) and Q(1, 2, 3). 6. Prove that the points (1, 2, 3), (3, 1, 7) and (7, -1, 15) are collinear.

Find all points of discontinuity of f, where f(x)= {((sin x)/(x)",","if " x lt 0),(x+1",","if" x ge 0):}

Find the locus of the incentre of the triangle formed by xy-4x-4y+16=0 and x+y=a (agt4,a nesqrt(2) " ""and a is the parameter").

Find the area bounded by the curve y=l nxthe X-axis and the straight line x=e

Which of the following scatterplots shows a relationship that is appropriately modeled with the equation y=ax^(b) , where a is positive and b is negative?

Match the following

If the distances from the origin to the centres of three circles x^(2)+y^(2)-2kix=c^(2), (i=1,2,3) are in G.P, then the length of the tangents drawn to them from any point on the circle x^(2)+y^(2)=c^(2)" are in "

Find the equation of the curve passing through the point (0,1), if the slope of the tangent to the curve at any point (x,y), is equal to the sum of x coordinate and product of x coordinate and y coordinate of that point.

A committee consisting of atleast three members is to be formed from a group of 6 boys and 6 girls, such that it always has a boy and a girl. Number of ways to form such committee is………

Find the centre of gravity of the first are of the cycloid: x=a (t - sin t), y= a (1- cos t) (0 le t le 2pi).

Differentiate.(sqrtx+1/sqrtx)x tanx

If x=CiSθ, then find the value of (x^6+(1/x^6))

If the solution of cosec^(2)x(dy)/(dx) - (1)/(y) = 0 is ax + b sin 2x + cy^(2) = k, a gt 0 then ascending order of a,b,c is

Evaluate int_(a)^(b) x dx as the limit of a sum.

Match the following

Expand f(x) = (1)/(x) about x =2 upto four terms

If y = log (log x) then (d^(2)y)/(dx^(2)) is equal to

A(2, 1) and B(2, 3) are two points.If Pis a point such that PA + PB - 2, then the locus of P is

Differentiate.sec^(-1)(e^x+x)

int_((-pi)/(4))^((pi)/(4)) (dx)/(1+e^(tan x))

Differentiate.sin^2(cos^(-1)x)

Evaluate the definite integral in exercise overset(1)underset(0)int (2x+3)/(5x^(2)+1)

For complex number z, the minimum value of |z| + |Z- cos alpha-I sin alpha| + |z-2 (cos alpha +I sin alpha)| is

Find the relationship between a and b so that the function f defined by f(x)={{:(ax+1-2," if "x le 3),(bx+3," if "x gt 3):} is continuous at x=3.

If isqrt(-1), then 4+5(-(1)/(2)+(isqrt(3))/(2))^(334)+3(-(1)/(2)+(isqrt(3))/(2))^(365) is

Solve the differential equation : (x^(3) + x^(2) + x +1)(dy)/(dx) = 2x^(2) + x.

If the straight line y=x meets y=f(x) at P, where f(x) is a solution of the differential equation (dy)/(dx)=(x^(2)+xy)/(x^(2)+y^(2)) such that f(1)=3, then the value of f'(x) at the point P is

Integrate the function in Exercise. sin^(-1)((2x)/(1+x^(2)))

Evalute the following integrals int (1)/((1 + sqrt(x)) sqrt(x - x^(2)) ) dx

Evaluate : int (2x+4)/(sqrt(x^(2)+4x+5))dx

Find the area of the parallelogram whose adjacent sides are determined by the vectors veca=hati-hatj+3hatk and vecb=2hati-7hatj+hatk

The region formed by the inequalities 2x+3y-5 le 0, 4x-3y+2 le 0" and " x ge 0 ………

Obtain the following integrals : int(1)/(sqrt(3t-2t^(2)))dt