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int (dx)/(tan x + cot x + sec x + cosec x) =

Integrate the following functions : (x)/(1-cosx)

Differentiate the following w.r.t. x: log (log x), x gt 1

Integrate the following : int2dx

Differentiate sin^(2)x w.r.t e^(cos x)

The values of theta in (0,(pi)/(2)) satisfying

Let veca,vecb,vecc be the three vectors such thatveca.(vecb+vecc)+vecb.(vecc+veca)+vecc.(veca+vecb)=0 and |veca|=1,|vecb|=4,|vecc|=8, then |veca+vecb+vecc| equals :

Let overset(n)underset(r=1)Sigma overset(r )underset(j=1)Sigma overset(j)underset(k=1)Sigma 1=364, then n =

int_(pi)^(10pi) |sin x| dx=

If alpha, beta , gamma are the roots of the equation x^(3)+4x+1=0, then (alpha+beta)^(-1)+(beta+gamma)^(-1)+(gamma+alpha)^(-1) is equal to

Ifz = cos 6^(@) + isin 6^(@), " then " underset(n = 1) overset(20)sum (z^(2n-1))=

The tangent and normal to the ellipse x^2+4y^2=4 at a point (theta)on its meets the major axis in Q and R respectively. If 0ltthetaltx/2and QR=2 , then show that theta=cos^-1(2/3) .

Find the number of ways to post 5 letters in 6 post boxes such that atleast two letters are posted in the same box.

Assertion (A) : (1)/(5)+(1)/(3.5^(3))+(1)/(5.5^(5))+(1)/(7.5^(7))+…(1)/(2)log((3)/(2)) Reason (R ) : If |x| lt 1 then log_(e )((1+x)/(1-x))=2(x+(x^(3))/(3)+(x^(5))/(5)+…)

Evalute the following integrals int tan^(5) xdx

A relation R is form set A to B , and a relation S is from set B to C . Then relation SOR is from ........

If the area bounded by f(x) = sqrt(1 + x^(2) (x gt 0) the line y = x,y axis and y = - x + a (a > - 1) is k, then area bounded by the graph of f^(-1) (x), the line y = x between the lines y = - x + 1 and y = - x + a is

If the matrix [{:(0,-1,3x),(1,y,-5),(-6,5,0):}] is skew- symmetric, then

If the inequality (m-2)x^(2) + 8x + m + 4 gt 0 is satisfied for all x in R, then the least integral value of m is:

Find X, if Y=[(3,2),(1,4)] and 2X+Y=[(1,0),(-3,2)]

{ xcos ((y)/(x)) + y sin ((y)/(x))} y dx = { y sin ((y)/(x)) - x cos ((y)/(x))}x dy

If C is arbitrary constant then inte^(sin^(-1)x)((lnx)/(sqrt(1-x^(2)))+1/x)dx is equal to

[sqrt2(cos56^(@)15+isin56^(@)15')]^(8)

sin A ( cot A +3( 3 cot A + 1) -1 " cosec " A -10 cos A =

Find the area of the smaller part of the circle x^(2) + y^(2) = a^(2) cut off by the line x=a/sqrt2.

Unit vectors vec(a),vec(b),vec(c ) are coplanar. A unit vector vec(d) is perpendicular to them. If (vec(a)xxvec(b))xx(vec(c )xxvec(d))=(1)/(6)i-(1)/(3)hat(j)+(1)/(3)hat(k) and the angle between vec(a) and vec(b) is 30^(@), then vec(c ) is/are :

Consider the points A(-2, -3) and B(1,6).﻿ i. Find the equation of the line passing through A and B.﻿ ii. Find the equation of the line passing through (2, 1) and perpendicular to AB iii. Find the foot of the above perpendicular to AB.

If a, b and c are perpendicular to b+c, c+a and a+b respectively and if |a+b|=6, |b+c|=8 and |c+a|=10, then |a+b+c| is equal to

Show that the homogenous system of equations x - 2y + z = 0, x + y - z = 0, 3 x + 6y - 5z = 0has a non-trivial solution. Also find the solution

If cosalpha+cosbeta+cosgamma=0=sin alpha+sinbeta+singamma then cos^2alpha+cos^2beta+cos^2gamma=

If the equation has no real root, then lamda lies in the interval

Consider the triangle formed by the lines y+3x+2=0, 3y-2x-5=0, 4y+x-14=0 Match the following lists:

int_(0)^(pi//4)(sinx + cos x)/(7+9 sin 2x) dx =

Verify mean value theorem, if f(x) = x^(2) - 4x in the interval [1, 4].

A straight line L intersects perpendicularly both the lines : (x+2)/(2)=(y+6)/(3)=(z-34)/(-10) and (x+6)/(4)=(y-7)/(-3)=(z-7)/(-2), then the square of perpendicular distance of origin from L is

What is the vital capacity of our lungs ?

The non-zero vectors vec(a),vec(b),vec(c) are related by vec(a)=8vec(b)andvec(c)=-7vec(b). The angle betweenis vec(a)andvec(c) is

Match the following : Let f be a function defined on {(m,n) : m and n are positive integers }Satisfying (i) f(m,m+1) = 1/3for all m (ii) f(m,n) = f (m,k) +f (k,n)-2f (k,n) for all such that m lt k lt n {:(,P,Q,R,S),((A),1,2,3,4),((B),4,3,1,2),((C ),3,2,4,1),((D),2,1,4,3):}

A chess game between two grandmasters X and Y is won by whoever first wins a total of two games. X's chances of winning or loosing any perticular game are a, b and c, respectively. The games are independent and a+b+c=1. The probability that X wins the match after (n+1)th game (nge1), is

Show that the lines 5x+3y-9=0,2x+y=0,x+3y=0 and x+4y+2=0 taken in order form a cyclic quadrilateral.

x dy - y dx = sqrt(x^(2) + y^(2)) dx

If A is a square matrix of order 3 such that |A|=13, then |adj A| is equal to

Let vec(a)=hati+4hatj+2hatk,vec(b)=3hati-2hatj+7hatk and vec( c )=2hati-hatj+4hatk. Find a vector vec(d) which is perpendicular to both vec(a) and vec(b), and vec( c ).vec(d)=15.

int_(0)^(1) x^(3//2) sqrt(1-x) dx is equal to

Derive the reduction formula for int tan^(n) xdx.

On putting (y)/(x)=v the differential equation (dy)/(dx)(2xy-y^(2))/(2xy-x^(2)) is transfored to

Solve the following differential equations (i) (dy)/(dx) - y tan x = e^(x)sec x (ii) (dy)/(dx) + y tan x = sin x

Find two positive numbers x and y, such that x+y=64 and x^(3)+y^(3) is minimum.

Consider the points A(-2, -3) and B(1,6). i. Find the equation of the line passing through A and B. ii. Find the equation of the line passing through (2, 1) and perpendicular to AB iii. Find the foot of the above perpendicular to AB.