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Find the vectors from the origin to the intersection of the medians of the triangle whose vertices are A(5,2,1), B(-4,7,0) and C(5, -3,5)

Differentiate (sinx)^(cosx) with respect to x.

The solution of the differential equation (dy)/(dx)=(x-2y+1)/(2x-4y) is

Find the cartesian equation fo the following planes. vecr.(hati+hatj-hatk)=2

If|z _1 |= 2and(1-i)z_2+(1+i)barz_2= 8sqrt2, then the minimumvalue of|z_1 - z_2| is ______.

Integrate the rational functions in exercise. (cosx)/((1-sinx)(2sinx))

The number of real solutions of the equation : cos^(7)x + sin^(4)x = 1 in the interval [-pi,pi] is :

If |{:(a+ib,c+id),(-c+id,a-ib):}|xx|{:(alpha-ibeta,gamma-idelta),(-gamma-idelta,alpha+ibeta):}|=|{:(A-iB,C-iD),(-C-iD,A+iB):}|, write down the values of A, B, C, (i=sqrt(-1)). Hence show that, the product of two sums, each of four squares, can be expressed as the sum of four squares.

Five dice are tossed. What is the probability that the five numbers shown will be different?

IF A be the area bounded by the x-axis and one are of the curve y= acos 3xbetween (0,0) and ((pi)/(6),0)and B be the area bounded by the x-axis and one are of the curve y=acos ^((x)/(4)) between (0,0) and (2pi,0)show that ,A : B =1: 12

Using the properties of determinants in Exercise 1 to 6, evaluate |{:(0,xy^2,xz^2),(x^2y,0,yz^2),(x^2z,y^2z,0):}|

The value of cos^(2)45^(@)-sin^(2)15^(@) is

The maximum value of sin x.cos x is ………….

The shortest distance of the lines bar r_1 = 4 hat i -3 hat j - hat k + lambda (2 hat i - 3 hat j + 8 hat k ) is.......

If from any point on the circle x^(2)+y^(2)+3gx+2fy+c=0 tangents are drawn to the circle x^(2)+y^(2)+2gx+2fy+c sin^(2)alpha+(g^(2)+f^(2)) cos^(2)alpha=0,(0 lt alpha lt(pi)/(2)) then the angle between those tangents is

Solved the following system of linear equations by matrix inversion method.2x+5y=-2, x+2y=-3

Prove that :int_(0)^(oo) log (x+(1)/(x)). (dx)/(1+x^(2)) = pi log_(e) 2

Two hunters A and B set out to hunt ducks. Each of them hits as often as he misses when shooting at ducks. Hunter A shoots at 50 ducks and hunter B shoots at 51 ducks. The probabilitythat B bags more ducks than A can be expressed as (p)/(q) in its lowest form. Find the value of (p+q).

How many numbers between 6000 and 10000 can be formed using the digits 2, 3, 4, 6, 7, 9 without repetition.

If the angle between the lines having direction ratios (alpha, 3,5) and (2, -1, 2) is (pi)/(4) then alpha is:

If [x] denotes the greatest integer le x , then underset( n rarr oo ) ( "Lim") (1)/( n^(4)) ( [1^(3) x ]+ [2^(3) x ] +"....." +[n^(3) x ] )equals

Compute the magnitude of the following vectors : (i) vec(a)=2hati+3hatj+sqrt(3)hatk (ii) vec(b)=3hati-4hatk (iii) vec( c )=hati+hatj-4hatk

A biased coin with probability of getting head is twice that of tail, is tossed 4 times If a random variable X is number of heads obtained, then expected value ofX is :

If x=8+3sqrt(7)" and "xy=1, then the value of (1)/(x^2)+(1)/(y^2) is

For any two vectorsbar(a)andbar(b) , show that(1+|bara|^(2))(1+|barb|^2)=|1-bar(a).bar(b)|^(2)+|bar(a)+bar(b)+bar(a)xxbar(b)|^(2)

If veca=hati+lamdahatj+2hatk,vecb=muhati+hatj-hatk are orthogonal and |veca|=|vecb| , then (lamda,mu)=

Using the Rolle's theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions : (i) f(x)=x^(2)-x, x in [0,1] (ii) f(x)=(x^(2)-2x)/(x+2), x in [-1, 6] (iii) f(x)=sqrtx-(x)/(3), x in [0,9]

2[7^(-1)+3^(-1)7^(-3)+5^(-1)7^(-5)+…]=

Transform 12x^(2)+7xy-12y^(2)-17x-31y-7=0 to rectangular through the point (1, -1) inclined at an angle tan^(-1)((4)/(3)) to the original axes.

State which of the following statements are true (T) or false(F) The line (x-1)/2=(y-1)/2=(z-1)/2 pass though the origin.

Integrate the function (sec^(2)x)/(sqrt(tan^(2)x+4))

Differentiate (5x+8)(x^(3)+7x+9) in the way mentioned below : by expanding the product to obtain a single polynomial.

For all real values of x, the minimum value of (1+x+x^(2))/(1+x+x^(2)) is

Using calculus prove that log_(3) gt log_(3)5gt log_(4)7.

Evaluate (2005int_(0)^(1002)dx(sqrt(1002^(2)-x^(2))+sqrt(1003^(2)-x^(2)))+int_(1002)^(1003)sqrt(1003^(32)-x^(2))dx)/(int_(0)^(1)sqrt(1-x^(2))dx)=k, then find the sum of squares of digits of naturalnumber k.

If A_(1),A_(2)….., A_(n) are theverticesof a regularplanepolygonwith n sidesand o isits centre. Thenshowthat Sigma_(n-1)^(i-1) (overset(to)(OA)_(i) xx overset(to)(OA)_(i+1) ) =(1-n)(overset(to)(OA)_(2)xxoverset(to)(OA)_(1))

If f(x)=unerset(x^2)overset(x^(2+1))e^(-t^2) dt then f(x) increases on

Differentiate w.r.t.x the function. cot^(-1)[(sqrt(1+sin x)+sqrt(1-sin x))/(sqrt(1+sin x)-sqrt(1-sin x))],0 lt x lt (pi)/(2).

Let I = int_1^2 (dx)/(sqrt(1 + x^2)), J = int_1^2 (dx)/(x), then

Let f(x) = {{:(int_(0)^(x){5+|1-t|}dt",","if",x gt 2),(5x + 1",","if",x le 2):} Test f(x) for continuity and differentiability for all real x.

Express the following matrices as the sum of a symmetric and a skew symmetric matrix : (i) [{:(3,5),(1,-1):}](ii)[{:(6,-2,2),(-2,3,-1),(2,-1,3):}](iii)[{:(3,3,-1),(-2,-2,1),(-4,-5,2):}](iv)[{:(1,5),(-1,2):}]

log_(4)2-log_(8)2+log_(16)2-....=

For real numbers x and y, define xRY if and only if x - y + sqrt(2) is an irrational number.Then the relation R is

Find derivatives of the following functions. 1/(x^3 + sinx)^2

Sum of the squares of the first 3 terms of the given series is

If A is a matrix of order 3, such that A(adj A) = 10 I, then find |adj A| =