`x_1a n dx_2` are the roots of `a x^2+b x+c=0a n dx_1x_2<0.` Roots of `x_1(x-x_2)^2+x_2(x-x_1)^2()_()=0` are a. real and of opposite sign b. negative c. positive d. none real

True or false The equation 2x^2+3x+1=0 has an irrational root. If a

If +-2,quad 3 are the roots of ax^(4)+bx^(3)+cx^(2)+dx+e=0 then the roots of a(x-1)^(3)+b(x-1)^(2)+c(x-1)+d=0

int x ^ (2n-1) cos x ^ (n) dx

LetI_(1)=int_(0)^(n)[x]dx and I_(2)=int_(0)^(n){x}dx where [x] and {x} are integral and fractionalparts of x and n in N-{1}. Then (I_(1))/(I_(2)) is equal to

Find the angle of intersection of the following curves : y^2=xa n dx^2=y y=x^2a n dx^2+y^2=20 2y^2=x^3a n dy^2=32 x x^2+y^2-4x-1=0a n dx^2+y^2-2y-9=0 (x^2)/(a^2)+(y^2)/(b^2)=1a n dx^2+y^2=a b x^2+4y^2=8a n dx^2-2y^2=2 x^2=27ya n dy^2=8x x^2+y^2=2xa n dy^2=x y=4-x^2a n dy=x^2

Differentiate the following with respect of x:a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)++a_(n-1)x+a_(n)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n)," prove that " + 3^(2) *C_(3) + …+ n^(2) *C_(n) 1^(2)*C_(1) + 2^(2) *C_(2) = n(n+1)* 2^(n-2) .

int_( then the value of n is )^( If )dx=((x+sqrt(1+x^(2)))^(n))/(n)+C

If m gt 0, n gt 0 , the definite integral l=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx depends upon the vlaues of m and n and is denoted by beta(m,n) , called the beta function. E.g. int_(0)^(1)x^(4)(1-x)^(5)dx=int_(0)^(1)x^(5-1)(1-x)^(6-1)dx=beta(5, 6) and int_(0)^(1)x^(5//2)(1-x)^(-1//2)dx=int_(0)^(1)x^(7//2-1)(1-x)^(1//2-1)dx=beta((7)/(2),(1)/(2)) . Obviously, beta(n, m)=beta(m, n) . If int_(0)^(n)(1-(x)/(n))^(n)x^(k-1)dx=R beta(k, n+1) , then R is equal to