If by change of axes without change of origin, the expression `ax^(2)+2hxy+by^(2)` becomes `a_(1)x_(1)^(2)+2h_(1)x_(1)y_(1)+b_(1)y_(1)^(2)`, prove that
`a+b=a_(1)+b_(1)`

If y= (tan^(-1) x)^(2) show that (x^(2) + 1)^(2) y_(2) + 2x (x^(2) + 1)y_(1) = 2

Differentiate a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)+...+a_(n-1)x+a_(n)

If y= (x + sqrt(x^(2) + 1))^(m) then prove that, (x^(2)+1) y_(2) + xy_(1)= m^(2)y

If ax^(2)+by^(2)=1 cute a'x^(2)+b'y^(2)=1 orthogonally, then

If a_(1), a_(2), a_(3) ,…., a_(n) are the terms of arithmatic progression then prove that (1)/(a_(1)a_(2)) + (1)/(a_(2)a_(3)) + (1)/(a_(3)a_(4)) + ….+ (1)/(a_(n-1) a_(n)) = (n-1)/(a_(1)a_(n))

If x_(1),x_(2) "and" y_(1),y_(2) are the roots of the equations 3x^(2) -18x+9=0 "and" y^(2)-4y+2=0 the value of the determinant |{:(x_(1)x_(2),y_(1)y_(2),1),(x_(1)+x_(2),y_(1)+y_(2),2),(sin(pix_(1)x_(2)),cos (pi//2y_(1)y_(2)),1):}| is

Two curves a_(i)x^(2) + b_(i)y^(2)= 1, i=1, 2 where a_(1) ne a_(2), b_(1) ne b_(2), a_(1), a_(2), b_(1), b_(2) ne 0 may intersect orthogonally If ______

If 4a^(2)+9b^(2)+16c^(2)=2(3ab+6bc+4ca) , where a,b,c are non-zero real numbers, then a,b,c are in GP. Statement 2 If (a_(1)-a_(2))^(2)+(a_(2)-a_(3))^(2)+(a_(3)-a_(1))^(2)=0 , then a_(1)=a_(2)=a_(3),AA a_(1),a_(2),a_(3) in R .

Two balls , having linear momenta vec(p)_(1) = p hat(i) and vec(p)_(2) = - p hat(i) , undergo a collision in free space. There is no external force acting on the balls. Let vec(p)_(1) and vec(p)_(2) , be their final momenta. The following option(s) is (are) NOT ALLOWED for any non -zero value of p , a_(1) , a_(2) , b_(1) , b_(2) , c_(1) and c_(2) (i) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) + c_(1) hat(k) , vec(p)_(2) = a_(2) hat(i) + b_(2) hat(j) (ii) vec(p)_(1) = c_(1) vec(k) , vec(p)_(2) = c_(2) hat(k) (iii) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) + c_(1) hat(k) ,vec(p)_(2) = a_(2) hat(i) + b_(2) hat(j) - c_(1) hat(k) (iv) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) , vec(p)_(2) = a_(2) hat(i) + b_(1) hat(j)