By using properties of determinants. Show that: (i) `|x+4 2x2x2xx+4 2x2x2xx+4|=(5x-4)(4-x)^2` (ii) `|y+k y y y y+k y y y y+k|=k^2(2ydotk)^2`

By using properties of determinants. Show that: (i) |(x+4,2x,2x),(2x,x+4,2x),(2x,2x,x+4)|=(5x-4)(4-x)^2 (ii) |(y+k,y,y),(y,y+k,y),(y,y,y+k)|=k^2(3y+k)

Using properties of determinants, prove the following |[x,x+y,x+2y],[x+2y,x,x+y],[x+y,x+2y,x]|=9y^2(x+y)

5x + 2y = 2k , 2 (k + 1 ) x + ky = (3k + 4)

Find each of the following products: (i) (4x + 5y) (4x - 5y) (ii) (3x^(2) + 2y^(2)) (3x^(2) - 2y^(2))

{:(2x + (k - 2)y = k),(6x + (2k - 1) y = 2k + 5):}

{:(2x + 3y = 2),((k + 2)x + (2k + 1)y = 2(k - 1)):}

If the normal at four points P_(i)(x_(i), (y_(i)) l, I = 1, 2, 3, 4 on the rectangular hyperbola xy = c^(2) meet at the point Q(h, k), prove that x_(1) + x_(2) + x_(3) + x_(4) = h, y_(1) + y_(2) + y_(3) + y_(4) = k x_(1)x_(2)x_(3)x_(4) =y_(1)y_(2)y_(3)y_(4) =-c^(4)

The equation of tangent to the curve y=x+(4)/(x^2) , that is parallel to the x-axis is y = k. Then the value of k is?

I. y^(2) + y - 1 = 4 - 2y - y^(2) II. x^(2)/2 - 3/2 x = x - 3