Let `C` be the curve `y=x^3` (where `x` takes all real values). The tangent at `A` meets the curve again at `Bdot` If the gradient at `B` is `K` times the gradient at `A ,` then `K` is equal to (a) 4 (b) 2 (c) `-2` (d) `1/4`

If the tangent at (1,1) on y^2=x(2-x)^2 meets the curve again at P , then find coordinates of P .

The slope of the tangent to the curve y=sqrt(4-x^2) at the point where the ordinate and the abscissa are equal is (a) -1 (b) 1 (c) 0 (d) none of these

A curve is represented by the equations x=sec^2ta n dy=cott , where t is a parameter. If the tangent at the point P on the curve where t=pi/4 meets the curve again at the point Q , then |P Q| is equal to (5sqrt(3))/2 (b) (5sqrt(5))/2 (c) (2sqrt(5))/3 (d) (3sqrt(5))/2

If x+y=k is normal to y^2=12 x , then k is (a)3 (b) 9 (c) -9 (d) -3

If X−k divides x^3−6x^2 +11x−6 =0, then k can't be equal to, (a) 1. (b) 2. (c) 3. (d) 4

The two curves x=y^2,x y=a^3 cut orthogonally at a point. Then a^2 is equal to 1/3 (b) 3 (c) 2 (d) 1/2

The line tangent to the curves y^3-x^2y+5y-2x=0 and x^2-x^3y^2+5x+2y=0 at the origin intersect at an angle theta equal to (a) pi/6 (b) pi/4 (c) pi/3 (d) pi/2

If the line x-1=0 is the directrix of the parabola y^2-k x+8=0 , then one of the values of k is (a) 1/8 (b) 8 (c) 4 (d) 1/4

For the curve y=4x^3-2x^5, find all the points at which the tangent passes through the origin.