(a) Find the value of t if the point `(4,t^2-4t+4)` lie on the x-axis.

(vi) Find the value of t if the point (2,t^2-5t-6) lie on the x-axis.

(ii) Find the value of t if the point (-t,t^2+t+1) is equidistant from the axes.

(b) For what value of t the point (t^2-(2+sqrt3)t+2sqrt3,8) lie on the y-axis ?

Find the locus of the point (t^2+t+1,t^2-t+1),t in Rdot

Find the locus of the point (t^2-t+1,t^2+t+1),t in R .

Find the value of each of the following polynomials for the indicated value of variables: (i) p(x) = 4x ^(2) - 3x + 7 at x =1 (ii) q (y) = 2y ^(3) - 4y + sqrt11 at y =1 (iii) r (t) = 4t ^(4) + 3t ^(3) -t ^(2) + 6 at t =p, t in R (iv) s (z) = z ^(3) -1 at z =1 (v) p (x) = 3x ^(2) + 5x -7 at x =1 (vi) q (z) = 5z ^(3) - 4z + sqrt2 at z =2

Find the equation of the tangent to the parabola y^(2)=8x at the point (2t^(2),4t) . Hence find the equation of the tangnet to this parabola, perpendicular to x+2y+7=0

Prove that the length of the intercept on the normal at the point P(a t^2,2a t) of the parabola y^2=4a x made by the circle described on the line joining the focus and P as diameter is asqrt(1+t^2) .

If t is a real number satisfying the equation 2t^3-9t^2+30-a=0, then find the values of the parameter a for which the equation x+1/x=t gives six real and distinct values of x .

Find the equation of the normal at the specified point to each of the following curves x= a( 2 cos t+ cos 2t), y= a (2 sin t- sin 2t) at t=(pi)/(2)