Question

Find the equations of tangents to the curve y = (x²– 1) (x – 2) at the points, where the curve cuts the X-axis.

Let f : N → R be a function defined as f(x) = 4x² + 12x + 15. Then show that f : N → S, where S is range of f, is invertible. Also find the inverse of f.

Show that the equation of normal at any point t on the curve x = 3 cos t – cos³t and y = 3 sin t – sin³t is 4(y cos2t – x sin³t) = 3 sin 4t

Let A = R – {2}, B = R – {1}. If f : A → B is a function defined by $f(x)\,=\,\left({\frac{x-1}{x-2}}\right)\!,$ show that f is one-one and onto. Hence find $f^{-1}.$

Evaluate : $\large\textstyle\int\!{\frac{d x}{x(x^{3}+8)}}\,.$

Find : $\large\textstyle\int(3x+1)\large{\sqrt{4-3x-2x^{2}}}d x\,.$

Show that the equation of tangent to the parabola y² = 4ax at (x₁, y₁) is yy₁ = 2a(x + x₁).