The normal to the curve \(y=x^3+\frac32x^2-2x+1\) at the point where \(x=0\) cuts the curve again at two other points.

Find the \(x\)-coordinates of these two points.

The normal to the curve \(y=\frac{4}{x^{2}}+a x+7\) at the point where \(x=2\) has equation \(x+4 y=b\). Find the values of \(a\) and \(b\).

The point \(P\) lies on the curve \(y=(5x+2)^{\frac23}\).

The \(x\)-coordinate of \(P\) is \(5\).

The normal to the curve at \(P\) intersects the line \(x+y=11\) at the point \(Q\).

The point \(R\) is the reflection of \(Q\) in the tangent to the curve at \(P\).

Find the coordinates of \(R\).

A curve has equation \(y=2 x \cos x\). The normal to the curve at ( \(\pi,-2 \pi\) ) meets the \(x\)-axis and \(y\)-axis at points \(P\) and \(Q\). Find the exact area of triangle \(P O Q\).

The tangent to the curve \(y=\frac{\sqrt{x+1}}{x}\) at the point where \(x=3\) meets the line \(y=x-16\) at the point \(A\). Find the coordinates of \(A\).

The curve \(C\) has equation \(y=\ln \left(x^{3}+3\right)\). The normal to \(C\) at the point where \(x=1\) meets the line \(y=x\) at the point \(P\). Find the exact coordinates of \(P\).

The tangent to the curve \(y=(3 x-1)^{\frac{1}{3}}\) at the point where \(x=3\) meets the coordinate axes at the points \(A\) and \(B\). The point with coordinates \((a, a)\) lies on the perpendicular bisector of the line \(A B\). Find the exact value of \(a\).

The line \(L\) is the normal to the curve \(y=3(5 x+6)^{\frac{1}{2}}\) at the point where \(x=2\). The point \((-2, k)\), where \(k\) is a constant, lies on \(L\). Find the exact value of \(k\).

In this question \(p\) and \(q\) are constants.

The normal to the curve

\(\displaystyle y=\frac{p}{x^2}+5x-2\)

at the point where \(x=1\), has equation \(y=-x+q\).

Find the values of \(p\) and \(q\).

(a) Find the equation of the normal to the curve

\(y=x^3-7x^2+12x-5\)

at the point \((1,1)\).

(b) Find the \(x\)-coordinates of the two points where the normal cuts the curve again. Give your answers in the form \(x=a\pm\sqrt b\), where \(a\) and \(b\) are integers.

The diagram shows the line \(y=1-4x\) meeting the curve \(y=4x^2-6x-5\) at the points \(A\) and \(B\).

The tangent to the curve at \(B\) meets the horizontal line through \(A\) at the point \(C\).

Find the \(x\)-coordinate of \(C\), giving your answer correct to 2 decimal places.

The normal to the curve \(y=1+\tan3x\) at the point \(P\) with \(x\)-coordinate \(\frac{\pi}{12}\) meets the \(x\)-axis at the point \(Q\).

The line \(x=\frac{\pi}{12}\) meets the \(x\)-axis at the point \(R\). Find the area of the triangle \(PQR\).

The normal to the curve

\(y=\sin(4x-\pi)\)

at the point \(A(a,0)\), where \(\frac{\pi}{2}\lt a\lt \pi\), meets the \(y\)-axis at the point \(B\).

Find the exact area of triangle \(OAB\), where \(O\) is the origin.

The normal to the curve \(y=\tan\left(3x+\frac\pi2\right)\) at \(P(p,-1)\), where \(0\lt p\le\frac\pi6\), meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). Find the exact coordinates of the midpoint of \(AB\).

The normal to the curve

\(y=\frac{\ln(3x^2+2)}{x+1}\)

at the point \(A\) on the curve where \(x=0\), meets the \(x\)-axis at point \(B\). Point \(C\) has coordinates \((0,3\ln2)\). Find the gradient of the line \(BC\) in terms of \(\ln2\).

(a) Find the coordinates of the point on the curve

\(y=\sqrt{1+3x}\)

where the gradient of the normal is \(-\frac83\).

(b) Find the equation of the normal to the curve

\(y=\sqrt{1+3x}\)

at the point \((8,5)\), in the form \(y=mx+c\).

In this question \(a\) and \(b\) are constants.

The normal to the curve

\(y=\frac{a}{x}+3x-2\)

at the point where \(x=1\) has equation

\(y=-\frac14x+b.\)

Find the values of \(a\) and \(b\).

The tangent to the curve \(y=ax^2-5x+2\) at the point where \(x=2\) has equation \(y=7x+b\).

Find the value of \(a\) and the value of \(b\).

A curve has equation \(y=p(x)\), where

\(p(x)=x^3-4x^2+6x-1.\)

(a) Find the equation of the tangent to the curve at the point \((3,8)\). Give your answer in the form \(y=mx+c\).

(b)

(i) Given that \(p^{-1}\) exists, write down the gradient of the tangent to the curve \(y=p^{-1}(x)\) at the point \((8,3)\).

(ii) Find the coordinates of the point of intersection of these two tangents.

The curve

\(y=\frac{\ln(x^2+2)}{2x-3}\)

has a normal at the point where \(x=2\). This normal meets the \(y\)-axis at \(P\).

Find the coordinates of \(P\).

A curve has equation

\(y=\frac{2+\sin 3x}{x+1}.\)

(a) Show that the exact value of \(\frac{dy}{dx}\) when \(x=\frac{\pi}{6}\) can be written in the form

\(\frac{k}{\left(\frac{\pi}{6}+1\right)^2},\)

where \(k\) is an integer to be found.

(b) Find the equation of the normal to the curve at the point where \(x=0\).

(a) Find the equation of the normal to the curve

\(y=x^3+x^2-4x+6\)

at the point \((1,4)\).

(b) Without using a calculator, find the exact \(x\)-coordinate of each of the two points where the normal cuts the curve again.

It is given that

\(x=2+\operatorname{sec}\theta,\qquad y=5+\tan^2\theta.\)

(a) Express \(y\) in terms of \(x\).

(b) Find \(\frac{dy}{dx}\) in terms of \(x\).

(c) A curve has the equation found in part (a). Find the equation of the tangent to the curve when \(\theta=\frac{\pi}{3}\).

The tangent to the curve

\(y=\ln(3x^2-4)-\frac{x^3}{6}\)

at the point where \(x=2\), meets the \(y\)-axis at the point \(P\). Find the exact coordinates of \(P\).

Find the equation of the tangent to the curve

\(y=\frac{\ln(3x^2-1)}{x+2}\)

at the point where \(x=1\). Give your answer in the form \(y=mx+c\), where \(m\) and \(c\) are constants correct to 3 decimal places.

(a) Find the equation of the tangent to the curve

\(2y=\tan 2x+7\)

at the point where \(x=\frac{\pi}{8}\). Give your answer in the form \(ax-y=\frac{\pi}{b}+c\), where \(a\), \(b\) and \(c\) are integers.

(b) This tangent intersects the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the length of \(PQ\).

Do not use a calculator in this question.

(a) Find the equation of the tangent to the curve

\(y=x^3-6x^2+3x+10\)

at the point where \(x=1\).

(b) Find the coordinates of the point where this tangent meets the curve again.

A curve has equation

\(y=x\cos x.\)

(a) Find \(\dfrac{dy}{dx}\).

(b) Find the equation of the normal to the curve at the point where \(x=\pi\).

(c) Find the exact value of

\(\int_0^{\frac{\pi}{6}} x\sin x\,dx.\)

It is given that \(y=(x^2+1)(2x-3)^{1/2}\).

(i) Show that

\(\frac{dy}{dx}=\frac{Px^2+Qx+1}{(2x-3)^{1/2}},\)

where \(P\) and \(Q\) are integers.

(ii) Hence find the equation of the normal to the curve at the point where \(x=2\), giving your answer in the form \(ax+by+c=0\), where \(a\), \(b\) and \(c\) are integers.

The normal to the curve \(y=(x-2)(3x+1)^{2/3}\) at the point where \(x=\frac73\), meets the \(y\)-axis at the point \(P\). Find the exact coordinates of \(P\).

Find the equation of the normal to the curve

\(y=\sqrt{8x+5}\)

at the point where \(x=\frac12\), giving your answer in the form \(ax+by+c=0\), where \(a\), \(b\) and \(c\) are integers.

At the point where \(x=1\) on the curve \(\displaystyle y=\frac{k}{(x+1)^2}\), the normal has a gradient of \(\frac13\).

(i) Find the value of the constant \(k\).

(ii) Using your value of \(k\), find the equation of the tangent to the curve at \(x=2\).

Find the equation of the normal to the curve

\(y=\frac{\ln(3x^2+1)}{x^2}\)

at the point where \(x=2\), giving your answer in the form \(y=mx+c\), where \(m\) and \(c\) are correct to 2 decimal places. You must show all your working.

The line \(y=kx-5\), where \(k\) is a positive constant, is a tangent to the curve \(y=x^2+4x\) at the point \(A\).

(i) Find the exact value of \(k\).

(ii) Find the gradient of the normal to the curve at \(A\), giving your answer in the form \(a+b\sqrt5\), where \(a\) and \(b\) are constants.

The normal to the curve

\(y=\sqrt{4x+9},\)

at the point where \(x=4\), meets the \(x\)- and \(y\)-axes at the points \(A\) and \(B\). Find the coordinates of the midpoint of the line \(AB\).

The point \(P\) lies on the curve \(y=3x^2-7x+11\).

The normal to the curve at \(P\) has equation \(5y+x=k\).

Find the coordinates of \(P\) and the value of \(k\).