Tangents & normals practice · Easy

The points on the curve $y=12 x-x^3$ at which the gradient is zero are

Let slope of the tangent line to a curve at any point $P\,(x,\,y)$ be given by $\frac{{x{y^2} + y}}{x}$. If the curve intersects the line $x + 2y = 4$ at $x = - 2$, then the value of $y$, for which the point $(3,\,y)$ lies on the curve, is

The subtangent, ordinate and subnormal to the parabola ${y^2} = 4ax$ at a point (different from the origin) are in

If at each point on the curve $y=x^3-a x^2+x+1$ the tangent is inclined at an acute angle with the positive direction of $x$-axis, then

If the rate of change of sine of an angle $\theta$ is $k$, then the rate of change of its tangent is

The lengths of tangent, subtangent, normal and subnormal for the curve $y=x^2+x-1$ at $(1,1)$ are $A, B, C$ and $D$ respectively, then their increasing order is

The equation of the normal to the curve $y=\sin x$ at $(0,0)$, is

Equation of normal to $x{\text{ = }}2{e^t},y = {e^{ - t}}{\text{ }}at{\text{ }}t = 0$ is

If the function $f(x)=x^2+\alpha / x$ has a local minimum at $x=2$, then the value of $\alpha$ is

The gradient of the curve passing through $(4,0)$ is given by $\frac{dy}{dx}-\frac{y}{x}+\frac{5x}{\left( x+2 \right)\left( x-3 \right)}=0$, if the point $(5,a)$ lies on the curve, then the value of $a$ is

The equation of the tangent to the curve $y=(2 x-1) e^{2(1-x)}$ at the point of its maximum, is

The angle between the curves ${y^2} = 4x + 4$ and ${y^2} = 36\left( {9 - x} \right)$ is

If the curve $y=a x^2+b x+c$ passes through the point $(1,2)$ such that the slope of the tangent at the origin is equal to 1, then $(a, b, c)=$

The angle between the curves $y = {a^x}$ and $y = {b^x}$ is equal to

Let y be the number of people in a village at time $t$. Assume that the rate of change of the population is proportional to the number of people in the village at any time and further assume that the population never increase in time. Then, the population of the village at any fixed $t$ is given by

The curves $x=y^2$ and $x y=a^3$ cut orthogonally at a point, then $a=$

The curve $y=a x^3+b x^2+c x+5$ touches the $\mathrm{x}$-axis at $\mathrm{P}(-2,0)$ and cut the $\mathrm{y}$-axis at a point $\mathrm{Q}$ where its gradient is 3 . Then, $a+b+c=$

The equation of the tangent to the curve $x = t\cos t,\;y = t\sin t$ at the origin is

Point of contact of tangents to the curve $y^{2}-2 x^{3}-4 y+8=0$ from the point $(1,2)$ is/are :

The number of points on the curve $y = {x^3} - 2{x^2} + x - 2$ where tangents are parallel to $x$-axis, is

The curve $y=x^{\frac{1}{5}}$ at $(0,0)$ has

The subtangent at any point on the curve $x^m y^n=a^{m+n}$ varies as

If the tangent to the curve, $y = {x^3} + ax - b\,$ at the point (1, -5) is perpendicular to the line, $- x + y + 4 = 0,\,$ then which one of the following points lies on the curve?

The point at which the tangent to the curve $y=x^2-4 x$ is parallel to $x$-axis, is

The slope of the normal to the curve $y=2 x^2+3 \sin x$ at $x=0$ is

The equation of tangent to the curve $\frac{{{x^2}}}{3} - \frac{{{y^2}}}{2} = 1$, which is parallel to $y = x$, is

The slope of normal at any point $(x,y),x > 0,y > 0$ on the curve $y = y(x)$ is given by $\frac{{{x^2}}}{{xy - {x^2}{y^2} - 1}}.$ If the curve passes through the point (1, 1) then $e.y(e)$ is equal to

The equation of the tangent to the curve $\left( {1 + {x^2}} \right)y = 2 - x$, where it crosses the $x$-axis, is

If $y = 4x - 5$ is a tangent to the curve ${y^2} = p{x^3} + q$ at (2, 3) then

The tangent to the curve $y=x e^{x^2}$ passing through the point $(1, e)$ also passes through the point