Question 1 :
The equation of normal to the curve $\displaystyle y=\tan x $ at the point $(0, 0)$ is -
Question 2 :
The normal at the point $(1,1)$ on the curve $2y+{x}^{2}=3$ is
Question 3 :
The slope of tangent to the curve $y=\int_{0}^{x}\displaystyle \frac{dx}{1+x^{3}}$ at the point where $x=1$ is
Question 4 :
The slope of the tangent to the curve $y = \int_{0}^{x} \dfrac {dt}{1 + t^{3}}$ at the point where $x = 1$ is
Question 5 :
Which one of the following be the gradient of the hyperbola $xy=1$ at the point $\left(t,\dfrac{1}{t}\right)$
Question 6 :
The sum of the intercepts made by a tangent to the curve $\displaystyle \sqrt{x}+\sqrt{y}=4 $ at point $(4, 4)$ on coordinate axes is -
Question 7 :
If tangent to curve at a point is perpendicular to $x$ - axis then at that point -
Question 8 :
If a tangent to the curve $\displaystyle y=6x-{ x }^{ 2 }$ is parallel to the line $\displaystyle 4x-2y-1=0$, then the point of tangency on the curve is:
Question 9 :
<div>A curve $y=me^{mx}$ where $m > 0$ intersects y-axis at a point $P$.<br/></div>What is the equation of tangent to the curve at $P$ ?
Question 10 :
The intercept on x-axis made by tangent to the curve, $\displaystyle y=\int _{ 0 }^{ x }{ \left| t \right| } dt,x\in R$, which are parallel to the line $y=2x$, are equal to
Question 11 :
The angle made by the tangent line at (1, 3) on the curve $y=4x-{ x }^{ 2 }$ with $\overset { - }{ OX } $ is
Question 12 :
The normal drawn at the point $\displaystyle P\left ( at_{1}^{2},2at_{1} \right )$ on the parabola meets the curve again at$\displaystyle Q\left ( at_{2}^{2},2at_{2} \right ).$ then $\displaystyle t_{2} =?$
Question 13 :
If tangent to the curve $\displaystyle x={ at }^{ 2 },y=2at$ is perpendicular to $x$-axis, then its point of contact is:
Question 14 :
What is/are the tangents to $\displaystyle y=(x^{3}-1)(x-2)$ at the points where the curve cuts the x-axis
Question 16 :
If $\displaystyle \frac{x}{a}+\frac{y}{b}=1$ is a tangent to the curve $\displaystyle x=Kt,y=\frac{K}{t},K> 0$ than
Question 17 :
Find the equation of a line passing through $(-2,3)$ and parallel to tangent at origin for the circle $\displaystyle x^{2}+y^{2}+x-y=0$
Question 18 :
The equation of the normal to the curve $\displaystyle 2y=3-x^{2}$ at the point $(1,1)$
Question 19 :
Find the tangents and normal to the curve $y(x-2)(x-3)-x+7=0,$ at point (7,0) are
Question 20 :
Normal to the parabola $\displaystyle y^{2}=4ax$ where $m$ is the slope of the normal is
Question 21 :
The slope of the tangent to the curve $\displaystyle y=-x^{3}+3x^{2}+9x-27$ is maximum when x equals.
Question 22 :
The values of $x$ for which the tangents to the curves $y=x\cos{x},y=\cfrac{\sin{x}}{x}$ are parallel to the axis of $x$ are roots of (respectively)
Question 23 :
The equation of normal to the curve $y=\left| { x }^{ 2 }-\left| x \right| \right| $ at $x=-2$ is
Question 24 :
The chord of the curve $y = x^{2} + 2ax + b$, joining the points where $x = \alpha$ and $x = \beta$, is parallel to the tangent to the curve at abscissa $x =$
Question 25 :
At what point the tangent to the curve $\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{a}$ is perpendicular to the $x$ - axis
Question 26 :
Find the equation of the tangent to the curve $\displaystyle y=x^{2}+1$ at the point $(1,2).$
Question 27 :
If tangent at a point of the curve $y = f(x)$ is perpendicular to $2x - 3y = 5$ then at that point $\displaystyle \dfrac{dy}{dx}$ equals
Question 28 :
The gradient of the tangent line at the point $(a cos \alpha, a sin \alpha)$ to the circle $x^2 + y^2 = a^2$, is
Question 29 :
The equation of the normal to the curve $\displaystyle 2y=3-x^{2}$ at $(1, 1)$ is -
Question 30 :
If the tangent to the curve $\displaystyle 2y^{3}=ax^{2}+x^{3}$ at a point $(a, a)$ cuts off intercepts p and q on the coordinates axes where $\displaystyle p^{2}+q^{2}=61$ then $a$ equals to
Question 31 :
Match the points on the curve $2y^{2}=x+1$ with the slope of normals at those points and choose <div><span>the correct answer.</span><div><table class="wysiwyg-table"><tbody><tr><td>Point<br/></td><td>Slope of normal<br/></td></tr><tr><td>I : $(7, 2)$<br/></td><td><br/>$a){-4\sqrt{2}}$<br/><br/></td></tr><tr><td>II: $(0, \displaystyle \frac{1}{\sqrt{2}})$<br/><br/></td><td>$b) -8$<br/></td></tr><tr><td>III : $(1, 1)$<br/></td><td>$c) -4$<br/></td></tr><tr><td>IV: $(3, \sqrt{2})$<br/><br/><br/></td><td>$d){-2\sqrt{2}}$<br/><br/><br/><br/></td></tr></tbody></table></div></div>
Question 32 :
If the tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ at any points on it cuts the axes $OX$ and $OY$ at $P$ and $Q$ respectively then $OP+OQ$ is
Question 33 :
The slope of the tangent and normal to $y={ x }^{ 2 }-3x+5$ at $(2,3)$ are ______ and _____ respectively.
Question 34 :
The slope of the normal to the curve $ y^3 - xy-8=0$ at the point $(0,2)$ is equal to :
Question 35 :
The slope of normal to the curve y= log (logx) at x = e is
Question 36 :
The curve $\displaystyle \frac{x^{n}}{a^{n}}+\frac{y^{n}}{b^{n}}=2$ touches the line $\displaystyle \frac{x}{a}+\frac{y}{b}=2$ at the point
Question 37 :
If $ f(x) = \displaystyle \int_{1}^{x} e^{t^2/2}(1-t^2)dt, $ then $\dfrac{d}{dx} f(x) $ at x=1 is
Question 38 :
The equation of tangent to the curve $y = 3x^{2} - x + 1$ at $P(1, 3)$ is ____
Question 39 :
The equation of tangent to the curve ${ \left( \cfrac { x }{ a } \right) }^{ n }+{ \left( \cfrac { y }{ b } \right) }^{ n }=2\quad $ at the point $(a,b)$ is
Question 40 :
Find the condition that the line $\displaystyle Ax+By= 1$ may be a normal to the curve $\displaystyle a^{n-1}y=x^{n}.$
Question 41 :
The equation of the normal at $x$ $=$ $2a$ for the curve $\displaystyle y=\frac{8a^{3}}{4a^{2}+x^{2}}$ is<br/>
Question 42 :
If the slope of the curve $y=\cfrac { ax }{ b-x } $ at the point $(1,1)$ is $2$, then the values of $a$ and $b$ are respectively
Question 43 :
The normal to the curve given by $\displaystyle x= a\left ( \cos \theta +\theta \sin \theta \right ), y= a\left ( \sin \theta -\theta \cos \theta \right )$ at any point $\displaystyle \theta $ is such that it<br>
Question 44 :
For the curve represented parametrically by the equations, $x = 2 ln \cot( t) + 1$ & $y = \tan( t) + \cot( t)$<br/>
Question 45 :
The equation of the normal to the curve $\displaystyle (\frac {x}{a})^n + (\frac {y}{b})^n = 2 (n \epsilon N)$ at the point with abscissa equal to 'a' can be:
Question 46 :
If the line $ax+by+c=0$ is a normal to the rectangular hyperbola $xy=1$ then
Question 47 :
Find the co-ordinates of the point (s) on the curve $\displaystyle y= \frac{x^{2}-1}{x^{2}+1}, x> 0$ such that tangent at these point (s)have the greatest slope.
Question 48 :
The equation of the common tangent to the curves $\displaystyle y^{2}= 8x$ and $ \displaystyle xy= -1$ is<br/>
Question 49 :
For the curve $x=t^2-1$, $y=t^2-t$, the tangent is perpendicular to $x$-axis then
Question 50 :
What is the minimum intercept made by the axes on the tangent to the ellipse $ \displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ ?
Question 51 :
The tangent of the acute angle between the curves $y=|x^2-1| $ and $y=\sqrt {7-x^2}$ at their points of intersection is
Question 52 :
The number of different points on the curve $y^2=x(x+1)^2$, where the tangent to the curve drawn at (1, 2) meets the curve again, is
Question 53 :
The angle made by the tangent of the curve $x=a (t+\sin t \cos t)$, $y=a(1+sint)^2$ with the $x- axis$ at any point on it is
Question 54 :
The point of intersection of the tangents drawn to the curve $\displaystyle x^{2}y= 1-y$ at the point where it is intersected by the curve xy$=1-y,$ is given by<br>
Question 55 :
The equation of the tangents to $\displaystyle 4x^{2}-9y^{2}=36$ which are perpendicular to the straight line $\displaystyle 2y+5x= 10$ are<br>
Question 56 :
The point of contact of vertical tangent to the curve given by the equations $\mathrm{x}=3-2\cos\theta, \mathrm{y}=2+3\sin\theta$ is <br>
Question 57 :
Let tangent at a point P on the curve $x^{2m}\: Y^{\frac{n}{2}}=a^{\frac{4m+ n}{2}}(m, \: n\in \: N, \: n \: is \: even)$, meets the x-axis and y-axis at A and B respectively, if $AP : PB \:is \:\dfrac{n}{\lambda m}$, where P lies between A and B, then find the value of $\lambda$<br/>
Question 58 :
The point on the curve $y^{2} = x ,$ the tangent at which makes an angle of $45^{0}$ with positive direction of $x -$ axis will be given by<br>
Question 59 :
If the tangent at P of the curve $y^2=x^3$ intersects the curve again at Q and the straight lines OP, OQ ma angles $\alpha, \beta$ with the x-axis where 'O' is the origin then $\tan\alpha/\tan\beta$ has the value equal to?
Question 60 :
The sum of the intercepts on the coordinate axis by any tangent to the curve $\sqrt{x} + \sqrt{y} = 2$ is<br>
Question 61 :
The portion of the tangent to the curve $x=\displaystyle \sqrt{a^{2}-y^{2}}+\frac{a}{2}\log\dfrac{a-\sqrt{a^2-y^2}}{a+\sqrt{a^{2}-y^{2}}}$ intercepted between the curve and $x-$axis, is of length<br>
Question 62 :
An equation for the line that passes through $(10, -1)$ and is perpendicular to $y \displaystyle = \frac{x^2}{4} - 2$ is
Question 63 :
The equation of the normal to the curve $y = e^{-2|x|}$ at the point where the curve cuts the line $x=\displaystyle \frac{1}{2}$ is<br>
Question 64 :
The real number $\displaystyle '\alpha'$ such that the curve $\displaystyle f(x) = e^x$ is tangent to the curve $\displaystyle g(x) = \alpha x^2$.
Question 65 :
Find the equations of tangents to the curve y=x$^{4}$ which are drawn from the point $(2,0)$
Question 66 :
Find the point of intersections of the tangets drawn to the curve $\displaystyle x^{2}y=1-y$ at the points where it is intersected by the curve $xy = 1 - y$
Question 67 :
Tangents are drawn from the origin to the curve $y=\cos { x } $. Their points of contact lie on
Question 68 :
A function $y=f(x)$ has a second-order derivative $f''(x)=6(x-1)$. If its graph passes through the point $(2,1)$ and at the point tangent to the graph is $y=3x-5$, then the value of $f(0)$ is
Question 69 :
The co-ordinates of the point P on the graph of the function $\displaystyle y = e^{-|x|}$ where the portion of the tangent intercepted between the co-ordinate axes has the greatest area, is
Question 70 :
The abscissas of points $P$ and $Q$ on the curve $y=e^x+e^{-x}$ such that tangents at $P$ and $Q$ make $60^{\circ}$ with the $x$-axis are
Question 71 :
At the point $P(a,a'')$ on the graph of $y=x^n$, $(n \epsilon N)$, in the first quadrant , a normal is drawn. The normal intersects the $y$-axis at the point $(0,b)$. If $\lim_{a\rightarrow 0}b=\displaystyle \frac{1}{2}$, then n equals
Question 72 :
The tangent to the curve $2a^2y=x^3-3ax^2$ is parallel to the x-axis at the points
Question 73 :
If the tangent to the conic, $y - 6 = x^2$ at (2, 10) touches the circle, $x^2 + y^2 + 8x - 2y = k$ (for some fixed k) at a point $(\alpha, \beta)$; then $(\alpha, \beta)$ is;
Question 74 :
If the tangent at $({x_1},{y_1})$ to the curve ${x^3} + {y^3} = {a^3}$ meets the curve again at $({x_2},{y_2})$, then<br/>
Question 75 :
The perpendicular distance between the point (1, 1) and the tangent to the curve y $=e^{2x}+x^2$ drawn at the point x $=$ 0 is<br/>
Question 76 :
Suppose that the equation $f(x)={x}^{2}+bx+c=0$ has two distinct real roots $\alpha,\beta$. The angle between the tangent to the curve $y=f(x)$ at the point $\left( \cfrac { \alpha +\beta }{ 2 } ,f\left( \cfrac { \alpha +\beta }{ 2 } \right) \right) $ and the positive direction of the x-axis is
Question 77 :
If $OT$ and $ON$ are perpendiculars dropped from the origin to the tangent and normal to the curve $x=a\sin^{3}t, y=a\cos^{3}t$ at an arbitrary point, then which of the following is/are correct?<br/>
Question 79 :
Find the slope of tangent of the curve$x = a\,{\sin ^3}t,y = b\,\,{\cos ^3}t$ at $t = \frac{\pi }{2}$