The semi-vertical angle of a right circular cone is $45^{\circ} \%$ If the radius of the base of the cone is measured as 14 cm with an error of $\left(\frac{\sqrt{2}-1}{11}\right) \mathrm{cm}$, then the approximate error in measuring its total surface area is (in sq cm)
If a man of height 1.8 mt , is walking away from the foot of a light pole of height 6 mt , with a speed of 7 km per hour on a straight horizontal road opposite to the pole, then the rate of change of the length of his shadow is (in kmph )
If the curves $2 x^2+k y^2=30$ and $3 y^2=28 x$ cut each other orthogonally, then $k$ is equal to
If $$3 f(\cos x)+2 f(\sin x)=5 x$$, then $$f^{\prime}(\cos x)+f^{\prime}(\sin x)=$$
If the normal drawn at a point $$P$$ on the curve $$3 y=6 x-5 x^3$$ passes through $$(0,0)$$, then the positive integral value of the abscissa of the point $$P$$ is
The line joining the points $$(0,3)$$ and $$(5,-2)$$ is a tangent to the curve $$y=\frac{c}{x+1}$$, then $$c=$$
If $$a, b>0$$, then minimum value of $$y=\frac{b^2}{a-x}+\frac{a^2}{x}, 0< x< a$$ is
The point on the curve $$y=x^2+4 x+3$$ which is closest to the line $$y=3 x+2$$ is
The number of those tangents to the curve $$y^2-2 x^3-4 y+8=0$$ which pass through the point $$(1,2)$$ is
If the straight line $$x \cos \alpha+y \sin \alpha=p$$ touches the curve $$\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2$$ at the point $$(a, b)$$ on it and $$\frac{1}{a^2}+\frac{1}{b^2}=\frac{k}{p^2}$$, then $$k=$$
Condition that 2 curves $$y^2=4 a x, x y=c^2$$ cut orthogonally is
A closed cylinder of given volume will have least surface area when the ratio of its height and base radius is
Two particles $$P$$ and $$Q$$ located at the points $$P\left(t, t^3-16 t-3\right), Q\left(t+1, t^3-6 t-6\right)$$ are moving in a plane, the minimum distance between the points in their motion is
If $$x^3-2 x^2 y^2+5 x+y-5=0$$, then at $$(\mathrm{l}, \mathrm{l}), y^{\prime \prime}(\mathrm{l})=$$
If the curves $$y=x^3-3 x^2-8 x-4$$ and $$y=3 x^2+7 x+4$$ touch each other at a point $$P$$, then the equation of common tangent at $$P$$ is
The maximum value of $$f(x)=\frac{x}{1+4 x+x^2}$$ is
The minimum value of $$f(x)=x+\frac{4}{x+2}$$ is
The condition that $$f(x)=a x^3+b x^2+c x+d$$ has no extreme value is
At any point $$(x, y)$$ on a curve if the length of the subnormal is $$(x-1)$$ and the curve passes through $$(1,2)$$, then the curve is a conic. A vertex of the curve is
If $$y=4 x-6$$ is a tangent to the curve $$y^2=a x^4+b$$ at $$(3,6)$$, then the values of $$a$$ and $$b$$ are
Find the positive value of $$a$$ for which the equality $$2 \alpha+\beta=8$$ holds, where $$\alpha$$ and $$\beta$$ are the points of maximum and minimum, respectively, of the function $$f(x)=2 x^3-9 a x^2+12 a^2 x+1$$.
If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area.
The diameter and altitude of a right circular cone, at a certain instant, were found to be 10 cm and 20 cm respectively. If its diameter is increasing at a rate of 2 cm/s, then at what rate must its altitude change, in order to keep its volume constant?
Given, $$f(x)=x^3-4x$$, if x changes from 2 to 1.99, then the approximate change in the value of $$f(x)$$ is
If the curves $$\frac{x^2}{a^2}+\frac{y^2}{4}=1$$ and $$y^3=16 x$$ intersect at right angles, then $$a^2$$ is equal to
Let $$x$$ and $$y$$ be the sides of two squares such that, $$y=x-x^2$$. The rate of change of area of the second square with respect to area of the first square is
If $$f^{\prime \prime}(x)$$ is a positive function for all $$x \in R, f^{\prime}(3)=0$$ and $$g(x)=f\left(\tan ^2(x)-2 \tan (x)+4\right)$$ for $$0 < x <\frac{\pi}{2}$$, then the interval in which $$g(x)$$ is increasing is
The line which is parallel to X-axis and crosses the curve $$y=\sqrt x$$ at an angle of 45$$\Upsilon$$ is
If the error committed in measuring the radius of a circle is 0.05%, then the corresponding error in calculating its area would be
The stationary points of the curve $$y=8 x^2-x^4-4$$ are
The distance between the origin and the normal to the curve $$y=e^{2 x}+x^2$$ drawn at $$x=0$$ is units