Application Of Derivatives - Calculus - Mathematics Previous Year Questions

If an error of $0.02 \mathrm{sq} . \mathrm{cm}$ is found in the surface area of a sphere when its radius is measured as 10 cm , then the approximate error that occurs in the volume of the sphere, in cubic centimeters, is

The local maximum value of the function $f(x)=-(x-2)^3(x+2)^2$ is

If $\theta$ is the angle between the curves $y^2=4 x$ and $x^2+y^2=5$, then $|\tan \theta|=$

The area of the triangle formed by the tangent and the normal drawn to the curve $y^2=4 x$ at $(1,2)$ with $Y$-axis is (in square units)

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta>0$ be a very small real number. Let $c \in(a, b)$ be such that $f(c-\delta)0$. Let $(f(\alpha-\delta)-f(\alpha))(f(\alpha+\delta))<0 \forall \alpha \in(a, b)$ and $\alpha \neq c$. Then,

Consider two families of curves $y^2=4 a x$ ( $a$ is a parameter) and $x^2+\frac{y^2}{2}=c^2(c$ is parameter). If one curve from each family is chosen, then the angle between those two curves is

Electric current $(I)$ is measured by galvanometer, the current being proportional to the tangent of the angle ( $\theta$ ) of deflection. If the deflection is read as $45^{\circ}$ and an error of $1 \%$ is made in reading it, the percentage error in the current is

For all real values of $x$, the minimum value of $\frac{1-x+\lambda^2}{1+x+x^2}$ is

If the equation of a tangent drawn to the curve $y=\cos (x+y),-1 \leq x \leq 1+\pi$ is $x+2 y=k$, then $k=$

$f: R \rightarrow R$ is a function defined by $f(x)=\frac{1}{e^x+2 e^{-x}}$

Assertion (A) : $f(c)=\frac{1}{3}$ for some values of $c \in R$

Reason (R) : $0 < f(x) \leq \frac{1}{2 \sqrt{2}}$ for all $x \in R$

Then, which of the following options is correct?

If the absolute maximum and absolute minimum values of the function $f(x)=x^3-2 x^2+x-3$ defined on $[0,2]$ are $M$ and $m$ respectively, then $M+m=$

The diameter of a sphere is measured as 42 cm . If there is an error of $1 / 77 \mathrm{~cm}$ in measuring it, then the error involved in the volume of that sphere (in cubic centimeters) is

For $h, k \in N$, let $P(h, k)$ be the point of intersection of the curves $x^2 y-x^3=8$ and $y^3-x y^2=32$. If $\theta$ is the acute angle between these two curves at $P$, then $\tan \theta=$

If the slope of the tangent drawn at any point $(x, y)$ to the curve $y=f(x)$ is $3 x^2-5$ and $f(1)=2$, then the tangent at $(1,2)$ to the curve $y=f(x)$ intersects the curve at the point

The nearest approximate value of $\sqrt{2023}$ is (let $\Delta x=87$ ).

The slope of the normal drawn at a point $P$ to the curve $y=x^3-10 x^2+31 x-30$ is $-\frac{1}{14}$. If the co-ordinates of $P$ are integers, then the $X$-intercept of the tangent drawn at $P$ to the given curve is

$x$ and $y$ are two positive integers such that $2 x+3 y=50$. If $x^2 y^3$ is maximum for $x=\alpha$ and $y=\beta$, then $\frac{\alpha}{2}+\frac{\beta}{5}=$

An angle between the curves $x^2-y^2=4$ and $x^2+y^2=4 \sqrt{2}$ is

A ladder of length 13 m has one end resting against a vertical wall and the other on the ground. If the lower end moves away from the wall at a speed of $2 \mathrm{~m} / \mathrm{min}$ then the speed (in $\mathrm{m} / \mathrm{min}$ ) at which upper end falls when the bottom is 5 m away from the wall is

The maximum volume (in cu. units) of the cylinder which can be inscribed in a sphere of radius 12 units is