If local maximum of $f(x)=\frac{a x+b}{(x-1)(x-4)}$ exists at $(2,-1)$, then $a+b=$

For the curve $\frac{x^n}{a^n}+\frac{y^n}{b^n}=2,(n \in N$ and $n>1)$ the line $\frac{x}{a}+\frac{y}{b}=2$ is

The height of a cone with semi-vertical angle $\frac{\pi}{3}$ is increasing at the rate of 2 units $/ \mathrm{min}$. The rate at which the radius of the cone is to be decreased so as to have a fixed volume always is

The function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ where $a>0$ attains its local maximum and local minimum at $p$ and $q$ respectively. If $p^2=q$, then $a=$

Consider all functions given in List I in the interval [1,3]. The list II has the value of ' $c$ ' obtained by applying Lagrange's mean value theorem on the function of List I . Match the function and values of ' c '

$ \begin{array}{llll} \hline & \text { List I } & & \text { List II } \\ \hline \text { A } & |x-1| & \text { I } & 2 \log \left(e^3+e^2\right) \\ \hline \text { B } & \log x & \text { II } & 2 \\ \hline \text { C } & x^2+x+1 & \text { III } & \log _3 e^2 \\ \hline \text { D } & e^x & \text { IV } & \sqrt{2} \\ \hline & & \text { V } & \log \left(\frac{e^3-e}{2}\right) \\ \hline \end{array} $

If the percentage error in the radius of a circle is 3 , then the percentage error in its area is

If the extreme values of the function $f(x)=(2 \sqrt{6}+1) \cos x+(2 \sqrt{2}-\sqrt{3}) \sin x-6$ are $m$ and $M$ then $\sqrt{\left|M^2-m^2\right|}=$

If $x=2 \sqrt{2} \sqrt{\cos 2 \theta}$ and $y=2 \sqrt{2} \sqrt{\sin 2 \theta}, 0<\theta<\frac{\pi}{4}$, then the value of $\frac{d y}{d x}$ at $\theta=22 \frac{1}{2}^{\circ}$ is

If the curves $y^2=12 x-3$ and $y^2=12-k x$ cut each other orthogonally, then the length of the sub-tangent at $(1, b)$ on the curve $y^2=12-k x$ is

A rod of length 41 m with an end $A$ on the floor and another end $B$ on the wall perpendicular to the floor is sliding away horizontally from the wall at the rate of $3 \mathrm{fit} / \mathrm{min}$. When the end $B$ is at the height of 9 ft from the floor, then the rate at which the area of the triangle formed by the rod with wall and floor changes at that instant is (in $\mathrm{ft} / \mathrm{min}$ )

There is a possible error of 0.02 cm in measuring the base diameter of a right circular cone as 14 cm . If the semi-vertical angle of the cone is $45^{\circ}$, then the approximate error in its volume is (in $\mathrm{cu} . \mathrm{cm}$ )

The real valued function $f(x)=\frac{x^2}{2}-\log \left(x^2+x+1\right)$ is

If $x$ and $y$ are two positive real numbers such that $x y=4$, then the minimum value of $\left(\sqrt{x}+\frac{y^2}{2}\right)$ is

If the tangent and the normal drawn to the curve $x y^2+x^2 y=12$ at the point $(1,3)$ meet the X -axis in $T$ and $N$ respectively, then $T N=$

A man of 5 feet height is walking away from a light fixed at a height of 15 feet at the rate of of $K$ miles/hour. If the rate of increase of his shadow is $\frac{11}{5}$ feet $/ \mathrm{sec}$, then $K=($ Take 1 mile $=5280$ feet $)$

There is a possible error of 0.03 cm in a scale of length 1 foot with which the height of a closed right circular cylinder and the diameter of a sphere are measured as 3.5 feet each. If the radii of both cylinder and sphere are same, then the approximate error in the sum of the surface areas of both cylinder and sphere is (in square feet)

If the point $P\left(x_1, y_1\right)$ lying on the curve $y=x^2-x+1$ is the closest point to the line $y=x-3$, then the perpendicular distance from $P$ to the line $3 x+4 y-2=0$ is

If the normal drawn at the point $P$ on the curve $y^2=x^3-x+1$ makes equal intercepts on the coordinate axes, then the equation of the tangent drawn to the curve at $P$ is

If a balloon lying at an altitude of 30 m from an observed at a particular instant is moving horizontally. At the rate of $1 \mathrm{~m} / \mathrm{s}$ away from him, then the rate at which the balloon is moving away directly from the observer at the 40 th second is (in m/s) .

The approximate value of $\sqrt{6560}$ is

The radius of a cone of height 9 units is changed from 2 units to 2.12 units. The exact change and approximate change in the volume of the cone are respectively

The local maximum value $l$ and local minimum value $m$ of $f(x)=\frac{x^2+2 x+2}{x+1}$ in $R-\{-1\}$ exist at $\alpha, \beta$ respectively, then $\frac{l+m}{\alpha+\beta}=$

$P(5,2)$ is a point on the curve $y=f(x)$ and $\frac{7}{2}$ is the slope of the tangent to the curve at $P$. The area of the triangle (in sq. units) formed by the tangent and the normal to the curve at $P$ with $X$-axis is

If a particle is moving in a straight line so that after $t$ seconds its distance $S$ (in cms) from a fixed point on the line is given by $S=f(t)=t^3-5 t^2+8 t$, then the acceleration of the particle at $t=5 \mathrm{sec}$ is (in $\mathrm{cm} / \mathrm{sec}^2$ )

If $f:[a, b] \rightarrow[c, d]$ is a continuous and strictly increasing function, then $\frac{d-c}{b-a}$ is

The acute angle between the curves $y=3 x^2-2 x-1$ and $y=x^3-1$ at their point of intersection which lies in the first quadrant is

If the rate of change of the slope of the tangent drawn to the curve $y=x^3-2 x^2+3 x-2$ at the point $(2,4)$ is $k$ times the rate of change of its abscissa, then $k=$

If $f(x)=x+\log \left(\frac{x-1}{x+1}\right)$ is a well-defined real valued function, then $f$ is

A real valued function $f(x)=\left|x^2-3 x+2\right|+2 x-3$ is defined on $[-2,1]$. If $m$ and $M$ are absolute minimum and absolute maximum values of $f$ respectively, then $M-4 m=$

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

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

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

If a line having slope 2 is a tangent to the curve $y=x^4-6 x^3+13 x^2-12 x+5$ at points $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right), x_1, x_2 \in N$, then $x_1 x_2-y_1 y_2=$

Let $m$ be the slope of the normal $L$ drawn at $(1,2)$ to the curve $x=t^2-7 t+7, y=t^2-4 t-10$ and $a x+b y+c=0$ be the equation of the normal $L$. If GCD of $(a, b, c)$ is 1 , then $m(a+b+c)=$

If the function $f(x)=x e^{-x}, x \in R$ attains its maximum value $\beta$ at $x=\alpha$, then $(\alpha, \beta)=$

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 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 $M+m=$

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 $\frac{\alpha}{2}+\frac{\beta}{5}=$

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

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?

The equation of the tangent to the curve $x^2+y-7=4 x$ at the point $(1,10)$ is

If $\theta$ is the angle between the curves $x^2-y^2=4$ and $y^2=3 x$, then $\tan \theta=$

The absolute maximum value of the function $f(x)=2 x^3-3 x^2-36 x+9$ defined on $[-3,3]$ is

The approximate value of $\sqrt[3]{28}$ rounded up to 3 decimal places is

$y=x^2$ is the given curve. Imagine that this curve is dragged along the positive $X$-axis to a distance of ' $a$ ' units. If the acute angle between the curves at two positions is $\theta$, then

If $x$ and $y$ are two positive integers such that $x+2 y=10$ and $x^2 y^3$ is maximum, then $x^2+2 y^3=$

The equation of the normal to the curve $\sin y=\sqrt{3} x \sin \left(\frac{\pi}{6}+y\right)$ at $x=0$, is

Assertion (A) The curves $y^2=4 x$ and $x^2=-2 y$ intersect at $(1,2)$ orthogonally.

Reason (R) If the product of the slopes of the tangents drawn to two curves at their point of intersection is -1 , then the curves are said to cut each other orthogonally.

Let $f(x)=\left\{\begin{array}{cc}1+6 x-3 x^2 & x \leq 1 \\ x+\log _2\left(b^2+7\right) & x>1\end{array}\right.$. Then, the set of all possible values of $b$ such that $f(1)$ is the maximum value of $f(x)$ is

If $\theta$ is the acute angle between the curves $x^2+y^2=4$ and $y^2=3 x$, then $\tan \theta=$

Let $\sqrt{3}$ be the radius and $\frac{\pi}{3}$ be the semi-vertical angle of the given cone. Then, the height of the right circular cylinder of maximum volume that can be inscribed in the given cone is

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

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

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

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)

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

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,

The radius of a sphere is changing. At an instant of time the rate of change in its volume and its surface area are equal. Then the value of radius at that instant is?

The volume of a sphere is increasing at the rate of $4 \pi \mathrm{cc} / \mathrm{sec}$. When its volume is $288 \pi \mathrm{cc}$, the rate of increase (in $\mathrm{cm} / \mathrm{sec}$ ) in its radius is

Assertion (A) The function $f(x)=x-\log \left(\frac{1+x}{x}\right), x>0$ has no maximum.

Reason (R) If a function $f(x)$ is strictly increasing in an interval $(a, b)$, then at any point in $(a, b) f^{\prime}(x) \neq 0$

The correct option among the following is

If $\alpha$ is a root of multiplicity 3 of the equation $x^5-8 x^4+25 x^3-38 x^2+28 x-8=0$, then $\alpha^2-5 \alpha+6=$

The angle $A$ of $\triangle A B C$ is found by measurement to be $67 \frac{1^{\circ}}{2}$ and the area of $\triangle A B C$ is calculated from the measurements of $b, c, A$. In measuring $A$, an error of 9 min is made then the percentage error in the area of the triangle is

Let $f: R \rightarrow R$ be a bijection. A curve represented by $y=f(x)$ is such that $f^{\prime}(x)>0 \forall x \in \mathbf{R}$. The tangent and normal drawn at $P(\alpha, 1)$ on the curve cuts the $X$-axis at $A, B$ respectively and $C$ is the foot of the perpendicular from $P$ onto the $X$-axis. If $P(\alpha, 1)$ is such a point that $A C+C B$ is minimum, then the tangent at $P$ is parallel to the line

The $x$-coordinate changes on the curve $y=3 x^5+15 x-8$ at the rate of $\frac{1}{5}$ units/sec. $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ are the points on the curve at which the $y$-coordinate changes at the rate of 6 units/sec, then the slope of $A B=$

In $\triangle A B C, \angle B=90^{\circ}$ and $(b+a)$ is always a constant. In order that $\triangle A B C$ encloses the maximum area, $\angle C=$

If $\frac{k}{\alpha^3}$ is the length of the sub normal at any point $P(\alpha, y)$ on the curve $x^2-a^2=\frac{x^2 y^2}{a^2}$, then $k=$

A tank in the shape of a rectangular parallelopiped has volume 27 cubic meters. This tank is filled with water such that the rate of change of level of the water is thrice the rate of change water quantity falling in the tank, then the height of the tank (in meters) is

$ \text { Match the functions of List I with the items of List II. } $

If the area of a circle increases at the rate of $\frac{1}{\sqrt{\pi}}$ sq. units/sec, then the rate (in units/sec) at which the perimeter of the circle changes, when perimeter is $\sqrt{\pi}$ units, is

Let $a$ be a fixed positive real number and $n$ be an arbitrary constant. For the curve $y=\frac{x^n}{a^{n-1}}$, if the length of the subnormal at any point $(\alpha, \beta)$ is proportional to $a^2$, then $n=$

Let $P(x)$ be a polynomial of degree 3 having extreme value at $x=1$. If $\mathop {\lim }\limits_{x \to 0}\left(\frac{P(x)+4}{x^2}+2\right)=6$, then $\left(\frac{d P}{d x}\right)_{x=\frac{1}{2}}=$