TS EAMCET 2020 (Online) 11th September Evening Shift

The dimension of angular momentum in mass $(M)$, length $(L)$ and time $(T)$ is

A particle of mass $m=1 \mathrm{~kg}$ moves in the $x y$-plane. The force on it at time $t$ is $F(t)=[2 \sin (\alpha t) \hat{\mathbf{i}}+3 \cos (\alpha t) \hat{\mathbf{j}}] \mathrm{N}$, where $\alpha=1 \mathrm{~s}^{-1}$. At time $t=0$, the particle is at rest at the origin. Calculate the magnitude of its position vector $\mathbf{r}$ (in m ) and velcoity vector $\mathbf{v}$ (in m/s) at time $t=\frac{\pi}{2} \mathrm{~s}$.

A particle aimed at a target, projected with an angle $15^{\circ}$ with the horizontal is short of the target by 10 m . If projected with an angle of $45^{\circ}$ is away from the target by 10 m , then the angle of projection to hit the target is

Consider that a truck is moving initially with $54 \mathrm{~km} / \mathrm{h}$. It has stopped by the driver after looking at an obstacle with a deceleration of $10 \mathrm{~m} / \mathrm{s}^2$. The distance travelled by truck before coming to rest is

A ball is thrown vertically upwards with an initial velcoity $u$ reaches maximum height in 5 s . The ratio of distance travelled by the ball in the 2nd and 7th second is (assume, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A circular freeway entrance and exit are commonly banked to control a moving car at $14 \mathrm{~m} / \mathrm{s}$. To design similar ramp for $28 \mathrm{~m} / \mathrm{s}$ one should

A cyclist leans with the horizontal at angle $30^{\circ}$, while negotiating round a circular road of radius $20 \sqrt{3} \mathrm{~m}$. The speed of the cycle should be

Under action of force, a 2 kg body moves such that its position $x$ as function of time $t$ is given by $x=\alpha t^2 / 2$, where $x$ is in metre, $t$ is in seconds and $\alpha=1 \mathrm{~m} / \mathrm{s}^2$. The work done by the force in the first two seconds is

The block starts from rest as shown in the figure. Find the work done by force of 10 N and friction in the time 0 to 4 s . [Take, $g=10 \mathrm{~m} / \mathrm{s}^2$ ]

Consider a thin metal strip of mass l kg and length 5 m . Calculate its moment of inertia about an axis perpendicular to strip and located at 100 cm on strip from one its end. (Assume the breadth as the strip is negligible)

A solid cylinder is released from rest from the top of an inclined plane of inclination $30^{\circ}$ and length 60 cm . If the cylinder rolls without slipping, then the speed when it reaches the bottom is

The long range force experienced by a neutral particle with a finite mass

If the radius of the earth shrinks by $1 \%$, its mass remaining the same, then the acceleration due to gravity on the earth surface would

A stiff spring having spring constant $k=400 \mathrm{~N} / \mathrm{m}$ is attached to the floor vertically. A mass $m=10 \mathrm{~kg}$ is placed on top of the spring. The block oscillates if it is pressed downward and released. Find the extension in the spring at which the block loses contact with spring. (Take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

The change in surface energy when a big spherical drop fo radius $R$ is split into $n$ spherical droplets of radius $r$ is ( $T=$ surface tension)

Young's modulus is proportionality constant that relates the force per unit area applied perpendicularly at the surface of an object to

Two trucks heading in opposite directions each with speed $0.1 u$, approach each other. The speed of the sound is $u$. The driver of first truck sounds his horn of frequency 495 Hz . Let $v_1$ and $v_2$ are the frequencies heard by the driver of second truck, when the trucks approach each other and when the trucks have passed each other. The magnitude of $v_1-v_2$ is

A sheet of steel at $20^{\circ} \mathrm{C}$ has size as shown in figure below. If the co-efficient of linear expansion for steel is $10^{-5}{ }^{\circ} \mathrm{C}^{-1}$, then what is the change in the area at $60^{\circ} \mathrm{C}$ ?

Different material of two identical long bars $A$ and $B$ are coated with wax and have their one end immersed in a hot oil bath. When the steady state is reached, the lengths for which wax melt are $l_A$ and $l_B$. If $k_A$ and $k_B$ are thermal conductivities of materials, then

Certain amount of heat supplied to an ideal gas under isothermal condition will result in

A gas is at constant pressure $4 \times 10^5 \mathrm{~N} / \mathrm{m}^2$. When a heat energy of 2000 J is supplied to the gas, its volume changes by $3 \times 10^{-3} \mathrm{~m}^3$. What is the increase in its internal energy?

A prism is made of a glass having refractive index $\sqrt{2}$. If the angle of minimum deviation is equal to angle of the prism, then the angle of prism is

A thin glass prism of angle $9^{\circ}$ with refractive index 1.4 is combined with another glass prism of refractive index 1.6 as shown in the figure. The combination of the prism provides dispersion without deviation. Determine the angle $(A)$ of the second prism.

Wavelength of light used in an optical instruments are $\lambda_1=4000 \mathop {\rm{A}}\limits^{\rm{o}}$ and $\lambda_2=5000 \mathop {\rm{A}}\limits^{\rm{o}}$, then the ratio of their respective resolving powers (corresponding to $\lambda_1$ and $\lambda_2$ ) is

105. The electric flux from a cube of edge $l$ is $\phi$ in an enclosed charge. If the edge of the cube is made $\frac{2}{3} l$ and the charge enclosed in the cube is doubled, then the electric flux value will be

106. If the dielectric constant of a substance $K=\frac{4}{3}$, then the electric susceptibility $\chi$ in terms of vacuum permittivity $\varepsilon_0$ is

Four $4 \Omega$ resistors are connected together along the edges of a square. A 12 V battery with internal resistance of $2 \Omega$ is connected across a pair of the diagonally opposite corners of the square. The power dissipated in the circuit is

Find potential difference points $A \& F$ and $F \& B$.

A straight wire of mass 0.2 kg and length 1.5 m carries a current 2A is shown in the figure. It is suspended in mid-air by a uniform magnetic field $B$ pointing to the plane of paper. The magnitude of magnetic field is (ignore, earth's magnetic field and assume $g=10 \mathrm{~m} / \mathrm{s}^2$ )

The ratio of the magnetic field inside a solenoid at an axial point well inside and at an axial end point is

An infinite long wire lying along the $Y$-axis, is carrying a current $I$ as shown in the figure. The magnetic flux through a circular loop of radius $R$ in the $x y$-plane is [assume, $\mu_0=$ magnetic in free space permeability]

A solenoid has a core of a material with relative permeability $\frac{800}{\pi}$. The windings of the solenoid are insulated from the core and carry current of 2 A . If the number of turns is 1000 per metre, find the magnetic field $B$.

For an $R$ - $L-C$ circuit, driven with voltage of amplitude $V_m$ and frequency $\omega_0=\frac{1}{\sqrt{L C}}$, the current exhibits resonance. The quality factor $Q$ is

The typical wavelength of X-ray is

The binding energy (BE) per nucleon for an element is 7.14 MeV . If the BE of element is 28.6 MeV , then the number of nucleons in the element is

In a photoelectric effect experiment if the frequency of light is doubled, the stopping potential will

A monochromatic light of wavelength $\lambda$ ejects photoelectrons from a metal surface with work function ( $\phi) 2.4 \mathrm{eV}$. These photoelectrons are made to collide with hydrogen atoms in ground state. The maximum value of $\lambda$ for which hydrogen atom may be ionised is [take, $h c=1240 \mathrm{eV}-\mathrm{nm}$ ]

In $p-n-p$ transistor, the collector current is

The output of a NOR gate is HIGH when

The electromagnetic waves of frequency 6 GHz are used in

How much volume of 1 N aqueous solution of $\mathrm{H}_2 \mathrm{SO}_4$ should be taken, which will contain 0.2 moles of $\mathrm{H}_2 \mathrm{SO}_4$ ?

The uncertainty in position and velocity of a particle in motion are $1 \times 10^{-8} \mathrm{~m}$ and $6.627 \times 10^{-20} \mathrm{~m} / \mathrm{s}$, respectively. The mass of the particle is ( $h=6.627 \times 10^{-34} \mathrm{Js}$ )

The maximum number of possible electrons in a subshell with $n=3$ and $l=2$ is

Equal amounts of two gases of molecular weights 4 and 40 are mixed. The pressure of the mixture is 1.1 atm . What will be the partial pressure of the lighter gas in the mixture?

Which of the curve ( $Z v s p$ ) will be followed by a real gas?

Which of the following statements regarding the first law of thermodynamics is correct?

Aqueous solution of ferric nitrate when mixed with aqueous solution of potassium thiocyanate gives red colour solution. The intensity of red colour becomes constant on attaining equilibrium.

Choose the correct statement when the following chemical is added to the above solution at equilibrium.

I. Oxalic acid

II. Mercuric chloride

The pH of the solution, when

(i) sodium acetate is dissolved in water.

(ii) ammonium chloride is dissolved in water.

A 1.17 % solution of solute $A$ is isotonic with 7.2 % solution of glucose. If the molecular weight of solute $A$ is 58.5, the value of van't Hoff factor, ' $i$ ' is

A mixture of 3.0 moles of $\mathrm{Na}_2 \mathrm{O}$ and 1.5 mol of $\mathrm{KO}_2$ is dissolved in 1000 mL of water. The vapour pressure of the solution in Torr, at $100^{\circ} \mathrm{C}$ is

The weight of potassium dichromate (molecular weight $=294$ ) required to prepare 0.04 N of 250 mL solution is

To resist the coagulation of 100 cc gold sol; 1 cc of $10 \% \mathrm{NaCl}$ is added to it in the presence of $10^{-4} \mathrm{~g}$ gelatin. The gold number of gelatin is

A compound can crystallise in two forms $\alpha$ and $\beta$ which are fcc and bcc, respectively. The $\alpha$-form has side length of 2 pm and the $\beta$-form has side length of 4 pm . The ratio of their density $\frac{\rho_\alpha}{\rho_\beta}$ is

A solution of $\mathrm{Fe}^{2+}$ is titrated potentiometrically using $\mathrm{Ce}^{4+}$ solution. When $80 \% \mathrm{Fe}^{2+}$ is titrated, the EMF of the system in $V$ is

(Given, $E^{\circ} \mathrm{Fe}^{3+} / \mathrm{Fe}^{2+}=0.77 \mathrm{~V}$ and $\left.\mathrm{Fe}^{2+}+\mathrm{Ce}^{4+} \longrightarrow \mathrm{Fe}^{3+}+\mathrm{Ce}^{3+}\right) (\log 2=0.3, \log 3=0.5, \log 4=0.6)$

What is the unit for the zero order rate constant?

The basic difference in approach between Mendeleev's periodic law and modern periodic law is the change on the basis of classification of elements from

Which of the following pairs shows diagonal relationship?

The correct set of symbols of the molecular orbitals given below is

Find out the correct order of repulsive interaction of electron pairs in the following systems.

(I) Lone pair - lone pair

(II) Lone pair- bond pair

(III) Bond pair-bond pair

The geometry of $\mathrm{XeOF}_4$ is

Dihydrogen can be prepared by which of the following reactions.

(I) Reaction of granulated Zn with dil. HCl

(II) Reaction of Zn with $a q . \mathrm{NaOH}$

(III) By heating calcium hydrogen carbonate

The carbonates of alkaline earth metals decompose on heating to give

I. $\mathrm{CO}_2$

II. Metal oxide

III. $\mathrm{H}_2 \mathrm{O}$

IV. CO

Buckminister fullerene contains the following $X$ number of six and $Y$ number of five member rings. What is the value of $X$ and $Y$ ?

Borax is converted into crystalline boron by the following steps :

$$ \text { Borax } \xrightarrow[\mathrm{H}_2 \mathrm{O}]{X} \mathrm{H}_3 \mathrm{BO}_3 \xrightarrow{\Delta} \mathrm{~B}_2 \mathrm{O}_3 \xrightarrow[\Delta]{Y} B $$

Identify $X$ and $Y$ respectively.

The number of dissociable protons in "orthophosphoric acid" is

Which of the following ions will exhibit colour in aqueous solution?

In the following reactions, identify $P, Q$ and $R$, respectively

I. $3 \mathrm{Fe}_2 \mathrm{O}_3+\mathrm{CO} \longrightarrow 2 P+\mathrm{CO}_2 \uparrow$

II. $\mathrm{Fe}_3 \mathrm{O}_4+4 \mathrm{CO} \longrightarrow 3 Q+4 \mathrm{CO}_2 \uparrow$

III. $\mathrm{Fe}_2 \mathrm{O}_3+\mathrm{CO} \longrightarrow 2 R+\mathrm{CO}_2 \uparrow$

Which of the following correctly represents the order of ligands in spectrochemical series?

The average atmospheric residence time is lowest for which of the given the greenhouse gas?

The major products $P$ and $Q$ from the below reactions are :

$$ \mathrm{CH}_3 \mathrm{CH}=\mathrm{CH}_2 \xrightarrow{\mathrm{HBr}} P $$

$$ \text { The major product formed in the following reactions is } $$

An alkene $A\left(\mathrm{C}_4 \mathrm{H}_8\right)$ exhibits cis/trans isomerism. $A$ on ozonolysis gives $B$, which when reacted with NaOH followed by hydroxylamine gave $C$. What are $B$ and $C$ ?

$$ \text { The major product in the following reaction is } $$

$$ \text { The major product formed in the following reaction is } $$

$$ \text { The major product in the following reactions, is } $$

$$ \text { Monomeric units of melamine polymer are } $$

Xerophthalmia disease is caused by the deficiency of

Artificial sweetening agent from below is

Which one of the following methods is suitable to separate a mixture of $n$-pentane and toluene?

The number of points $z$ on the Argand plane which satisfy the conditions $\operatorname{Re}\left(\frac{z-2}{z-4 i}\right)=0$ and $\lim \left(\frac{z-2}{z-4 i}\right)=1$ simultaneously is

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4+x^2+1=0$, then $\frac{\alpha^3+\beta^3+\gamma^3+\delta^3}{\alpha^6+\beta^6+\gamma^6+\delta^6}=$

Let the roots of the equation $E_1 \equiv x^3+x^2+l x+n=0$ be $x_i,(i=1,2,3)$ and the roots of $E_2 \equiv x^3+a x^2+b x+c=0$ be $\frac{x_i-1}{2}$. If the equation $E_2=0$ is a equation of class one, then the roots of these two equations excluding the common roots are

If $(\sqrt{3}+i)^{10}=a+b i, a, b \in \mathbf{R}$, then the values of $a$ and $b$ are respectively

If $z$ is a complex number such that $z^2+z+1=0$, then $\left(z+\frac{1}{z}\right)^3+\left(z^2+\frac{1}{z^2}\right)^3+\left(z^3+\frac{1}{z^3}\right)^3+\ldots . .+\left(z^{2020}+\frac{1}{z^{2020}}\right)^3=$

For $n>2$ and $n \in \mathbf{N}$, the product of the roots of $(x-n)\left(\left(x^2-2 n x\right)^2+\left(2 n^2-5\right)\left(x^2-2 n x\right)\right. \left.+\left(n^4-5 n^2+4\right)\right)=0$ is divisible by

If $\alpha, \beta$ are the roots of $a x^2+b x+c=0$ then $\left(\frac{\alpha}{a \beta+b}\right)^3-\left(\frac{\beta}{a \alpha+b}\right)^3=$

The maximum value of $\left\{x \in \mathbf{R} / \sqrt{x+2}>\sqrt{8-x^2}\right\}=$

For $n=1,2,3, \ldots .50$, let

$$ A=\left\{a_n / a_n=\left\{\begin{array}{ll} (-1)^{\frac{n}{2}}\left(\frac{n}{2}\right), & \text { if } n \text { is even } \\ (-1)^{\frac{n-1}{2}}\left(\frac{n-1}{2}\right), & \text { if } n \text { is odd } \end{array}\right\}\right\} $$

and $B$ is the set of all distinct elements of $A$. The number of permutations all the elements of set $B$ such that even integers are in increasing order, is

If $\alpha$ represents the number of arrangements of $p$ men and $q$ women in a row such that all men are together and $\beta$ represents the number of circular arrangements of the same people with the same condition, then $\alpha: \beta$ is

If $E_1, E_2 \ldots, E_n$ are an independent events such that $P\left(E_r\right)=\frac{1}{1+r},(r=1,2, \ldots, n)$, then the probability that atleast one of $E_1, E_2, \ldots, E_n$ happens is

If the probability function of a random variable $X$ is given by $P(X=n)=\frac{k(n+1)}{3 n}$ for $n \in \mathbf{N} \cup\{0\}$ where $k$ is a constant, then $P(X<2)=$

An urn contains five balls. Two balls are drawn at random and they are found to be white. The probability that all the balls in the urn are white, is

4-digit numbers are formed using the digits 4, 5, 6, 7, 8, 9 allowing repetition of the given digits. If a number is chosen at random from those numbers thus formed, then the probability that it is exactly divisible by 3 is

An observer counts 240 vehicles per hour at a specific location on a highway. Assuming that the arrival of vehicles at the location follows Poisson distribution, the probability that more than two vehicles arrive over a 30 sec time interval is

If sum of the coefficients of $x^r(r=0,1,2, \ldots, 2 n)$ in the expansion of $\left(1+3 x-2 x^2\right)^n$ is 128 , then $\sum_{r=1}^{2 n} r \frac{(2 n)_{C_r}}{(2 n)_{C_{r-1}}}=$

If ${ }^n C_0,{ }^n C_1,{ }^n C_2, \ldots,{ }^n C_n$ respectively are the binomial coefficients in the expansion of $(1+x)^n$, then when $n=10, \sum_{r=1}^{10}{ }^n C_r \cdot r(r-4)=$

The approximate value of $\left(3 \sqrt{126}+\sin 61^{\circ}\right)$ correct to three decimal places, obtained by taking $1^{\circ}=0.0174$ radians, is

Let $\mathbf{a}=2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=-\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If $\mathbf{c}$ is a vector such that $\mathbf{a} \cdot \mathbf{c}=|\mathbf{c}|,|\mathbf{c}-\mathbf{a}|=2 \sqrt{2}$ and the angle between $\mathbf{a} \times \mathbf{b}$ and $\mathbf{c}$ is $\frac{\pi}{3}$, then $|(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}|=$

The equation of the plane in normal form passing through the point $A(\bar{a})$, parallel to a vector $\bar{b}$ and containing a vector $\bar{c}$ is

If $A(4,7,8), B(2,3,4)$ and $C(2,5,7)$ are the vertices of $\triangle A B C$, then the length of the internal bisector of the angle $A$ is

The position vectors of the points $A$ and $B$ are respectively $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ and $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If the points $P$ and $Q$ are respectively the orthogonal projections of $A$ and $B$ on the plane $x+y+z=3$, then $P Q=$

For scalars $\lambda, \mu$ if the vector equation of a plane is $\mathbf{r}=(2+3 \lambda-\mu) \hat{\mathbf{i}}+(1-2 \lambda+3 \mu) \hat{\mathbf{j}}+(-2+2 \lambda+\mu) \hat{\mathbf{k}}$, then its Cartesian equation is

If $A(4,3,2), B(5,4,6), C(-1,-1,5)$ are the vertices of a triangle, then the coordinates of the point in which the bisector of the angle $A$ meet the side $B C$ is

Assertion (A) The direction ratios of line $L_1$ are 2, 5, 7 and those of line $L_2$ are $\frac{4}{\sqrt{19}}, \frac{10}{\sqrt{19}}, \frac{14}{\sqrt{19}}$. The lines $L_1, L_2$ are parallel.

$\boldsymbol{\operatorname { R e a s o n }}(R)$ The direction ratios of a line $L_1$ are $a_1, b_1, c_1$ and those of another line $L_2$ are $a_2, b_2, c_2$. The lines $L_1$ and $L_2$ are parallel if $a_1 a_2+b_1 b_2+c_1 c_2=0$

The correct option among the following is

If $\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-7}{2}$ lies in the plane $a x+b y+z=7$, then $a+b=$

Let $I$ be a unit matrix of order 6 . Let $A=\left(a_{i j}\right)$ be a square matrix of order 6 such that $a_{i j}=\left\{\begin{array}{l}1, \text { if } i+j=7 \\ 0, \text { if } i+j \neq 7\end{array}\right.$ then $\left(A(\operatorname{adj} A) A^{-1}\right) A^2=$

$A$ is a singular matrix of order five. $B$ is another matrix having the rank $\rho(B)$ equal to the $\operatorname{rank} \rho(A)$ and $B$ has a non-zero minor of order 3. Then which one of the following is true?

Let $a, b, c \notin\{0,1\}$. If the system of equations

$$ \begin{aligned} & \Pi_1 \equiv x+a y+a z=0 \\ & \Pi_2 \equiv b x+y+b z=0 \\ & \Pi_3 \equiv c x+c y+z=0 \end{aligned} $$

has a non-trivial solution, then the system of equations $\Pi_1=a, \Pi_2=b, \Pi_3=c$ has

If $S_1$ and $S_2$ are the variances of the first $2 k$ and $k(k>1)$ natural numbers respectively, then ( $S_1 / S_2$ ) lies in the interval

The standard deviations of two sets of observations $X=\left\{x_i\right\}$ and $Y=\left\{y_i\right\}(i=1,2, \ldots, 100)$ are respectively 5 and 6 . If $\bar{x}, \bar{y}$ are their means and $\sum_{i=1}^{100}\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)=600$, then the standard deviation of $Z=\left\{z_i / z_i=x_i-y_i\right)$ is

For the least possible value of $n \in \mathbf{Z}$ the solution $(x, y)$ of the equations $\cos ^{-1} x+\left(\sin ^{-1} y\right)^2=\frac{n \pi^2}{4}$ and $\cos ^{-1} x\left(\sin ^{-1} y\right)^2=\frac{\pi^4}{16}$, is

If $x=\left(\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{8}\right)$, then $\frac{\sin x+\cos x}{\tan x}=$

If for $|x|>1, \tanh ^{-1}\left(\frac{1}{x}\right)+\operatorname{coth}^{-1}(x)=\log _e(f(x))$, then $f(-5)=$

$$ \text { Match the items of List-I with those of List-II } $$

$$ \text { The correct match is } $$

The period of $\cos (3 x+5)+7$ is

If $\cos \left(\frac{\alpha-\beta}{2}\right)=2 \cos \left(\frac{\alpha+\beta}{2}\right)$, then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}=$

In a triangle $A B C$, if $c=9, s=10$ and $\Delta=10 \sqrt{2}$ then $b\left[1+\sqrt{2} \tan \left(\frac{A-B}{2}\right)\right]=$

In a triangle $A B C$, if $a

In a $\triangle A B C, \cot A+\cot B+\cot C=$

Match the functions of List-I with derivates given in List-II

If $f(x)=\frac{x-1}{e^x}$, then $f^{\prime}(0)+f^{\prime \prime}(0)=$

$$ \begin{aligned} & \text { If }\left(\frac{d y}{d x}\right)=\frac{1}{\left(\frac{d x}{d y}\right)} \text { and } \frac{d^2 x}{d y^2}\left(\frac{d y}{d x}\right)^3+\frac{d^2 y}{d x^2}=k \text {, then } \\ & e^{k f(x)}-k f(x)= \end{aligned} $$

At $x=0, f(x)=\left\{\begin{array}{l}\frac{x}{|x|+2 x^2}, x \neq 0 \\ k, \quad x=0\end{array}\right.$ is

$$\mathop {\lim }\limits_{x \to 0} \frac{1-\cos (1-\cos x)}{\sin ^4 x}= $$

If

$$ f(x)=\left|\begin{array}{ccc} 1+\sin x+\sin 2 x+\sin 3 x & \frac{3+\sin 2 x}{2} & \frac{-2+\sin 3 x}{3} \\ 3+4 \sin x & \frac{3}{2} & \frac{4}{3} \sin x \\ 1+\sin x & \frac{1}{2} \sin x & \frac{1}{3} \end{array}\right| $$

then $\int_0^{\pi / 2}\left(f(x)+f^{\prime}(x)\right) d x=$

$$ \lim\limits_{n \rightarrow \infty} \frac{1}{n}\left[\frac{1}{n} \sin ^{-1} \frac{1}{n}+\frac{2}{n} \sin ^{-1} \frac{2}{n}+\ldots+\frac{\pi}{2}\right]= $$

The area (in sq. units) enclosed by the curves $y=2 x-x^2$ and $y=x^2-2 x-6$ is

If $\alpha$ and $\beta$ are respectively the order and degree of the differential equation for which $a x^2+b y^2=1$ is the general solution, then the eccentricity of the ellipse $\alpha x^2+\beta y^2=1$ is

If the solution $y(x)$ of the differential equation $\sin x \frac{d y}{d x}+y \cos x=e^{2 x}, x \in(0, \pi)$ satisfies $y\left(\frac{\pi}{2}\right)=0$, then $y\left(\frac{\pi}{6}\right)=$

The solution of the differential equation $x d y-y d x=\sqrt{x^2+y^2} d x$, given that $y=1$ when $x=\sqrt{3}$, is

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?

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

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

For each $n \in \mathbf{N}$, let $A_n=\{(n+1) k / k \in \mathbf{N}\}$ and $X=\bigcup_{n \in \mathbf{N}} A_n \cdot A$ mapping $f: X \rightarrow N$ defined by $f(x)=x$, $\forall x \in \mathbf{X}$, is

The number of bijective functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ such that $f(x+y)=f(x)+f(y) \forall x, y \in \mathbf{Z}$, is

$$ \int \frac{x^2}{\left(\sqrt{4-x^2}\right)^3} d x= $$

If the partial fractions decomposition of $\frac{x^4+24 x^2+28}{\left(x^2+1\right)^3}$ is $\frac{A}{x^2+1}+\frac{B}{\left(x^2+1\right)^2}+\frac{C}{\left(x^2+1\right)^3}$ then $B-2 A+C=$

Let $I_n=\int \sec ^n x d x$. If $5 I_6-4 I_4=f(x)$, then $f\left(\frac{\pi}{4}\right)$ is equal to

If $I_m=\int x^m \cos n x d x=g(x)-\frac{m(m-1)}{n^2} I_{m-2}$, then $g(x)=$

$$ \int \frac{d x}{x \ln (x) \ln ^2(x) \ln ^3(x) \ldots \ln ^m(x)}=\frac{(\ln (x))^K}{K}+C \Rightarrow 2 K= $$

If $x^2+y^2-a^2+\lambda(x \cos \alpha+y \sin \alpha-p)=0$ is the smallest circle through the points of intersection of $x^2+y^2=a^2$ and $x \cos \alpha+y \sin \alpha=p, 0

Let $a=1+i$ and $z=x+i y$. If the curve $z \bar{z}+a z+\bar{a} \bar{z}-4=0$ is cut by the straight line $(z+\bar{z})-i(z-\bar{z})+2=0$ at two points $A$ and $B$, then the equation of the circle passing through the origin, $A$ and $B$ is

A point $P$ moves so that distance from $(0,2)$ to $P$ is $\frac{1}{\sqrt{2}}$ times the distance of $P$ from $(-1,0)$. Then the locus of the point is

The point/points of intersection of the common tangents of the two circles $x^2+y^2-8 x-6 y+21=0$ and $x^2+y^2-2 y-15=0$ is/are

If $P A$ and $P B$ are the tangents drawn from the point $P(1,1)$ to the circle $x^2+y^2+g x+g y-2=0$ with $C$ as the centre, then the area (in sq. units) of the quadrilateral $P A C B$ is

The centre of the smallest circle which cuts the circles $x^2+y^2-2 x-4 y-4=0$ and $x^2+y^2-10 x+12 y+52=0$ orthogonally is

$L_1$ and $L_2$ are two common tangents to two circles. If $L_1$ touches the two circles at $A(1,1)$ and $B(0,1)$ and $L_2$ touches the two circles at $C\left(\frac{3}{5}, \frac{4}{5}\right), D\left(\frac{-1}{5}, \frac{7}{5}\right)$, then the equation of the radical axis of the two circles is