TS EAMCET 2023 ONLINE 13TH MAY MORNING SHIFT

If the angle between the circles $x^2+y^2-2 x+2 y+1=0$ and $x^2+y^2+2 x-2 y+k=0$ is $\frac{\pi}{3}$, then

Let a chord $A B$ subtend an angle of $60^{\circ}$ at the centre $C(2,3)$ of a circle $S$. If the equation of $A B$ is $x+y+1=0$, then the equation of the circle $S$ is

Which of the following is the strongest reducing agent?

A body of weight 50 N is placed on a horizontal surface as shown in the figure. The minimum force required to move the body is 28.28 N . The frictional force and the normal reaction are respectively

A stone projected from the ground with a velocity $50 \mathrm{~ms}^{-1}$ at an angle of $30^{\circ}$ with the horizontal crosses a wall after a time of 3 s . Then the horizontal distance beyond the wall that the stone strikes the ground is (acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

A body starts rest with uniform acceleration. If its velocity after $n^{\text {th }}$ second (last second) is $v$ then its displacement in the last two seconds is

Velocities $(v)$ and accelerations (a) in two systems of units 1 and 2 are related as $v_2=\frac{n}{m^2} v_1$ and $a_2=\frac{a_1}{m n}$ respectively. Here $m$ and $n$ are constants. Dimensionally relations between distances ( $s_1$ and $s_2$ ) and times ( $t_1$ and $t_2$ ) in the two systems are respectively

The percentage increase in the energy for an artificial satellite to shift it from an orbit of radius $r$ to an orbit of radius $3 r / 2$ is

Gravitational forces operate among which of the following?

A body is rolling without slipping on a horizontal plane. If the rotational kinetic energy of the body is $50 \%$ of its total kinetic energy, then the body is

A block of mass $M$ moving on a frictionless horizontal surface collides with a spring of spring constant $K$, as shown in the figure. If the spring compresses by a length $L$, then the maximum momentum of the block after the collision is

Two blocks of equal masses are tied to the ends of a light string. The string passes over a mass less pulley fixed on frictionless surface as shown in the figure. The acceleration of the centre of mass of the blocks is ( $g=$ acceleration due to gravity)

A body falls freely from a height $h$ on a fixed horizontal plane and rebounds. If $e$ is the coefficient of restitution, the total distance travelled before it comes to rest is

The length of four wires $A, B, C$ and $D$ made of same materials are $1 \mathrm{~m}, 2 \mathrm{~m}, 3 \mathrm{~m}$ and 4 m respectively. The radii of the wires $A, B, C$ and $D$ are $0.2 \mathrm{~mm}, 0.4 \mathrm{~mm}$, 0.6 mm and 0.8 mm respectively. For the same applied tension, the elongation is more in the wire

An observer moves towards a stationary source of sound, with a speed of one fifth of the speed of sound. The apparent increase in the frequency heard by the observer is

A vessel having small hole in the bottom has to hold water without leakage, if water is poured into if upto a height of 7 cm . Then the radius of the hole is (surface tension of water is $0.07 \mathrm{Nm}^{-1}$, angle of contact is $0^{\circ}$ and $g=10 \mathrm{~ms}^{-2}$ )

If the seventh harmonic of a closed pipe is in unison with fourth harmonic of an open organ pipe, then the ratio of length of closed pipe to that of open pipe is

A liquid is taken in a long vertical cylindrical vessel and the cylinder is rotated about its vertical axis as shown in figure. During rotation, the liquid rises along its sides. If the radius of vessel is 0.05 m and speed of rotation is $10 \mathrm{rads}^{-1}$, then the height difference between the liquid at the centre of the vessel and its sides is $\left(g=10 \mathrm{~ms}^{-2}\right)$

A pendulum has a time period $T$ in air. Whạt it is made to oscillate in water its time period is $\sqrt{2} T$. Then the relative density of the material of the bob of the pendulum is (neglect damping)

Two slits are made one millimetre apart and the screen is placed one metre away from the slits. The fringe width when light of wavelength 500 nm is used is

For an ideal gas at a temperature of $27^{\circ} \mathrm{C}$ and at constant pressure, the coefficient of volume expansion is nearly

Refractive index of a medium is $\mu$. If the angle of incidence is twice that of the angle of refracation, then the angle of incidence is

Air is filled at $60^{\circ} \mathrm{C}$ in a vessel of open mouth. The vessel is heated to a temperature $f^{\circ} \mathrm{C}$ so that $1 / 4$ th of the air is escaped from the vessel. Assuming air as ideal gas and the volume of the vessel remaining constant, then the value of $t$ is

Two identical containers $A$ and $B$ with frictionless pistons contain the same ideal gas at the same temperature and same volume $V$. The mass of the gas in $A$ is $m_A$ and that in $B$ is $m_B$. The gas in each cylinder is now allowed to expand isothermally to the same final volume $2 V$. The changes in the pressure of the gases in $A$ and $B$ are found to be $2 \Delta p$ and $3 \Delta p$ respectively. Then the relation between $m_A$ and $m_B$ is

The temperature of 100 g of water is to be raised from $24^{\circ} \mathrm{C}$ to $90^{\circ} \mathrm{C}$ by adding steam at $100^{\circ} \mathrm{C}$ to it. The mass of the steam required in this process is (latent heat of steam is $540 \mathrm{cal} \mathrm{g}^{-1}$ )

The flux of the electric field $\mathbf{E}=24 \hat{\mathbf{i}}+30 \hat{\mathbf{j}}+28 \hat{\mathbf{k}} \mathrm{NC}^{-1}$ through an area of $20 \mathrm{~m}^2$ on the $Y Z$-plane is

In a galvanometer, $5 \%$ of the total current in the circuit passes through it. If the resistance of the galvanometer is $G$, the shunt resistance $S$ connected to the galvanometer is

The displacement current through the plates of a parallel plate capacitor of capacitance $30 \mu \mathrm{~F}$ is $150 \mu \mathrm{~A}$. The capacitor is charged by a source of varying potential at the rate of

The electric resistance of a certain wire of iron is $R$. If its length and radius are both doubled, then

The effective capacitance between points $A$ and $B$ shown in the figure is

In a meter bridge experiment the ratio of the left gap resistance to the right gap resistance is $2: 3$. The balance length from left end is

In hydrogen spectrum, the shortest and longest wavelengths of Balmer series are $\lambda_1$ and $\lambda_2$ respectively. The Rydberg constant of hydrogen is

A series $L-C-R$ circuit is connected to an AC source of voltage $150 \sin (80 \pi t)$ volt. If the resistance of the resistor in the circuit is $25 \Omega$ and the impedance in the circuit is $75 \Omega$, the average power dissipated per cycle in the circuit is

A charge $q$ moves with a velocity $2 \mathrm{~ms}^{-1}$ along $X$-axis in a uniform magnetic field $\mathbf{B}=(2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ T, then charge will experience a force

The expression for the magnetic energy stored in a solenoid of length $L$, in terms of magnetic field $B$ and area $A$ is

materials suitable for permanent magnets, must have which of the following properties?

An aeroplane is travelling horizontally towards west with a speed of $540 \mathrm{kmh}^{-1}$. The wing span of the plane is 20 m . If the horizontal component of the earth's magnetic field at the location is $2.5 \sqrt{3} \times 10^{-4} \mathrm{~T}$ and the dip angle is $30^{\circ}$, the potential difference developed between the ends of the wing is

The loss of strength of a signal while propagating through a medium is known as

$\alpha$-decay of a parent nucleus $X$ results in a daughter nucleus $Y$. If $m_x, m_y$ and $m_\alpha$ are the masses of the parent nucleus, the daughter nucleus and the $\alpha$-particles respectively, then the net kinetic energy gained in the process is

The de-Broglie wavelength of a particle moving with a speed of $0.8 c$ is equal to the wavelength of a photon. If $c$ is speed of the photon in vacuum, the ratio of the energy of the photon and the kinetic energy of the particle is

The built-in potential of a $p-n$ junction diode is 0.7 V . If the diode is forward biased and the applied voltage is 0.3 V , the effective barrier height is

In the nuclear fission of one nucleus of $\mathrm{U}^{235}$ the energy released is 188 MeV . The energy released in the nuclear fission of 235 g of $\mathrm{U}^{235}$ is nearly

(Avogadro number $=6.02 \times 10^{23} \mathrm{~mol}^{-1}$ )

The phase difference between the input voltage and the output voltage in a common emitter amplifier is

The radius of first Bohr orbit of hydrogen atom is same as that of orbit $(n)$ of hydrogen like species $X .(n)$ and $X$ respectively are

The hybridisations of the central atom in the molecules $\mathrm{BF}_3, \mathrm{BeF}_2, \mathrm{BrF}_3$ are respectively.

When compared with alkaline earth metals, the alkali metals have

Identify the impossible quantum number set for the electron from the following.

The ratio of rates of diffusion of gases $X$ and $Y$ of molecular weights 36 and 64 is

$\mathrm{Cr}^{2+}$ and $\mathrm{Mn}^{3+}$ do possess $d^4$ electronic configuration. So,

' $x^{\prime} \mathrm{g}$ of urea (molar mass $60 \mathrm{gmol}^{-1}$ ) is completely dissolved in ' $y^{\prime} \mathrm{g}$ of pure water and the solution boiled at 373.202 K . If the boiling point of pure water at $1.01^3$ bar is 373.15 K , then $x: y$ is $\left(K_b\left(\mathrm{H}_2 \mathrm{O}\right)=0.52 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)$

The graph given below is showing the relation between the extent of adsorption $(x / \mathrm{m})$ and pressure at different temperatures. The correct order of temperatures for curves i , ii and iii is

A body centred cubic lattice is made up of two different types of atoms $X$ and $Y$. Atom $X$ occupies the body centre and atoms $Y$ occupy the corner positions. One of the corners is left unoccupied per unit cell. The empirical formula of it is

The conductivity of a solution of concentration $0.1 \mathrm{~mol} \mathrm{~L}^{-1}$ of a weak monobasic acid $(\mathrm{HA})$ (in $\mathrm{S} \mathrm{cm}^{-1}$ ) is (Given : $\Lambda^{\circ}{ }_{\mathrm{HA}}=400 \mathrm{Scm}^2 \mathrm{~mol}^{-1}$ and degree of dissociation ( $\alpha$ ) of $\mathrm{H} A=0.02$ )

Enthalpy of formation of $\mathrm{CO}(g), \mathrm{CO}_2(g)$ are -110 , $-393 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively. The enthalpy of combustion of CO (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) is

At $T(\mathrm{~K})$ when one mol of $X$ and one mol of $Y$ are heated in a 1 L flask, 0.5 moles of $Z$ is formed at the equilibrium. The $K_C$ value of the reaction is

$$ X(g)+Y(g) \rightleftharpoons Z(g)+A(g) $$

From the following, identify the ions with same bond order.

I. $\mathrm{CN}^{-} \quad$ II. $\quad \mathrm{N}_2^{+}$

III. $\mathrm{O}_2^{2-}$ IV. $\mathrm{NO}^{+}$

The rate of a first order reaction doubles when the temperature changes from 300 K to 310 K . The activation energy of the reaction (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) is ( $R=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}, \log 2=0.3$ )

Which one of the following compounds is having maximum 'lone pair-lone pair' electron repulsions?

$$ \text { Match the following. } $$

The correct answer is

Choose the correct statement from the following with reference to inert pair effect.

Assertion (A) Alkali metals and their salts impart characteristic flame colours.

Reason (R) Alkali metals have low ionisation enthalpy values. So, electron excitation is possible. The correct option among the following is

Identify the correct statements about the hydrides.

I. In saline hydrides oxidation state of hydrogen is +1 .

II. La H is an example of interstitial hydride.

III. $\mathrm{NH}_3, \mathrm{H}_2 \mathrm{O}$ have the tendency to form hydrogen bonds.

IV. Electrolysis of molten sodium hydride liberates $\mathrm{H}_2$ gas at cathode.

Identify the correct orders from the following with respect to the property associated with

A. $\mathrm{NH}_3>\mathrm{PH}_3>\mathrm{AsH}_3>\mathrm{SbH}_3-$ Bond angle

B. $\mathrm{NH}_3>\mathrm{PH}_3>\mathrm{AsH}_3>\mathrm{SbH}_3-$ Basic character

C. $\mathrm{PH}_3>\mathrm{AsH}_3>\mathrm{SbH}_3>\mathrm{NH}_3-$ Thermal stability

D. $\mathrm{SbH}_3>\mathrm{NH}_3>\mathrm{AsH}_3>\mathrm{PH}_3-$ Boiling point

Kernite and cryolite are the minerals of two elements $X$ and $Z$. Respectively $X$ and $Z$ are

Ellingham diagram is the plot of $X$ vs $Y$, what are $X$ and $Y$ ?

• According to Werner's theory, the number of groups bonded to the central metal atom/ion in a coordination complex represent.

An alkene $(X)$ on ozonolysis gives propanal and ethanal. What is $X$ ?

Choose the correct decreasing order of reactivity of alkyl halides towards $\mathrm{S}_{\mathrm{N}} 1$ reaction.

Match the following.

The correct answer is

$$ \text { The correct IUPAC name of the given structure is } $$

1-propanol can be distinguished from 2-propanol by which test?

In the following reaction ' $C$ ' is an aromatic compound having substitutents $D$ and $E$. What are $D$ and $E$ ?

Statement I Aldehyde on reaction with HCN gives cyanohydrin.

Statement II Cyanohydrin is a compound which consists of hydroxy and cyano groups on the same carbon.

Choose the correct answer from the following with reference to above statements.

The product( $s$ ) formed when $m$-chlorobenzaldehyde is heated with concentrated NaOH is/are

Which of the following reactions are used in the preparation of aliphatic primary amines?

I. Gabriel's phthalimide reaction

II. Hoffmann bromamide reaction

III. Carbylamine reaction

IV. Sandmeyer reaction

$$ \text { In the given reaction, what is } X \text { ? } $$

$$ \text { The Haworth projection shown below represents } $$

A polymer sample contains 3 molecules of molar mass $10^3, 3$ molecules of molar mass 500 and 4 molecules of molar mass 200 . What is it's weight average molecular mass?

Which one of the following will improve the lathering property of soap?

If the value of $\sqrt{-5-12 i}+\sqrt{7+24 i}$ is a negative real number $k$, then $k=$

Let $z=x+i y$ be a point in the argand plane. If the amplitude of $\left(\frac{z-3}{z+2 i}\right)$ is $\frac{\pi}{2}$, then the locus of $z$ is

If $i$ is the root of the equation $x^2+1=0$, then

$$ (1+\sqrt{3} i)^{2023}+(1-\sqrt{3} i)^{2023}= $$

If a point $P$ denotes the complex number $z=x+i y$ in the argand plane and if $\frac{z-(2+i)}{z+(1-2 i)}$ is purely real, then the locus of $P$ is

One of the values of $(\sqrt{3}-i)^{\frac{1}{6}}$ is

In the following sequence of reaction, identify the major product $B$.

If $a x^2-x y-3 y^2-5 x+20 y+c=0$ represents a pair of lines passing through the point $(2,3)$, then $a-c=$

If $x^2+3 x-2 k=0$ and $x^2-2 x-7 k=0$ have a non-zero common root, then the positive root of the equation $k x^2+(k+2) x-(k+1)=0$ is

If $n, r$ are two positive integers such that $1 \leq r

The values of $\frac{x^2-2 x+1}{x^2+x-1}$ do not lie in the interval

If the sum of two particular roots of the equation $x^4-4 x^3-7 x^2+22 x+24=0$ is equal to the sum of the remaining two roots, then the sum of the cubes of all the roots of this equation is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ and $\alpha<\beta<\gamma$, then $\alpha+2 \beta+3 \gamma=$

Among the positive divisors of the number 12600 , if $n_1$ is the number of divisors which are multiples of 3 and $n_2$ is the number of divisors which are multiples of 14 , then $n_1+n_2=$

The number of ways in which $n$ boys and $n$ girls can be arranged in a row such that all the boys are together and all the girls are also together is equal to

If three unbiased dice are rolled simultaneously, then the probability that all the three dice show distinct numbers is

Three persons $A, B$ and $C$ attended a recruitment test, The ratio of the chances of $A, B, C$ in getting through the test is $1: 2: 3$ and their probabilities to face the interview successfully are $0.8,0.7,0.6$, respectively. If one of them is to be selected for the post, then the probability that $A$ gets the post is

If two cards are drawn at random simultaneously from a pack of 52 playing cards, then the probability of getting a face card and a spade card other than the face card is

Two cards are drawn at random one after the other with replacement from a pack of 52 playing cards. Then, the variance of the random variable of the number of spade cards among the drawn cards is

Let $C_0, C_1, C_2, \ldots, C_n$ be the binomial coefficients in the expansion of $(1+x)^n$. If $S_{n+1}=5 \cdot C_0+8 \cdot C_1+11 \cdot C_2+\ldots(n+1)$ terms, then $S_{11}=$

In the expansion of $(x-2 y+3 z)^5$, if the total number of terms is $p$ and the coefficient of $x^2 y z^2$ is $q$, then $\frac{q}{p}=$

Let $\mathbf{O A}=\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{O B}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{O C}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ be the position vectors of three points, $A, B$ and $C$. Let $P$ be the point which divides $A B$ in the ratio $2: 1$. If $l, m, n$ are the direction cosines of the vector $\mathbf{P C}$, then $l+3 m+2 n=$

The point of intersection of the line passing through the point $\hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and the plane passing through the points $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}, 2 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+2 \hat{\mathbf{k}}$ is

If the vectors $\mathbf{B C}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{C D}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ represent two adjacent sides of a parallelogram ABCD and $\theta$ is the angle between its diagonals $\mathbf{A C}$ and $\mathbf{B D}$, then $\tan \theta=$

$(1,-2,1)$ is a point on a plane $\pi$ and $\pi$ is parallel to the plane $x-y-z=0$. If the equation of $\pi$ is $a x+b y+c z-2=0$, then $b-2 c=$

If $|x|$ is so small that $x^3$ and higher powers of $x$ can be neglected, then an approximate value of $\frac{1}{\sqrt{4-x}(2+x)^3}$ is

Let $\mathbf{a}=\lambda \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ and $\mathbf{c}=\lambda \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ be three vectors for some integer $\lambda$. If the volume of the parallelopiped with $\mathbf{a}, \mathbf{b}, \mathbf{c}$ as coterminous edges is 61 cubic units, then the number of possible values of $\lambda$ is

If $A=\left[\begin{array}{ccc}k & 5 & 2 \\ 2 & -k & 5 \\ 5 & 2 & -k\end{array}\right]$ and $\operatorname{det} A=190$, then $\operatorname{adj} A=$

If the unique solution of the simultaneous linear equations $3 x-2 y+z=5 k, 2 x+3 y-2 z=-5 k$, $x+4 y+3 z=k$ is $x=\alpha, y=\beta, z=3$, then $k=$

If $A=\left[\begin{array}{ccc}1 & 2 & -1 \\ -1 & 0 & 2 \\ 1 & 2 & 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}-3 & -2 & 4 \\ 2 & 2 & -1 \\ -2 & 0 & 3\end{array}\right]$, then $A^2=$

$$ \left|\begin{array}{lll} 2 & 3 & 5 \\ 3 & 5 & 2 \\ 5 & 2 & 3 \end{array}\right|+\left|\begin{array}{ccc} 1 & 1 & 1 \\ 7 & 11 & 13 \\ 49 & 121 & 169 \end{array}\right|= $$

A plane $\pi$ passing through the point $3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ is parallel to the plane which passes through the point $\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and perpendicular to the vector $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$. Then, the cartesian equation of $\pi$ is

Let the direction cosines of two lines satisfy the equations $3 l+2 m+n=0$ and $2 m n-3 n l+5 l m=0$. If $\theta$ is the angle between these two lines, then $\cos \theta=$

In $\triangle A B C$, if $A$ is an acute angle, $b=6, c=9$ and $\sin A=\frac{2 \sqrt{14}}{9}$, then $3 a(\cos B+\cos C)=$

If $A+B+C+D=2 \pi$, then $\cos A-\cos B+\cos C-\cos D=$

If $540^{\circ}<\theta<630^{\circ}$ and $\tan \theta=5 / 12$, then

$$ \frac{\cos \frac{\theta}{2}-5 \sin \frac{\theta}{2}}{\sqrt{-(12 \sec \theta+5 \operatorname{cosec} \theta)}}= $$

If the variance of the data $2,3,5,8,12$ is $\sigma^2$ and the mean deviation from the median for this data is $M$, then $\sigma^2-M=$

If $\cot \theta=-\frac{2}{3}$ and $\theta$ does not lie in the 4 th quadrant, then $\frac{(5 \sin \theta+\cos \theta)^2}{\tan \theta+\cot \theta}=$

If $\cosh x=\frac{4}{3}$, then $3 \cosh x+3^2 \cosh 2 x+3^3 \cosh 3 x=$

If $f(x)=|x-1|+|x-2|$, then

$$ f^{\prime}(-2023)+f^{\prime}\left(\frac{2024}{2023}\right)+f^{\prime}(2023)= $$

If the roots of the equation $x^3-11 x^2+36 x-36=0$ are the ex-radii of a $\triangle A B C$, then the perimeter of the $\triangle A B C$ is

$$\mathop {\lim }\limits_{x \to 2 + }\left([x]^2-[x]-2\right)+\mathop {\lim }\limits_{x \to- 3 - }\left([x]^2-4[x]+3\right)= $$

If $f(x)=x^{\tan x}+(\tan x)^x$, then $f^{\prime}\left(\frac{\pi}{4}\right)=$

If $f(x)=\frac{e^{2 x}-e^{-2 x}}{e^{3 x}+e^{-3 x}}$, then $f^{\prime}(0)=$

$$ \lim _{x \rightarrow 0} \frac{\left(3^{2 x}-\sqrt{x+1}\right) \sin 5 x}{1-\cos 4 x}= $$

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

The general solution of the differential equation $\frac{d y}{d x}=\sin (x-y)+\cos (x-y)$ is

The general solution of the differential equation $x^2 d y-\left(x y-y^2\right) d x=0$ is

$$ \int_0^2 x^{\frac{5}{2}} \sqrt{2-x} d x= $$

$$ \int_0^3\left|x^2-3 x+2\right| d x= $$

$$ \int_{-\frac{\pi}{8092}}^{\frac{\pi}{8092}} \frac{\sec (2023 x)}{1+(2023)^{(2023 x)}} d x= $$

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

The range of the real valued function $f(x)=\frac{1}{x-|x|}$ 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 $\frac{\alpha}{2}+\frac{\beta}{5}=$

If $\frac{6 x^4+13 x^3+2 x^2-x+3}{2 x^2+3 x-2}=f(x)+\frac{A}{a x-1}+\frac{B}{x+b}$, then $f(\mathrm{l})+a \cdot B+b \cdot A=$

If the function $f: R \rightarrow R$ is defined by

$$ f(x)= \begin{cases}2 x-3, & \text { if } x<-2 \\ x^2-1, & \text { if }-2 \leq x \leq 2 \\ 3 x+2, & \text { if } x>2\end{cases} $$

then $f$ is

The domain of the real valued function

$$ f(x)=\frac{\sqrt{\log _{10}\left(\frac{x}{x-2}\right)}}{\sqrt{[x]^2-5[x]+6}} \text { is } $$

(Here, $[x]$ denotes the greatest integer function)

$$ \int \frac{\sec ^2 x}{(\sec x+\tan x)^2} d x= $$

$$ \int \frac{1}{16-7 \sin ^2 x} d x= $$

$$ \int \frac{1}{3 \cos x-4 \sin x+5} d x= $$

Let 6,8 be the $X$ and $Y$-intercepts made by the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$, respectively. If $g x+f y+1=0$ is a line passing through the point $(1,-1)$, then the radius of the circle $S=0$ is

If $(3,1)$ and $(-2,4)$ are points on a circle $S$ whose centre lies on the line $x-y+1=0$, then the parametric equations of $S$ are

$$ \int \frac{1}{(x-2)\left(x^2+1\right)} d x= $$

If $y=m x+4(m>0)$ is a tangent to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$, then the point of contact of this tangent is

Let the line $x-y+1=0$ intersect the circle $x^2+y^2+2 x+2 y+1=0$ in two points $A$ and $B$. If $A B$ is the diameter of the circle $x^2+y^2+2 g x+2 f y+c=0$, then $g+f=$

If $\alpha$ is the angle made by the perpendicular drawn from origin to the line $3 x-4 y+5=0$ with positive $X$-axis in positive direction and $a x+b y=1$ is the equation of a line passing through the point $(1,-1)$ with $\tan \alpha$ as its slope, then $a+a b+b=$

Let $S \equiv x^2+y^2-8 x+10 y+5=0$ be a circle. Let $P(1,1)$ and $Q(1,-1)$ be two points. Then, the point of intersection of the polar of $P$ with respect to $S=0$ and the chord with $Q$ as mid-point to $S=0$ is

If the focal distance of a point $P\left(2, y_1\right)$ on the parabola $y^2=k x$ is 3 , then the equation of the tangent drawn at $P$ to the given parabola is

Normals are drawn from the point $P(8,0)$ to the parabola $y^2=12 x$. If $\theta$ is the acute angle between two non-horizontal normals among them, then $\tan \theta=$

Let the line $L_1$ passing through the point of intersection of the lines $2 x+3 y-5=0$ and $4 x-5 y+7=0$ divide the line segment joining the points $(2,3)$ and $(1,-1)$ in the ratio $2: 1$. If the equation of $L_1$ is $a x+b y=1$, then $33(a-b)=$

Let $A(4,3,5), B(1,-2,1), C(3,2,1)$ be the vertices of a $\triangle A B C$. If the internal bisector of $\angle B A C$ meet the side $B C$ at $D$, then $C D=$

If $L_1$ is a line passing through the point $P(4,-3)$ and perpendicular to the line $3 x-4 y+k=0$ then the distance of $P$ from the line $5 x-3 y-2=0$ measured along the line $L_1$ is

Let $P$ be the point to which origin has to be shifted by the translation of axes, so as to remove the first degree terms from the equation $3 x^2+y^2-6 x+4 y+4=0$. If the origin is shifted to $P$ by the translation of axes, then the transformed equation of $2 x^2+3 x y-5 y^2+2 x-23 y-24=0$ is

Let $A=(2,0)$ and $B=(0,-2)$. Let $P$ be any point such that the sum of the distance of $P$ from $A$ and $B$ is 4 . Then, the equation of the locus of the point $P$ is

Let $A B C$ be a triangle and $A=(1,2)$. If $x-3 y-5=0$ the and $x+5 y-9=0$ are the perpendicular bisectors of the sides $A B$ and $B C$ respectively, then the length of the side $A C$ is

Let $S$ and $S^{\prime}$ be the foci of an ellipse $E$ and $B$ be one end of its minor axis. Let $\angle S^{\prime} S B=\pi / 6$ and $(2 \sqrt{3}, 1)$ be a point on $E$. If $X$-axis is the major axis and $Y$-axis is the minor axis of the ellipse $E$, then the sum of the squares of the lengths of major and minor axis is

If $A$ and $B$ are arbitrary constants, then the differential equation having $y=A e^x+B \sin 2 x$ as its general solution is

Area of the region bounded by the curve $y=2-x-3 x^2$, the $X$-axis, the $Y$-axis and the line $x=-2$ is

If $4 x+2 y+n=0$ is a normal to the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ then $n=$

The masses of carbondioxide and water (in g) respectively formed during complete combustion of 10 g of glucose at STP are

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ and $\alpha<\beta<\gamma$, then $\alpha+2 \beta+3 \gamma=$