TS EAMCET 2022 ONLINE 18TH JULY MORNING SHIFT

If $x>0$ and $x \neq(2 n+1) \frac{\pi}{2}$, then $\int\left(x \sqrt{x}-e^{\log (\sec x \tan x)}+\frac{3 x^2-2 x+1}{x^2}\right) d x=$

A projectile is launched with an initial speed of $40 \mathrm{~m} / \mathrm{s}$ at an angle $30^{\circ}$ above the ground. The projectile lands on a hillside 2.0 s later. The net displacement from where the projectile lands on hillside 2.0 s later. The net displacement from where the projectile was launched to where it hits the target is (take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A lens is made of glass having an index of refraction 1.5 . One side of the lens is flat and the other side is convex with a radius $R$. If an object is placed 60 cm , towards the convex side of the lens, the image is formed at

A body starts from the rest and acquires a velocity of $10 \mathrm{~m} / \mathrm{s}$ in 2s. What is the acceleration of the body and the distance travelled?

A time varying force acts on a ball of mass 100 g for 2 ms . The force versus time curve is shown below. If the initial speed of the ball is $10 \mathrm{~m} / \mathrm{s}$, then the speed of ball after 2 ms is

The potential energy of an object is $U(x)=\left(5 x^2-4 x^3\right) \mathrm{J}$, where $x$ is the position in metre. The position at which the force becomes zero is

Two blocks of masses 1 kg and 2 kg connected by a light rod and the system is slipping down a rough incline angle $45^{\circ}$ with the horizontal. The frictional coefficient at both the contacts is 0.4 . If the acceleration of the system is $\alpha \sqrt{2}$, the value of $\alpha$ is (use, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

An ant starts from the origin and crawls 10 cm along the $X$-axis and then 20 cm along the $Y$-axis. The dot product of the ant's displacement vector with the position vector of a point that makes $45^{\circ}$ with the $X$-axis and has a magnitude of $\sqrt{2} \mathrm{~cm}$ is

A bullet fired into a target losses one-third of its velocity after travelling a distance $x$ metre into the target. If the bullet comes to rest by travelling a further distance $x^{\prime}$, then the ratio $\frac{x^{\prime}}{x}$ is

An air bubble of radius 1 mm is at a depth of 8 cm below the free surface of a liquid column. If the surface tension and density of the liquid is $0.1 \mathrm{~N} / \mathrm{m}$ and 2000 $\mathrm{kg} / \mathrm{m}^3$, respectively, by what amount is the pressure inside the bubble greater than the atmospheric pressure? (take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

Three particles, each of mass $M$, situated at the vertices of an equilateral triangle of side length $l$. The only forces acting on the particles are their mutual gravitational forces. It is desired that each particle moves in a circle while maintaining the original separation $l$. The initial speed that should be given to each particle is

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

The correct match is

A body starting at $t=0$ from origin oscillates simple harmonically with a period of 4 s . After what time will its kinetic energy by $75 \%$ of its total energy?

A large storage tank, open to the atmosphere at top and filled with water, develops a small hole in its side at a point 20.0 m below the water level. If the rate of flow from the hole is $3.08 \times 10^{-5} \mathrm{~m}^3 / \mathrm{s}$, then the diameter of the hole is (take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

Four masses are arranged along a circle of radius 1 m as shown in the figure. The centre of mass of this system of masses is at

If the root mean square (rms) speed of nitrogen molecules at room temperature is $100 \mathrm{~m} / \mathrm{s}$, then the rms speed of helium molecule at the same temperature is

An object cools from $100^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$ in 10 min , when the surrounding temperature is $10^{\circ} \mathrm{C}$. Then the time taken by the object to cool from $70^{\circ} \mathrm{C}$ to $20^{\circ} \mathrm{C}$ is (take, $\ln 2=0.7, \ln 3=11, \ln 6=18$ )

Two waves of amplitudes $A_1$ and $A_2$ respectively, are superimposed. The ratio between the maximum and minimum intensities of the resultant waves is $9: 4$.

The value of $\frac{A_2}{A_1}$ is (assume $A_1>A_2$ )

Find the ratio of the length of a steel rod and a copper rod, if the steel rod is 4 cm longer, then the copper rod at any temperature.

(The coefficient of linear expansion for steel and copper are $1.1 \times 10^{-5} /{ }^{\circ} \mathrm{C}$ and $1.7 \times 10^{-5} /{ }^{\circ} \mathrm{C}$, respectively)

1.00 kg of liquid water at $100^{\circ} \mathrm{C}$ undergoes a phase change into steam at $100^{\circ} \mathrm{C}$ at 1.0 atm (take it to be $1.00 \times 10^5 \mathrm{~Pa}$ ). The initial volume of the liquid water was $1.00 \times 10^{-3} \mathrm{~m}^3$ which is changed to $2.001 \mathrm{~m}^3$ of steam. Find the change in the internal energy of the system.

(Use heat of vaporisation $\simeq 2000 \mathrm{~kJ} / \mathrm{kg}$ )

A monoatomic gas does 100 J of work, when it is expanded isobarically. How much of heat is given to the gas in the process?

A Young's double slit experiment apparatus has slits separated by 0.2 mm and a screen 60 cm away from the slits. The whole apparatus is immersed in a liquid medium of refractive index $11 / 9$ and the slits are illuminated with green light ( $\lambda=550$ nm in vacuum). Find the fringe width of the pattern formed on the screen.

A metal wire of length $L$ and radius $r$ has a resistance $R$. If a wire of the same metal of length $2 L$ and radius $3 r$ is taken, then what will be its resistance?

A current $I=5 \mathrm{~A}$ flows along a thin wire shaped as shown in figure. The radius of curved part of the wire is equal to $R=100 \mathrm{~mm}$, the angle $2 \phi=90^{\circ}$. The magnitude of magnetic field at the point $O$ is approximately

$$ \left(\text { use, } \frac{\mu_0}{4 \pi}=10^{-7} \mathrm{~T} \mathrm{~mA}^{-1}\right) $$

An electron is released from a distance of 4 m from a stationary point charge 20 nC . What will be the speed of the electron, when it is

2 m away from the point charge?

(Charge of electron $=1.6 \times 10^{-19} \mathrm{C}$, mass of electron

$$ =9 \times 10^{-31} \mathrm{~kg}, \frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \text { SI unit) } $$

The following figure shows a 9 V battery and 3 uncharged capacitors of capacitances $C_1=C_2=C_3=1 \mu \mathrm{~F}$. The switch is thrown to the right side until capacitor $C_1$ is fully charged, then the switch is thrown to the left. The final charge on capacitor $C_2$ is

Balancing point of a potentiometer shifts from a length of 60 cm to 40 cm by shunting the cell with a $4 \Omega$ resistance. What is the internal resistance of the cell?

A toroid has a core (non-ferro magnetic) of inner radius 24 cm and outer radius 26 cm around which 2000 turns of a wire is wound. If the current in the wire is 12 A , the magnetic field inside the core of the toroid is

An AC current is given by the expression, $I(t)=50 \sin (200 \pi t)$ in amperes. The frequency and rms value of the current, respectively are

A planet has magnetic dipole moment of $27 \times 10^{22} \mathrm{~A}-\mathrm{m}^2$. If the radius of the planet is 300 km , what would be the magnetic field at its equator? $\left(\right.$ use,$\left.\frac{\mu}{4 \pi}=10^{-7}\right)$

An electromagnetic wave is propagating in vacuum along $-\hat{\mathbf{j}}$ direction. The magnetic field of the wave is given by $\mathbf{B}=\left(2 \times 10^{-8}\right) \cos \left[\pi \times 10^{15}\left(t+\frac{y}{c}\right)\right] \hat{\mathbf{k}} \mathrm{T}$. The electric field $\mathbf{E}$ of this wave is ( $c \equiv$ speed of light)

A long solenoid has 20 turns per cm. A small loop of area $4 / \pi \mathrm{cm}^2$ is placed inside the solenoid normal to its axis. If the current carried by the solenoid changed steadily from 1.0 A to 3.0 A in 0.2 s , what is the magnitude of the induced emf in the loop while the current is changing?

As the mass number $A$ increases, which of the following quantities related to a nucleus does not change?

A TV transmission antenna is 40 m tall. How much service area is can cover, if the receiving antenna is at the ground level?

(radius of the earth $=6400 \mathrm{~km}$ )

The light emitted in the transition $n=3$ to $n=2$, (where $n$ is the principal quantum number of the state) in hydrogen is called $\mathrm{H}_\alpha$-light. Find the maximum work function that a metal can have, so that $\mathrm{H}_\alpha$-light can emit photoelectrons from it.

In a NAND gate, $A$ and $B$ are inputs and $Y$ is the output, then the correct option is

The velocity $(v)$ of de-Broglie wave is given by

$$ \left[\begin{array}{l} v=\text { frequency } \\ m=\text { mass } \\ C=\text { velocity } \text { of } \text { light } \end{array}\right] $$

Gases deviate from ideal behaviour at high pressures because the gas molecules

If 92 g Na reacts with water in open vessel at 300 K . What is the value of work done?

[Assume ideal nature of the gaseous product.]

According to kinetic molecular theory of gases, which of the following statements are correct?

(A) The actual volume of the molecules is negligible in comparison to the empty space between them.

(B) Collisions of gas molecules are inelastic.

(C) At any particular time, different particles in the gas have same speed and same kinetic energies.

(D) Pressure is exerted by the gas as a result of collision of the particles with the walls of the container.

In hydrogen atom, the minimum energy required to excite an electron from 2nd orbit to the 3rd orbit is

At $0^{\circ} \mathrm{C}$ urea solution has an osmotic pressure of 400 mm . On dilution by $x$ times, its osmotic pressure decreased to 100 mm at $20^{\circ} \mathrm{C}$. The dilution factor $x$ is approximately

Identify the pair of reactions undergoing disproportionation from the following.

Iron crystalises in FCC with an edge length of 400 pm . If it contains $0.1 \%$ Schottky defects, calculate its approximate density

[Atomic weight of $\mathrm{Fe}=56 \mathrm{~g} / \mathrm{mol}$ ]

The pH of pure water at $80^{\circ} \mathrm{C}$ is

For $\mathrm{As}_2 \mathrm{~S}_3$ sol, the most effective coagulating agent is

Which of the following are correct for an ideal solution?

(A) $\Delta V_{\text {mix }}=0$

(B) $V_{\text {solvent }}+V_{\text {solute }}=V_{\text {solution }}$

(C) $\Delta H_{\text {mix }}=0$

(D) $\mathrm{H}_2 \mathrm{O}+\mathrm{CO}_2 \longrightarrow \mathrm{H}_2 \mathrm{CO}_3$ is an example of ideal solution

The correct order of $\mathrm{C}-\mathrm{O}$ bond length is

The compound with more covalent character in the following is

If the definition of the temperature coefficient of the reaction holds good for a reaction between $27^{\circ} \mathrm{C}$ and $37^{\circ} \mathrm{C}$, the activation energy for the reaction in $\mathrm{kJ} \mathrm{mol}^{-1}$ is

How many of the following statements are correct?

(A) ' $\mathrm{He}^{\prime}$ ' is the second most abundant element in the universe.

(B) The symbol for the element with atomic number 110 is Ds.

(C) Osmium has the highest density among all elements.

(D) Francium is the most electropositive element in the periodic table.

The correct order of the first ionisation enthalpies of the following elements is

In which of the following reactions there is no liberation of nitrogen gas

The relative order of electronegativity of $\mathrm{C}, \mathrm{Ge}$, and Pb is

On passing electric current over molten ionic hydrides of $s$-block elements,

How many of the following species have the bond order 2? $\mathrm{C}_2, \mathrm{~B}_2^{2-}, \mathrm{N}_2^{2+}, \mathrm{CN}^{+}, \mathrm{NO}^{-}, \mathrm{O}_2, \mathrm{C}_2^{+}$

The correct order of boiling points of $\mathrm{H}_2 \mathrm{O}, \mathrm{H}_2 \mathrm{~S}, \mathrm{H}_2 \mathrm{Se}$ and $\mathrm{H}_2$ Te respectively is

Lithium nitrate on heating gives

The correct order of decreasing field strength of the below given ligands is

Aluminium when treated with aqueous NaOH , liberates a gaseous molecule majorly. The gas is

Which of the following is not a mineral of fluorine?

The correct order of the stability of the following compounds based upon hyperconjugation is

The element that even can diffuse through silica glass is

The correct order of rate of addition of $\mathrm{Br}_2$ /water to the following alkenes is

The reagent that can reduce the carboxylic acid group to the corresponding alcohol is

The major products $P$ and $Q$ formed in the following reactions schemes are

The major aromatic product of the following reaction sequence is

The order of reactivity of the following compounds towards the esterification with acetic acid is

154. The major product formed in the following reaction sequence is

     

Which of the following reactions will give benzophenone as major product?

(A) Benzoyl chloride + benzene $+\mathrm{AlCl}_3$ (anhyd).

(B) Benzoyl chloride + phenylmagnesium bromide

(C) Benzoyl chloride + diphenyl cadmium

$$ \sqrt{(-3+4 i)(8+6 i)}= $$

If $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^m=1,2022 < m < 2029$, then $m=$

The starting material that produce pentanamine by Hoffmann bromamide reaction is

Assertion (A) The denaturation of proteins can destroy all $1^{\circ}, 2^{\circ}$ and $3^{\circ}$ protein structures.

Reason (R) Curdling of milk is due to denaturation of proteins.

The correct option among the following is

If $1, \omega, \omega^2$ are the cube roots of unity, $n \in N$ and $n>2$ then the least value of $n$ such that $1+\omega$ is a root of $x^n-x=0$ is

Consider following reaction, where

(A) the change in the functional group and

(B) the corresponding change in the hybridisation from starting to the final product $A$ and $B$ are

If $\alpha, \beta, \gamma$ are the roots of the equation $4 x^3+12 x^2-7 x+165=0$ and $\alpha+5, \beta+5, \gamma+5$ are the roots of the equation $a x^3+b x^2+c x+d=0$ then the product of the roots of the second equation is

If $\alpha, \beta, \gamma$ are the roots of the equation $3 x^3-26 x^2+52 x-24=0$ such that $\alpha, \beta, \gamma$ are in geometric progression and $\alpha<\beta<\gamma$, then $3 \alpha+2 \beta+\gamma=$

The number of 3-digit odd numbers divisible by 3 that can be formed using the digits $1,2,3,4,5,6$ when repetition is not allowed, is

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

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-5 x^2-2 x+24=0$, then $\frac{\beta \gamma}{\alpha}+\frac{\gamma \alpha}{\beta}+\frac{\alpha \beta}{\gamma}=$

Let $p(x)$ be a quadratic polynomial with real coefficients. If $p(x)=0$ has only purely imaginary roots, then the zeroes of the polynomial $p(p(x))$ are

A cube having edge of length 5 cm is painted on all faces and then it is cut into equal cubes of unit volume. A small cube is selected at random and found that a face of it is painted, then the probability that two more faces of it are also painted is

A pair of dice is thrown twice in succession. The probability of getting prime number on both the dice in first throw and composite numbers on both the dice in second throw is

If the mean and variance of a binomial distribution are 4 and $\frac{4}{3}$ respectively, then $P(X=2)=$

3 balls are drawn one after the other without replacement from an urn containing 4 red, 5 blue and 6 yellow balls. The probability of getting three different coloured balls is

Two balls are drawn at random from a bag containing 5 black balls and 3 white balls. If the random variable $X$ denotes the number of white balls drawn, then the mean of $X$ is

If $A=\left\{x \in R / \sqrt{x^2-8 x+15} \in R\right\}$ and $B=\left\{x \in R / \frac{x-3}{2 x-5}<\frac{x-6}{2 x-11}\right\}$, then $A \cap B=$

In a $\triangle A B C, D$ and $E$ divide the sides $B C$ and $C A$ in the ratio $2: 1$ respectively. If $P$ is the point of intersection of $A D$ and $B E$, then the ratio in which $P$ divides $A D$ is

Let $\mathbf{a}$ be a vector in the plane containing vectors $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{c}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If $\mathbf{a}$ is perpendicular to $\hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and its projection on $\mathbf{b}$ is $3 \sqrt{6}$, then $|\mathbf{a}|^2=$

If the points with position vectors $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-4 \hat{\mathbf{k}},-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ and $a \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ are coplanar, then $a=$

Let $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$, $\mathbf{d}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ be four vectors and let $l=\mathbf{b} \cdot \mathbf{c}$ and $m=\mathbf{c} \cdot \mathbf{a}$. Then, $[m \mathbf{b}+l \mathbf{a} \mathbf{b d}]=$

18. If $L$ and $M$ are respectively the coefficient of $x^{-7}$ in $\left(a x+\frac{b}{x^2}\right)^{11}$ and the coefficient of $x^7$ in $\left(b x^2+\frac{a}{x^2}\right)^{11}$, then $L+M=$

Let $A(1,2,3), B(-1,4,6), C(0,-6,4)$ and $D(1,1,1)$ be the vertices of a tetrahedron, $G$ be its centroid and $G_1$ be the centroid of its face $B C D$. Then, $\frac{A G_1}{A G}=$

$\left|\begin{array}{ccc}1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3\end{array}\right|=$

The cartesian equation of the plane passing through the point $(1,-2,3)$ and perpendicular to the vector $-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$, is

If a line $L$ is common to the planes $x-y+z+2=0$ and $2 x+y-2 z+5=0$ then the direction cosines of the line $L$ are

If $P$ is a point on the line parallel to the vector $2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}$ and passing through the point $A$ whose position vector is $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $A P=21$, then the position vector of $P$ can be

Let the foot of the perpendicular drawn from the point $(1,2,3)$ to a plane be $(-1,3,-2)$. Then, the perpendicular distance from the origin to the plane is

If $[\alpha \beta \gamma]\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & 3 & -5\end{array}\right]=[352]$, then $\alpha^3+\beta^3+\gamma^3=$

If $\bar{x}$ is the mean of $n$ observations $x_1, x_2, \ldots ., x_n$ then the mean of the absolute deviations of these observations from $\bar{x}$ is

If $\tan A=\frac{2}{3}$, then $\sin 4 A=$

If $A$ and $B(A>B)$ are acute angles, $\sin (A-B)=\frac{16}{65}$ and $\sin B=\frac{5}{13}$, then $\tan A+\cot A=$

$$ \frac{\sqrt{2} \cos 45^{\circ}+\cos 56^{\circ}+\cos 58^{\circ}-\cos 66^{\circ}}{\sqrt{2} \cos 28^{\circ} \cos 29^{\circ} \sin 33^{\circ}} $$

Let $\alpha, \beta, \gamma$ be real numbers. If $A=\left[\begin{array}{ccc}7 & 3 & \alpha \\ \beta & 1 & -11 \\ -5 & \gamma & 19\end{array}\right]$ is a $3 \times 3$ matrix satisfying $A\left[\begin{array}{c}5 \\ -13 \\ 11\end{array}\right]=\left[\begin{array}{c}-290 \\ -119 \\ 210\end{array}\right]$, then $(\operatorname{adj} A)^{-1}+\operatorname{adj} A^{-1}=$

If $\theta=\frac{\pi}{12}$ and $x=\log \left(\cot \left(\frac{\pi}{4}+\theta\right)\right)$, then $\cosh x=$

$2 \cosh (x+y) \sinh (x-y)+\sinh 2 y=$

In a $\triangle A B C$, if $(b+c)^2 \sin ^2 \frac{A}{2}+(b-c)^2 \cos ^2 \frac{A}{2}=K(1-\cos 2 A)$, then $K=$

In a $\triangle A B C$, if $b=7, c=4 \sqrt{3}$ and $A=\frac{\pi}{6}$ then a $\sin B \sin C=$

In $\triangle A B C$, if $B C$ is the hypotenuse, then $r_2+r_3=$

If $f(x)=\log _e\left(e^{2 x}\left(\frac{3 x+5}{5-3 x}\right)^{2 / 3}\right), x \neq \frac{-5}{3}, \frac{5}{3}$, then the value of $\frac{d f}{d x}$ at $x=1$, is

If $x=\operatorname{cosec} \theta-\sin \theta, y=\operatorname{cosec}^{2022} \theta-\sin ^{2022} \theta$ and $\left(\frac{d y}{d x}\right)^2=\frac{k\left(y^2+4\right)}{g(x)}$ where $k \in R$, then $10+k-g(2022)=$

If $f(x)=\left\{\begin{array}{l}\frac{x^2-16}{x-4} \text { if } x>4 \\ 2 x \quad \text { if } x \leq 4\end{array}\right.$ then $f^{\prime}\left(4^{-}\right)+f^{\prime}\left(4^{+}\right)=$

$$ \int_1^4\left(x+\sqrt{x}+\frac{1}{x}\right) d x-\int_1^{2 \log 2} d x= $$

$$ \lim _{x \rightarrow 0} \frac{2^{2 x}-2^{x+1}+2-\cos 2 x}{x^2}= $$

$$ \lim _{x \rightarrow 3^{-}} \frac{x^3-3 x^2-4 x+12}{2 x^3-7 x^2+2 x+3}= $$

Let $I=\int_{-\pi / 4}^{\pi / 4} \frac{1}{2-\cos 2 x}\left(\frac{\beta}{\pi}+\log \left(\frac{4+\sin x}{4-\sin x}\right)\right) d x$. Given that $\int \frac{d x}{1+k x^2}=\frac{1}{\sqrt{k}} \tan ^{-1}(\sqrt{k} x)+c, \tan ^{-1}(0)=0$ and $\tan ^{-1}(\sqrt{3})=\pi / 3$. Then, $3 I^2=$

The curve that satisfies the differential equation $x y d y-\left(1+y^2\right) d x=0$ passes through $(1,0)$ and intersects the curve $x^2+3 y^2=3$ at an angle $\theta$. Then, $\frac{2 \theta}{\pi}=$

The domain of the real valued function $f(x)=\frac{\sqrt{6 x^2+5 x-6}}{\sqrt{4-x}-\sqrt{x+4}}$ 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 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)

The differential equation of the family of circles with fixed radius $r$ units and centre on the line $y=3$, is

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

If $[x]$ represents the greatest integer $\leq x$, then the range of the real valued function $f(x)=\frac{1}{\sqrt{[x]^2+[x]-2}}$ is

If $\frac{x^2-3 x+2}{(x-4)(x-3)^2}=\frac{A}{x-4}+\frac{B}{x-3}+\frac{C}{(x-3)^2}$ then $A+B+C=$

Let the slope of a diameter $A C$ of a circle of radius 25 units be $\frac{3}{4}$. If $(3,2)$ is the centre of the circle, $A=\left(x_1, y_1\right)$ and $C=\left(x_2, y_2\right)$, then $\frac{x_1 x_2}{y_1 y_2}=$

Let $f(x)=\int \frac{2 x^3-3 x^2+4 x-5}{x^2} d x$ and $f(1)=1$. Then, $f(5)=$

A circle passes through the points $(1,2)$, $(3,4)$. If its centre lines on the line $x-y+3=0$, then its radius is equal to

If $\frac{x^2+3}{\left(x^2+1\right)\left(x^2+2\right)}=\frac{A x+B}{x^2+1}+\frac{C x+D}{x^2+2}$ then $A+B+C+D=$

A line drawn through the point $A(5,7)$ cut the circle $x^2+y^2-36=0$ at the points $P$ and $Q$. Then, $A P \cdot A Q=$

If $y^2=16 x$ is the given parabola, then the point of intersection of the focal chord through the point $(2,2)$ and the double ordinate of length 24 is

If the circles $C_1: x^2+y^2+2 x+4 y-20=0$, $C_2: x^2+y^2+6 x-8 y+9=0$ have $n$ common tangents and the length of the tangent drawn from the centre of similitude to the circle $C_2$ is $l$, then $\frac{l}{n^2}=$

If the common chord of the circles $x^2+y^2+4 y=0$ and $x^2+y^2-4 x-5=0$ is the diameter of the circle $S=0$, then the abscissa of the centre of the circle $S=0$ is

Let $P$ be any point on the circle $x^2+y^2-2 x-1=0$ and $C$ be its centre. Let $A B$ be the chord of contact of $P$ with respect to the circle $x^2+y^2-2 x=0$. Then, the locus of the circumcentre of the $\triangle C A B$ is

If a circle $C$ passing through $(4,0)$ touches the circle $x^2+y^2+4 x-6 y-12=0$ externally at the point $(1$, -1 ), then the radius of $C$ is

Let $P Q$ and $R T$ be two focal chords of the parabola $y^2=16 x$. If $P=(4,8)$ are $R=(16,16)$, then $Q T=$

If $\theta$ is the acute angle between the lines $\frac{x}{a}+\frac{y}{b}=1, \frac{x}{b}+\frac{y}{a}=1$, then $\sin \theta=$

If the line $x-y+1=0$ cuts the lines $2 x+2 y+3=0$ and $3 x+3 y+2=0$ at the points $A$ and $B$ respectively, then $A B=$

Let $A\left(\theta_1\right)$ and $B\left(\theta_2\right)$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $S$ be the focus of the hyperbola, If $A, S, B$ are collinear and

a $\cos \left(\frac{\theta_1+\theta_2}{2}\right)=k \cos \left(\frac{\theta_1-\theta_2}{2}\right)$, then $k=$

Let $A(5,-3), B(3,-2), C(-1,5)$ be three points. If $P$ is a point satisfying the condition $P A^2+2 P B^2=3 P C^2$, then a point that lies on the locus of $P$ is

If $\frac{x^2}{k-\frac{5}{2}}+\frac{y^2}{\frac{7}{3}-k}=1$ ( $k$ is a real number) represents a hyperbola, then the set of all values of $k$ is

If the incentre and the circumcentre of the triangle formed by the lines $x=2,4 x+3 y+7=0$ and $y=3$ are $I$ and $S$ respectively, then $I S=$

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi}{4}$, if the equation $49 x^2+25 y^2=1225$ is transformed to $p x^2+q x y+r y^2=t$ and the GCD of $p, q, r, t$ is 1 , then

The equation $x^2-y^2+a x+b=0$ represents a pair of lines for the ordered pair $(a, b)=$

$a x^2-4 x y-2 y^2=0$ represents a pair of lines. If $\theta$ is the angle between these lines, $\cos \theta=\frac{1}{5}$ and the possible values of ' $a$ ' are $a_1$ and $a_2\left(a_1

The parametric equations of the ellipse whose focii are $(-3,0),(9,0)$ and eccentricity is $\frac{1}{3}$, are

If the extreme value of $3 x-2 x^2+1$ is $k$, then the set of all real values of $x$ for which $k x^2+2 x+1>0$ is

If the eccentricity and the length of the latusrectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $\frac{\sqrt{3}}{2}$ and 1 respectively, then the sum of the lengths of major axis and minor axis of the ellipse is

The degree of the differential equation

$$ x\left(\frac{d^2 y}{d x^2}\right)^{1 / 3}+2 x^2\left(\frac{d^2 y}{d x^2}\right)^{5 / 3}+7 \frac{d y}{d x}+y=0 $$

Let $L_1, L_2$ be the lines represented by the equation $4 x^2-5 x y+3 y^2=0$. Let $L_3, L_4$ be two lines passing through the point $(4,3)$ such that $L_3$ and $L_4$ are perpendicular to $L_1$ and $L_2$ respectively. If the combined equation of $L_3$ and $L_4$ is $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$, and $a f+b g+c h=$

$$ \int(2 x-3) \sqrt{3 x+2} d x= $$

The amount of $50 \%(w / w)$ solution of hydrochloric acid required to react with 200 g of $\mathrm{CaCO}_3$ would be