TS EAMCET 2022 ONLINE 18TH JULY EVENING SHIFT

A projectile is given an initial velocity of $(3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}$ where $\hat{\mathbf{i}}$ is along the ground and $\hat{\mathbf{j}}$ is along the vertical. Assuming $g=10 \mathrm{~m} / \mathrm{s}^2$, if the equation of its trajectory can be written as $\frac{1}{9}\left[\beta x+\gamma x^2\right]$. Then the value of $\gamma$ is

A small object slides down with initial velocity equal to zero from the top of a smooth hill of height $H$. The other end of the hill is horizontal and is at height $H / 2$ as shown in the figure. The horizontal distance covered by the object from the end of the hill to the ground is

In a $R$ - $C$ circuit, where $R$ is resistance and $C$ is capacitance, which of the following has the dimension of time?

A solid cylinder of mass $m$ and radius $R$ rolls down on an inclined plane of height 30 m without slipping. The speed of its centre of mass, when the cylinder reaches the bottom is

[use $g=10 \mathrm{~m} / \mathrm{s}^2$ ]

A block is placed on a parabolic shape ramp given by equation, $y=\frac{x^2}{20}$. If the coefficient of static friction $\left(\mu_s\right)$ is 0.5 , then what is the maximum height above the ground at which the block can be placed without slipping?

A moving particle collides with a stationary particle of mass $\frac{1}{n}$ times the mass of moving particle, the fraction of its kinetic energy transferred to the stationary particle is

A river has a steady speed of $v$. A man swims upstream at a distance of $d$ and swims back to the starting point in total time $t$. The man can swim at a speed of $2 v$ in still water. If the time taken by the man in still water is $t_0$ to complete the same length of swim, then $\frac{t}{t_0}$ is

A particle starts from rest. Its acceleration (a) versus time $(t)$ graph is as shown in the figure. The maximum speed of the particle will be

Assertion (A) The zero velocity of a particle at any instant always implies zero acceleration at that instant.

Reason (R) A body is momentarily at rest when it reverses its direction of motion. The correct option among the following is

A hollow spherical body of outer and inner radii of 4 cm and 2 cm respectively, floats half submerged in a liquid of density $2.0 \mathrm{~g} / \mathrm{cm}^3$. The density of the material of the sphere is

What is the terminal velocity of a rain drop of radius 0.02 mm ?

[Note that the coefficient of viscosity of air is $1.8 \times 10^{-5} \mathrm{N} / \mathrm{m}^2$, density of water is $1000 \mathrm{~kg} / \mathrm{m}^3$. Use, $g=10 \mathrm{~m} / \mathrm{s}^2$ and density of air can be neglected in comparision with density of water]

A simple pendulum consists of a small sphere of mass $m$ suspended by a thread of length $l$. The sphere carries a positive charge $q$. The pendulum is allowed to do small oscillations in uniform electric field $E$ with direction vertically upwards. The time period of oscillation is

A swimming pool has a depth of 22 m and area $700 \mathrm{~m}^2$. Calculate fractional change $\Delta v / v$ of water at the bottom of the swimming pool, given that the bulk modulus of water is $2.2 \times 10^9 \mathrm{Nm}^{-2}, g=10 \mathrm{~m} / \mathrm{s}^2$, and density of water is $1000 \mathrm{~kg} / \mathrm{m}^3$

The distance between two successive minima of a transverse wave is 2.7 m . Five crests of the wave pass a given point along the direction of travel every 15.0 s . The speed of the wave is

A rocket is fired vertically with a speed of $4 \mathrm{~km} / \mathrm{s}$ from the earth's surface. How far from the earth does the rocket go before returning to the earth?

(Take, radius of earth $=6.4 \times 10^6 \mathrm{~m}$ and $g=10 \mathrm{~m} / \mathrm{s}^2$ )

Statement I Gas thermometers are less sensitive than liquid thermometers.

Statement II The ratio of universal gas constant and avogadro's number is called Boltzmann's constant.

Statement III The density of a given mass of a gas at constant pressure is inversely proportional to its absolute temperature.

The correct option among the following is

A hole of diameter 5 cm is drilled in a metal sheet at $30^{\circ} \mathrm{C}$. The linear expansion of metal is $2 \times 10^{-5} \mathrm{~K}^{-1}$. The diameter of the hole when the temperature is raised to $230^{\circ} \mathrm{C}$, is equal to

A diatomic gas $\left(C_p=\frac{7}{2} R\right)$ does 200 J of work when it is expanded isobarically. The heat given to the gas in the process is

The angular width of a fringe in a double slit experiment is found to be $0.2^{\circ}$ on a screen 1 m away the wavelength of light used is 600 nm . The change in angular width of the fringe, if the entire measurement system is immersed in water is [use refractive index of water as $4 / 3$ ]

The net work done by an ideal gas going through the cycle as shown in the $p-V$ diagram below is

A metal cube absorbs 2100.0 J of heat when its temperature is raised by $2^{\circ} \mathrm{C}$. If the specific heat of the metal is $900 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$, then the mass of the cube is

The resistivity of a material is found to be $10^8 \Omega-\mathrm{m}$, then the material would be

A cylindrical metallic wire is stretched to increase its length in such a way that the metallic wire changes its resistance by $6 \%$. The percentage increase in its length is

Find the current in the three resistors as shown in the following figure?

A large metal plate has a surface charge density of $8.85 \times 10^{-6} \mathrm{C} / \mathrm{m}^2$. An electron having initial kinetic energy of $8 \times 10^{-17} \mathrm{~J}$ is moving towards the centre of the plate. If the electron stops just before reaching the plate, then the initial distance between the electron and the plate is

[take $\varepsilon_0=8.85 \times 10^{-12} \mathrm{C}^2 / \mathrm{N}-\mathrm{m}^2$ ]

A horizontal wire carries 160 A current below which another wire of linear density $10 \mathrm{~g} / \mathrm{m}$ carrying a current is kept at 4 cm distance. If the wire kept below hangs in air, what is the current in this wire, when the direction of current in both the wires is same? $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right.$ and $\left.\mu_0=4 \pi \times 10^{-7}\right)$

The equivalent capacitance between points $A$ and $B$ is

A beam of white light is incident normally on a plane surface absorbing 70\% of the light and reflecting the rest. If the incident beam carries 10 W of power, the force exerted by it on the surface is

A flat circular coil has 100 turns of wire of radius 10 cm . A uniform magnetic field exists in a direction perpendicular to the plane of the coil and it grows at a rate of $0.1 \mathrm{~T} / \mathrm{s}$. The induced emf in the coil is

A long solenoid has 70 turns / cm and carries current $I$. An electron moves within the solenoid in a circle of radius 2.5 cm perpendicular to the solenoid axis. If the speed of the electron is $4.4 \times 10^6 \mathrm{~m} / \mathrm{s}$, then the current $I$ in the solenoid is

(take $\mu_0=4 \pi \times 10^{-7}$ Si unit, mass electron $=9 \times 10^{-31}$

kg , charge of electron $1.6 \times 10^{-19} \mathrm{C}$ )

An electromagnetic wave has its electric and magnetic fields given by

$$ \mathbf{E}(t)=\mathbf{E}_m \sin (k x-\omega t) ; \quad \mathbf{B}(t)=\mathbf{B}_m \sin (k x-\omega t) $$

If the direction of $\mathbf{E}_m$ and $\mathbf{B}_m$ are in the direction of $(\hat{\mathbf{i}}+\hat{\mathbf{j}})$ and $(\hat{\mathbf{i}}-\hat{\mathbf{j}})$ respectively, the unit vector that gives the direction of propagation of the wave is

A $2 \mu \mathrm{~F}$ capacitor is charged to 50 V by a battery. The battery is removed after capacitor if fully charged. At time $t=0$, a 10 mH coil is connected in series with the capacitor. The maximum rate at which the current changes in the circuit is

A message signal of frequency 15 kHz is used to modulate a carrier of frequency $v_c$. If the side bands produced are 1515 kHz and 1485 kHz , then $v_c$ is

Which of the following interaction is responsible for beta decay?

The value of planck's constant, if the slope of the graph of stopping potential versus frequency of incident light is $4 \times 10^{-15} \mathrm{~V}$-s is (given charge of an electron $=1.6 \times 10^{-19} \mathrm{C}$ )

If the series limit frequency of Balmer series is $v_B$, then the series limit frequency of the Brackett series is

Consider a nucleus ${ }_{30}^{60} \mathrm{X}$. Its approximate density is (take, $1 \mathrm{amu}=1.6 \times 10^{-27} \mathrm{~kg}, R_0=1.2 \times 10^{-15} \mathrm{~m}$ )

The behaviour of the circuit is like $\_\_\_\_$ gate

The total number of spectral lines observed when electron returns from the 6th shell to the 2nd shell in hydrogen atom is

Among the following given substances, the one with zero $\Delta_f H^{\circ}$ is

Dipole-induced dipole interactions are present between which of the following pairs?

A gaseous mixture of 2 moles of $A, 3$ moles of $B$, 5 moles of $C$ and 10 moles of $D$ is contained is contained in a vessel. Assuming that gases are ideal and partial pressure of $C$ is 1.5 atm , the total pressure is

The change in enthalpy $[\Delta H]$ in $\mathrm{kJ} \mathrm{mol}^{-1}$ for the reaction, $\mathrm{Mg}+2 \mathrm{~F} \longrightarrow \mathrm{MgF}_2$ is

Given, electron affinity of $\mathrm{F}=328 \mathrm{~kJ} \mathrm{~mol}^{-1}$,

IE ${ }_1$ of $\mathrm{Mg}=737 \mathrm{kJmol}^{-1}, \mathrm{IE}_2$ of $\mathrm{Mg}=1451 \mathrm{kJmol}^{-1}$

The orbital angular momentum of an electron in $d$-orbital is equal to

For the formation of ammonia gas from its constituent elements, the $K_p / K_C$ is

Which of the following is/are "not correct" for $\mathrm{CH}_3 \mathrm{OH}+\mathrm{CH}_3 \mathrm{COOH}$ mixture solution?

1. 1. $\Delta H_{\text {mix }}<0$

2. 2. Does not obey Raoult's law.

3. 3. $\Delta H_{\text {mix }}>0$

4. 4. An example of ideal solution.

A certain mass of a gas was brought from state $A$ to $B$ by following three different paths, namely 1,2 and 3 , respectively. Which of the following relations is correct for the work done?

Henry's law is valid for

(A) ammonia gas dissolution in water

(B) $\mathrm{O}_2$ gas dissolution in unsaturated blood

(C) $\mathrm{O}_2$ dissolution in water

(D) $\mathrm{CO}_2$ dissolution in water

Catalysts in the following reactions are

(I) $\mathrm{CH}_3 \mathrm{COOCH}_3(l)+\mathrm{H}_2 \mathrm{O}(l)\longrightarrow \mathrm{CH}_3 \mathrm{COOH}(a q)+\mathrm{CH}_3 \mathrm{OH}(l)$

(II) $2 \mathrm{SO}_2(\mathrm{~g}) \longrightarrow 2 \mathrm{SO}_3(\mathrm{~g})$

(III) $2 \mathrm{SO}_2(\mathrm{~g})+\mathrm{O}_2(\mathrm{~g}) \longrightarrow 2 \mathrm{SO}_3(\mathrm{~g})$

(IV) $\mathrm{N}_2(\mathrm{~g})+3 \mathrm{H}_2(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_3(\mathrm{~g})$

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

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

A cube of edge length 1 cm is divided into smaller cubes of uniform size of length 1 nm . Assuming that, no voids are present, the ratio of total surface area of all the cubes of 1 nm edge length to the surface area of the initial cube is

The correct option for axial distances and axial angles for hexagonal crystal system is

The rate constant of a reaction is increased by 4 times after addition of catalyst to the reaction mixture at the same temperature of $27^{\circ} \mathrm{C}$. The change in the activation energy of this reaction is (Take $\ln (1 / 4)=-1386, R=8.314$ )

Half-life periods for a reaction at initial concentrations of 0.1 M and 0.01 M are 5 and 50 minutes, respectively. The order of reaction is

On passing a current of 1.2 A through a solution of salt of copper for $40 \mathrm{~min}, 0.96 \mathrm{~g}$ of copper was deposited. The equivalent weight of copper in g is

Potassium superoxide on hydrolysis gives

The most effective water softening method is

When aluminium chloride is dissolved in water, it gives

The linear molecule among the following is

The main products $P$ and $Q$ of the following unbalanced disproportionation reaction $\mathrm{Se}_2 \mathrm{Cl}_2 \longrightarrow P+Q$ are

According to the Lewis formula of $\mathrm{O}_3$, the correct option is

The total number of paramagnetic gaseous products formed in all the following reactions $[A+B+C]$ are

(A) $\mathrm{NH}_4 \mathrm{NO}_3 \xrightarrow{\Delta}$ Products

(B) $\mathrm{NaNO}_2+\mathrm{H}_2 \mathrm{SO}_4 \longrightarrow$ Products

(C) $\mathrm{Pb}\left(\mathrm{NO}_3\right)_2 \xrightarrow{673 \mathrm{~K}}$ Products

The most suitable solvent for Wurtz reaction is

The correct order of acidity of $\mathrm{HClO}, \mathrm{HBrO}$ and HIO is

Among the given complexes the possess " $\mathrm{CO}^{\prime \prime}$ as a bridged ligands are

I. $\left[\mathrm{Co}_2(\mathrm{CO})_8\right]$

II. $\left[\mathrm{Fe}_3(\mathrm{CO})_{12}\right]$

III. $\left[\mathrm{Mn}_2(\mathrm{CO})_{10}\right]$

IV. $\left[\mathrm{Fe}_2(\mathrm{CO})_9\right]$

Assertion (A) In general, transition metals have high melting points.

Reason (R) More number of electrons from ' $(n-1) d^{\prime}$ and ' $n s$ ' are involved in interatomic metallic bonding.

The correct option among the following is

Identify the chiral molecule among the following.

Which of the following statements are correct for phenol?

(A) In general, phenol is more acidic than alcohol.

(B) Phenol is used in the production of melamine plastic.

(C) Phenol gives violet colour with neutral ferric chloride solution.

(D) Phenol when heated with acetyl chloride gives phenetole.

Propyne was subjected to a reaction with $\mathrm{HgSO}_4 /$ dil. $\mathrm{H}_2 \mathrm{SO}_4$, which resulted in a product $P$. The product $P$ was heated with $\mathrm{Ba}(\mathrm{OH})_2$ to give the product $Q$. The molecular formula of the product $Q$ is

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

$$ A, B, C, D \text { in the following reactions are } $$

$n$ - propanol on treatment with concentrated HBr gives $P$. The product $P$ on reaction with KCN gave the product $Q$. The product $Q$ on heating with aqueous acidic solution, furnished the product $R$. The product ' $R$ ' is

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

One of the values of $(-32 i)^{\frac{2}{5}}$ is

1. If $\cos \alpha+\cos \beta+\cos \gamma=0$ and $\sin \alpha+\sin \beta+\sin \gamma=0$ then $\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=$

The amount of sucrose needed to produce 1 mole of glucose using acid hydrolysis is

The major product of the following synthetic sequence is

If the quadratic equations $x^2-7 x+3 c=0$ and $x^2+x-5 c=0$ have a common root, then for non-zero real value of $c$ the sign of the expression $x^2-3 x+c$ is

If the point $(x, y)$ satisfies the equation $\frac{x+i(x-2)}{3+i}-i =\frac{2 y+i(1-3 y)}{i-3}$, then $x+y=$

If $1-\sqrt{2}$ and $2+i$ are the roots of the equation $x^4+b x^3+c x^2+d x+e=0$ where $b, c, d, e$ are rational numbers, then the roots of the equation $b x^2+c x+d=0$ are

II. Let $f(x)=\frac{6 x^2-18 x+21}{6 x^2-18 x+17}$. If $m$ is the maximum value of $f(x)$ and $f(x)>n \forall x \in R$. Then, $14 m-7 n=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+x^2+x+r=0$ and $\alpha^3+\beta^3+\gamma^3=5$, then $r=$

18. If $\frac{5}{2}$ is the sum of two roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$ then the sum of all non-real roots of the equation is

Let the transformed equation of $2 x^4-8 x^3+3 x^2-1=0$ so that the term containing the cubic power of $x$ is absent be $2 x^4+b x^2+c x+d=0$. Then, $b=$

If $\tan 15^{\circ}$ and $\tan 30^{\circ}$ are the roots of equation $x^2+p x+q=0$, then $p q=$

The exponent of 6 in 72 ! is

Let $A, B, C$ be three pairwise independent events of a random experiment. If $P(\bar{B} \cup \bar{C})=\frac{1}{2}, P(A)>0, P(B)=b$ and $P(C)=c, P((\bar{B} \cap \bar{C} \mid A)=$

$a, b, c$ are three particular speakers among the 10 speakers of a meeting. The number of ways of arranging all 10 speakers on the dias in a row so that all the three speakers $a, b, c$ do not sit together is

Two dice are thrown and the sum of the numbers appearing on the dice is observed to be a multiple of 4 . If $p$ is the conditional probability that number 4 has appeared atleast once, then $3 p+2=$

When two dice are thrown, the probability of getting a prime number on die and a composite number on the other is

In a random experiment of throwing 5 coins, the number of heads is defined as a random variable. The mean of the random variable is

The variance of a Poisson variate $X$ is 2 . Then, $P(X \geq 3)=$

If $3 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}, \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $-7 \hat{\mathbf{i}}-17 \hat{\mathbf{j}}+16 \hat{\mathbf{k}}$ are position vectors of the points $A, B, C$ and $D$ respectively, then the angle between $\mathbf{A B}$ and $\mathbf{C D}$ is

If the 4 th term in the expansion of $\left(\frac{x}{2}-\frac{2 y}{3}\right)^6$ is -20, then $x y=$

If $A(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), B(\lambda \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}), C(-4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ and $D(-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ are four points in space such that $\mathbf{A B}=x \mathbf{A C}+y \mathbf{A D}$ for some real number $x \neq 0, y \neq 0$, then $17(\lambda+9)=$

Let $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{b}$ be two vectors such that $\mathbf{a} \cdot \mathbf{b}=1$, $\cos (\mathbf{a} \cdot \mathbf{b})=\frac{1}{3}$ and the components of $\mathbf{b}$ w.r.t. $(\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}})$ be integers. Then, the number of possible vectors that represent $\mathbf{b}$ is

If $\mathbf{a}$ and $\mathbf{b}$ are two vectors such that $\mathbf{a}=2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+p \hat{\mathbf{k}}$, $|\mathbf{b}|=7, \mathbf{a} \cdot \mathbf{b}=4$ and $|\mathbf{a} \times \mathbf{b}|=5 \sqrt{17}$, then $p=$

Let $\pi_1^{\prime}$ be the plane passing through the point $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and perpendicular to the vector $a \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\pi_2$ be the plane passing through the point $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and perpendicular to the vector $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$. If $\theta$ is the angle between the planes $\pi_1$ and $\pi_2$ and $\cos \theta=-\sqrt{\frac{3}{7}}$, then the integral value of $a$ is

Let a plane $P$ has the points $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. Let $L$ be the line through the point $A$ and parallel to the vector $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If the plane $P$ and line $L$ intersect at a point $B(0,3,2)$ and the distance from $A$ to $B$ is 3 units, then equations of the normal to the plane $P$ through $A$ are

Let $A=(3,4,0), B=(4,4,4), C=(-6,2,3)$ and $D=(1,1,2)$. If $\theta$ is the acute angle between the lines $A B$ and $C D$, then $\cos \theta=$

If the points $A(1,3,5), B(2,4,6), C(4,5, k)$ form a right angled triangle then the number of possible values of $k$ is

A plane containing two lines whose direction ratios are $(-1,2,1)$ and $(1,3,2)$ passes through the point $(2,1, k)$. If this plane also passes through the point $(3,-1,4)$, then $k=$

If $A$ is a $3 \times 3$ matrix and $|A|=\frac{1}{2}$, then $\left|A^{-1}(\operatorname{Adj}(\operatorname{Adj} A))\right|^{-1}=$

Let $x=\alpha, y=\beta, z=\gamma$ be the unique solution of the system of simultaneous linear equations $2 x+3 y-2 z+4=0,3 x-4 y+3 z+5=0$, $k x-2 y+z+3=0$. If $\alpha=-2$, then $k=$

If $A, P, B$ are $3 \times 3$ matrices. If $|-B|=5,\left|B A^T\right|=15$, $\left|P^T A P\right|=-27$, then one of the values of $|P|$ is

20. If $\frac{x^2+7}{\left(x^2+1\right)(x-2)}=\frac{A}{x-2}+\frac{B x+C}{x^2+1}$, then the determinant of the matrix $\left[\begin{array}{ll}A & B \\ C & \frac{2}{5}\end{array}\right]$ is

If $A+B=\left[\begin{array}{lll}2 & 1 & 2 \\ 1 & 2 & 0 \\ 0 & 2 & 2\end{array}\right], A B=\left[\begin{array}{lll}1 & 2 & 2 \\ 1 & 1 & 0 \\ 1 & 2 & 1\end{array}\right]$, then $A^2+B(A+B)=$

If $\cos x+\cos y=p, \sin x+\sin y=q$, then $\cos \left(\frac{x-y}{2}\right)=$

$$ \frac{e^{4 x}+e^{-4 x}+14}{4\left(e^x-e^{-x}\right)^2}= $$

The mean deviation from the mean of the discrete data $1,3,4,7,11,18,29,47,78$ is

If $\tanh x=\frac{1}{2}$, then $\sinh 2 x-\operatorname{sech} 2 x=$

In $\triangle A B C$, if $A$ is acute, $C$ is obtuse, $\sin A=\frac{3 \sqrt{3}}{14}, a=3$ and $b=5$, then $c=$

If $A+B+C=\frac{3 \pi}{2}$, then $4 \sin A \sin B \sin C+\cos 2 A+\cos 2 B+\cos 2 C=$

Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix defined by $a_{i j}=\left\{\begin{array}{cc}k^i, & \forall i=j \\ 0, & \text { otherwise }\end{array}\right.$. If $m=$ trace of $A$ and $\lim _{k \rightarrow 1} \frac{n-m}{1-k}=171$, then the value of $n$ is

Let $A$ be the area of in-circle and $A_1, A_2, A_3$ be the area of ex-circles of a triangle. If $A_1=4, A_2=9, A_3=16$, then $A=$

If $f(x)=\sin \left(\cosh \left(\frac{x^2+1}{x^2+2}\right)\right)$, then $f^{\prime}(1)=$

If $a f(x)+b f\left(\frac{1}{x}\right)=x+1$, and $\frac{d}{d x}\left(x^2 f(x)\right)=2 x^2+2 x+\frac{1}{3}$, then $a-b$

If $\Delta$ denotes the area of $\triangle A B C$, then $(b \sin C+c \sin B)(b \cos C+c \cos B)=$

$$\mathop {\lim }\limits_{x \to \infty } {x^3}\left[\sqrt{x^2+\sqrt{x^4+1}}-\sqrt{2 x}\right]= $$

The differential equation corresponding to the family of curves given by $a x^2+b y^2=1$, where $a$ and $b$ are arbitrary constants is

If $[x]$ denotes the greatest integer function of $x$ and

$$ \int_{-3 / 2}^{3 / 2}[2 x-3] d x=k, \text { then }\left|k+\frac{1}{2}\right|= $$

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

$$ \int_1^2 x \sqrt{4-x^2} d x= $$

Let $f(x)=\left\{\begin{array}{ccc}3-x & \text { if } & x<-3 \\ 6 & \text { if } & -3 \leq x \leq 3 . \text { Let } \alpha \text { be the number } \\ 3+x & \text { if } & x>3\end{array}\right.$ of points of discontinuity of $f$ and $\beta$ be the number of points where $f$ is not differentiable. Then, $\alpha+\beta=$

For the differential equation

$$ \sqrt{\frac{d^2 y}{d x^2}}=\sqrt[3]{\left[y \frac{d y}{d x}+x \sin \left(\frac{d y}{d x}\right)\right]^2} $$

Let $R$ be the set of all real numbers. Let $f: R \rightarrow R$ be a function defined by

$$ f(x)=\left\{\begin{array}{rcc} 2 x-5, & \text { if } & x<-3 \\ x+2, & \text { if } & -3 \leq x<5 \\ 3 x+1, & \text { if } & x \geq 5 \end{array}\right. $$

Match the following

$$ \begin{array}{llll} \hline & \text { List I } & & \text { List II } \\ \hline \text { A } & f(-5)+f(0)+f(-1)= & \text { I } & 16 \\ \hline \text { B } & f(f(5)+10 f(-3))= & \text { II } & 40 \\ \hline \text { C } & f(|f(-4)|)= & \text { III } & -32 \\ \hline \text { D } & f(f(f(1)))= & \text { IV } & -12 \\ \hline & & \text { V } & 19 \\ \hline \end{array} $$

Let $R$ be the set of all real number

Statement I The function $f:\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow R$ defined by $f(x)=\sec x+\tan x$ is one-one function.

Statement II The function $f:[0, \infty) \rightarrow R$ defined by $f(x)=x^2$ is a one-one function

Which of the above statements is (are) true?

Let $x \neq \frac{-3}{5}, \frac{2}{5}$, if $f\left(\frac{2 x+1}{5 x+3}\right)=x+2$, then $\int f(x) d x=$

$$ \begin{aligned} & \text { If } \int \frac{(x+3)}{(x-1)^2(2 x-1)} d x \\ & =\frac{A}{x-1}+B \log (2 x-1)+C \log (x-1)+k, \text { then } A+B+C= \end{aligned} $$

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 $|\tan \theta|=$

If $\int \frac{1+\cos 8 x}{\tan 2 x-\cot 2 x} d x=f(x) \cdot \cos (g(x))+c$, then $f\left(\frac{1}{4}\right)+g\left(\frac{1}{4}\right)=$

If $\int e^x \cos x d x=\frac{e^x}{2}(\cos x+\sin x)$ and

$$ \int \frac{\cos \left(\log \left(\frac{2 x+3}{3-2 x}\right)\right)}{(3-2 x)^2} d x=\frac{f(x)}{24}[\cos (g(x))+\sin (g(x))]+c $$

then $g(1)=$

From a point $A(0,3)$ on the circle $(x+2)^2+(y-3)^2=4$, a chord $A B$ is drawn and it is extended to a point $Q$ such that $A Q=2 A B$. Then, the locus of $Q$ is

If $m_1, m_2$ are the slopes of the tangents drawn from a point $(1,-3)$ to the circle $x^2+y^2-6 x+4 y+12=0$, then $9\left(m_1^2+m_2^2\right)=$

The equation of the transverse common tangent of the circles $x^2+y^2-6 x-8 y+9=0$ and $x^2+y^2+2 x-2 y+1=0$

If $A, B$ are the points of contact of the tangents drawn from the point $P(-2,-3)$ to the circle $x^2+y^2-8 x-10 y+5=0$ and the chord $A B$ subtends an angle $\theta$ at $P$, then $\tan \theta=$

The vertex and the focus of the parabola $2 x^2+5 y-6 x+1=0$ respectively, are

The axis of a parabola is along the line $y=x$ and the distance of its vertex $A$ from $(0,0)$ is $\sqrt{2}$ and that of its focus $S$ from $(0,0)$ is $2 \sqrt{2}$. If $A$ and $S$ lie in first quadrant, then the equation of the parabola in parametric form is

If the perimeter of a triangle is 20 and two of its vertices are $(-5,0)$ and $(6,0)$, then the locus of the third vertex is

If $\left(0, \frac{3}{4}\right)$ is the radical centre of the circles $S \equiv x^2+y^2+\alpha x+6 y=0, S \equiv x^2+y^2+2 \alpha x+\alpha y+6=0$ and $S^{\prime \prime} \equiv x^2+y^2+6 \alpha x-\alpha y+3=0$, then the distance between the radical centre and the centre of the circle $S^{\prime}=0$ is

If $\theta$ is the angle between the circles

$x^2+y^2-2 x-4 y-4=0$ and $x^2+y^2-8 x-12 y+43=0$, then $|7 \sec \theta-18 \cos \theta|=$

Let $S$ be the focus of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ lying on the positive $X$ - axis and $P\left(5, y_1\right)$ be point on the hyperbola. Then $S P=$

If $P(\theta)=\left(x_1, \frac{3 \sqrt{5}}{2}\right), 0<\theta<\frac{\pi}{2}$ is a point on the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$, where $\theta$ is the parameter in its parametric form, then $2 x_1+9 \sin ^2 \theta=$

If the slope of one of the lines represented by $5 x^2+\frac{40}{3} x y+k y^2=0$ is 3 , then the angle between the pair of lines is

The centroid of the triangle formed by the lines $x-3 y+3=0, x+3 y+3=0 x+y-1=0$ is

If the line $2 x-3 y+4=0$ divides the line segment joining the points $A(-2,3)$ and $B(3,-2)$ in the ratio $m: n$, then the point which divides $A B$ in the ratio $-4 m: 3 n$ is

The transformed equation of $3 X^2+4 X Y+Y^2-8 X-4 Y-4=0$ is $f(X, Y)=a X^2+2 h X Y+b Y^2+c=0$ when the origin is shifted to a new point by the translation of axes. Then, $f(1,1)=$

If $Q$ is the image of the point $P(1,1)$ with respect to the straight line $x+y+1=0$, then the length of the perpendicular drawn from $Q$ to the line $3 x-4 y+3=0$ is

Let $S \equiv \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0, S \equiv \frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}-1=0$ be two intersecting ellipses. If $P(a \cos \theta, b \sin \theta)$ and $Q\left(a \cos \left(\frac{\pi}{2}+\theta\right), b \sin \left(\frac{\pi}{2}+\theta\right)\right)$ are their points of intersection then $\frac{1}{2}\left(a^2 \beta^2+b^2 \alpha^2\right)=$

If the lines $L_1 \equiv 2 x+y+3=0, L_2 \equiv k x+2 y-3=0$ and $L_3 \equiv 3 x-2 y+1=0$ are concurrent then the cosine of the acute angle between the lines $L_2=0$ and $2 x-5 y+7=0$ is

If $L$ is a line passing through the point $(-1,1)$ and parallel to the common line of the pairs of lines $6 x^2-x y-12 y^2=0$ and $15 x^2+14 x y-8 y^2=0$, then the equation of pair of lines joining the origin to the points of intersection of the curve $2 x^2-x y-y^2+x-y=0$ and the line $L$ is

$P\left(\theta_1\right)$ and $Q\left(\theta_2\right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $e$. If $P S Q$ is a focal chord and $\tan \left(\frac{\theta_1}{2}\right) \tan \left(\frac{\theta_2}{2}\right)=-(2 \sqrt{2}+3)$, then $e$ and $S$ are

A convex lens focusses an object 20 cm from it on a screen placed 5 cm away from it. A glass plate (refractive index $=7 / 5$ ) of thickness 1.4 cm is inserted between the lens and the screen. What is the distance of the object from the lens, so that its image is again focused on the screen?

If a line $L$ is common to the pairs of lines $6 x^2-x y-12 y^2=0$ and $15 x^2+14 x y-8 y^2=0$ then the combined equation the other two lines is

The general solution of the differential equation $\frac{d y}{d x}=\frac{x y+x-2 y-2}{x y-2 x+y-2}$ is

Calculate the number of moles of NaOH required to completely neutralise 100 g of $118 \%$ oleum.