AP EAPCET 2023 - 15th May Morning Shift

Two vectors of same magnitude act at a point. Twice the product of the magnitudes of two vectors is equal to the square of the magnitude of their resultant. The angle between the two vectors is

The time of flight of a vertically projected stone is 8 s . the position of the stone after 6 s from the ground is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

A stone thrown with velocity $u$ at angles $\theta$ and $\left(90^{\circ}\right.$ $-\theta$ ) with the horizontal reaches to maximum heights $H_1$ and $H_2$ respectively. Its horizontal range is

A car of mass 1500 kg is moving with $20 \mathrm{~ms}^{-1}$ velocity. If the breaks are applied, it comes to rest in 5 s, then the retarding force is

A body is made to move up along an inclined plane of inclination $30^{\circ}$ and the coefficient of friction is 0.5 , then its retardation is

( $g$ is acceleration due to gravity)

A machine gun fires 300 bullets per minute each with a velocity of $500 \mathrm{~ms}^{-1}$. If the mass of each bullet is 4 g . The power of the machine gun is

A body falls freely from a height of 100 m on to the ground and rebounds to a maximum height of 36 m after collision. The coefficient of restitution between ground and the body is

A force of $(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \mathrm{N}$ acts on a particle whose position vector with respect to the origin of the coordinate system is $(6 \hat{\mathbf{i}}+b \hat{\mathbf{j}}+12 \hat{\mathbf{k}}) \mathrm{m}$. If the angular momentum of the body is constant, the value of $b$ is

The maximum possible height of a mountain on earth is approximately

(elastic limit of mountain rock $=30 \times 10^7 \mathrm{Nm}^{-2}$, average density of mountain rock $=3 \times 10^3 \mathrm{kgm}^{-3}$, $g=10 \mathrm{~ms}^{-2}$ )

Two springs $A$ and $B$ fixed at the top and are stretched by 8 cm and 16 cm respectively, when loads of 20 N and 10 N suspended at the lower ends. The ratio of the spring constant of the springs $A$ and $B$ is

A person of height 1.65 m is standing upright. The additional external force required by blood vessel of length 1 cm , diameter 1 mm at feet to balance the pressure compared to similar blood vessel in head is (Density of blood $=1.1 \times 10^3 \mathrm{~kg} \mathrm{~m}^{-3}, g=10 \mathrm{~ms}^{-2}$ )

2 moles of a monatomic gas requires heat energy $Q$ to be heated from $30^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$ at constant volume. The heat energy required to raise the temperature of 4 moles of a diatomic gas from $28^{\circ} \mathrm{C}$ to $33^{\circ} \mathrm{C}$ at constant volume is

A block of steel of mass 2 kg slides down a rough inclined plane of inclination of $\sin ^{-1}\left(\frac{3}{5}\right)$ at a constant speed. The temperature of the block as it slides through 80 cm , assuming that the mechanical energy lost is used to increase the temperature of the block is nearly (Specific heat capacity of steel $=420 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}$ and Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

The specific heat capacity of a copper block of mass $M$ is $s$. If the mass of the copper block is doubled, the specific heat capacity will be

Isothermal bulk modulus of a gas at a pressure $p$ is ( $\gamma$-ratio of specific heat capacities of the gas)

A vessel of volume 10 litres is filled with $\mathrm{H}_2$ gas. The total average translational kinetic energy of its molecules is $4.5 \times 10^5 \mathrm{~J}$. The pressure of hydrogen in the vessel is

The amplitudes of a damped harmonic oscillator after 2 and 4 seconds are $A_1$ and $A_2$ respectively. If the initial amplitude of the oscillator is $A_0$, then

Small amplitude progressive wave in a stretched string has a speed of $100 \mathrm{cms}^{-1}$ and frequency 100 Hz . The phase difference between two points 2.75 cm apart on the string in radians, is

The property of light which cannot be explained by Huygen's construction of wavefront is

The value of acceleration due to gravity at a height of $4 R_E$ from surface of earth is

( $R_E$ is radius of earth and acceleration due to gravity on the surface of the earth $=10 \mathrm{~ms}^{-2}$ )

$$ \text { Blue colour of the sky is due to } $$

The electric field ( E in $\mathrm{NC}^{-1}$ ) in a region is given by $\mathrm{E}=3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}$. The net electric flux through a square area of side 2 m parallel to $y z$-plane is

In an AC generator, if coils of $N$ turns and area $A$ is rotated at $v$ revolutions per second in a uniform magnetic field $B$, then the motional emf produced is equal to (At $t=0 \mathrm{~s}$, the coil is perpendicular to the field)

Two thin conducting concentric shells of radii $R$ and $2 R$ are shown in the figure. The outer shell carries a charge $+Q$ and the inner shell is neutral. Then the correct statement/s is

(a) When the switch is closed, the potential on the inner shell becomes zero

(b) With the switch closed, the charge on the inner sphere is $-\frac{Q}{2}$

A wire is first bent in the form of a circular coil of 5 turns and the same wire is then bent in the form of another circular coil of 10 turns. If same current is passed in both the coils, then the ratio of the magnetic fields at their centres is

A magnetic field intensity at the centre of a circular wire of radius 0.1 m carrying a current 0.2 A is

A solenoid has a core of a material with relative permeability 400 . The windings of the solenoid are insulated from the core and carry of 4 A . If the number of turns is 500 per metre, then the magnetising field is

When a resistance $R_1$ is connected across a cell, the current is $I_1$ and if the resistance $R_1$ is replaced by $R_2$, the current is $I_2$. Then the internal resistance of the cell is

The average power output of a point source of an electromagnetic radiation is 1080 W . The maximum value of the rms value of the electric field at a distance of 3 m from the source is

Charge passing through a conductor of cross-section $0.3 \mathrm{~m}^2$ is given by $q=\left(3 t^2+5 t+2\right) \mathrm{C}$ where $t$ is in seconds. The drift velocity at $t=2 \mathrm{~s}$ is (Concentration of electrons in the conductor $=2 \times 10^{25} \mathrm{~m}^{-3}$ )

Two square shaped metal plates of side 1 m , kept 0.01 m apart in air from a parallel plate capacitor. It is connected to a battery of 500 V . The plates of the capacitor are then immersed in an insulating oil by lowering the plates vertically with a speed of $0.001 \mathrm{~ms}^{-1}$. If the dielectric constant of the oil is 11 , the current drawn from the battery during this process is

The frequency of AC at which $16 \mu \mathrm{~F}$ capacitor and $\frac{10}{\pi^2} \mathrm{mH}$ inductor will have same reactance is

The relation between the mean-life time $\tau$ and the half-life time $T_{1 / 2}$ of a radioactive substance is

The transition of an electron in hydrogen atom that emits a photon whose wavelength lies in the ultraviolet region of the electromgnetic spectrum is

The velocity of a particle $A$ is 3 times the velocity of proton. If the ratio of the de- Broglie wavelengths of the particle $A$ and the proton is $3: 2$, the mass of the particle $A$ is

( $m_p=$ mass of proton)

A $p-n$ junction diode is reverse biased with a voltage of 8 V . If the resistance of the diode is $4 \times 10^7 \Omega$, then the reverse saturation current is

$$ \text { In a } p-n-p \text { transistor } $$

A carrier wave of peak voltage 15 V is used to transmit a message signal. If the modulation index is $60 \%$, then the peak of voltage of the modulating signal is

20 mL of 0.1 M HCl is added to the 30 mL of 0.1 M NaOH . To this solution extra 50 mL of water was added. What is the molarity of the final solution formed?

The minimum number of quantum numbers required to specify an orbital in an atom is

Vijaywada FM radio station broadcasts at frequency of 103.4 MHz . The wavelength of the corresponding radio wave (in m ) is

An ideal gas $(X)$ present in a vessel of volume $V \mathrm{~L}$ exerted a pressure of 16.4 atm at 200 K . What is its concentration in $\mathrm{mol} \mathrm{L}^{-1}$ ?

(Given, $R=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ )

A 100 L cylinder containing $\mathrm{H}_2$ exerted a pressure of 4 atm at 300 K . It was accidentally opened and some $\mathrm{H}_2$ was escaped. When it was closed, it exerted a pressure of 3 atm at 300 K . The number of moles of $\mathrm{H}_2$ remaining in the cylinder is equal to

(Assume, $\mathrm{H}_2$ as an ideal gas; $R=$ gas constant)

If the standard enthalpy change $\left(\Delta_r H^{-}\right)$for a certain reaction at 298 K and constant pressure $-1860 \mathrm{~kJ} \mathrm{~mol}^{-1}$, the standard entropy change $\left(\Delta_{\text {sys }} \Omega^{\circ}\right)$ of the same reaction is $-550 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$. Which one of the following statements is correct?

If the standard enthalpy change $\left(\Delta_r H^{\circ}\right)$ for reaction $\mathrm{H}_2(\mathrm{~g})+\mathrm{Br}_2(l) \longrightarrow 2 \mathrm{HBr}(\mathrm{g})$ is -72.8 kJ , the standard enthalpy of formation $\left(\Delta_f H^{\circ}\right)$ of $\mathrm{HBr}(g)$ (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) is

One mole $\mathrm{H}_2 \mathrm{O}(g)$ and one mole $\mathrm{CO}(g)$ are taken in 1 L flask and heated to 725 K . At equilibrium, $40 \%$ (by mass) of water reacted with $\mathrm{CO}(g)$ as follows

$$ \mathrm{H}_2 \mathrm{O}(g)+\mathrm{CO}(g) \rightleftharpoons \mathrm{H}_2(g)+\mathrm{CO}_2(g) $$

Its $K_c$ value is

At $T(\mathrm{~K})$, the equilibrium constant for $A(g) \rightleftharpoons B(g)$ is $10^{-2}$. If rate of forward reaction is $0.025 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$. The rate of backward reaction (in $\mathrm{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}$ ) is

The molarity of one molal glucose solution having density of $1.2 \mathrm{~g} / \mathrm{mL}$ is

Observe the following reaction,

$$ \mathrm{Cl}_2(g)+2 \mathrm{OH}^{-}(a q) \longrightarrow \mathrm{ClO}^{-}(a q)+\mathrm{Cl}^{-}(a q)+\mathrm{H}_2 \mathrm{O}(l) $$

Identify the correct statements about this reaction,

[I.] $\mathrm{Cl}_2(g)$ is oxidised to $\mathrm{ClO}^{-}(a q)$.

[II.] $\mathrm{Cl}_2(g)$ is oxidised to $\mathrm{Cl}^{-}(a q)$.

[III.] $\mathrm{Cl}_2(g)$ is reduced to $\mathrm{ClO}^{-}(a q)$.

[IV.] $\mathrm{Cl}_2(g)$ is reduced to $\mathrm{Cl}^{-}(a q)$.

The correct answer is

The correct statements about zeolites are

They are good shape selective catalysts.

They contain Al-O-Si framework.

They are not found in nature.

They are used as catalysts for cracking hydrocarbons, in petrochemical industry.

Which of the following is not the application of adsorption?

An example for network solid is

The conductivity of a solution containing 2.08 g of anhydrous barium chloride in 200 mL solution is $6 \times 10^{-3} \mathrm{ohm}^{-1} \mathrm{~cm}^{-1}$. The molar conductivity of the solution (in $\mathrm{ohm}^{-1} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$ ) is $\underline{x} \times 10^2$. The value of $\underline{x}$ is (Atomic mass of $\mathrm{Ba}=137, \mathrm{Cl}=35.5$ )

Consider the following two statements.

Statement I Among $\mathrm{Mg}, \mathrm{Al}, \mathrm{Mg}^{2+}, \mathrm{Al}^{3+}$ the one having smallest size is $\mathrm{Al}^{3+}$.

Statement II Eu is having exceptionally high atomic radii among lanthanide elements.

The correct answer is

Identify the pair of elements with the highest and lowest electronegativity respectively

For the reaction $A+B \rightarrow C$, the following data were obtained

The order of reaction with respect to $A$ and $B$ are respectively

If the bond order in $C_2$ is ' $x^{\prime}$ then the bond order in $B_2$ and $\mathrm{O}_2$, respectively are

Identify incorrectly matched set from the following.

How many of the following can be purified by using zone refining method?

Ni, B, Ti, In, Ge, Mn, Ag, Si, Ga

The $\mathrm{O}-\mathrm{H}$ bond length in $\mathrm{H}_2 \mathrm{O}$ in gas phase is

Which of the following alloys are correctly matched with their uses?

$\mathrm{Li} / \mathrm{Mg}$ Armour plates

$\mathrm{Cu} / \mathrm{Be}$ High strength springs

$\mathrm{Mg} / \mathrm{Al}$ Aircraft construction

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

Choose the acidic oxide from the following.

From the list given below, the number of lanthanides which exhibit +4 state in their oxides is Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy

For ozone molecule consider the following.

It is a linear molecule with bond angle $180^{\circ}$.

It is an angular molecule with bond angle $117^{\circ}$.

The bond lengths of both $\mathrm{O}-\mathrm{O}$ bonds are same.

With respect to oxygen it is thermodynamically more stable.

The correct options are

Given below are two statements.

Assertion (A) All Cu (II) halides are known except the iodide.

Reason (R) $\mathrm{Cu}^{2+}$ oxidises $\mathrm{I}^{-}$to $\mathrm{I}_2$.

$$ \text { Which of the following molecule is not aromatic? } $$

The optically inactive compound from the following is

In the Kolbe electrolysis of sodium propanoate, the products $X$ and $Y$ are formed at respected electrodes. What are $X$ and $Y$ ?

$$ \begin{aligned} &\text { Which is the correct option for the following reaction? }\\ &2 \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{Br} \xrightarrow{2 \mathrm{Na} / \text { Dry ether }} \text { ? } \end{aligned} $$

The number of products obtained in the following reaction is

What are ' $X$ ' and ' $Y$ ' respectively in the following reaction?

$$ \mathrm{C}_6 \mathrm{H}_5 \mathrm{CHO}+\mathrm{C}_6 \mathrm{H}_5 \mathrm{COCH}_3 \xrightarrow[293 \mathrm{~K}]{\mathrm{OH}^{-}} \underset{\text { Major }}{X} \xrightarrow{\mathrm{NaBH}_4} Y $$

Which of the following is used in perfumery?

Arrange the following in increasing order of their boiling points.

[I.] N-ethylethanamine

[II.] Butanamine

[III.] N,N-dimethyl ethanamine

$$ \text { Monomers of nylon 2-nylon } 6 \text { are } $$

$$ \text { The group present in prontosil is } $$

$$ \text { The structure of Gly-Ala is } $$

Given below are two statements

Assertion (A) Hydrolysis of DNA does not form equal number of $A$ and $T, G$ and $C$.

Reason (R) In DNA, adenine forms hydrogen bonds with thymine and cytosine forms hydrogen bonds with guanine.

The correct answer is

If the values of $k$ for which the euqation $x^2+2(k+2) x+6 k+7=0$ has equal roots are $k_1$ and $k_2$, then $k_1^2+k_2^2=$

If $(3+2 \sqrt{2})^{x^2-4}+(3-2 \sqrt{2})^{x^2-4}=6$, then $x^4+x^2+5=$

If -1 is a twice repeated root of the equation $a\left(x^3+x^2\right)+b x+c=0$, then $a: b: c=$

If the equation $x^4+a x^3+b x^2+c x+d=0$ has three equal roots, then that root is

The number of natural numbers less than 10000 which are divisible by 5 and that no digit is repeated in the same number, is

A team of 5 students is to be selected from 12 students. If two particular students are to be included in that team, then the number of ways that such team can be selected is

If the number of diagonals of a regular polygon of $n$ sides is 104 , then $n=$

If two subsets $A$ and $B$ are selected at random from a set $S$ containing $n$ elements, then the probability that $A \cap B=\phi$ and $A \cup B=S$, is

In a game, a pair of dice is rolled 24 times. If a person whis the game by not getting 6 on both the dice in any one of the 24 rolls, then the probability that a person wins the game is

The range of a random variable $X$ is $\{0,1,2\}$. If $P(X=0)=3 C^3, P(X=1)=4 C-10 C^2$ and $P(X=2)=5 C-1$, then the value of $C$ is

If four dice are thrown simultaneously, then the probability that none of the dice shows the number 1 on its face, is

Bag $B_1$ contains 4 white and 2 black balls. Bag $B_2$ contains 3 white and 4 black balls. A bag is chosen at random and a ball is drawn from it at random, then the probability that the ball drawn is white, is

If eight coins are tossed simultaneously, then the probability of getting atleast six heads is

If $(2-5 x)^{\frac{-1}{5}}=a_0+a_1 x+a_2 x^2+\ldots$, then $\frac{a_1}{a_2}=$

If $C_j$ stands for ${ }^n C_j$, then

$$ \frac{C_1}{C_0}+\frac{2 \times C_2}{C_1}+\frac{3 \times C_3}{C_2}+\ldots+\frac{n \times C_n}{C_{n-1}}= $$

$3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}},-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $-\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ are the position vectors of the vertices $A, B$ and $C$ of a $\triangle A B C$ respectively. If $H$ is its orthocentre, then $\mathbf{H A}+\mathbf{H B}+\mathbf{H C}=$

a and b are two vectors such that a is not parallel to b. If $\mathbf{p}=(x+2 y+3) \mathbf{a}+(5 x-y+2) \mathbf{b}$ and $\mathbf{q}=(2 x+3 y+5) \mathbf{a}+(x-5 y-2) \mathbf{b}$ are two vectors such that $\mathbf{p}=2 \mathbf{q}$, then $x-2 y=$

If $\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}, \mathbf{b}=3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $\mathbf{c}=5 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}$ are three vectors, then a vector which is perpendicular to $\mathbf{a}$ and $\mathbf{b} \times \mathbf{c}$ is

Let $\mathbf{O A}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{O B}=\hat{\mathbf{i}}-4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\mathbf{O C}=-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ be the position vectors of three points $A, B, C$ respectively. If $G$ is the centroid of $\triangle A B C$, then $B C^2+C A^2+A B^2+9(O G)^2=$

Let $\pi_1$ be the plane determined by the vectors $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ and $3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. Let $\pi_2$ be the plane determined by the vectors $\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{k}}-2 \hat{\mathbf{i}}$. If $\theta$ is the angle between $\pi_1$ and $\pi_2$, then $\cos \theta=$

$\triangle A B C$ is formed by $A(1,8,4), B(0,-11,4)$ and $C(2,-3,1)$. If $D$ is the foot of the perpendicular drawn from $A$ to $B C$, then the coordinates of $D$ are

The distance between two parallel planes $a x+b y+c z+d_1=0, a x+b y+c z+d_2=0$ is given by $\frac{\left|d_1-d_2\right|}{\sqrt{a^2+b^2+c^2}}$. If the plane $2 x-y+2 z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4 x-2 y+4 z+\lambda=0$ and $2 x-y+2 z+\mu=0$ respectively, then the maximum value of $\lambda+\mu$ is

If the line passing through the points $(5,1, a)$ and $(3, b, 1)$ crosses the $Y Z$-plane at the point $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$, then $a+b=$

Let $A=\left(\begin{array}{l}0 \\ -6 \\ 8\end{array}\right), B=\left(\begin{array}{ccc}3 & 5 & -7 \\ 0 & -1 & 8 \\ 6 & -1 & 0\end{array}\right)$ and $X=\left(\begin{array}{l}x \\ y \\ z\end{array}\right)$. If $D=[\alpha \beta \gamma]^T$ is the solution of $X^T B^T=A^T$, then $D^T A=$

If $S=\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right]$ and $A=\frac{1}{2}\left[\begin{array}{lll}b+c & c-a & b-a \\ c-b & c+a & a-b \\ b-c & a-c & a+b\end{array}\right]$, then $S A S^{-1}=$

The variance of 20 observations is 5. If each one of the observations is multiplied by 2 , then the variance of the resulting observations is

If $A=\left[\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right], B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$ and $C=\left[\begin{array}{cc}0 & i \\ i & 0\end{array}\right]$, then

If $z_1=(2,-1)$ and $z_2=(6,3)$, then $\operatorname{amp}\left(\frac{z_1-z_2}{z_1+z_2}\right)=$

The number of all possible solutions of the equation $z^3+\bar{z}=0$ is

$\left[\sqrt{2}\left(\cos 56^{\circ} 15^{\prime}+i \sin 56^{\circ} 15^{\prime}\right)\right]^8=$

$(1+i)^{2024}+(1-i)^{2024}=$

$\sin 21^{\circ} \cos 9^{\circ}-\cos 84^{\circ} \cos 6^{\circ}=$

If $(1+\sqrt{1+a}) \sin \alpha=(1+\sqrt{1-a}) \cos \alpha$ and $0

If $A=\frac{\pi}{24}$, then $\frac{\cos A+\cos 3 A+\cos 5 A+\cos 7 A}{\sin A+\sin 3 A+\sin 5 A+\sin 7 A}=$

If $\sec (\theta+\alpha), \sec \theta$ and $\sec (\theta-\alpha)$ are in arithmetic progression, then $\sin ^2 \theta=$

If $\cos \alpha+\cos \beta=a$ and $\sin \alpha+\sin \beta=b$, then match the items given in List-A with those of their values in List-B

If $\alpha=\log _e(2+\sqrt{3})$, then $\frac{\cosh \alpha}{1-\tanh \alpha}+\frac{\sinh \alpha}{1-\operatorname{coth} \alpha}=$

In $\triangle A B C$, if $\cot \frac{A}{2}: \cot \frac{B}{2}: \cot \frac{C}{2}=3: 7: 9$, then $a: b: c=$

In $\triangle A B C$, if $\frac{1}{r_1}, \frac{1}{r_2}$ and $\frac{1}{r_3}$ are in arithmetic progression, then $r_2: r=$

If $P_1, P_2$ and $P_3$ are the lengths of the altitudes drawn from the vertices $A, B$ and $C$ of $\triangle A B C$ respectively, then $\frac{\cos A}{P_1}+\frac{\cos B}{P_2}+\frac{\cos C}{P_3}=$

If $f(x)=x^3-x$ and $g(x)=\sin 2 x$, then $f\left(g\left(\frac{\pi}{12}\right)\right)=$

For $x \in R$ if $f(x)=\sqrt{\log _{10}\left(\frac{3-x}{x}\right)}$, then the domain of $f$ is

If $f(x)=\left\{\begin{array}{cl}1+6 x-3 x^2, & x \leq 1 \\ x+\log _2\left(b^2+7\right), & x>1\end{array}\right.$ is continuous at all real $x$, then $b=$

If $f: R \rightarrow R$ defined by

$$ f(x)= \begin{cases}\frac{\sin x-\sin \frac{X}{2}}{x}, & x<0 \\ \frac{\sqrt{x^2+x}-\sqrt{x}}{x^{3 / 2}}, & x>0\end{cases} $$

is continuous on $R$, then $f(0)=$

Let $f(x)$ be a real valued function. If $f^{\prime}(x)$ is a constant for all $x \in R, f(0)=2$ and $f^{\prime}(0)=1$, then

If $y=\frac{\log x}{x}$, then the value of $x^2 \frac{d^2 y}{d x^2}+3 x \frac{d y}{d x}+y$ at the point $(\sqrt[3]{e}, \sqrt{e})$ is

If $f(x)$ is a differentiable function, $f^{\prime}(x) \geq 5 \forall x \in[2,6]$, $f(2)=4$ and $f(3)=15$, then a possible value of $f(6)$

The number of points on the curve $y=2 t^2+3 t-5$ and $x=t^3-4 t^2-3 t$ such that the normals drawn at them on the curve are parallel to $X$-axis is

If the tangent drawn to the curve $y=x^3-a x^2+x+1$ at each point $x \in R$, is inclined at an acute angle with the positive direction of $X$-axis, then the set of all possible values of ' $a$ ' is

Let $f(x)$ be a differentiable function, $A(0, \alpha)$ and $B(8, \beta)$ be two points on the curve $y=f(x)$. Given $f(0)=2$ and $f^{\prime}(4)=\frac{-3}{4}$. If the chord $A B$ of the curve is parallel to the tangent drawn at the point $(4, f(4))$, then $\beta=$

If $f(x)=\sqrt{x+\sin x}$, then all the points of the set $\left\{(x, f(x)) / f^{\prime}(x)=0\right\}$ lie on

If $A=\left\{9 x \geq x^2+20\right\}$ and $f: A \rightarrow R$ is defined by $f(x)=2 x^3-15 x^2+36 x-48$, then the maximum value of $f(x)$ is

If $\int \frac{4 e^x+6 e^{-x}}{9 e^x-4 e^{-x}} d x=A x+B \log \left|\left(9 e^{2 x}-4\right)\right|+C$, then $(A, B)=$

If $f(x)=\int \frac{d x}{\left(x^2+2\right)}$ and $f(\sqrt{2})=0$, then $f(0)=$

If $\frac{2 x^2+5 x+6}{(x+2)^3}=\frac{a}{x+2}+\frac{b}{(x+2)^2}+\frac{c}{(x+2)^3}$, then $a \cdot b+b \cdot c+c \cdot a=$

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

If $I=\int_{-a}^a\left(x^4-2 x^2\right) d x$, then $I$ is minimum at $a=$

Assertion (A) $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{(\sin x)^{\sqrt{2}} d x}{(\sin x)^{\sqrt{2}}+(\cos x)^{\sqrt{2}}}=\frac{\pi}{12}$

Reason (R) $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{f(x) d x}{f(x)+f\left(\frac{\pi}{2}-x\right)}=\frac{\pi}{12}$

If $[x]$ is the greatest integer not exceeding $x$, then $$ \int_{-0.5}^{1.5} x^2[x] d x= $$

$$ \int_0^{50 \pi} \sqrt{1-\cos 2 x} d x= $$

If $f(x)=\frac{x^3+5}{\sqrt{12+x}}$ and $\int_{-5}^5 f(x) d x=\int_0^5(f(x)+g(x)) d x$, then $g(x)=$

The area (in sq units) bounded by the curves $x^2=9 y$, $(x-6)^2=9 y$ and the $X$-axis is

The substitution $x=v y$ converts which one of the following differential equation to an equation solvable by variable separable method?

The order and degree of the differential equation $\left(\frac{d^3 y}{d x^3}\right)^{\frac{1}{2}}-2\left(\frac{d y}{d x}\right)^{\frac{1}{4}}+x y=0$ are respectively

If $\frac{d y}{d x}=f(x, y)$ is a homogeneous differential equation, then the general form of $f(x, y)$ is

The locus of a point which is at a distance of 2 units from the line $2 x-3 y+4=0$ and at a distance of $\sqrt{13}$ units from a point $(5,0)$, is

Let the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ cut the circles $x^2+y^2-2 x+2 y-2=0$ and $x^2+y^2+4 x-6 y+9=0$ orthogonally. If the centre of the circle $S=0$ lies on the line $2 x+3 y-2=0$, then $2 g+f=$

If the chord of contact of the point $P(h, k)$ with respect to the circle $x^2+y^2-4 x-4 y+8=0$ meets the circle in two distinct points and it also makes an angle $45^{\circ}$ with the positive $X$-axis in the positive direction, then $(h, k)$ cannot be

The equation of the pair of tangents drawn from the point $(1,1)$ to the circle $x^2+y^2+2 x+2 y+1=0$ is

If the coordinates of point of contact of the circles $x^2+y^2-4 x+8 y+4=0$ and $x^2+y^2+2 x=0$ is $(a, b)$, then $a+2 b=$

Let $S$ be the circumcircle of the triangle formed by the line $x-2 y-4=0$ with the coordinate axes. If $P(-2,-4)$ is a point in the plane of the circle $S$ and $Q$ is a point on $S$ such that the distance between $P$ and $Q$ is the least, then $P Q=$

If $\theta$ is the acute angle between the tan

from the point $(2,3)$ to the hyperbola from the point $(2,3)$ to the hyperbola $5 x^2-6 y^2-30=0$, then $\tan \theta=$

The difference between the focal distances of any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is 6. If $(\sqrt{13}, k)$ is an end point of a latusrectum of this hyperbola, then $k=$

Let the equation of the tangent at a point $P$ on the parabola $x^2-4 x-4 y+16=0$ be $2 x-y-5=0$. If the equation of the normal drawn at $P$ to this parabola is $a x+y+c=0$, then $a c=$

The equal sides of an isosceles triangle are given by equations $7 x-y+3=0$ and $x+y-3=0$. If the slope $m$ of the third side is an integer, then $m=$

In $\triangle A B C$ the coordinates of the vertex $A$ are $(-3,1)$. If the equation of the median through $B$ is $2 x+y-3=0$ and the equation of the bisector of $\angle C$ is $7 x-4 y-1=0$, then the equation of the side $B C$ is

The combined equation of the lines passing through the point $(3,4)$ and each making an angle $45^{\circ}$ with the line $x+y+1=0$ is-

If the lines given by $\left(x^2+y^2\right) \sin ^2 \alpha=(x \cos \alpha-y \sin \alpha)^2$ are perpendicular to each other, then $\sin ^2 \alpha+\tan ^2 \alpha=$

If the area of the triangle formed by the lines $y=x+c$ and $2 x^2+5 x y+3 y^2=0$ is $\frac{1}{20}$ sq units, then $c=$