AP EAPCET 2023 - 15th May Evening Shift

Two forces whose magnitudes are in the ratio $5: 3$ are acting at a point at angle $60^{\circ}$ simultaneously. If the resultant of the two forces is 35 N , then the magnitudes of two forces respectively are

The following is not the method of reducing friction

A block of mass $m$ and charge $q$ is connected to a point owith an inextensible string. This system is on a horizontal table. An electric field $E$ is applied perpendicular to the string and in the plane of the horizontal table. The tension in the string when it becomes parallel to the electric field is

A block of mass 5 kg moving on a rough surface with a velocity of $4 \mathrm{~ms}^{-1}$ is stopped by the friction in 2 seconds. Then, the coefficient of friction between the contact surfaces is

(Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

In the case of non-conservative forces, the following statement is correct.

A body of mass 2 kg is moving with a constant acceleration of $(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}}) \mathrm{ms}^{-2}$. If the displacement made by the body is $(3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}) \mathrm{m}$, then the work done is

The angular speed of a rigid body rotating about a fixed axis is $(8-2 t) \mathrm{rad} \mathrm{s}^{-1}$. The angle through which the body rotates before it comes to rest is

Two particles of masses 5 g and 3 g are separated by a distance of 40 cm . The centre of mass of the system of these two particles

If $\mathbf{F}=(4 \hat{\mathbf{i}}-10 \hat{\mathbf{j}}) \mathrm{N}$ and $\mathbf{r}=(-5 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}) \mathrm{m}$, then $(\mathbf{r} \times \mathbf{F})$ is

A spherical ball of volume $2000 \mathrm{~cm}^3$ is subjected to a hydraulic pressure of 15 atm . If the change in volume is $5 \times 10^{-2} \mathrm{~cm}^3$, the bulk modulus of the material of the spherical ball is

$$ \left(1 \mathrm{~atm}=10^5 \mathrm{Nm}^{-2}\right) $$

The temperature of the sink of a Carnot's engine is 300 K and the efficiency of the engine is 0.25 . If the temperature of the source of the engine is increased by 100 K , the efficiency of the engine increased by

The emissivity of a perfect black body is increased to 16 times by increasing its temperature. If the initial temperature is $T$, then final temperature of that black body is

The gauge pressure at a depth of 50 m in a sea is (Density of sea water is $1025 \mathrm{~kg} \mathrm{~m}^{-3}$ and $g=10 \mathrm{~ms}^{-2}$ )

A monatomic gas of volume $V$ and pressure $p$ expands isothermally to a volume $27 V$ and then compressed adiabatically to a volume $V$. The final pressure of the gas is

The specific heat capacity of a monatomic gas at constant volume is $x \%$ of its specific heat capacity at constant pressure. Then, $x=$

The number of rotational degrees of freedom of a monatomic molecule is

An object of mass 3 kg is executing simple harmonic motion with an amplitude $\frac{2}{\pi} \mathrm{~m}$. If the kinetic energy of the object when it crosses the mean position is 6 J , the time period of oscillation of the object is

As shown in the figure, a block of weight 20 N is connected to the top of a smooth inclined plane by massless spring of constant $8 \pi^2 \mathrm{Nm}^{-1}$. If the block is pulled slightly from its mean position and released, the period of oscillations is

(Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

A string of length $L$ is stretched by $L / 20$ and the speed of transverse waves along it is $v$. The speed of wave when it is stretched by $L / 10$ will be

(assume that Hooke's law is applicable)

In a Young's double slit experiment, if the wavelength of light is increased by 50\% and the distance between the slits is doubled, then the percentage change in fringe width is

The orbital velocity of a body near the surface of a planet $A$ is equal to escape velocity of a body from the planet $B$. If the masses of planets $A$ and $B$ are same, the ratio of their radii is

The magnifying power of a telescope with tube length 60 cm is 5 . Then, the focal length of its eye piece is

The net electric flux due to a uniform electric field of $3 \times 10^3 \hat{\mathrm{i}} \mathrm{NC}^{-1}$ through a cube of side 20 cm oriented such that its faces are parallel to the coordinate planes is

The Lenz's law is associated with

The magnetic force $q[\mathbf{v} \times \mathbf{B}]$ is

In the given circuit, if the current flowing through $5 \Omega$ resistor is 0.5 A , then the value of $E$ is

One of the following substances having the tendency to move from stronger region to the weaker region of the magnetic field is

Among the following the unit of permeability is NOT represented by

The drift velocity of electrons in a conducting wire connected to a cell is $v_d$. If the length of the wire is doubled and area of cross-section is halved, then the drift velocity of electrons becomes

The device used to detect infrared radiations is

Two capacitors of capacity $4 \mu \mathrm{~F}$ and $6 \mu \mathrm{~F}$ are connected in series to a 500 V battery. The potential difference across $4 \mu \mathrm{~F}$ capacitor is

The relation between permittivity of free space, the permeability of free space and speed of light is

An inductor and a resistor are connected in series to an AC source. If the power factor of the circuit is 0.5 , the ratio of the resistance of the resistor and the reactance of the inductor is

The centripetal acceleration $a$ of an electron in an orbit of hydrogen and the principal quantum number $n$ of the orbit are related by

The radius of the nucleus of an atom whose mass number is 125 is

The de-Broglie wavelength of the most energetic photoelectrons emitted from a photosensitive metal of work function $\phi$, when light frequency $v$ incidents on it is $\lambda$. Then, $v=$

In a transistor, when the emitter current changes by 9.85 mA , the collector current changes to 9.5 mA . Then, the base current is

Bandwidth of an optical fibre is

The material used in the fabrication of infrared LED's is

Which of the following electron arrangement does not obey Hund's rule?

At 300 K , one mole of a gas present in a 10 L flask exerted a pressure of 2.706 atm. What is its compressibility factor $(Z)$ ?

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

The de Broglie wavelength of an electron travelling with $20 \%$ of velocity of light is

$$ \left(h=6.626 \times 10^{-34} \mathrm{Js} ; m_e=9.1 \times 10^{-31} \mathrm{~kg}\right) $$

At. $T(\mathrm{~K})$, equal weights of $\mathrm{H}_2, \mathrm{D}_2$ and $\mathrm{T}_2$ are present in closed vessel. The pressure exerted by this gaseous mixture is $p \mathrm{~atm}$. The ratio of partial pressure of $\mathrm{T}_2, \mathrm{D}_2$ and $\mathrm{H}_2$ is approximately

(H, D and T are isotopes of hydrogen).

At 300 K the enthalpy change for the following reaction is $-2800 \mathrm{~kJ} \mathrm{~mol}^{-1}$.

$$ \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6(s)+6 \mathrm{O}_2(g) \longrightarrow 6 \mathrm{CO}_2(g)+6 \mathrm{H}_2 \mathrm{O}(l) $$

What is the $\Delta U$ for the same reaction at 300 K ( $\mathrm{kJ} \mathrm{mol}^{-1}$ )?

If the work done by 2 mol of an ideal gas during isothermal reversible expansion from 5 L to 50 L is -189.1 L atm at constant pressure, the temperature of the gas (in ${ }^{\circ} \mathrm{C}$ ) is

At $T(\mathrm{~K})$, the equilibrium constant for the reaction $a A(g) \rightleftharpoons b B(g)$ is $K_c$. If the reaction takes place in the following form $2 a A(g) \rightleftharpoons 2 b B(g)$, its equilibrium constant is $K^{\prime}{ }_c$. The correct relationship between $K_c$ and $K^{\prime}{ }_C$ is

Observe the following solutions.

I. Black coffee

II. 0.2 M NaOH

III. Lemon juice

IV. Lime water

V. Human saliva

VI. Tomato juice

The number of solutions having pH range of $1-7$ and 7-14, in the above list, is respectively

What is the van't Hoff factor of ferric sulphate? (Assume $100 \%$ ionisation)

Which of the following does not belong to an ideal solution?

6 g of urea (molar mass $=60 \mathrm{~g} \mathrm{~mol}^{-1}$ ) and 9 g of glucose $\left(\right.$ molar mass $\left.=180 \mathrm{~g} \mathrm{~mol}^{-1}\right)$ were dissolved in 35 g of water. The mass per cent of urea and glucose is respectively

In $A_x B_y$ crystal structure, $A$ atoms occupied all

Octahedral as well as tetrahedral voids and $B$ atoms are at fcc centres. What is the formula of compound $A_x B_y$.

The most effective coagulating agent for antimony sulphide sol is

The plot of $\log \frac{x}{m}$ ( $y$-axis) and $\log p$ ( $x$-axis) is a straight line inclined at an angle of $45^{\circ}$. When the intercept, $k$ is 10 and pressure is 0.3 atm the amount of solute in grams absorbed per gram of adsorbent ( $\log 3=0.4771$ )?

The number of Faradays required to completely deposit magnesium from 1 L of $0.1 \mathrm{M} \mathrm{MgCl}_2$ aqueous solution is

The correct order of covalent radii of $\mathrm{Si}, \mathrm{Ge}, \mathrm{Sn}$ is

The rate constant, $k$ of a zero order reaction $2 \mathrm{NH}_3(g) \xrightarrow[1130 \mathrm{~K}]{\mathrm{Pt}} \mathrm{N}_2(g)+3 \mathrm{H}_2(g)$ is $y \times 10^{-4} \mathrm{molL}^{-1} \mathrm{~s}^{-1}$. The rate of formation of hydrogen (in $\mathrm{molL}^{-1} \mathrm{~s}^{-1}$ ) is

The first ionisation enthalpies of $\mathrm{Na}, \mathrm{Mg}$ and Si are respectively 496, 737 and $786 \mathrm{~kJ} \mathrm{~mol}^{-1}$. What would be the first ionisation enthalpy of Al in $\mathrm{kJ} \mathrm{mol}^{-1}$ ?

$$ \text { Which of the following sets is correct? } $$

Which of the following is not the property of covalent substances?

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

Among the following, the correctly matched pair is

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

The correct answer is

Among the following the correct statements are

I. Germanium exists only in traces.

II. $\mathrm{PbF}_4$ molecule is tetrahedral in shape.

III. $\mathrm{GeX}_2$ is more stable than $\mathrm{GeX}_4$.

The correct answer is

The correct order of melting points of Al, Ga, In is

Identify the reaction in which oxygen is not one of the products.

Identify the correctly matched pairs,

i. TiO - pigment industry.

ii. $\mathrm{MnO}_2$ - dry battery cells.

iii. Cu/Ni alloy - UK 'copper' coins.

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

Arrange the following in order of increasing number of unpaired electrons in them

i. $\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{3-}$

ii. $\left[\mathrm{MnCl}_6\right]^{3-}$

iii. $\left[\mathrm{FeF}_6\right]^{3-}$

iv. $\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_6\right]^{3+}$

In $\mathrm{S}_{\mathrm{N}} 2$ reaction, the carbon in the transition state is

Two statements are given below.

Statement I Chlorobenzene on nitration gives 1-chloro-4-nitrobenzene as major product.

Statement II Chlorobenzene undergoes nitration slowly than benzene.

Identify the correct answer

$$ \text { Which of the following reaction is not feasible? } $$

Which of the following has more $\mathrm{p} K_a$ value?

Vinegar and butter are the sources for which of the following carboxylic acids respectively?

$$ \begin{aligned} &\text { Identify } X, Y \text { and } Z \text { in the following reactions sequence }\\ &\mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{CH}_2 \mathrm{OH} \xrightarrow{X} \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{COOH} \xrightarrow{Y} \mathrm{Z} \end{aligned} $$

In which of the following pairs the polymer is correctly matched with the forces possessed by them.

A. Neoprene - Weak intermolecular forces

B.Terylene - Hydrogen bonding

C. Polystyrene - Very weak intermolecular forces

D. Polythene - Hydrogen bonding

Which of the following produces nitrogen gas after the reaction with nitrous acid?

Hypothyroidism is due to

[A.] high level of iodine in the diet.

[B.] enlargement of thyroid gland.

[C.] low levels of iodine in the diet.

[D.] increased levels of thyroxine.

In nucleoside, the base is attached to which position of sugar molecule?

ortho-sulphobenzimide is used as

The minimum value of $f(x)=\frac{x^2-2 x+3}{x^2-4 x+7}$ is

If -1 is a twice repeated root of the equation $a x^3+b x^2+c x+1=0$, then

$\alpha$ and $\beta$ are the roots of the equation $x^2-a x+b=0$. If $\alpha^2+\beta^2$ and $\alpha^3+\beta^3$ are the roots of the equation $A x^2+B x+C=0$, then $C=$

If one root of the equation $a x^3+b x+c=0$ is twice another root, then

The number of six digit natural numbers that can be formed with the digits $2,3,4,0,5,6,7,8$ is

The number of non-negative integral solutions of $x_1+x_2+x_3+x_4=10$ is

Three screws are drawn at random from a lot of 50 screws containing 5 defective ones. Then, the probability of the event that all 3 screws drawn are non-defective, assuming that the drawing is (a) with replacement (b) without replacement respectively is

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96 is

If $P(X=x)=k\left(\frac{3}{8}\right)^x, x=1,2,3 \ldots$ is the probability distribution function of a discrete random variable $X$, then $k=$

If $A$ and $B$ are events of a random experiment with $P(A)=0.5, P(B)=0.4$ and $P(A \cap B)=0.3$, then the probability that neither $A$ nor $B$ occurs is

From a collection of eight cards numbered 1 to 8 , if two cards are drawn at random, one after the other with replacement, then the probability that the product of numbers that appear on the cards is a perfect square is

A coin is tossed three times. Let $A$ be the event of "getting three heads" and $B$ be the event of "getting a head on the first toss". Then, $A$ and $B$ are

The coefficient of the highest power of $x$ in the expansion of $\left(x+\sqrt{x^2-1}\right)^8+\left(x-\sqrt{x^2-1}\right)^8$ is

If $C_j={ }^n C_j$, then $C_0 C_r+C_1 C_{r+1}+C_2 C_{r+2}+\ldots+C_{n-r} C_n=$

If $7 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ is the position vector of the vertex $A$ of a tetrahedron $A B C D$ and $-\hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ is the position vector of the centroid of the $\triangle B C D$, then the position vector of the centroid of the tetrahedron $A B C D$ is

If $\mathbf{a}$ and $\mathbf{b}$ are two vectors such that $|\mathbf{a}|=|\mathbf{b}|=\sqrt{14}$ and $\mathbf{a} \cdot \mathbf{b}=-7$, then $\frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a} \cdot \mathbf{b}|}=$

Let $(\mathbf{a}, \mathbf{b})$ denote the angle between vectors $\mathbf{a}$ and $\mathbf{b}$. If $\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}, \mathbf{a} \cdot \mathbf{b}=4$ and $(\mathbf{a}, \mathbf{b})=\cos ^{-1}\left(\frac{4}{21}\right)$, then $\mathbf{a}+\mathbf{b}=$

Let $\mathbf{O A}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{O B}=-2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ be the position vectors of two points $A$ and $B$. If $C$ is a point on the bisector $\angle A O B$ and $O C=\sqrt{42}$, then $\mathbf{O C}=$

The distance of a point $\mathbf{a}$ from the plane $\mathbf{r} \cdot \mathbf{m}=q$ is given by $\frac{|\mathbf{a} \cdot \mathbf{m}-9|}{|\mathbf{m}|}$. If the distance of the point $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ from the plane $\mathbf{r} \cdot(2 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}-9 \hat{\mathbf{k}})=-1$ is $p$ and the distance of the origin from this plane is $q$, then $p-q=$

Let $A B C D$ be a tetrahedron in which the coordinates of each of its vertices are in arithmetic progression with same common difference. If the centroid $G$ of the tetrahedron is $(2,3, k)$, then the distance of $G$ from the origin is

If $A(3,-1,11), B(0,2,3)$ and $C(4,8,11)$ are three points, then the coordinates of the foot of the perpendicular drawn from the point $A$ to the line joining the points $B$ and $C$ is

If $S$ is the set of all real values of ' $a$ ' such that a plane passing through the points $\left(-a^2, 1,1\right),\left(1,-a^2, 1\right)$ and $\left(1,1,-a^2\right)$ also passes through the point $(-1,-1,1)$, then $S=$

If $\operatorname{det}(A B)=(\operatorname{det} A)(\operatorname{det} B)$ and $A$ is a non-singular matrix of order $3 \times 3$, then $\operatorname{det}(\operatorname{adj} A)=$

3. If the co-factors of the elements 3, 7 and 6 of the matrix $\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & -1 & 7 \\ 2 & 4 & 6\end{array}\right]$ are $a, b$ and $c$ respectively, then $\left[\begin{array}{lll}a & b & c\end{array}\right]\left[\begin{array}{l}1 \\ 4 \\ 2\end{array}\right]+\left[\begin{array}{lll}a & b & c\end{array}\right]\left[\begin{array}{l}3 \\ 7 \\ 6\end{array}\right]=$

Let $B=\left(\begin{array}{ccc}2 & 6 & 4 \\ 1 & 0 & 1 \\ -1 & 1 & -1\end{array}\right)$ and $C=\left(\begin{array}{ccc}-1 & 0 & 1 \\ 1 & 1 & 3 \\ 2 & 0 & 2\end{array}\right)$. If a matrix $A$ is such that $B A C=I$, then $A^{-1}=$

If $A=\left\{\left(\begin{array}{ll}a & b \\ c & d\end{array}\right): a, b, c, d \in\{-1,1\}\right\}$, then the number of singular matrices in $A$ is

The mean of 5 observations is 4.4 and their variance is 8.24. If three of those observations are 1,2 and 6 , then the other two observations are

If $\left|z-\frac{2}{z}\right|=2$, then the greatest value of $|z|$ is

$$ S=\{z \in C /|z-1+i|=1\} \text { represents } $$

One of the 15 th roots of -1 is

The product of the four values of $(1+i \sqrt{3})^{3 / 4}$ is

If $\cot x \cot y=a$ and $x+y=\frac{\pi}{6}$, then the quadratic equation satisfying $\cot x$ and $\cot y$ is

If $\tan A+\tan B=x$ and $\cot A+\cot B=y$, then $\tan (A+B)=$

$$ \frac{1}{\cos 290^{\circ}}+\frac{1}{\sqrt{3} \sin 250^{\circ}}= $$

$$ \begin{aligned} & \text { If }\left[1-\cos \left(\frac{\pi}{2}+\alpha\right)+\sin \left(\frac{3 \pi}{2}+\alpha\right)\right]^2 +\left[1-\sin \left(\frac{3 \pi}{2}-\alpha\right)-\cos \left(\frac{3 \pi}{2}+\alpha\right)\right]^2 \\ & =a+b \sin ^2\left(\frac{\pi}{4}+\alpha\right), \text { then } a^2+b^2= \end{aligned} $$

The range of $\frac{1}{\sin ^2 x+3 \sin x \cos x+5 \cos ^2 x}$ is

If $\cos h x=\frac{5}{4}$, then $\tan h 3 x=$

In $\triangle A B C$, if $a \cos ^2 \frac{C}{2}+c \cos ^2 \frac{A}{2}=\frac{3 b}{2}$, then $a+c: b=$

In $\triangle A B C$, if $\sin ^2 B=\sin C$ and $3 \cos ^2 B=2 \cos ^2 C$, then $\triangle A B C$ is

If $\triangle A B C$ is a right angled isosceles triangle and $\angle C=90^{\circ}$, then $r: r_3=$

If $f(a)=\log \left|\frac{1-a}{1+a}\right|$ for $a \neq\{-1,1\}$, then the set of values of all ' $a$ ', for which $f\left(\frac{2 a}{1+a^2}\right)>0$ is

If a real valued function $f$ is defined by $f(x)=\frac{a x+\sqrt{a^2-x^2}}{b x}$, then $f$ is

The function $f(x)=\sqrt{\frac{3 x^2-5 x-2}{2 x^2-7 x+5}}$ has discontinuous points at $x=$

If $f(x)=\left\{\begin{array}{cc}\frac{x-[x]}{x-2}, & x > 2 \\ b, & x=2 \\ \frac{\left|x^2-x-2\right|}{a\left(2+x-x^2\right)}, & -1 < x \leq 2 \\ 2 a-b, & x \leq-1\end{array}\right.$

is continuous on $R$, then $\lim _{x \rightarrow 0} \frac{\sin ^2 a x+x \tan b x}{x^2}=$

Let $[x]$ represents the greatest integer not more than $x$. The discontinuous points of the function $f(x)=\frac{5+[x]}{\sqrt{11+[x]-6 \sqrt{2+[x]}}}$ lies in the interval

If $x^x y^y=e^e$, then $\left(\frac{d^2 y}{d x^2}\right)_{(e, e)}=$

If $f(x)=|x-5|+|x+5|+|x-4|+|x+4|$, then $\frac{f^{\prime}(1)-f^{\prime}(-6)}{f^{\prime}(-1)+f^{\prime}(6)}=$

If $f(x+a y)+g(x-a y)=0$, then $a \frac{d y}{d x}=$

If $(2, a)$ and $(b, 19)$ are two stationary points of the curve $y=2 x^3-15 x^2+36 x+c$, then $a+b+c=$

If the angle between the curves $y=e^{2(1+x)-4}$ and $x^2 y=1$ at the point $(1,1)$ is $\theta$, then $|\sin \theta|+|\cos \theta|=$

If the points of contact of the tangents drawn from $(0,0)$ to the curve $y=x^2+3 x+4$ are $(\alpha, \beta)$ and $(\gamma, \delta)$, then $\beta+\delta=$

$\int \frac{d x}{4+5 \cos x}=$

$$ \int x^3(\log x)^2 d x= $$

If $f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+2 x^7+1\right)^2} d x(x \geq 0)$ and $f(0)=0$, then the value of $f(1)=$

If $\frac{x+2}{x^2-3}$ is one of the partial fractions of $\frac{3 x^3-x^2-2 x+17}{x^4+x^2-12}$, then the other partial fraction of it is

If $\int \frac{1}{x\left[(\log x)^2+4 \log x-1\right]} d x=A \log \left[\frac{\log x+B}{\log x+C}\right]+K$,

where $K$ is the constant of integration, then

If $f(x)=\int_0^x\left[(a+1)(t+1)^2-(a-1)\left(t^2+t+1\right)\right] d t$, then a possible positive value of ' $a$ ', for which $f^{\prime}(x)=0$ has equal roots, is

$$ \int_0^1(\sqrt{10})^{2 x} d x= $$

$$ \int_0^{\frac{\pi}{2}}\left(\frac{\sin \left(\frac{\pi}{4}+x\right)+\sin \left(\frac{3 \pi}{4}+x\right)}{\cos x+\sin x}\right) d x= $$

$$ \int_0^{\frac{\pi}{4}} \frac{\cos ^2 x}{\cos ^2 x+4 \sin ^2 x} d x= $$

If $\alpha$ and $\beta$ are respectively, the order and degree of the differential equation $y=e^{\left(\frac{d y}{d x}+\frac{d^2 y}{d x^2}\right)}$, then the value of $\alpha+\alpha^\beta+\alpha^{2 \beta}+\ldots+\alpha^{2023 \beta}=$

The general soluiton of the differential equation $\tan x \tan y d x+\cos ^2 x \operatorname{cosec}^2 y d y=0$ is

The area (in sq units) of the region bounded by the curves $y=4|\cos x|$ and $y=-|\cos x|$ from $x=-\frac{\pi}{2}$ to $\frac{\pi}{2}$ is

If $x^\alpha \frac{d y}{d x}=y^\beta(\gamma \log x+\delta \log y+1)$ is a homogeneous differential equation, then

If $P\left(\frac{\pi}{3}\right)$ and $Q\left(\frac{2 \pi}{3}\right)$ represent two points on the circle $x^2+y^2-4 x+6 y-12=0$ in parametric form, then the length of the chord $P Q$ is

The distance between the centres of similitude of the circles $x^2+y^2+6 x-8 y+16=0$ and $x^2+y^2-2 x-2 y+1=0$ is

Let $A(2,3), B(3,-1)$ and $C(-3,2)$ be three points. If the centre of the circle passing through $A, B$ and $C$ is $(h, k)$, then $2 k-4 h=$

Let $A(1,2)$ be the centre and 3 be the radius of a circle $S$. Let $B(-1,-1)$ be the centre and $r$ be the radius of another circle $S^{\prime}$. If $\frac{\pi}{3}$ is the angle between the circles $S$ and $S^{\prime}$, then the number of possible values of $r$ is

Let $P$ and $Q$ be the inverse points with respect to the circle $S \equiv x^2+y^2-4 x-6 y+k=0$ and $C$ be the centre of the circle $S=0$ such that $C P \cdot C Q=4$. If $P=(1,2)$ and $Q=(a, b)$, then $2 a=$

If $A(4,0)$ and $B(-4,0)$ are two points, then the locus of a point $P$ such that $P A-P B=4$ is

If $S \equiv \frac{x^2}{k-7}+\frac{y^2}{11-k}-1=0, k \in R-\{7,11\}$, then which one of the following statements is incorrect?

The perpendicular distance from the origin to the focal chord drawn through the point $(4,5)$ to the parabola $y^2-4 y-3 x+7=0$ is

Let $(1,2)$ be the focus and $x+y+1=0$ be the directrix of a hyperbola $H$. If $\sqrt{3}$ is the eccentricity of $H$, then its equation is

The diagonals $A C$ and $B D$ of a rhombus $A B C D$ intersect at the point $(3,4)$. If $B D=2 \sqrt{2}, A=(1,2), B=(\alpha, \beta)$, $C(5,6), D=(\gamma, \delta)$ and $\alpha<\delta<\gamma<\beta$, then $\beta+\gamma-\delta=$

The lines $p\left(p^2+1\right) x-y+q=0$ and

$\left(p^2+1\right)^2 x+\left(p^2+1\right) y+2 q=0$ are perpendicular to a line $L$ for

If $2 x^2-3 x y+y^2=0$ represents two sides of a triangle and $x+y-1=0$ is its third side, then the distance between the orthocentre and the circumcentre of that triangle is

Let $P=(-1,0), Q=(0,0)$ and $R=(3,3 \sqrt{3})$ be three Points. Then, the equation of the bisector of the $\angle P Q R$ is

Assertion (A) The difference of the slopes of the lines represented by $y^2-2 x y \sec ^2 \alpha+\left(3+\tan ^2 \alpha\right)$ $\left(-1+\tan ^2 \alpha\right) x^2=0$ is 4

Reason (R) The difference of the slopes represented by $a x^2+2 h x y+b y^2=0$ is $\frac{2 \sqrt{h^2-a b}}{|b|}$

If a line is moving between the coordinate axes such that the sum of the intercepts made by it on the coordinate axes is always 12, then the equation of that line which forms a triangle of maximum area with the coordinate axes is

Let the length of the latusrectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be equal to the length of its semi-major axis. If the radius of its director circle is $\sqrt{3}$ and $e$ is its eccentricity, then the length of its latusrectum is