Consider the following compound (X)

The most stable and least stable carbon radicals, respectively, produced by homolytic cleavage of corresponding $\mathrm{C}-\mathrm{H}$ bond are :

$$\mathrm{CaCO}_3(\mathrm{~s})+2 \mathrm{HCl}(\mathrm{aq}) \rightarrow \mathrm{CaCl}_2(\mathrm{aq})+\mathrm{CO}_2(\mathrm{~g})+\mathrm{H}_2 \mathrm{O}(\mathrm{l})$$

Consider the above reaction, what mass of $\mathrm{CaCl}_2$ will be formed if 250 mL of 0.76 M HCl reacts with 1000 g of $\mathrm{CaCO}_3$ ?

(Given : Molar mass of $\mathrm{Ca}, \mathrm{C}, \mathrm{O}, \mathrm{H}$ and Cl are $40,12,16,1$ and $35.5 \mathrm{~g} \mathrm{~mol}^{-1}$, respectively)

If equal volumes of $A B_2$ and $X Y$ (both are salts) aqueous solutions are mixed, which of the following combination will give a precipitate of $\mathrm{AY}_2$ at 300 K ? (Given $\mathrm{K}_{\mathrm{sp}}\left(\right.$ at 300 K ) for $\mathrm{AY}_2=5.2 \times 10^{-7}$ )

Given below are two statements :

Statement (I) : The metallic radius of Al is less than that of Ga .

Statement (II) : The ionic radius of $\mathrm{Al}^{3+}$ is less than that of $\mathrm{Ga}^{3+}$.

In the light of the above statements, choose the most appropriate answer from the options given below :

Designate whether each of the following compounds is aromatic or not aromatic.

Choose the correct tests with respective observations.

(A) $\mathrm{CuSO}_4$ (acidified with acetic acid) $+\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right] \rightarrow$ Chocolate brown precipitate.

(B) $\mathrm{FeCl}_3+\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right] \rightarrow$ Prussian blue precipitate.

(C) $\mathrm{ZnCl}_2+\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]$, neutralised with $\mathrm{NH}_4 \mathrm{OH} \rightarrow$ White or bluish white precipitate.

(D) $\mathrm{MgCl}_2+\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right] \rightarrow$ Blue precipitate.

(E) $\mathrm{BaCl}_2+\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]$, neutralised with $\mathrm{NaOH} \rightarrow$ White precipitate.

Choose the correct answer from the options given below :

A solution is made by mixing one mole of volatile liquid $A$ with 3 moles of volatile liquid $B$. The vapour pressure of pure A is 200 mm Hg and that of the solution is 500 mm Hg . The vapour pressure of pure B and the least volatile component of the solution, respectively, are:

A molecule with the formula $\mathrm{AX}_4 \mathrm{Y}$ has all it's elements from p-block. Element A is rarest, monoatomic, non-radioactive from its group and has the lowest ionization enthalpy value among $\mathrm{A}, \mathrm{X}$ and Y . Elements X and Y have first and second highest electronegativity values respectively among all the known elements. The shape of the molecule is:

Given below are two statements :

Statement (I) : In octahedral complexes, when $\Delta_0<\mathrm{P}$ high spin complexes are formed. When $\Delta_0>P$ low spin complexes are formed.

Statement (II) : In tetrahedral complexes because of $\Delta_t

In the light of the above statements, choose the most appropriate answer from the options given below :

On complete combustion 1.0 g of an organic compound $(\mathrm{X})$ gave 1.46 g of $\mathrm{CO}_2$ and 0.567 g of $\mathrm{H}_2 \mathrm{O}$. The empirical formula mass of compound $(\mathrm{X})$ is __________ g. (Given molar mass in $\mathrm{g} \mathrm{mol}^{-1} \mathrm{C}: 12, \mathrm{H}: 1, \mathrm{O}: 16$ )

Given below are two statements:

Statement I : Vanilin will react with NaOH and also with Tollen's reagent.

Statement II : Vanilin will undergo self aldol condensation very easily.

In the light of the above statements, choose the most appropriate answer from the options given below :

An optically active alkyl halide $\mathrm{C}_4 \mathrm{H}_9 \mathrm{Br}[\mathrm{A}]$ reacts with hot KOH dissolved in ethanol and forms alkene $[B]$ as major product which reacts with bromine to give dibromide $[C]$. The compound [C] is converted into a gas [D] upon reacting with alcoholic $\mathrm{NaNH}_2$. During hydration 18 gram of water is added to 1 mole of gas [D] on warming with mercuric sulphate and dilute acid at 333 K to form compound [E]. The IUPAC name of compound [ E ] is :

Two vessels A and B are connected via stopcock. The vessel A is filled with a gas at a certain pressure. The entire assembly is immersed in water and is allowed to come to thermal equilibrium with water. After opening the stopcock the gas from vessel A expands into vessel B and no change in temperature is observed in the thermometer. Which of the following statement is true ?

Among $\mathrm{SO}_2, \mathrm{NF}_3, \mathrm{NH}_3, \mathrm{XeF}_2, \mathrm{ClF}_3$ and $\mathrm{SF}_4$, the hybridization of the molecule with nonzero dipole moment and highest number of lone-pairs of electrons on the central atom is

Which of the following graph correctly represents the plots of $\mathrm{K}_{\mathrm{H}}$ at 1 bar for gases in water versus temperature?

According to Bohr's model of hydrogen atom, which of the following statement is incorrect?

Identify the correct statement among the following :

The correct order of basic nature in aqueous solution for the bases $\mathrm{NH}_3, \mathrm{H}_2 \mathrm{~N}-\mathrm{NH}_2, \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{NH}_2,\left(\mathrm{CH}_3 \mathrm{CH}_2\right)_2 \mathrm{NH}$ and $\left(\mathrm{CH}_3 \mathrm{CH}_2\right)_3 \mathrm{~N}$ is :

The property/properties that show irregularity in first four elements of group-17 is/are

(A) Covalent radius

(B) Electron affinity

(C) Ionic radius

(D) First ionization energy

Choose the correct answer from the options given below :

Consider the following molecules:

The correct order of rate of hydrolysis is :

For the reaction A $\to$ products.

The concentration of A at 10 minutes is _________ $\times 10^{-3} \mathrm{~mol} \mathrm{~L}^{-1}$ (nearest integer). The reaction was started with $2.5 \mathrm{~mol} \mathrm{~L}^{-1}$ of A .

0.1 mol of the following given antiviral compound $(\mathrm{P})$ will weigh ________ $\times 10^{-1} \mathrm{~g}$ (nearest integer).

(Given : molar mass in $\mathrm{g} \mathrm{mol}^{-1} \mathrm{H}: 1, \mathrm{C}: 12, \mathrm{~N}: 14, \mathrm{O}: 16, \mathrm{~F}: 19, \mathrm{I}: 127$ )

Consider the following electrochemical cell at standard condition.

$$\mathrm{Au}(\mathrm{~s})\left|\mathrm{QH}_2, \mathrm{Q}\right| \mathrm{NH}_4 \mathrm{X}(0.01 \mathrm{M})| | \mathrm{Ag}^{+}(1 \mathrm{M}) \mid \mathrm{Ag}(\mathrm{~s}) \mathrm{E}_{\text {cell }}=+0.4 \mathrm{~V}$$

The couple $\mathrm{QH}_2 / \mathrm{Q}$ represents quinhydrone electrode, the half cell reaction is given below:

$$\left[\text { Given : } \mathrm{E}_{\mathrm{Ag}^{+} / \mathrm{Ag}}^0=+0.8 \mathrm{~V} \text { and } \frac{2.303 \mathrm{RT}}{\mathrm{~F}}=0.06 \mathrm{~V}\right]$$

The $\mathrm{pK}_{\mathrm{b}}$ value of the ammonium halide salt $\left(\mathrm{NH}_4 \mathrm{X}\right)$ used here is __________ . (nearest integer)

Consider the following equilibrium,

$$\mathrm{CO}(\mathrm{~g})+2 \mathrm{H}_2(\mathrm{~g}) \rightleftharpoons \mathrm{CH}_3 \mathrm{OH}(\mathrm{~g})$$

0.1 mol of CO along with a catalyst is present in a $2 \mathrm{dm}^3$ flask maintained at 500 K . Hydrogen is introduced into the flask until the pressure is 5 bar and 0.04 mol of $\mathrm{CH}_3 \mathrm{OH}$ is formed. The $K_p^\theta$ is __________ $\times 10^{-3}$ (nearest integer).

Given : $\mathrm{R}=0.08 \mathrm{~dm}^3$ bar $\mathrm{K}^{-1} \mathrm{~mol}^{-1}$

Assume only methanol is formed as the product and the system follows ideal gas behaviour.

A transition metal (M) among $\mathrm{Mn}, \mathrm{Cr}, \mathrm{Co}$ and Fe has the highest standard electrode potential $\left(\mathrm{M}^{3+} / \mathrm{M}^{2+}\right)$. It forms a metal complex of the type $\left[\mathrm{M}(\mathrm{CN})_6\right]^{4-}$. The number of electrons present in the $\mathrm{e}_{\mathrm{g}}$ orbital of the complex is ___________.

Let $\mathrm{A}=\left[\begin{array}{cc}\alpha & -1 \\ 6 & \beta\end{array}\right], \alpha>0$, such that $\operatorname{det}(\mathrm{A})=0$ and $\alpha+\beta=1$. If I denotes $2 \times 2$ identity matrix, then the matrix $(I+A)^8$ is :

Let the vertices Q and R of the triangle PQR lie on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}, \mathrm{QR}=5$ and the coordinates of the point $P$ be $(0,2,3)$. If the area of the triangle $P Q R$ is $\frac{m}{n}$ then :

Let $a \in R$ and $A$ be a matrix of order $3 \times 3$ such that $\operatorname{det}(A)=-4$ and $A+I=\left[\begin{array}{lll}1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2\end{array}\right]$, where $I$ is the identity matrix of order $3 \times 3$. If $\operatorname{det}((a+1) \operatorname{adj}((a-1) A))$ is $2^{\mathrm{m}} 3^{\mathrm{n}}, \mathrm{m}$, $\mathrm{n} \in\{0,1,2, \ldots, 20\}$, then $\mathrm{m}+\mathrm{n}$ is equal to :

If $\theta \in[-2 \pi, 2 \pi]$, then the number of solutions of $2 \sqrt{2} \cos ^2 \theta+(2-\sqrt{6}) \cos \theta-\sqrt{3}=0$, is equal to:

If $S$ and $S^{\prime}$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and P be a point on the ellipse, then $\min \left(S P \cdot S^{\prime} P\right)+\max \left(S P \cdot S^{\prime} P\right)$ is equal to :

Let the focal chord PQ of the parabola $y^2=4 x$ make an angle of $60^{\circ}$ with the positive $x$ axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the $y$-axis at the point $(0, \alpha)$, then $5 \alpha^2$ is equal to:

If the function $f(x)=2 x^3-9 a x^2+12 \mathrm{a}^2 x+1$, where $\mathrm{a}>0$, attains its local maximum and local minimum values at p and q , respectively, such that $\mathrm{p}^2=\mathrm{q}$, then $f(3)$ is equal to :

Let $A B C D$ be a tetrahedron such that the edges $A B, A C$ and $A D$ are mutually perpendicular. Let the areas of the triangles $\mathrm{ABC}, \mathrm{ACD}$ and ADB be 5,6 and 7 square units respectively. Then the area (in square units) of the $\triangle B C D$ is equal to :

Let $z$ be a complex number such that $|z|=1$. If $\frac{2+\mathrm{k}^2 z}{\mathrm{k}+\bar{z}}=\mathrm{k} z, \mathrm{k} \in \mathbf{R}$, then the maximum distance of $\mathrm{k}+i \mathrm{k}^2$ from the circle $|z-(1+2 i)|=1$ is :

Let one focus of the hyperbola $\mathrm{H}: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ be at $(\sqrt{10}, 0)$ and the corresponding directrix be $x=\frac{9}{\sqrt{10}}$. If $e$ and $l$ respectively are the eccentricity and the length of the latus rectum of H , then $9\left(e^2+l\right)$ is equal to :

The largest $\mathrm{n} \in \mathbf{N}$ such that $3^{\mathrm{n}}$ divides 50 ! is :

Let $\mathrm{P}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}$. If $\mathrm{P}_{10}=123, \mathrm{P}_9=76, \mathrm{P}_8=47$ and $\mathrm{P}_1=1$, then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is :

If $\overrightarrow{\mathrm{a}}$ is a nonzero vector such that its projections on the vectors $2 \hat{i}-\hat{j}+2 \hat{k}, \hat{i}+2 \hat{j}-2 \hat{k}$ and $\hat{k}$ are equal, then a unit vector along $\overrightarrow{\mathrm{a}}$ is :

If the system of linear equations

$$ \begin{aligned} & 3 x+y+\beta z=3 \\ & 2 x+\alpha y-z=-3 \\ & x+2 y+z=4 \end{aligned} $$

has infinitely many solutions, then the value of $22 \beta-9 \alpha$ is :

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x \sin y)(f(2 x+2 y)+f(2 x-2 y))$, for all $x, y \in \mathbf{R}$. If $f^{\prime}(0)=\frac{1}{2}$, then the value of $24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)$ is :

Let A be the set of all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ and R be a relation on A such that $\mathrm{R}=\{(\mathrm{f}, \mathrm{g}): f(0)=\mathrm{g}(1)$ and $f(1)=\mathrm{g}(0)\}$. Then R is :

The number of sequences of ten terms, whose terms are either 0 or 1 or 2 , that contain exactly five 1 s and exactly three 2 s , is equal to :

For $\alpha, \beta, \gamma \in \mathbf{R}$, if $\lim _\limits{x \rightarrow 0} \frac{x^2 \sin \alpha x+(\gamma-1) \mathrm{e}^{x^2}}{\sin 2 x-\beta x}=3$, then $\beta+\gamma-\alpha$ is equal to :

The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2 / 3}+1-x^{1 / 3}\right)}-\frac{(x-1)}{\left(x-x^{1 / 2}\right)}\right)^{10}, x>1$, is :

Let $a_1, a_2, a_3, \ldots$ be in an A.P. such that $\sum_\limits{k=1}^{12} a_{2 k-1}=-\frac{72}{5} a_1, a_1 \neq 0$. If $\sum_\limits{k=1}^n a_k=0$, then $n$ is:

Three distinct numbers are selected randomly from the set $\{1,2,3, \ldots, 40\}$. If the probability, that the selected numbers are in an increasing G.P., is $\frac{m}{n}, \operatorname{gcd}(m, n)=1$, then $m+n$ is equal to __________ .

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a thrice differentiable odd function satisfying $f^{\prime}(x) \geq 0, f^{\prime}(x)=f(x), f(0)=0, f^{\prime}(0)=3$. Then $9 f\left(\log _e 3\right)$ is equal to __________ .

If the area of the region $\left\{(x, y):\left|4-x^2\right| \leq y \leq x^2, y \leq 4, x \geq 0\right\}$ is $\left(\frac{80 \sqrt{2}}{\alpha}-\beta\right), \alpha, \beta \in \mathbf{N}$, then $\alpha+\beta$ is equal to _________.

The absolute difference between the squares of the radii of the two circles passing through the point $(-9,4)$ and touching the lines $x+y=3$ and $x-y=3$, is equal to ________ .

Let [.] denote the greatest integer function. If $\int_\limits0^{e^3}\left[\frac{1}{e^{x-1}}\right] d x=\alpha-\log _e 2$, then $\alpha^3$ is equal to _________.

Consider two infinitely large plane parallel conducting plates as shown below. The plates are uniformly charged with a surface charge density $+\sigma$ and $-2 \sigma$. The force experienced by a point charge +q placed at the mid point between two plates will be:

A monochromatic light is incident on a metallic plate having work function $\phi$. An electron, emitted normally to the plate from a point A with maximum kinetic energy, enters a constant magnetic field, perpendicular to the initial velocity of electron. The electron passes through a curve and hits back the plate at a point $B$. The distance between $A$ and $B$ is : (Given : The magnitude of charge of an electron is e and mass is $\mathrm{m}, \mathrm{h}$ is Planck's constant and c is velocity of light. Take the magnetic field exists throughout the path of electron)

A spherical surface separates two media of refractive indices 1 and 1.5 as shown in figure. Distance of the image of an object ' O ', is :

( C is the center of curvature of the spherical surface and R is the radius of curvature)

A river is flowing from west to east direction with speed of $9 \mathrm{~km} \mathrm{~h}^{-1}$. If a boat capable of moving at a maximum speed of $27 \mathrm{~km} \mathrm{~h}^{-1}$ in still water, crosses the river in half a minute, while moving with maximum speed at an angle of $150^{\circ}$ to direction of river flow, then the width of the river is :

Moment of inertia of a rod of mass ' M ' and length ' L ' about an axis passing through its center and normal to its length is ' $\alpha$ '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is :

Let $B_1$ be the magnitude of magnetic field at center of a circular coil of radius $R$ carrying current I. Let $\mathrm{B}_2$ be the magnitude of magnetic field at an axial distance ' $x$ ' from the center. For $x: \mathrm{R}=3: 4, \frac{\mathrm{~B}_2}{\mathrm{~B}_1}$ is :

A light wave is propagating with plane wave fronts of the type $x+y+z=$ constant. Th angle made by the direction of wave propagation with the $x$-axis is :

Match List I with List II.

Choose the correct answer from the options given below:

A particle is subjected to two simple harmonic motions as : $$ x_1=\sqrt{7} \sin 5 \mathrm{tcm} $$ and $x_2=2 \sqrt{7} \sin \left(5 t+\frac{\pi}{3}\right) \mathrm{cm}$ where $x$ is displacement and $t$ is time in seconds. The maximum acceleration of the particle is $x \times 10^{-2} \mathrm{~ms}^{-2}$. The value of $x$ is :

A square Lamina OABC of length 10 cm is pivoted at ' $\mathrm{O}^{\prime}$. Forces act at Lamina as shown in figure. If Lamina remains stationary, then the magnitude of F is :

A point charge $+q$ is placed at the origin. A second point charge $+9 q$ is placed at ($\mathrm{d}, 0,0$) in Cartesian coordinate system. The point in between them where the electric field vanishes is:

A slanted object $A B$ is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is :

Considering Bohr's atomic model for hydrogen atom :

(A) the energy of H atom in ground state is same as energy of $\mathrm{He}^{+}$ion in its first excited state.

(B) the energy of H atom in ground state is same as that for $\mathrm{Li}^{++}$ ion in its second excited state.

(C) the energy of H atom in its ground state is same as that of $\mathrm{He}^{+}$ion for its ground state.

(D) the energy of $\mathrm{He}^{+}$ion in its first excited state is same as that for $\mathrm{Li}^{++}$ion in its ground state.

Choose the correct answer from the options given below :

In an adiabatic process, which of the following statements is true?

The equation for real gas is given by $\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^2}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}$, where $\mathrm{P}, \mathrm{V}, \mathrm{T}$ and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $\mathrm{ab}^{-2}$ is equivalent to that of :

The relationship between the magnetic susceptibility $(x)$ and the magnetic permeability $(\mu)$ is given by :

( $\mu_0$ is the permeability of free space and $\mu_T$ is relative permeability)

The battery of a mobile phone is rated as $4.2 \mathrm{~V}, 5800 \mathrm{~mAh}$. How much energy is stored ir it when fully charged?

A zener diode with 5 V zener voltage is used to regulate an unregulated dc voltage input of 25 V . For a $400 \Omega$ resistor connected in series, the zener current is found to be 4 times load current. The load current $\left(I_L\right)$ and load resistance $\left(R_L\right)$ are :

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m , would be:

A small bob of mass 100 mg and charge $+10 \mu \mathrm{C}$ is connected to an insulating string of length 1 m . It is brought near to an infinitely long non-conducting sheet of charge density ' $\sigma$ ' as shown in figure. If string subtends an angle of $45^{\circ}$ with the sheet at equilibrium the charge density of sheet will be.

(Given, $\epsilon_0=8.85 \times 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}$ and acceleration due to gravity, $\mathrm{g}=10 \frac{\mathrm{~m}}{\mathrm{~s}^2}$ )

A steel wire of length 2 m and Young's modulus $2.0 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$ is stretched by a force. If Poisson ratio and transverse strain for the wire are 0.2 and $10^{-3}$ respectively, then the elastic potential energy density of the wire is __________ $\times 10^5$ (in SI units).

$\gamma_{\mathrm{A}}$ is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. $\gamma_B$ is the specific heat ratio of polyatomic gas $B$ having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If $\frac{\gamma_A}{\gamma_B}=\left(1+\frac{1}{n}\right)$, then the value of $n$ is ________ .

A vessel with square cross-section and height of 6 m is vertically partitioned. A small window of $100 \mathrm{~cm}^2$ with hinged door is fitted at a depth of 3 m in the partition wall. One part of the vessel is filled completely with water and the other side is filled with the liquid having density $1.5 \times 10^3 \mathrm{~kg} / \mathrm{m}^3$. What force one needs to apply on the hinged door so that it does not get opened ?

$$\text { (Acceleration due to gravity }=10 \mathrm{~m} / \mathrm{s}^2 \text { ) }$$

If the measured angular separation between the second minimum to the left of the central maximum and the third minimum to the right of the central maximum is $30^{\circ}$ in a single slit diffraction pattern recorded using 628 nm light, then the width of the slit is _______ $\mu$m.

A person travelling on a straight line moves with a uniform velocity $v_1$ for a distance $x$ and with a uniform velocity $v_2$ for the next $\frac{3}{2} x$ distance. The average velocity in this motion is $\frac{50}{7} \mathrm{~m} / \mathrm{s}$. If $v_1$ is $5 \mathrm{~m} / \mathrm{s}$ then $v_2=$ __________ $\mathrm{m} / \mathrm{s}$.