Major product of the following reaction is -

Given below are two statements:

Statement I : Group 13 trivalent halides get easily hydrolyzed by water due to their covalent nature.

Statement II : $\mathrm{AlCl}_3$ upon hydrolysis in acidified aqueous solution forms octahedral $\left[\mathrm{Al}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]^{3+}$ ion.

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

The azo-dye $(Y)$ formed in the following reactions is

Sulphanilic acid $+\mathrm{NaNO}_2+\mathrm{CH}_3 \mathrm{COOH} \rightarrow \mathrm{X}$.

Given below are two statements:

Statement I : $\mathrm{S}_8$ solid undergoes disproportionation reaction under alkaline conditions to form $\mathrm{S}^{2-}$ and $\mathrm{S}_2 \mathrm{O}_3{ }^{2-}$.

Statement II : $\mathrm{ClO}_4^{-}$ can undergo disproportionation reaction under acidic condition.

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

Identify structure of 2,3-dibromo-1-phenylpentane.

The correct order of reactivity in electrophilic substitution reaction of the following compounds is :

Choose the correct statements from the following

A. $\mathrm{Mn}_2 \mathrm{O}_7$ is an oil at room temperature

B. $\mathrm{V}_2 \mathrm{O}_4$ reacts with acid to give $\mathrm{VO}_2{ }^{2+}$

C. $\mathrm{CrO}$ is a basic oxide

D. $\mathrm{V}_2 \mathrm{O}_5$ does not react with acid

Choose the correct answer from the options given below :

Which of the following is least ionic?

$\mathrm{A}_{(\mathrm{g})} \rightleftharpoons \mathrm{B}_{(\mathrm{g})}+\frac{\mathrm{C}}{2}(\mathrm{g})$ The correct relationship between $\mathrm{K}_{\mathrm{P}}, \alpha$ and equilibrium pressure $\mathrm{P}$ is

Identify major product 'P' formed in the following reaction.

Choose the correct statements from the following

A. All group 16 elements form oxides of general formula $\mathrm{EO}_2$ and $\mathrm{EO}_3$, where $\mathrm{E}=\mathrm{S}, \mathrm{Se}, \mathrm{Te}$ and $\mathrm{Po}$. Both the types of oxides are acidic in nature.

B. $\mathrm{TeO}_2$ is an oxidising agent while $\mathrm{SO}_2$ is reducing in nature.

C. The reducing property decreases from $\mathrm{H}_2 \mathrm{~S}$ to $\mathrm{H}_2$ Te down the group.

D. The ozone molecule contains five lone pairs of electrons.

Choose the correct answer from the options given below:

Select the option with correct property -

Let $A$ be a $3 \times 3$ real matrix such that

$$A\left(\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right)=2\left(\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right), A\left(\begin{array}{l} -1 \\ 0 \\ 1 \end{array}\right)=4\left(\begin{array}{l} -1 \\ 0 \\ 1 \end{array}\right), A\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right)=2\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right) \text {. }$$

Then, the system $(A-3 I)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$ has :

The four quantum numbers for the electron in the outer most orbital of potassium (atomic no. 19) are

Let $(\alpha, \beta, \gamma)$ be the mirror image of the point $(2,3,5)$ in the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$. Then, $2 \alpha+3 \beta+4 \gamma$ is equal to

If $a=\sin ^{-1}(\sin (5))$ and $b=\cos ^{-1}(\cos (5))$, then $a^2+b^2$ is equal to

The shortest distance, between lines $L_1$ and $L_2$, where $L_1: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}$ and $L_2$ is the line, passing through the points $\mathrm{A}(-4,4,3), \mathrm{B}(-1,6,3)$ and perpendicular to the line $\frac{x-3}{-2}=\frac{y}{3}=\frac{z-1}{1}$, is

Let $f, g:(0, \infty) \rightarrow \mathbb{R}$ be two functions defined by $f(x)=\int\limits_{-x}^x\left(|t|-t^2\right) e^{-t^2} d t$ and $g(x)=\int\limits_0^{x^2} t^{1 / 2} e^{-t} d t$. Then, the value of $9\left(f\left(\sqrt{\log _e 9}\right)+g\left(\sqrt{\log _e 9}\right)\right)$ is equal to :

Let $2^{\text {nd }}, 8^{\text {th }}$ and $44^{\text {th }}$ terms of a non-constant A. P. be respectively the $1^{\text {st }}, 2^{\text {nd }}$ and $3^{\text {rd }}$ terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to -

Let $z_1$ and $z_2$ be two complex numbers such that $z_1+z_2=5$ and $z_1^3+z_2^3=20+15 i$ Then, $\left|z_1^4+z_2^4\right|$ equals -

\left|\frac{120}{\pi^3} \int_\limits0^\pi \frac{x^2 \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x\right| \text { is equal to } ________.

A block of mass $5 \mathrm{~kg}$ is placed on a rough inclined surface as shown in the figure. If $\overrightarrow{F_1}$ is the force required to just move the block up the inclined plane and $\overrightarrow{F_2}$ is the force required to just prevent the block from sliding down, then the value of $\left|\overrightarrow{F_1}\right|-\left|\overrightarrow{F_2}\right|$ is : [Use $\left.\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\right]$

The speed of sound in oxygen at S.T.P. will be approximately: (given, $R=8.3 \mathrm{~JK}^{-1}, \gamma=1.4$)

The measured value of the length of a simple pendulum is $20 \mathrm{~cm}$ with $2 \mathrm{~mm}$ accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is $\mathrm{N} \%$. The value of $\mathrm{N}$ is:

A body of mass $2 \mathrm{~kg}$ begins to move under the action of a time dependent force given by $\vec{F}=\left(6 t \hat{i}+6 t^2 \hat{j}\right) N$. The power developed by the force at the time $t$ is given by:

The mass of the moon is $\frac{1}{144}$ times the mass of a planet and its diameter is $\frac{1}{16}$ times the diameter of a planet. If the escape velocity on the planet is $v$, the escape velocity on the moon will be :

Two blocks of mass $2 \mathrm{~kg}$ and $4 \mathrm{~kg}$ are connected by a metal wire going over a smooth pulley as shown in figure. The radius of wire is $4.0 \times 10^{-5} \mathrm{~m}$ and Young's modulus of the metal is $2.0 \times 10^{11} \mathrm{~N} / \mathrm{m}^2$. The longitudinal strain developed in the wire is $\frac{1}{\alpha \pi}$. The value of $\alpha$ is _________. [Use $g=10 \mathrm{~m} / \mathrm{s}^2$]

In the following circuit, the battery has an emf of $2 \mathrm{~V}$ and an internal resistance of $\frac{2}{3} \Omega$. The power consumption in the entire circuit is _________ W.

Light from a point source in air falls on a convex curved surface of radius $20 \mathrm{~cm}$ and refractive index 1.5. If the source is located at $100 \mathrm{~cm}$ from the convex surface, the image will be formed at ________ $\mathrm{cm}$ from the object.

Consider the following elements.

Which of the following is/are true about $\mathrm{A}^{\prime}, \mathrm{B}^{\prime}, \mathrm{C}^{\prime}$ and $\mathrm{D}^{\prime}$ ?

A. Order of atomic radii: \mathrm{B}^{\prime}<\mathrm{A}^{\prime}<\mathrm{D}^{\prime}<\mathrm{C}^{\prime}

B. Order of metallic character: \mathrm{B}^{\prime}<\mathrm{A}^{\prime}<\mathrm{D}^{\prime}<\mathrm{C}^{\prime}

C. Size of the element: \mathrm{D}^{\prime}<\mathrm{C}^{\prime}<\mathrm{B}^{\prime}<\mathrm{A}^{\prime}

D. Order of ionic radii: \mathrm{B}^{\prime+}<\mathrm{A}^{1^{+}}<\mathrm{D}^{\prime+}<\mathrm{C}^{+}

Choose the correct answer from the options given below :

Identify A and B in the following reaction sequence.

Given below are two statements :

Statement I : Aniline reacts with con. $\mathrm{H}_2 \mathrm{SO}_4$, followed by heating at $453-473 \mathrm{~K}$ gives $\mathrm{p}$-aminobenzene sulphonic acid, which gives blood red colour in the 'Lassaigne's test'.

Statement II : In Friedel - Craft's alkylation and acylation reactions, aniline forms salt with the $\mathrm{AlCl}_3$ catalyst. Due to this, nitrogen of aniline aquires a positive charge and acts as deactivating group.

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

Let $P$ be a parabola with vertex $(2,3)$ and directrix $2 x+y=6$. Let an ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$, of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then, the square of the length of the latus rectum of $E$, is

The temperature $T(t)$ of a body at time $t=0$ is $160^{\circ} \mathrm{F}$ and it decreases continuously as per the differential equation $\frac{d T}{d t}=-K(T-80)$, where $K$ is a positive constant. If $T(15)=120^{\circ} \mathrm{F}$, then $T(45)$ is equal to

Let $y=y(x)$ be the solution of the differential equation

$\sec ^2 x d x+\left(e^{2 y} \tan ^2 x+\tan x\right) d y=0,0< x<\frac{\pi}{2}, y(\pi / 4)=0$.

If $y(\pi / 6)=\alpha$, then $e^{8 \alpha}$ is equal to ____________.

Let $A(-2,-1), B(1,0), C(\alpha, \beta)$ and $D(\gamma, \delta)$ be the vertices of a parallelogram $A B C D$. If the point $C$ lies on $2 x-y=5$ and the point $D$ lies on $3 x-2 y=6$, then the value of $|\alpha+\beta+\gamma+\delta|$ is equal to ___________.

If \lim _\limits{x \rightarrow 0} \frac{a x^2 e^x-b \log _e(1+x)+c x e^{-x}}{x^2 \sin x}=1, then $16\left(a^2+b^2+c^2\right)$ is equal to ________.

Let A be a $3 \times 3$ matrix and $\operatorname{det}(A)=2$. If $n=\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots . .(\operatorname{adj} A))}_{2024-\text { times }}))$, then the remainder when $n$ is divided by 9 is equal to __________.

Given below are two statements:

Statement I: Electromagnetic waves carry energy as they travel through space and this energy is equally shared by the electric and magnetic fields.

Statement II: When electromagnetic waves strike a surface, a pressure is exerted on the surface.

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

Consider two physical quantities $A$ and $B$ related to each other as $E=\frac{B-x^2}{A t}$ where $E, x$ and $t$ have dimensions of energy, length and time respectively. The dimension of $A B$ is

By what percentage will the illumination of the lamp decrease if the current drops by 20%?

A light string passing over a smooth light fixed pulley connects two blocks of masses $m_1$ and $m_2$. If the acceleration of the system is $g / 8$, then the ratio of masses is:

The resistance per centimeter of a meter bridge wire is $r$, with $X \Omega$ resistance in left gap. Balancing length from left end is at $40 \mathrm{~cm}$ with $25 \Omega$ resistance in right gap. Now the wire is replaced by another wire of $2 r$ resistance per centimeter. The new balancing length for same settings will be at

The output of the given circuit diagram is -

A body of mass '$m$' is projected with a speed '$u$' making an angle of $45^{\circ}$ with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as $\frac{\sqrt{2} m u^3}{X g}$. The value of '$X$' is _________.

Two circular coils $P$ and $Q$ of 100 turns each have same radius of $\pi \mathrm{~cm}$. The currents in $P$ and $R$ are $1 A$ and $2 A$ respectively. $P$ and $Q$ are placed with their planes mutually perpendicular with their centers coincide. The resultant magnetic field induction at the center of the coils is $\sqrt{x} ~m T$, where $x=$ __________.

[Use $\mu_0=4 \pi \times 10^{-7} \mathrm{~TmA}^{-1}$]

A nucleus has mass number $A_1$ and volume $V_1$. Another nucleus has mass number $A_2$ and Volume $V_2$. If relation between mass number is $A_2=4 A_1$, then $\frac{V_2}{V_1}=$ __________.

Identify the name reaction.

The fragrance of flowers is due to the presence of some steam volatile organic compounds called essential oils. These are generally insoluble in water at room temperature but are miscible with water vapour in vapour phase. A suitable method for the extraction of these oils from the flowers is -

Match List I with List II

Choose the correct answer from the options given below:

From the vitamins $\mathrm{A}, \mathrm{B}_1, \mathrm{~B}_6, \mathrm{~B}_{12}, \mathrm{C}, \mathrm{D}, \mathrm{E}$ and $\mathrm{K}$, the number of vitamins that can be stored in our body is _________.

Number of moles of $\mathrm{H}^{+}$ ions required by $1 \mathrm{~mole}$ of $\mathrm{MnO}_4^{-}$ to oxidise oxalate ion to $\mathrm{CO}_2$ is _________.

The molarity of $1 \mathrm{~L}$ orthophosphoric acid $\left(\mathrm{H}_3 \mathrm{PO}_4\right)$ having $70 \%$ purity by weight (specific gravity $1.54 \mathrm{~g} \mathrm{~cm}^{-3}$) is __________ $\mathrm{M}$.

(Molar mass of $\mathrm{H}_3 \mathrm{PO}_4=98 \mathrm{~g} \mathrm{~mol}^{-1}$)

$\mathrm{r}=\mathrm{k}[\mathrm{A}]$ for a reaction, $50 \%$ of $\mathrm{A}$ is decomposed in 120 minutes. The time taken for $90 \%$ decomposition of $\mathrm{A}$ is _________ minutes.

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

If the function $f:(-\infty,-1] \rightarrow(a, b]$ defined by $f(x)=e^{x^3-3 x+1}$ is one - one and onto, then the distance of the point $P(2 b+4, a+2)$ from the line $x+e^{-3} y=4$ is :

If for some $m, n ;{ }^6 C_m+2\left({ }^6 C_{m+1}\right)+{ }^6 C_{m+2}>{ }^8 C_3$ and ${ }^{n-1} P_3:{ }^n P_4=1: 8$, then ${ }^n P_{m+1}+{ }^{\mathrm{n}+1} C_m$ is equal to

Consider the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=e^{-\left|\log _e x\right|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m+n$ is

Let the mean and the variance of 6 observations $a, b, 68,44,48,60$ be $55$ and $194$, respectively. If $a>b$, then $a+3 b$ is

Let $\vec{a}=3 \hat{i}+2 \hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}) \times \vec{c}=2(\vec{a} \times \vec{b})+24 \hat{j}-6 \hat{k}$ and $(\vec{a}-\vec{b}+\hat{i}) \cdot \vec{c}=-3$. Then $|\vec{c}|^2$ is equal to ________.

Let $a, b, c$ be the lengths of three sides of a triangle satistying the condition $\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0$. If the set of all possible values of $x$ is the interval $(\alpha, \beta)$, then $12\left(\alpha^2+\beta^2\right)$ is equal to __________.

Let the coefficient of $x^r$ in the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots \ldots \ldots .+(x+2)^{n-1}$ be $\alpha_r$. If \sum_\limits{r=0}^n \alpha_r=\beta^n-\gamma^n, \beta, \gamma \in \mathbb{N}, then the value of $\beta^2+\gamma^2$ equals _________.

A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$. The point $P(a, b, c)$, where $a, b, c$ are non-negative integers, on the line $A B$ lies at a distance of 21 units, from the point $A$. The distance between the points $P(a, b, c)$ and $Q(4,-12,3)$ is equal to __________.

The mass number of nucleus having radius equal to half of the radius of nucleus with mass number 192 is :

A uniform magnetic field of $2 \times 10^{-3} \mathrm{~T}$ acts along positive $Y$-direction. A rectangular loop of sides $20 \mathrm{~cm}$ and $10 \mathrm{~cm}$ with current of $5 \mathrm{~A}$ is in $Y-Z$ plane. The current is in anticlockwise sense with reference to negative $X$ axis. Magnitude and direction of the torque is:

If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at angle $\theta$, then

Two identical spheres each of mass $2 \mathrm{~kg}$ and radius $50 \mathrm{~cm}$ are fixed at the ends of a light rod so that the separation between the centers is $150 \mathrm{~cm}$. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is $\frac{x}{20} \mathrm{~kg} \mathrm{m^{2 }}$, where the value of $x$ is ___________.

The magnetic flux $\phi$ (in weber) linked with a closed circuit of resistance $8 \Omega$ varies with time (in seconds) as $\phi=5 t^2-36 t+1$. The induced current in the circuit at $t=2 \mathrm{~s}$ is __________ A.

A sample of $\mathrm{CaCO}_3$ and $\mathrm{MgCO}_3$ weighed $2.21 \mathrm{~g}$ is ignited to constant weight of $1.152 \mathrm{~g}$. The composition of mixture is :

(Given molar mass in $\mathrm{g} \mathrm{~mol}^{-1} \mathrm{CaCO}_3: 100, \mathrm{MgCO}_3: 84$)

A diatomic molecule has a dipole moment of $1.2 \mathrm{~D}$. If the bond distance is $1 \mathrm{~A}^{\circ}$, then fractional charge on each atom is _________ $\times 10^{-1}$ esu.

(Given $1 \mathrm{~D}=10^{-18}$ esucm)

Number of isomeric products formed by monochlorination of 2-methylbutane in presence of sunlight is ________.

In the reaction of potassium dichromate, potassium chloride and sulfuric acid (conc.), the oxidation state of the chromium in the product is $(+)$ _________.

A compound $(x)$ with molar mass $108 \mathrm{~g} \mathrm{~mol}^{-1}$ undergoes acetylation to give product with molar mass $192 \mathrm{~g} \mathrm{~mol}^{-1}$. The number of amino groups in the compound $(x)$ is ___________.

If 5 moles of an ideal gas expands from $10 \mathrm{~L}$ to a volume of $100 \mathrm{~L}$ at $300 \mathrm{~K}$ under isothermal and reversible condition then work, $\mathrm{w}$, is $-x \mathrm{~J}$. The value of $x$ is __________.

(Given R = 8.314 J K$^{-1}$ mol$^{-1}$)

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

The number of solutions, of the equation $e^{\sin x}-2 e^{-\sin x}=2$, is :

The area of the region enclosed by the parabolas $y=4 x-x^2$ and $3 y=(x-4)^2$ is equal to :

Let $f: \rightarrow \mathbb{R} \rightarrow(0, \infty)$ be strictly increasing function such that \lim _\limits{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1. Then, the value of \lim _\limits{x \rightarrow \infty}\left[\frac{f(5 x)}{f(x)}-1\right] is equal to

Let a variable line passing through the centre of the circle $x^2+y^2-16 x-4 y=0$, meet the positive co-ordinate axes at the points $A$ and $B$. Then the minimum value of $O A+O B$, where $O$ is the origin, is equal to

Let $A(a, b), B(3,4)$ and $C(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2 a+3,7 b+5)$ from the line $2 x+3 y-4=0$ measured parallel to the line $x-2 y-1=0$ is

Let $A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}$. Let $R$ be a relation on $\mathrm{A}$ defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_1$ be a symmetric relation on $A$ such that $R \subset R_1$ and the number of elements in $R_1$ is $\mathrm{n}$. Then, the minimum value of $\mathrm{n}$ is _________.

In a photoelectric effect experiment a light of frequency 1.5 times the threshold frequency is made to fall on the surface of photosensitive material. Now if the frequency is halved and intensity is doubled, the number of photo electrons emitted will be:

A gas mixture consists of 8 moles of argon and 6 moles of oxygen at temperature T. Neglecting all vibrational modes, the total internal energy of the system is:

A small spherical ball of radius $r$, falling through a viscous medium of negligible density has terminal velocity '$v$'. Another ball of the same mass but of radius $2 r$, falling through the same viscous medium will have terminal velocity: