Q JEE Advanced 2021

Let $\overline{\mathrm{u}}, \overline{\mathrm{v}}$ and $\overline{\mathrm{w}}$ be vectors in three-dimensional space, where $\overline{\mathrm{u}}$ and $\overline{\mathrm{v}}$ are unit vectors which are not...

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In a triangle $A B C$, let $A B=\sqrt{23}, B C=3$ and $C A=4$. Then the value of $...

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For $x \in \mathbb{R}$, then number of real roots of the equation $3 x^2-4\left|x^2-1\right|+x-1=0$ is $\_\_\_\_$ .

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For any complex number $\mathrm{w}=\mathrm{c}+\mathrm{id}$, let $\arg (\mathrm{w}) \in(-\pi, \pi]$, where $\mathrm{i}=\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such...

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Q JEE Advanced 2021

For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$. If $...

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Let $E$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$. For any three distinct points $P, Q$ and $Q^{\prime}$ on $E$, let $M(P, Q)$...

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Q [JEE Advanced 2021

For any positive integer n , let $\mathrm{S}_{\mathrm{n}}:(0, \infty) \rightarrow \mathbb{R}$ be defined by $\mathrm{S}_{\mathrm{n}}(\mathrm{x})=\sum_{\mathrm{k}=1}^{\mathrm{n}} \cot ^{-1}\left(\frac{1+\mathrm{k}(\mathrm{k}+1) \mathrm{x}^2}{\mathrm{x}}\right)$, where for...

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Q JEE Advanced 2021

A number is chosen at random from the set {1, 2, 3, ... , 2000}. Let p be the probability...

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Q JEE Advanced 2021

Let $\psi_1:[0, \infty) \rightarrow \mathbb{R}, \psi_2:[0, \infty) \rightarrow \mathbb{R}, \mathrm{f}:[0, \infty) \rightarrow \mathbb{R}$ and $\mathrm{g}:[0, \infty) \rightarrow \mathbb{R}$ be functions...

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Q JEE Advanced 2021

For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $I$ be the $...

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Q JEE Advanced 2021

Let $\psi_1:[0, \infty) \rightarrow \mathbb{R}, \psi_2:[0, \infty) \rightarrow \mathbb{R}, f:[0, \infty) \rightarrow \mathbb{R}$ and $g:[0, \infty) \rightarrow \mathbb{R}$ be functions...

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Q JEE Advanced 2021

Let $E F$ and $G$ be three events having probabilities $P(E)=\frac{1}{8}, P(F)=\frac{1}{6}$ and $P(G)=\frac{1}{4}$, and let $...

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Let $\quad M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^2+y^2 \leq r^2\right\}$, where $\mathrm{r}>0$. Consider the geometric progression $\mathrm{a}_{\mathrm{n}}=\frac{1}{2^{\mathrm{n}-1}}, \mathrm{n}=1,2,3 \ldots$....

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Q JEE Advanced 2021

Let $f: R \rightarrow R$ be defined by $f(x)=\frac{x^2-3 x-6}{x^2+2 x+4}$. Then which of the following statements is (are) JRUE...

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Q JEE Advanced 2021

Let $\quad M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^2+y^2 \leq r^2\right\}$, where $\mathrm{r}>0$. Consider the geometric progression $\mathrm{a}_{\mathrm{n}}=\frac{1}{2^{\mathrm{n}-1}}, \mathrm{n}=1,2,3 \ldots$....

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Q JEE Advanced 2021

Let $\mathrm{g}_{\mathrm{i}}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, \mathrm{i}=1,2$ and $\mathrm{f}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}$ be functions such that $\mathrm{g}_1(\mathrm{x})=1, \mathrm{~g}_2(\mathrm{x})=|4 \mathrm{x}-\pi|$...

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Q JEE Advance2021

Ozonolysis of $\mathrm{ClO}_2$ produces an oxide of chlorine. The average oxidation state of chlorine in this oxide is $\_\_\_\_$ .

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For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $...

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Consider a helium (He) atom that absorbs a photon of wavelength 330 nm . The change in the velocity (in...

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One mole of an ideal gas at 900 K , undergoes two reversible processes, I followed by II, as shown...

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The reaction of $\mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_8\right]$ with freshly prepared $\mathrm{FeSO}_4$ solution produces a dark blue precipitate called Turnbull's blue. Reaction of $\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]$...

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Q JEE Advance 2021

In the reaction given below, the total number of atoms having $\mathrm{sp}^2$ hybridization in the major product P is $\_\_\_\_$...

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The maximum number of possible isomers (including stereoisomers) which may be formed on mono-bromination of 1-methylcyclohex-1-ene using Br2 and UV...

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A small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is...

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The reaction of $\mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_8\right]$ with freshly prepared $\mathrm{FeSO}_4$ solution produces a dark blue precipitate called Turnbull's blue. Reaction of $\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]$...

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The amount of energy required to break a bond is same as the amount of energy released when the same...

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The amount of energy required to break a bond is same as the amount of energy released when the same...

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Q JEE Advanced 2021

Let $\mathrm{g}_{\mathrm{i}}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, \mathrm{i}=1,2$ and $\mathrm{f}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}$ be functions such that $\mathrm{g}_1(\mathrm{x})=1, \mathrm{~g}_2(\mathrm{x})=|4 \mathrm{x}-\pi|$...

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Let $f_1:(0, \infty) \rightarrow \mathbb{R}$ and $f_2:(0, \infty) \rightarrow \mathbb{R}$ be defined by $ f_1(x)=\int_0^x \prod_{j=1}^{21}(t-j)^j d t, x>0 $...

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Q JEE Advanced 2021

Let $f_1:(0, \infty) \rightarrow \mathbb{R}$ and $f_2:(0, \infty) \rightarrow \mathbb{R}$ be defined by $ f_1(x)=\int_0^x \prod_{j=1}^{21}(t-j)^j d t, x>0 $...

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Consider the region $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x} \geq 0\right.$ and $\left.\mathrm{y}^2 \leq 4-\mathrm{x}\right\}$. Let F be the...

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Q JEE Advanced 2021

Consider the region $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x} \geq 0\right.$ and $\left.\mathrm{y}^2 \leq 4-\mathrm{x}\right\}$. Let F be the...

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Q JEE Advance 2021

A thin rod of mass $M$ and length $a$ is free to rotate in horizontal plane about a fixed vertical...

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Q JEE Advance 2021

A mixture of two salts is used to prepare a solution S, which gives the following results:The correct option(s) for...

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A sample $(5.6 \mathrm{~g})$ containing iron is completely dissolved in cold dilute HCI to prepare a 250 mL of solution....

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Let $E$ denote the parabola $y^2=8 x$. Let $P=(-2,4)$, and let $Q$ and $Q^{\prime}$ be two distinct points on $E$...

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A sample $(5.6 \mathrm{~g})$ containing iron is completely dissolved in cold dilute HCl to prepare a 250 mL of solution....

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Q JEE Advanced 2021

Let O be the origin and $\overrightarrow{\mathrm{OA}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{OB}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{OC}}=\frac{1}{2}(\overrightarrow{\mathrm{OB}}-\lambda \overrightarrow{\mathrm{OA}})$ for some $\lambda>0$. If $...

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Q JEE Advance2021

Reaction of x g of Sn with HCl quantitatively produced a salt. Entire amount of the salt reacted with yg...

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Q JEE Advanced 2021

For any real numbers $\alpha$ and $\beta$, let $y_{\alpha, \beta}(x), x \in \mathbb{R}$, be the solution of the differential equation...

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Q JEE Advance 2021

The correct statement(s) related to the metal extraction processes is(are)

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Q JEE Advance2021

Reaction of $x g$ of Sn with HCl quantitatively produced a salt. Entire amount of the salt reacted with y...

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Q JEE Advance 2021

An $\alpha$-particle (mass 4 amu ) and a singly charged sulfur ion (mass 32 amu ) are initially at rest....

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In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals $P$, $Q$ and $R$ are $E_P, E_Q$ and...

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At 298 K , the limiting molar conductivity of a weak monobasic acid is $4 \times 10^2 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$....

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Consider the lines $L_1$ and $L_2$ defined by $L_1: x \sqrt{2}+y-1=0$ and $L_2: x \sqrt{2}-y+1=0$ For a fixed constant $...

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Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=1$ and $\int_0^{\frac{\pi}{3}} f(t) d t=0$. Then which of...

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At 298 K , the limiting molar conductivity of a weak monobasic acid is $4 \times 10^2 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$....

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Consider the lines $L_1$ and $L_2$ defined by $L_1: x \sqrt{2}+y-1=0$ and $L_2: x \sqrt{2}-y+1=0$ For a fixed constant $...

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Q JEE Advance 2021

The distance between two stars of masses 3 Ms and 6 Ms is 9 R . Here R is the...

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An ideal gas undergoes a reversible isothermal expansion from state 1 to state 11 followed by a reversible adiabatic expansion...

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A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with...

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Q JEE Advance2021

The correct statement(s) related to oxoacids of phosphorous is(are)

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Q JEE Advanced 2021

Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q and R,...

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Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations $$ \begin{aligned} & x+2 y+3...

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Q JEE Advance 2021

The correct statement(s) related to colloids is(are)

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In order to measure the internal resistance r 1 of a cell of emf E , a meter bridge of...

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Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations $$ \begin{aligned} & x+2 y+3...

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Let $ \begin{gathered} \mathrm{S}_1=\{(\mathrm{i}, \mathrm{j}, \mathrm{k}): \mathrm{i}, \mathrm{j}, \mathrm{k} \in\{1,2, \ldots, 10\}\} \\ \mathrm{S}_2=\{(\mathrm{i}, \mathrm{j}): 1 \leq \mathrm{i}

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Q JEE Advance 2021

The reaction of Q with PhSNa yields an organic compound (major product) that gives positive Carius test on treatment with...

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Q JEE Advance 2021

A long straight wire carries a current, $\mathrm{I}=2$ ampere. A semi-circular conducting rod is placed beside it on two conducting...

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Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3, \ldots, 100\}$. Let $p_1$ be...

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Some standard electrode potentials at 298 K are given below: To a solution containing 0.001 M of $\mathrm{X}^{2+}$ and 0.1...

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Given: The compound(s), which on reaction with $\mathrm{HNO}_3$ will give the product having degree of rotation, $[\alpha] \mathrm{D}=-52.7^{\circ}$ is(are)

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Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3, \ldots, 100\}$. Let $p_1$ be...

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A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure...

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Which of the following statement(s) is (are) correct about the spectrum of hydrogen atom ?

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The boiling point of water in a 0.1 molal silver nitrate solution (solution A ) is $\mathrm{x}^{\circ} \mathrm{C}$. To this...

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Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+\ldots+\theta_{10}=2 \pi$. Define the complex numbers $...

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Consider three sets $E_1=\{1,2,3\}, F_1=\{1,3,4\}$ and $G_1=\{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $\mathrm{E}_1$, and...

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For the following reaction $$ 2 X+Y \xrightarrow{K} P $$ the rate of reaction is $\frac{d[P]}{d t}=k[X]$. Two moles of...

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A particle of mass $M=0.2 \mathrm{~kg}$ is initially at rest in the $x y$-plane at a point $(x=-1, y=-h)$, where...

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The boiling point of water in a 0.1 molal silver nitrate solution (solution A ) is $\mathrm{x}^{\circ} \mathrm{C}$. To this...

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A special metal S conducts electricity without any resistance. A closed wire loop, made of S , does not allow...

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C orrect option(s) for the following sequence of reactions is(are)

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For the reaction, $\mathbf{X}(s) \rightleftharpoons \mathbf{Y}(s)+\mathbf{Z}(g)$, the plot of $\ln \frac{\mathrm{p}_{\mathrm{z}}}{\mathrm{p}^{\Theta}}$ versus $\frac{10^4}{\mathrm{~T}}$ is given below (in solid line), where...

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A special metal S conducts electricity without any resistance. A closed wire loop, made of S , does not allow...

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The area of the region $...

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The reaction sequence(s) that would lead to o-xylene as the major product is(are)

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Consider a triangle $\Delta$ whose two sides lie on the $x$-axis and the line $x+y+1=0$. If the orthocentre of $\Delta$...

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For the reaction, $\mathbf{X}(s) \rightleftharpoons \mathbf{Y}(s)+\mathbf{Z}(g)$, the plot of $\ln \frac{\mathrm{p}_{\mathrm{z}}}{\mathrm{p}^{\Theta}}$ versus $\frac{10^4}{\mathrm{~T}}$ is given below (in solid line), where...

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A wide slab consisting of two media of refractive indices $n_1$ and $n_2$ is placed in air as shown in...

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In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance $\mathrm{C} \mu \mathrm{F}$ across...

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In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance $\mathrm{C} \mu \mathrm{F}$ across...

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A horizontal force $F$ is applied at the center of mass of a cylindrical object of mass $m$ and radius...

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For the reaction, $\mathbf{X}(s) \rightleftharpoons \mathbf{Y}(s)+\mathbf{Z}(g)$, the plot of $\ln \frac{\mathrm{p}_{\mathrm{z}}}{\mathrm{p}^{\Theta}}$ versus $\frac{10^4}{\mathrm{~T}}$ is given below (in solid line), where...

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A pendulum consists of a bob of mass $\mathrm{m}=0.1 \mathrm{~kg}$ and a massless inextensible string of length $\mathrm{L}=1.0 \mathrm{~m}$. It...

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For the following reaction scheme, percentage yields are given along the arrow:$x g$ and $y g$ are mass of $R$...

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For the following reaction scheme, percentage yields are given along the arrow:$x g$ and $y g$ are mass of $R$...

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The calculated spin only magnetic moments of $\left[\mathrm{Cr}\left(\mathrm{NH}_3\right)_6\right]^{3+}$ and $\left[\mathrm{CuF}_6\right]^{3-}$ in BM , respectively, are (Atomic numbers of Cr and...

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Two point charges $-Q$ and $+Q / \sqrt{3}$ are placed in the $x y$-plane at the origin $(0,0)$ and a...

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Two point charges $-Q$ and $+Q / \sqrt{3}$ are placed in the $x y$-plane at the origin $(0,0)$ and a...

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For the given close packed structure of a salt made of cation $X$ and anion $Y$ shown below (ions of...

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A soft plastic bottle, filled with water of density $1 \mathrm{gm} / \mathrm{cc}$, carries an inverted glass test-tube with some...

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A soft plastic bottle, filled with water of density $1 \mathrm{gm} / \mathrm{cc}$, carries an inverted glass test-tube with some...

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In the circuit shown below, the switch S is connected to position P for a long time so that the...

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In the circuit shown below, the switch S is connected to position P for a long time so that the...

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Two concentric circular loops, one of radius $R$ and the other of radius $2 R$, lie in the $x y$-plane...

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A projectile is thrown from a point O on the ground at an angle $45^{\circ}$ from the vertical and with...

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A projectile is thrown from a point O on the ground at an angle $45^{\circ}$ from the vertical and with...

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Among the following, the conformation that corresponds to the most stable conformation of meso-butane-2,3-diol is

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A heavy nucleus N , at rest, undergoes fission $\mathrm{N} \rightarrow \mathrm{P}+\mathrm{Q}$, where P and Q are two lighter nuclei....

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A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%....

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T he major product formed in the following reaction is

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A physical quantity $\vec{S}$ is defined as $\vec{S}=(\vec{E} \times \vec{B}) / \mu_0$, where $\vec{E}$ is electric field, $\vec{B}$ is magnetic...

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For a prism of prism angle $\theta=60^{\circ}$, the refractive indices of the left half and the right half are, respectively,...

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An extended object is placed at point O, 10 cm in front of a convex lens L1 and a concave...

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An ideal gas undergoes a four step cycle as shown in the P – V diagram below. During this cycle,...

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A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a...

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The smallest division on the main scale of a Vernier calipers is 0.1 cm . Ten divisions of the Vernier...

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One end of a horizontal uniform beam of weight W and length L is hinged on a vertical wall at...

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$\mathrm{H}_2 \mathrm{~S}$ (5 moles) reacts completely with acidified aqueous potassium permanganate solution. In this reaction, the number of moles of...

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