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chem | JEE Adv 2022 Paper 1 | 90.39 | 37 | Numeric | 2 mol of $\mathrm{Hg}(g)$ is combusted in a fixed volume bomb calorimeter with excess of $\mathrm{O}_{2}$ at $298 \mathrm{~K}$ and 1 atm into $\mathrm{HgO}(s)$. During the reaction, temperature increases from $298.0 \mathrm{~K}$ to $312.8 \mathrm{~K}$. If heat capacity of the bomb calorimeter and enthalpy of formation ... |
chem | JEE Adv 2022 Paper 1 | 0.77 | 38 | Numeric | What is the reduction potential $\left(E^{0}\right.$, in $\left.\mathrm{V}\right)$ of $\mathrm{MnO}_{4}^{-}(\mathrm{aq}) / \mathrm{Mn}(\mathrm{s})$?
[Given: $\left.E_{\left(\mathrm{MnO}_{4}^{-}(\mathrm{aq}) / \mathrm{MnO}_{2}(\mathrm{~s})\right)}^{0}=1.68 \mathrm{~V} ; E_{\left(\mathrm{MnO}_{2}(\mathrm{~s}) / \mathrm{... |
chem | JEE Adv 2022 Paper 1 | 10.02 | 39 | Numeric | A solution is prepared by mixing $0.01 \mathrm{~mol}$ each of $\mathrm{H}_{2} \mathrm{CO}_{3}, \mathrm{NaHCO}_{3}, \mathrm{Na}_{2} \mathrm{CO}_{3}$, and $\mathrm{NaOH}$ in $100 \mathrm{~mL}$ of water. What is the $p \mathrm{H}$ of the resulting solution?
[Given: $p \mathrm{~K}_{\mathrm{a} 1}$ and $p \mathrm{~K}_{\math... |
chem | JEE Adv 2022 Paper 1 | 0.32 | 40 | Numeric | The treatment of an aqueous solution of $3.74 \mathrm{~g}$ of $\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}$ with excess KI results in a brown solution along with the formation of a precipitate. Passing $\mathrm{H}_{2} \mathrm{~S}$ through this brown solution gives another precipitate $\mathbf{X}$. What is the amount of... |
chem | JEE Adv 2022 Paper 1 | 2.38 | 41 | Numeric | Dissolving $1.24 \mathrm{~g}$ of white phosphorous in boiling NaOH solution in an inert atmosphere gives a gas $\mathbf{Q}$. What is the amount of $\mathrm{CuSO}_{4}$ (in g) required to completely consume the gas $\mathbf{Q}$?
[Given: Atomic mass of $\mathrm{H}=1, \mathrm{O}=16, \mathrm{Na}=23, \mathrm{P}=31, \mathrm{... |
chem | JEE Adv 2022 Paper 1 | AD | 45 | MCQ(multiple) | For diatomic molecules, the correct statement(s) about the molecular orbitals formed by the overlap of two $2 p_{z}$ orbitals is(are)
(A) $\sigma$ orbital has a total of two nodal planes.
(B) $\sigma^{*}$ orbital has one node in the $x z$-plane containing the molecular axis.
(C) $\pi$ orbital has one node in the pla... |
chem | JEE Adv 2022 Paper 1 | AD | 46 | MCQ(multiple) | The correct option(s) related to adsorption processes is(are)
(A) Chemisorption results in a unimolecular layer.
(B) The enthalpy change during physisorption is in the range of 100 to $140 \mathrm{~kJ} \mathrm{~mol}^{-1}$.
(C) Chemisorption is an endothermic process.
(D) Lowering the temperature favors physisorptio... |
chem | JEE Adv 2022 Paper 1 | BCD | 47 | MCQ(multiple) | The electrochemical extraction of aluminum from bauxite ore involves
(A) the reaction of $\mathrm{Al}_{2} \mathrm{O}_{3}$ with coke (C) at a temperature $>2500^{\circ} \mathrm{C}$.
(B) the neutralization of aluminate solution by passing $\mathrm{CO}_{2}$ gas to precipitate hydrated alumina $\left(\mathrm{Al}_{2} \mat... |
chem | JEE Adv 2022 Paper 1 | AD | 48 | MCQ(multiple) | The treatment of galena with $\mathrm{HNO}_{3}$ produces a gas that is
(A) paramagnetic
(B) bent in geometry
(C) an acidic oxide
(D) colorless |
chem | JEE Adv 2022 Paper 1 | D | 52 | MCQ | LIST-I contains compounds and LIST-II contains reactions
LIST-I
(I) $\mathrm{H}_2 \mathrm{O}_2$
(II) $\mathrm{Mg}(\mathrm{OH})_2$
(III) $\mathrm{BaCl}_2$
(IV) $\mathrm{CaCO}_3$
LIST-II
(P) $\mathrm{Mg}\left(\mathrm{HCO}_{3}\right)_{2}+\mathrm{Ca}(\mathrm{OH})_{2} \rightarrow$
(Q) $\mathrm{BaO}_{2}+\mathrm{H}_{2} \ma... |
chem | JEE Adv 2022 Paper 1 | A | 53 | MCQ | LIST-I contains metal species and LIST-II contains their properties.
LIST-I
(I) $\left[\mathrm{Cr}(\mathrm{CN})_{6}\right]^{4-}$
(II) $\left[\mathrm{RuCl}_{6}\right]^{2-}$
(III) $\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$
(IV) $\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\r... |
chem | JEE Adv 2022 Paper 1 | D | 54 | MCQ | Match the compounds in LIST-I with the observations in LIST-II, and choose the correct option.
LIST-I
(I) Aniline
(II) $o$-Cresol
(III) Cysteine
(IV) Caprolactam
LIST-II
(P) Sodium fusion extract of the compound on boiling with $\mathrm{FeSO}_{4}$, followed by acidification with conc. $\mathrm{H}_{2} \mathrm{SO}_{4... |
math | JEE Adv 2022 Paper 2 | 1 | 1 | Integer | Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4}<\beta<0<\alpha<\frac{\pi}{4}$. If $\sin (\alpha+\beta)=\frac{1}{3}$ and $\cos (\alpha-\beta)=\frac{2}{3}$, then what is the greatest integer less than or equal to
\[
\left(\frac{\sin \alpha}{\cos \beta}+\frac{\cos \beta}{\sin \alpha}+\frac{\cos \alpha}... |
math | JEE Adv 2022 Paper 2 | 8 | 2 | Integer | If $y(x)$ is the solution of the differential equation
\[
x d y-\left(y^{2}-4 y\right) d x=0 \text { for } x>0, \quad y(1)=2,
\]
and the slope of the curve $y=y(x)$ is never zero, then what is the value of $10 y(\sqrt{2})$? |
math | JEE Adv 2022 Paper 2 | 5 | 3 | Numeric | What is the greatest integer less than or equal to
\[
\int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}} d x
\]? |
math | JEE Adv 2022 Paper 2 | 1 | 4 | Integer | What is the product of all positive real values of $x$ satisfying the equation
\[
x^{\left(16\left(\log _{5} x\right)^{3}-68 \log _{5} x\right)}=5^{-16}
\]? |
math | JEE Adv 2022 Paper 2 | 5 | 5 | Integer | If
\[
\beta=\lim _{x \rightarrow 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin ^{2} x},
\]
then what is the value of $6 \beta$? |
math | JEE Adv 2022 Paper 2 | 3 | 6 | Integer | Let $\beta$ be a real number. Consider the matrix
\[
A=\left(\begin{array}{ccc}
\beta & 0 & 1 \\
2 & 1 & -2 \\
3 & 1 & -2
\end{array}\right)
\]
If $A^{7}-(\beta-1) A^{6}-\beta A^{5}$ is a singular matrix, then what is the value of $9 \beta$? |
math | JEE Adv 2022 Paper 2 | 7 | 7 | Integer | Consider the hyperbola
\[
\frac{x^{2}}{100}-\frac{y^{2}}{64}=1
\]
with foci at $S$ and $S_{1}$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle S P S_{1}=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having th... |
math | JEE Adv 2022 Paper 2 | 6 | 8 | Integer | Consider the functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ defined by
\[
f(x)=x^{2}+\frac{5}{12} \quad \text { and } \quad g(x)= \begin{cases}2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4} \\ 0, & |x|>\frac{3}{4}\end{cases}
\]
If $\alpha$ is the area of the region
\[
\left\{(x, y) \in \mathbb{R} \times... |
math | JEE Adv 2022 Paper 2 | AB | 9 | MCQ(multiple) | Let $P Q R S$ be a quadrilateral in a plane, where $Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}$ and $\angle P R S=40^{\circ}$. If $\angle R P S=\theta^{\circ}, P Q=\alpha$ and $P S=\beta$, then the interval(s) that contain(s) the value of $4 \alpha \beta \sin \theta^{\circ}$ is/are
(A) $(0, \... |
math | JEE Adv 2022 Paper 2 | ABC | 10 | MCQ(multiple) | Let
\[
\alpha=\sum_{k=1}^{\infty} \sin ^{2 k}\left(\frac{\pi}{6}\right)
\]
Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by
\[
g(x)=2^{\alpha x}+2^{\alpha(1-x)}
\]
Then, which of the following statements is/are TRUE ?
(A) The minimum value of $g(x)$ is $2^{\frac{7}{6}}$
(B) The maximum value of $g(... |
math | JEE Adv 2022 Paper 2 | A | 11 | MCQ(multiple) | Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
\[
(\bar{z})^{2}+\frac{1}{z^{2}}
\]
are integers, then which of the following is/are possible value(s) of $|z|$ ?
(A) $\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}... |
math | JEE Adv 2022 Paper 2 | CD | 12 | MCQ(multiple) | Let $G$ be a circle of radius $R>0$. Let $G_{1}, G_{2}, \ldots, G_{n}$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_{1}, G_{2}, \ldots, G_{n}$ touches the circle $G$ externally. Also, for $i=1,2, \ldots, n-1$, the circle $G_{i}$ touches $G_{i+1}$ externally, and $G_{n}$ touches $G_{1}$ exter... |
math | JEE Adv 2022 Paper 2 | BCD | 13 | MCQ(multiple) | Let $\hat{i}, \hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let
\[
\vec{a}=3 \hat{i}+\hat{j}-\hat{k},
\]
\[\vec{b}=\hat{i}+b_{2} \hat{j}+b_{3} \hat{k}, \quad b_{2}, b_{3} \in \mathbb{R},
\]
\[\vec{c}=c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}, \quad c_{1}, c_{2}, c_{3} \in \mathbb... |
math | JEE Adv 2022 Paper 2 | C | 14 | MCQ(multiple) | For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation
\[
\frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), \quad y(0)=0
\]
Then, which of the following statements is/are TRUE?
(A) $y(x)$ is an increasing function
(B) $y(x)$ is a decreasing function
(C) There exists a r... |
math | JEE Adv 2022 Paper 2 | A | 15 | MCQ | Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?
(A) 21816
(B) 85536
(C) 12096
(D) 156816 |
math | JEE Adv 2022 Paper 2 | A | 16 | MCQ | If $M=\left(\begin{array}{rr}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right)$, then which of the following matrices is equal to $M^{2022}$ ?
(A) $\left(\begin{array}{rr}3034 & 3033 \\ -3033 & -3032\end{array}\right)$
(B) $\left(\begin{array}{ll}3034 & -3033 \\ 3033 & -3032\end{array}\right)... |
math | JEE Adv 2022 Paper 2 | C | 17 | MCQ | Suppose that
Box-I contains 8 red, 3 blue and 5 green balls,
Box-II contains 24 red, 9 blue and 15 green balls,
Box-III contains 1 blue, 12 green and 3 yellow balls,
Box-IV contains 10 green, 16 orange and 6 white balls.
A ball is chosen randomly from Box-I; call this ball $b$. If $b$ is red then a ball is chosen ... |
math | JEE Adv 2022 Paper 2 | B | 18 | MCQ | For positive integer $n$, define
\[
f(n)=n+\frac{16+5 n-3 n^{2}}{4 n+3 n^{2}}+\frac{32+n-3 n^{2}}{8 n+3 n^{2}}+\frac{48-3 n-3 n^{2}}{12 n+3 n^{2}}+\cdots+\frac{25 n-7 n^{2}}{7 n^{2}}
\]
Then, the value of $\lim _{n \rightarrow \infty} f(n)$ is equal to
(A) $3+\frac{4}{3} \log _{e} 7$
(B) $4-\frac{3}{4} \log _{e}\le... |
phy | JEE Adv 2022 Paper 2 | 3 | 19 | Integer | A particle of mass $1 \mathrm{~kg}$ is subjected to a force which depends on the position as $\vec{F}=$ $-k(x \hat{i}+y \hat{j}) k g \mathrm{ks}^{-2}$ with $k=1 \mathrm{~kg} \mathrm{~s}^{-2}$. At time $t=0$, the particle's position $\vec{r}=$ $\left(\frac{1}{\sqrt{2}} \hat{i}+\sqrt{2} \hat{j}\right) m$ and its velocity... |
phy | JEE Adv 2022 Paper 2 | 2 | 20 | Integer | In a radioactive decay chain reaction, ${ }_{90}^{230} \mathrm{Th}$ nucleus decays into ${ }_{84}^{214} \mathrm{Po}$ nucleus. What is the ratio of the number of $\alpha$ to number of $\beta^{-}$particles emitted in this process? |
phy | JEE Adv 2022 Paper 2 | 4 | 22 | Integer | In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_{e}$, Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4 \pi \epsilon_{0}}$, where $\epsilon_{0}$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the... |
phy | JEE Adv 2022 Paper 2 | 6 | 25 | Integer | On a frictionless horizontal plane, a bob of mass $m=0.1 \mathrm{~kg}$ is attached to a spring with natural length $l_{0}=0.1 \mathrm{~m}$. The spring constant is $k_{1}=0.009 \mathrm{Nm}^{-1}$ when the length of the spring $l>l_{0}$ and is $k_{2}=0.016 \mathrm{Nm}^{-1}$ when $l<l_{0}$. Initially the bob is released fr... |
phy | JEE Adv 2022 Paper 2 | CD | 29 | MCQ(multiple) | A bubble has surface tension $S$. The ideal gas inside the bubble has ratio of specific heats $\gamma=$ $\frac{5}{3}$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a 1}$, the radius of the bubble is found to be $r_{1}$ and the temperature of the... |
phy | JEE Adv 2022 Paper 2 | ACD | 30 | MCQ(multiple) | A disk of radius $\mathrm{R}$ with uniform positive charge density $\sigma$ is placed on the $x y$ plane with its center at the origin. The Coulomb potential along the $z$-axis is
\[
V(z)=\frac{\sigma}{2 \epsilon_{0}}\left(\sqrt{R^{2}+z^{2}}-z\right)
\]
A particle of positive charge $q$ is placed initially at rest at... |
phy | JEE Adv 2022 Paper 2 | A | 34 | MCQ | When light of a given wavelength is incident on a metallic surface, the minimum potential needed to stop the emitted photoelectrons is $6.0 \mathrm{~V}$. This potential drops to $0.6 \mathrm{~V}$ if another source with wavelength four times that of the first one and intensity half of the first one is used. What are the... |
phy | JEE Adv 2022 Paper 2 | C | 35 | MCQ | Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is $0.5 \mathrm{~mm}$. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.
\begin{center}
\begin{tabular}{|... |
chem | JEE Adv 2022 Paper 2 | 6 | 37 | Integer | Concentration of $\mathrm{H}_{2} \mathrm{SO}_{4}$ and $\mathrm{Na}_{2} \mathrm{SO}_{4}$ in a solution is $1 \mathrm{M}$ and $1.8 \times 10^{-2} \mathrm{M}$, respectively. Molar solubility of $\mathrm{PbSO}_{4}$ in the same solution is $\mathrm{X} \times 10^{-\mathrm{Y}} \mathrm{M}$ (expressed in scientific notation). W... |
chem | JEE Adv 2022 Paper 2 | 5 | 38 | Integer | An aqueous solution is prepared by dissolving $0.1 \mathrm{~mol}$ of an ionic salt in $1.8 \mathrm{~kg}$ of water at $35^{\circ} \mathrm{C}$. The salt remains $90 \%$ dissociated in the solution. The vapour pressure of the solution is $59.724 \mathrm{~mm}$ of $\mathrm{Hg}$. Vapor pressure of water at $35{ }^{\circ} \ma... |
chem | JEE Adv 2022 Paper 2 | 4 | 40 | Integer | The reaction of $\mathrm{Xe}$ and $\mathrm{O}_{2} \mathrm{~F}_{2}$ gives a Xe compound $\mathbf{P}$. What is the number of moles of $\mathrm{HF}$ produced by the complete hydrolysis of $1 \mathrm{~mol}$ of $\mathbf{P}$? |
chem | JEE Adv 2022 Paper 2 | 6 | 41 | Integer | Thermal decomposition of $\mathrm{AgNO}_{3}$ produces two paramagnetic gases. What is the total number of electrons present in the antibonding molecular orbitals of the gas that has the higher number of unpaired electrons? |
chem | JEE Adv 2022 Paper 2 | BC | 45 | MCQ(multiple) | To check the principle of multiple proportions, a series of pure binary compounds $\left(\mathrm{P}_{\mathrm{m}} \mathrm{Q}_{\mathrm{n}}\right)$ were analyzed and their composition is tabulated below. The correct option(s) is(are)
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Compound & Weight \% of P & Weight \% ... |
chem | JEE Adv 2022 Paper 2 | BCD | 46 | MCQ(multiple) | The correct option(s) about entropy (S) is(are)
$[\mathrm{R}=$ gas constant, $\mathrm{F}=$ Faraday constant, $\mathrm{T}=$ Temperature $]$
(A) For the reaction, $\mathrm{M}(s)+2 \mathrm{H}^{+}(a q) \rightarrow \mathrm{H}_{2}(g)+\mathrm{M}^{2+}(a q)$, if $\frac{d E_{c e l l}}{d T}=\frac{R}{F}$, then the entropy change... |
chem | JEE Adv 2022 Paper 2 | ABC | 47 | MCQ(multiple) | The compound(s) which react(s) with $\mathrm{NH}_{3}$ to give boron nitride (BN) is(are)
(A) $\mathrm{B}$
(B) $\mathrm{B}_{2} \mathrm{H}_{6}$
(C) $\mathrm{B}_{2} \mathrm{O}_{3}$
(D) $\mathrm{HBF}_{4}$ |
chem | JEE Adv 2022 Paper 2 | ABC | 48 | MCQ(multiple) | The correct option(s) related to the extraction of iron from its ore in the blast furnace operating in the temperature range $900-1500 \mathrm{~K}$ is(are)
(A) Limestone is used to remove silicate impurity.
(B) Pig iron obtained from blast furnace contains about $4 \%$ carbon.
(C) Coke (C) converts $\mathrm{CO}_{2}$... |
chem | JEE Adv 2022 Paper 2 | BC | 50 | MCQ(multiple) | Among the following, the correct statement(s) about polymers is(are)
(A) The polymerization of chloroprene gives natural rubber.
(B) Teflon is prepared from tetrafluoroethene by heating it with persulphate catalyst at high pressures.
(C) PVC are thermoplastic polymers.
(D) Ethene at 350-570 K temperature and 1000-2... |
chem | JEE Adv 2022 Paper 2 | B | 51 | MCQ | Atom $\mathrm{X}$ occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in \%) of the resultant solid is closest to
(A) 25
(B) 35
(C) 55
(D) 75 |
chem | JEE Adv 2022 Paper 2 | C | 52 | MCQ | The reaction of $\mathrm{HClO}_{3}$ with $\mathrm{HCl}$ gives a paramagnetic gas, which upon reaction with $\mathrm{O}_{3}$ produces
(A) $\mathrm{Cl}_{2} \mathrm{O}$
(B) $\mathrm{ClO}_{2}$
(C) $\mathrm{Cl}_{2} \mathrm{O}_{6}$
(D) $\mathrm{Cl}_{2} \mathrm{O}_{7}$ |
chem | JEE Adv 2022 Paper 2 | C | 53 | MCQ | The reaction of $\mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}$ and $\mathrm{NaCl}$ in water produces a precipitate that dissolves upon the addition of $\mathrm{HCl}$ of appropriate concentration. The dissolution of the precipitate is due to the formation of
(A) $\mathrm{PbCl}_{2}$
(B) $\mathrm{PbCl}_{4}$
(C) $\left[\... |
math | JEE Adv 2023 Paper 1 | ACD | 1 | MCQ(multiple) | Let $S=(0,1) \cup(1,2) \cup(3,4)$ and $T=\{0,1,2,3\}$. Then which of the following statements is(are) true?
(A) There are infinitely many functions from $S$ to $T$
(B) There are infinitely many strictly increasing functions from $S$ to $T$
(C) The number of continuous functions from $S$ to $T$ is at most 12... |
math | JEE Adv 2023 Paper 1 | AC | 2 | MCQ(multiple) | Let $T_{1}$ and $T_{2}$ be two distinct common tangents to the ellipse $E: \frac{x^{2}}{6}+\frac{y^{2}}{3}=1$ and the parabola $P: y^{2}=12 x$. Suppose that the tangent $T_{1}$ touches $P$ and $E$ at the points $A_{1}$ and $A_{2}$, respectively and the tangent $T_{2}$ touches $P$ and $E$ at the points $A_{4}$ and $A_{3... |
math | JEE Adv 2023 Paper 1 | BCD | 3 | MCQ(multiple) | Let $f:[0,1] \rightarrow[0,1]$ be the function defined by $f(x)=\frac{x^{3}}{3}-x^{2}+\frac{5}{9} x+\frac{17}{36}$. Consider the square region $S=[0,1] \times[0,1]$. Let $G=\{(x, y) \in S: y>f(x)\}$ be called the green region and $R=\{(x, y) \in S: y<f(x)\}$ be called the red region. Let $L_{h}=\{(x, h) \in S: x \in[0,... |
math | JEE Adv 2023 Paper 1 | C | 4 | MCQ | Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=\sqrt{n}$ if $x \in\left[\frac{1}{n+1}, \frac{1}{n}\right)$ where $n \in \mathbb{N}$. Let $g:(0,1) \rightarrow \mathbb{R}$ be a function such that $\int_{x^{2}}^{x} \sqrt{\frac{1-t}{t}} d t<g(x)<2 \sqrt{x}$ for all $x \in(0,1)$. Then $\lim _{x \right... |
math | JEE Adv 2023 Paper 1 | A | 5 | MCQ | Let $Q$ be the cube with the set of vertices $\left\{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{1}, x_{2}, x_{3} \in\{0,1\}\right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the c... |
math | JEE Adv 2023 Paper 1 | B | 6 | MCQ | Let $X=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: \frac{x^{2}}{8}+\frac{y^{2}}{20}<1\right.$ and $\left.y^{2}<5 x\right\}$. Three distinct points $P, Q$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q$ and $R$ form a triangle whose area is a positive integer, is
(A) $\frac{71}{220}$
(B) $... |
math | JEE Adv 2023 Paper 1 | A | 7 | MCQ | Let $P$ be a point on the parabola $y^{2}=4 a x$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $P F Q$, where $F$ is the focus of the parabola, is 120 . If the slope $m$ of the normal and $a$ are both positive integers, then the pair $(a, m)$ is
(A) $(2,3)$... |
math | JEE Adv 2023 Paper 1 | 3 | 8 | Integer | Let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, for $x \in \mathbb{R}$. Then what is the number of real solutions of the equation
$\sqrt{1+\cos (2 x)}=\sqrt{2} \tan ^{-1}(\tan x)$ in the set $\left(-\frac{3 \pi}{2},-\frac{\pi}{2}\right) \cup\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \cup\left(\fra... |
math | JEE Adv 2023 Paper 1 | 8 | 9 | Integer | Let $n \geq 2$ be a natural number and $f:[0,1] \rightarrow \mathbb{R}$ be the function defined by
$$
f(x)= \begin{cases}n(1-2 n x) & \text { if } 0 \leq x \leq \frac{1}{2 n} \\ 2 n(2 n x-1) & \text { if } \frac{1}{2 n} \leq x \leq \frac{3}{4 n} \\ 4 n(1-n x) & \text { if } \frac{3}{4 n} \leq x \leq \frac{1}{n} \\ ... |
math | JEE Adv 2023 Paper 1 | 1219 | 10 | Integer | Let $7 \overbrace{5 \cdots 5}^{r} 7$ denote the $(r+2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5 . Consider the sum $S=77+757+7557+\cdots+7 \overbrace{5 \cdots 5}^{98}7$. If $S=\frac{7 \overbrace{5 \cdots 5}^{99}7+m}{n}$, where $m$ and $n$ are natural numbers less than 3... |
math | JEE Adv 2023 Paper 1 | 281 | 11 | Integer | Let $A=\left\{\frac{1967+1686 i \sin \theta}{7-3 i \cos \theta}: \theta \in \mathbb{R}\right\}$. If $A$ contains exactly one positive integer $n$, then what is the value of $n$? |
math | JEE Adv 2023 Paper 1 | 45 | 12 | Integer | Let $P$ be the plane $\sqrt{3} x+2 y+3 z=16$ and let $S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: \alpha^{2}+\beta^{2}+\gamma^{2}=1\right.$ and the distance of $(\alpha, \beta, \gamma)$ from the plane $P$ is $\left.\frac{7}{2}\right\}$. Let $\vec{u}, \vec{v}$ and $\vec{w}$ be three distinct vectors in $S$ suc... |
math | JEE Adv 2023 Paper 1 | 3 | 13 | Integer | Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^{5}$ in the expansion of $\left(a x^{2}+\frac{70}{27 b x}\right)^{4}$ is equal to the coefficient of $x^{-5}$ in the expansion of $\left(a x-\frac{1}{b x^{2}}\right)^{7}$, then the value of $2 b$ is |
math | JEE Adv 2023 Paper 1 | A | 14 | MCQ | Let $\alpha, \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations
$x+2 y+z=7$
$x+\alpha z=11$
$2 x-3 y+\beta z=\gamma$
Match each entry in List-I to the correct entries in List-II.
\textbf{List-I}
(P) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$, then the syste... |
math | JEE Adv 2023 Paper 1 | A | 15 | MCQ | Consider the given data with frequency distribution
$$
\begin{array}{ccccccc}
x_{i} & 3 & 8 & 11 & 10 & 5 & 4 \\
f_{i} & 5 & 2 & 3 & 2 & 4 & 4
\end{array}
$$
Match each entry in List-I to the correct entries in List-II.
\textbf{List-I}
(P) The mean of the above data is
(Q) The median of the above da... |
math | JEE Adv 2023 Paper 1 | B | 16 | MCQ | Let $\ell_{1}$ and $\ell_{2}$ be the lines $\vec{r}_{1}=\lambda(\hat{i}+\hat{j}+\hat{k})$ and $\vec{r}_{2}=(\hat{j}-\hat{k})+\mu(\hat{i}+\hat{k})$, respectively. Let $X$ be the set of all the planes $H$ that contain the line $\ell_{1}$. For a plane $H$, let $d(H)$ denote the smallest possible distance between the point... |
math | JEE Adv 2023 Paper 1 | B | 17 | MCQ | Let $z$ be a complex number satisfying $|z|^{3}+2 z^{2}+4 \bar{z}-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-I to the correct entries in List-II.
\textbf{List-I}
(P) $|z|^{2}$ is equal to
(Q) $|Z-\bar{Z}|^{2}$ is equal to
... |
phy | JEE Adv 2023 Paper 1 | A | 21 | MCQ | A bar of mass $M=1.00 \mathrm{~kg}$ and length $L=0.20 \mathrm{~m}$ is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m=0.10 \mathrm{~kg}$ is moving on the same horizontal surface with $5.00 \mathrm{~m} \mathrm{~s}^{-1}$ speed on a pa... |
phy | JEE Adv 2023 Paper 1 | A | 23 | MCQ | One mole of an ideal gas expands adiabatically from an initial state $\left(T_{\mathrm{A}}, V_{0}\right)$ to final state $\left(T_{\mathrm{f}}, 5 V_{0}\right)$. Another mole of the same gas expands isothermally from a different initial state $\left(T_{\mathrm{B}}, V_{0}\right)$ to the same final state $\left(T_{\mathrm... |
phy | JEE Adv 2023 Paper 1 | A | 24 | MCQ | Two satellites $\mathrm{P}$ and $\mathrm{Q}$ are moving in different circular orbits around the Earth (radius $R$ ). The heights of $\mathrm{P}$ and $\mathrm{Q}$ from the Earth surface are $h_{\mathrm{P}}$ and $h_{\mathrm{Q}}$, respectively, where $h_{\mathrm{P}}=R / 3$. The accelerations of $\mathrm{P}$ and $\mathrm{Q... |
phy | JEE Adv 2023 Paper 1 | 3 | 25 | Integer | A Hydrogen-like atom has atomic number $Z$. Photons emitted in the electronic transitions from level $n=4$ to level $n=3$ in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is $1.95 \mathrm{eV}$. If the photoelectric threshold... |
phy | JEE Adv 2023 Paper 1 | 1 | 27 | Integer | In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10 \pm 0.1 \mathrm{~cm}$ and the distance of its real image from the lens is $20 \pm 0.2 \mathrm{~cm}$. The error in the determination of focal length of the lens is $n \%$. What is the value of $n... |
phy | JEE Adv 2023 Paper 1 | 121 | 28 | Integer | A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas $(\gamma=5 / 3)$ and one mole of an ideal diatomic gas $(\gamma=7 / 5)$. Here, $\gamma$ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of 66 Joule when heate... |
phy | JEE Adv 2023 Paper 1 | 40 | 29 | Integer | A person of height $1.6 \mathrm{~m}$ is walking away from a lamp post of height $4 \mathrm{~m}$ along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is $60 \mathrm{~cm} \mathrm{~s}^{-1}$, then what is the speed of the tip of the person... |
phy | JEE Adv 2023 Paper 1 | A | 31 | MCQ | List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option.
\textbf{List-I}
(P) ${ }_{92}^{238} U \rightarrow{ }_{91}^{234} \mathrm{~Pa}$
(Q) ${ }_{82}^{214} \mathrm{~Pb} \r... |
phy | JEE Adv 2023 Paper 1 | C | 32 | MCQ | Match the temperature of a black body given in List-I with an appropriate statement in List-II, and choose the correct option.
[Given: Wien's constant as $2.9 \times 10^{-3} \mathrm{~m}-\mathrm{K}$ and $\frac{h c}{e}=1.24 \times 10^{-6} \mathrm{~V}-\mathrm{m}$ ]
\textbf{List-I}
(P) $2000 \mathrm{~K}$
(Q) $3... |
phy | JEE Adv 2023 Paper 1 | B | 33 | MCQ | A series LCR circuit is connected to a $45 \sin (\omega t)$ Volt source. The resonant angular frequency of the circuit is $10^{5} \mathrm{rad} \mathrm{s}^{-1}$ and current amplitude at resonance is $I_{0}$. When the angular frequency of the source is $\omega=8 \times 10^{4} \mathrm{rad} \mathrm{s}^{-1}$, the current am... |
chem | JEE Adv 2023 Paper 1 | BCD | 35 | MCQ(multiple) | The correct statement(s) related to processes involved in the extraction of metals is(are)
(A) Roasting of Malachite produces Cuprite.
(B) Calcination of Calamine produces Zincite.
(C) Copper pyrites is heated with silica in a reverberatory furnace to remove iron.
(D) Impure silver is treated with aqueous K... |
chem | JEE Adv 2023 Paper 1 | A | 39 | MCQ | Plotting $1 / \Lambda_{\mathrm{m}}$ against $\mathrm{c} \Lambda_{\mathrm{m}}$ for aqueous solutions of a monobasic weak acid (HX) resulted in a straight line with y-axis intercept of $\mathrm{P}$ and slope of $\mathrm{S}$. The ratio $\mathrm{P} / \mathrm{S}$ is
$\left[\Lambda_{\mathrm{m}}=\right.$ molar conductivity... |
chem | JEE Adv 2023 Paper 1 | B | 40 | MCQ | On decreasing the $p \mathrm{H}$ from 7 to 2, the solubility of a sparingly soluble salt (MX) of a weak acid (HX) increased from $10^{-4} \mathrm{~mol} \mathrm{~L}^{-1}$ to $10^{-3} \mathrm{~mol} \mathrm{~L}^{-1}$. The $p\mathrm{~K}_{\mathrm{a}}$ of $\mathrm{HX}$ is
(A) 3
(B) 4
(C) 5
(D) 2 |
chem | JEE Adv 2023 Paper 1 | 222 | 42 | Integer | The stoichiometric reaction of $516 \mathrm{~g}$ of dimethyldichlorosilane with water results in a tetrameric cyclic product $\mathbf{X}$ in $75 \%$ yield. What is the weight (in g) obtained of $\mathbf{X}$?
[Use, molar mass $\left(\mathrm{g} \mathrm{mol}^{-1}\right): \mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mat... |
chem | JEE Adv 2023 Paper 1 | 100 | 43 | Integer | A gas has a compressibility factor of 0.5 and a molar volume of $0.4 \mathrm{dm}^{3} \mathrm{~mol}^{-1}$ at a temperature of $800 \mathrm{~K}$ and pressure $\mathbf{x}$ atm. If it shows ideal gas behaviour at the same temperature and pressure, the molar volume will be $\mathbf{y} \mathrm{dm}^{3} \mathrm{~mol}^{-1}$. Wh... |
chem | JEE Adv 2023 Paper 1 | D | 48 | MCQ | Match the reactions (in the given stoichiometry of the reactants) in List-I with one of their products given in List-II and choose the correct option.
\textbf{List-I}
(P) $\mathrm{P}_{2} \mathrm{O}_{3}+3 \mathrm{H}_{2} \mathrm{O} \rightarrow$
(Q) $\mathrm{P}_{4}+3 \mathrm{NaOH}+3 \mathrm{H}_{2} \mathrm{O} \rig... |
chem | JEE Adv 2023 Paper 1 | D | 49 | MCQ | Match the electronic configurations in List-I with appropriate metal complex ions in List-II and choose the correct option.
[Atomic Number: $\mathrm{Fe}=26, \mathrm{Mn}=25$, $\mathrm{Co}=27$ ]
\textbf{List-I}
(P) $t_{2 g}^{6} e_{g}^{0}$
(Q) $t_{2 g}^{3} e_{g}^{2}$
(R) $\mathrm{e}^{2} \mathrm{t}_{2}^{3}$
... |
chem | JEE Adv 2023 Paper 1 | D | 51 | MCQ | The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match List-I with List-II and choose the correct option.
\textbf{List-I}
(P) Etard reaction
(Q) Gattermann reaction
(R) Gattermann-Koch reaction
(S) Rosenmund reduction
\textbf{... |
math | JEE Adv 2023 Paper 2 | C | 1 | MCQ | Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(1)=\frac{1}{3}$ and $3 \int_{1}^{x} f(t) d t=x f(x)-\frac{x^{3}}{3}, x \in[1, \infty)$. Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$ is
(A) $\frac{e^{2}+4}{3}$
(B) $\frac{\log _{e} 4+e}{3}$
(C) $\... |
math | JEE Adv 2023 Paper 2 | B | 2 | MCQ | Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$, then the probability that the experiment stops with head is
(A) $\frac{1}{3}$
(B) $\frac{5}{21}$
(C) $\frac{4}{21}$
(D) $\frac{2}... |
math | JEE Adv 2023 Paper 2 | C | 3 | MCQ | For any $y \in \mathbb{R}$, let $\cot ^{-1}(y) \in(0, \pi)$ and $\tan ^{-1}(y) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then the sum of all the solutions of the equation $\tan ^{-1}\left(\frac{6 y}{9-y^{2}}\right)+\cot ^{-1}\left(\frac{9-y^{2}}{6 y}\right)=\frac{2 \pi}{3}$ for $0<|y|<3$, is equal to
(A) $2 \s... |
math | JEE Adv 2023 Paper 2 | B | 4 | MCQ | Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}$, $\vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}$ and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$, respectively. Then which of the following statements is true?
(A) The points... |
math | JEE Adv 2023 Paper 2 | BC | 5 | MCQ(multiple) | Let $M=\left(a_{i j}\right), i, j \in\{1,2,3\}$, be the $3 \times 3$ matrix such that $a_{i j}=1$ if $j+1$ is divisible by $i$, otherwise $a_{i j}=0$. Then which of the following statements is(are) true?
(A) $M$ is invertible
(B) There exists a nonzero column matrix $\left(\begin{array}{l}a_{1} \\ a_{2} \\ a_{3}\... |
math | JEE Adv 2023 Paper 2 | AB | 6 | MCQ(multiple) | Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4 x]\left(x-\frac{1}{4}\right)^{2}\left(x-\frac{1}{2}\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?
(A) The function $f$ is discontinuous exactly at one point in ... |
math | JEE Adv 2023 Paper 2 | ABC | 7 | MCQ(multiple) | Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{d^{2} f}{d x^{2}}(x)>0$ for all $x \in(-1,1)$. For $f \in S$, let $X_{f}$ be the number of points $x \in(-1,1)$ for which $f(x)=x$. Then which of the following statements is(are) true?
(A) There exists a f... |
math | JEE Adv 2023 Paper 2 | 0 | 8 | Integer | For $x \in \mathbb{R}$, let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then what is the minimum value of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\int_{0}^{x \tan ^{-1} x} \frac{e^{(t-\cos t)}}{1+t^{2023}} d t$? |
math | JEE Adv 2023 Paper 2 | 16 | 9 | Integer | For $x \in \mathbb{R}$, let $y(x)$ be a solution of the differential equation $\left(x^{2}-5\right) \frac{d y}{d x}-2 x y=-2 x\left(x^{2}-5\right)^{2}$ such that $y(2)=7$.
Then what is the maximum value of the function $y(x)$? |
math | JEE Adv 2023 Paper 2 | 31 | 10 | Integer | Let $X$ be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of 20 given that it is a... |
math | JEE Adv 2023 Paper 2 | 512 | 11 | Integer | Let $A_{1}, A_{2}, A_{3}, \ldots, A_{8}$ be the vertices of a regular octagon that lie on a circle of radius 2. Let $P$ be a point on the circle and let $P A_{i}$ denote the distance between the points $P$ and $A_{i}$ for $i=1,2, \ldots, 8$. If $P$ varies over the circle, then what is the maximum value of the product $... |
math | JEE Adv 2023 Paper 2 | 3780 | 12 | Integer | $R=\left\{\left(\begin{array}{lll}a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0\end{array}\right): a, b, c, d \in\{0,3,5,7,11,13,17,19\}\right\}$. Then what is the number of invertible matrices in $R$? |
math | JEE Adv 2023 Paper 2 | 2 | 13 | Integer | Let $C_{1}$ be the circle of radius 1 with center at the origin. Let $C_{2}$ be the circle of radius $r$ with center at the point $A=(4,1)$, where $1<r<3$. Two distinct common tangents $P Q$ and $S T$ of $C_{1}$ and $C_{2}$ are drawn. The tangent $P Q$ touches $C_{1}$ at $P$ and $C_{2}$ at $Q$. The tangent $S T$ touche... |
math | JEE Adv 2023 Paper 2 | 1008 | 14 | Numeric | Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1. Let $a$ be the area of the triangle $A B C$. Then what is the value of ... |
math | JEE Adv 2023 Paper 2 | 0.25 | 15 | Numeric | Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1. Then what is the inradius of the triangle ABC? |
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Chem Questions Involving Calculation
This query retrieves questions related to performing calculations within chemistry, helping to identify specific topics in the dataset but providing limited analytical context.
Questions with "Calculate"
The query retrieves a sample of 100 questions containing the word "calculate," providing a basic filter but limited analytical context.