question stringlengths 39 1.32k | reference_answer stringlengths 1 17 | id stringlengths 1 29 | metadata dict | answer_content stringclasses 1
value | reasoning_content stringclasses 1
value | verifier_score float64 | community_score dict |
|---|---|---|---|---|---|---|---|
A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated $(P)$ by the metal. The sensor has a scale that displays $\log _{2}\left(P / P_{0}\right)$, where $P_{0}$ is a constant. When the metal surface is at a temperature of $487^{\circ} \mathrm{C}$, the sensor shows a value 1. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to $2767{ }^{\circ} \mathrm{C}$ ? | 9 | 14 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
The isotope ${ }_{5}^{12} \mathrm{~B}$ having a mass 12.014 u undergoes $\beta$-decay to ${ }_{6}^{12} \mathrm{C}$. ${ }_{6}^{12} \mathrm{C}$ has an excited state of the nucleus $\left({ }_{6}^{12} \mathrm{C}^{*}\right)$ at $4.041 \mathrm{MeV}$ above its ground state. If ${ }_{5}^{12} \mathrm{~B}$ decays to ${ }_{6}^{12} \mathrm{C}^{*}$, what is the maximum kinetic energy of the $\beta$-particle in units of $\mathrm{MeV}$?
$\left(1 \mathrm{u}=931.5 \mathrm{MeV} / c^{2}\right.$, where $c$ is the speed of light in vacuum). | 9 | 15 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A hydrogen atom in its ground state is irradiated by light of wavelength 970 A. Taking $h c / e=1.237 \times 10^{-6} \mathrm{eV} \mathrm{m}$ and the ground state energy of hydrogen atom as $-13.6 \mathrm{eV}$, what is the number of lines present in the emission spectrum? | 6 | 16 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Consider two solid spheres $\mathrm{P}$ and $\mathrm{Q}$ each of density $8 \mathrm{gm} \mathrm{cm}^{-3}$ and diameters $1 \mathrm{~cm}$ and $0.5 \mathrm{~cm}$, respectively. Sphere $P$ is dropped into a liquid of density $0.8 \mathrm{gm} \mathrm{cm}^{-3}$ and viscosity $\eta=3$ poiseulles. Sphere $Q$ is dropped into a liquid of density $1.6 \mathrm{gm} \mathrm{cm}^{-3}$ and viscosity $\eta=2$ poiseulles. What is the ratio of the terminal velocities of $P$ and $Q$? | 3 | 17 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Two inductors $L_{1}$ (inductance $1 \mathrm{mH}$, internal resistance $3 \Omega$ ) and $L_{2}$ (inductance $2 \mathrm{mH}$, internal resistance $4 \Omega$ ), and a resistor $R$ (resistance $12 \Omega$ ) are all connected in parallel across a $5 \mathrm{~V}$ battery. The circuit is switched on at time $t=0$. What is the ratio of the maximum to the minimum current $\left(I_{\max } / I_{\min }\right)$ drawn from the battery? | 8 | 18 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
The mole fraction of a solute in a solution is 0.1 . At $298 \mathrm{~K}$, molarity of this solution is the same as its molality. Density of this solution at $298 \mathrm{~K}$ is $2.0 \mathrm{~g} \mathrm{~cm}^{-3}$. What is the ratio of the molecular weights of the solute and solvent, $\left(\frac{M W_{\text {solute }}}{M W_{\text {solvent }}}\right)$? | 9 | 32 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
The diffusion coefficient of an ideal gas is proportional to its mean free path and mean speed. The absolute temperature of an ideal gas is increased 4 times and its pressure is increased 2 times. As a result, the diffusion coefficient of this gas increases $x$ times. What is the value of $x$? | 4 | 33 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
In neutral or faintly alkaline solution, 8 moles of permanganate anion quantitatively oxidize thiosulphate anions to produce $\mathbf{X}$ moles of a sulphur containing product. What is the magnitude of $\mathbf{X}$? | 6 | 34 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the total number of distinct $x \in \mathbb{R}$ for which $\left|\begin{array}{ccc}x & x^{2} & 1+x^{3} \\ 2 x & 4 x^{2} & 1+8 x^{3} \\ 3 x & 9 x^{2} & 1+27 x^{3}\end{array}\right|=10$? | 2 | 50 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $m$ be the smallest positive integer such that the coefficient of $x^{2}$ in the expansion of $(1+x)^{2}+(1+x)^{3}+\cdots+(1+x)^{49}+(1+m x)^{50}$ is $(3 n+1){ }^{51} C_{3}$ for some positive integer $n$. Then what is the value of $n$? | 5 | 51 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the total number of distinct $x \in[0,1]$ for which $\int_{0}^{x} \frac{t^{2}}{1+t^{4}} d t=2 x-1$? | 1 | 52 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $\alpha, \beta \in \mathbb{R}$ be such that $\lim _{x \rightarrow 0} \frac{x^{2} \sin (\beta x)}{\alpha x-\sin x}=1$.Then what is the value of $6(\alpha+\beta)$? | 7 | 53 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $z=\frac{-1+\sqrt{3} i}{2}$, where $i=\sqrt{-1}$, and $r, s \in\{1,2,3\}$. Let $P=\left[\begin{array}{cc}(-z)^{r} & z^{2 s} \\ z^{2 s} & z^{r}\end{array}\right]$ and $I$ be the identity matrix of order 2 . Then what is the total number of ordered pairs $(r, s)$ for which $P^{2}=-I$? | 1 | 54 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A drop of liquid of radius $\mathrm{R}=10^{-2} \mathrm{~m}$ having surface tension $\mathrm{S}=\frac{0.1}{4 \pi} \mathrm{Nm}^{-1}$ divides itself into $K$ identical drops. In this process the total change in the surface energy $\Delta U=10^{-3} \mathrm{~J}$. If $K=10^{\alpha}$ then what is the value of $\alpha$? | 6 | 8 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
An electron in a hydrogen atom undergoes a transition from an orbit with quantum number $n_{i}$ to another with quantum number $n_{f} . V_{i}$ and $V_{f}$ are respectively the initial and final potential energies of the electron. If $\frac{V_{i}}{V_{f}}=6.25$, then what is the smallest possible $n_{f}$? | 5 | 9 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A stationary source emits sound of frequency $f_{0}=492 \mathrm{~Hz}$. The sound is reflected by a large car approaching the source with a speed of $2 \mathrm{~ms}^{-1}$. The reflected signal is received by the source and superposed with the original. What will be the beat frequency of the resulting signal in Hz? (Given that the speed of sound in air is $330 \mathrm{~ms}^{-1}$ and the car reflects the sound at the frequency it has received). | 6 | 11 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
${ }^{131} \mathrm{I}$ is an isotope of Iodine that $\beta$ decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with ${ }^{131} \mathrm{I}$ is injected into the blood of a person. The activity of the amount of ${ }^{131} \mathrm{I}$ injected was $2.4 \times 10^{5}$ Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, $2.5 \mathrm{ml}$ of blood is drawn from the person's body, and gives an activity of $115 \mathrm{~Bq}$. What is the total volume of blood in the person's body, in liters is approximately? (you may use $e^{x} \approx 1+x$ for $|x| \ll 1$ and $\ln 2 \approx 0.7$ ). | 5 | 12 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge of $400 \mathrm{pm}$. If the density of the substance in the crystal is $8 \mathrm{~g} \mathrm{~cm}^{-3}$, then the number of atoms present in $256 \mathrm{~g}$ of the crystal is $N \times 10^{24}$. What is the value of $N$? | 2 | 26 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
The conductance of a $0.0015 \mathrm{M}$ aqueous solution of a weak monobasic acid was determined by using a conductivity cell consisting of platinized Pt electrodes. The distance between the electrodes is $120 \mathrm{~cm}$ with an area of cross section of $1 \mathrm{~cm}^{2}$. The conductance of this solution was found to be $5 \times 10^{-7} \mathrm{~S}$. The $\mathrm{pH}$ of the solution is 4 . The value of limiting molar conductivity $\left(\Lambda_{m}^{o}\right)$ of this weak monobasic acid in aqueous solution is $Z \times 10^{2} \mathrm{~S} \mathrm{~cm}^{-1} \mathrm{~mol}^{-1}$. What is the value of $Z$? | 6 | 27 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the sum of the number of lone pairs of electrons on each central atom in the following species?
$\left[\mathrm{TeBr}_{6}\right]^{2-},\left[\mathrm{BrF}_{2}\right]^{+}, \mathrm{SNF}_{3}$, and $\left[\mathrm{XeF}_{3}\right]^{-}$
(Atomic numbers: $\mathrm{N}=7, \mathrm{~F}=9, \mathrm{~S}=16, \mathrm{Br}=35, \mathrm{Te}=52, \mathrm{Xe}=54$ ) | 6 | 28 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Among $\mathrm{H}_{2}, \mathrm{He}_{2}{ }^{+}, \mathrm{Li}_{2}, \mathrm{Be}_{2}, \mathrm{~B}_{2}, \mathrm{C}_{2}, \mathrm{~N}_{2}, \mathrm{O}_{2}^{-}$, and $\mathrm{F}_{2}$, what is the number of diamagnetic species? (Atomic numbers: $\mathrm{H}=1, \mathrm{He}=2, \mathrm{Li}=3, \mathrm{Be}=4, \mathrm{~B}=5, \mathrm{C}=6, \mathrm{~N}=7, \mathrm{O}=8, \mathrm{~F}=9$ ) | 5 | 29 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
For how many values of $p$, the circle $x^{2}+y^{2}+2 x+4 y-p=0$ and the coordinate axes have exactly three common points? | 2 | 44 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=0, f\left(\frac{\pi}{2}\right)=3$ and $f^{\prime}(0)=1$. If
\[
g(x)=\int_{x}^{\frac{\pi}{2}}\left[f^{\prime}(t) \operatorname{cosec} t-\cot t \operatorname{cosec} t f(t)\right] d t
\]
for $x \in\left(0, \frac{\pi}{2}\right]$, then what is the $\lim _{x \rightarrow 0} g(x)$? | 2 | 45 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
For a real number $\alpha$, if the system
\[
\left[\begin{array}{ccc}
1 & \alpha & \alpha^{2} \\
\alpha & 1 & \alpha \\
\alpha^{2} & \alpha & 1
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right]
\]
of linear equations, has infinitely many solutions, then what is the value of $1+\alpha+\alpha^{2}$? | 1 | 46 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Words of length 10 are formed using the letters $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, what is the value of $\frac{y}{9 x}$? | 5 | 47 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side? | 6 | 48 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Two vectors $\vec{A}$ and $\vec{B}$ are defined as $\vec{A}=a \hat{i}$ and $\vec{B}=a(\cos \omega t \hat{i}+\sin \omega t \hat{j})$, where $a$ is a constant and $\omega=\pi / 6 \mathrm{rads}^{-1}$. If $|\vec{A}+\vec{B}|=\sqrt{3}|\vec{A}-\vec{B}|$ at time $t=\tau$ for the first time, what is the value of $\tau$, in seconds? | 2 | 7 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed $1.0 \mathrm{~ms}^{-1}$ and the man behind walks at a speed $2.0 \mathrm{~m} \mathrm{~s}^{-1}$. A third man is standing at a height $12 \mathrm{~m}$ above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency $1430 \mathrm{~Hz}$. The speed of sound in air is $330 \mathrm{~m} \mathrm{~s}^{-1}$. At the instant, when the moving men are $10 \mathrm{~m}$ apart, the stationary man is equidistant from them. What is the frequency of beats in $\mathrm{Hz}$, heard by the stationary man at this instant? | 5 | 8 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle $60^{\circ}$ with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $(2-\sqrt{3}) / \sqrt{10} s$, then what is the height of the top of the inclined plane, in metres?
Take $g=10 m s^{-2}$. | 0.75 | 9 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Sunlight of intensity $1.3 \mathrm{~kW} \mathrm{~m}^{-2}$ is incident normally on a thin convex lens of focal length $20 \mathrm{~cm}$. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. What is the average intensity of light, in $\mathrm{kW} \mathrm{m}{ }^{-2}$, at a distance $22 \mathrm{~cm}$ from the lens on the other side? | 130 | 13 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Among the species given below, what is the total number of diamagnetic species?
$\mathrm{H}$ atom, $\mathrm{NO}_{2}$ monomer, $\mathrm{O}_{2}^{-}$(superoxide), dimeric sulphur in vapour phase,
$\mathrm{Mn}_{3} \mathrm{O}_{4},\left(\mathrm{NH}_{4}\right)_{2}\left[\mathrm{FeCl}_{4}\right],\left(\mathrm{NH}_{4}\right)_{2}\left[\mathrm{NiCl}_{4}\right], \mathrm{K}_{2} \mathrm{MnO}_{4}, \mathrm{~K}_{2} \mathrm{CrO}_{4}$ | 1 | 25 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
The ammonia prepared by treating ammonium sulphate with calcium hydroxide is completely used by $\mathrm{NiCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}$ to form a stable coordination compound. Assume that both the reactions are $100 \%$ complete. If $1584 \mathrm{~g}$ of ammonium sulphate and $952 \mathrm{~g}$ of $\mathrm{NiCl}_{2} .6 \mathrm{H}_{2} \mathrm{O}$ are used in the preparation, what is the combined weight (in grams) of gypsum and the nickelammonia coordination compound thus produced?
(Atomic weights in $\mathrm{g} \mathrm{mol}^{-1}: \mathrm{H}=1, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{~S}=32, \mathrm{Cl}=35.5, \mathrm{Ca}=40, \mathrm{Ni}=59$ ) | 2992 | 26 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Consider an ionic solid $\mathbf{M X}$ with $\mathrm{NaCl}$ structure. Construct a new structure (Z) whose unit cell is constructed from the unit cell of $\mathbf{M X}$ following the sequential instructions given below. Neglect the charge balance.
(i) Remove all the anions (X) except the central one
(ii) Replace all the face centered cations (M) by anions (X)
(iii) Remove all the corner cations (M)
(iv) Replace the central anion (X) with cation (M)
What is the value of $\left(\frac{\text { number of anions }}{\text { number of cations }}\right)$ in $\mathbf{Z}$? | 3 | 27 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
For the electrochemical cell,
\[
\operatorname{Mg}(\mathrm{s})\left|\mathrm{Mg}^{2+}(\mathrm{aq}, 1 \mathrm{M}) \| \mathrm{Cu}^{2+}(\mathrm{aq}, 1 \mathrm{M})\right| \mathrm{Cu}(\mathrm{s})
\]
the standard emf of the cell is $2.70 \mathrm{~V}$ at $300 \mathrm{~K}$. When the concentration of $\mathrm{Mg}^{2+}$ is changed to $\boldsymbol{x} \mathrm{M}$, the cell potential changes to $2.67 \mathrm{~V}$ at $300 \mathrm{~K}$. What is the value of $\boldsymbol{x}$?
(given, $\frac{F}{R}=11500 \mathrm{~K} \mathrm{~V}^{-1}$, where $F$ is the Faraday constant and $R$ is the gas constant, $\ln (10)=2.30)$ | 10 | 28 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Liquids $\mathbf{A}$ and $\mathbf{B}$ form ideal solution over the entire range of composition. At temperature $\mathrm{T}$, equimolar binary solution of liquids $\mathbf{A}$ and $\mathbf{B}$ has vapour pressure 45 Torr. At the same temperature, a new solution of $\mathbf{A}$ and $\mathbf{B}$ having mole fractions $x_{A}$ and $x_{B}$, respectively, has vapour pressure of 22.5 Torr. What is the value of $x_{A} / x_{B}$ in the new solution? (given that the vapour pressure of pure liquid $\mathbf{A}$ is 20 Torr at temperature $\mathrm{T}$ ) | 19 | 30 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
The solubility of a salt of weak acid (AB) at $\mathrm{pH} 3$ is $\mathbf{Y} \times 10^{-3} \mathrm{~mol} \mathrm{~L}^{-1}$. The value of $\mathbf{Y}$ is (Given that the value of solubility product of $\mathbf{A B}\left(K_{s p}\right)=2 \times 10^{-10}$ and the value of ionization constant of $\left.\mathbf{H B}\left(K_{a}\right)=1 \times 10^{-8}\right)$ | 4.47 | 31 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the value of
\[
\left(\left(\log _{2} 9\right)^{2}\right)^{\frac{1}{\log _{2}\left(\log _{2} 9\right)}} \times(\sqrt{7})^{\frac{1}{\log _{4} 7}}
\]? | 8 | 43 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the number of 5 digit numbers which are divisible by 4 , with digits from the set $\{1,2,3,4,5\}$ and the repetition of digits is allowed? | 625 | 44 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11, \ldots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23, \ldots$. Then, what is the number of elements in the set $X \cup Y$? | 3748 | 45 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the number of real solutions of the equation
\[
\sin ^{-1}\left(\sum_{i=1}^{\infty} x^{i+1}-x \sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^{i}\right)=\frac{\pi}{2}-\cos ^{-1}\left(\sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^{i}-\sum_{i=1}^{\infty}(-x)^{i}\right)
\]
lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is
(Here, the inverse trigonometric functions $\sin ^{-1} x$ and $\cos ^{-1} x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$, respectively.) | 2 | 46 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
For each positive integer $n$, let
\[
y_{n}=\frac{1}{n}((n+1)(n+2) \cdots(n+n))^{\frac{1}{n}}
\]
For $x \in \mathbb{R}$, let $[x]$ be the greatest integer less than or equal to $x$. If $\lim _{n \rightarrow \infty} y_{n}=L$, then what is the value of $[L]$? | 1 | 47 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a} \cdot \vec{b}=0$. For some $x, y \in \mathbb{R}$, let $\vec{c}=x \vec{a}+y \vec{b}+(\vec{a} \times \vec{b})$. If $|\vec{c}|=2$ and the vector $\vec{c}$ is inclined the same angle $\alpha$ to both $\vec{a}$ and $\vec{b}$, then what is the value of $8 \cos ^{2} \alpha$? | 3 | 48 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $a, b, c$ be three non-zero real numbers such that the equation
\[
\sqrt{3} a \cos x+2 b \sin x=c, x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
\]
has two distinct real roots $\alpha$ and $\beta$ with $\alpha+\beta=\frac{\pi}{3}$. Then, what is the value of $\frac{b}{a}$? | 0.5 | 49 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A farmer $F_{1}$ has a land in the shape of a triangle with vertices at $P(0,0), Q(1,1)$ and $R(2,0)$. From this land, a neighbouring farmer $F_{2}$ takes away the region which lies between the side $P Q$ and a curve of the form $y=x^{n}(n>1)$. If the area of the region taken away by the farmer $F_{2}$ is exactly $30 \%$ of the area of $\triangle P Q R$, then what is the value of $n$? | 4 | 50 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m=0.4 \mathrm{~kg}$ is at rest on this surface. An impulse of $1.0 \mathrm{~N}$ is applied to the block at time $t=0$ so that it starts moving along the $x$-axis with a velocity $v(t)=v_{0} e^{-t / \tau}$, where $v_{0}$ is a constant and $\tau=4 \mathrm{~s}$. What is the displacement of the block, in metres, at $t=\tau$? Take $e^{-1}=0.37$ | 6.3 | 7 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A ball is projected from the ground at an angle of $45^{\circ}$ with the horizontal surface. It reaches a maximum height of $120 \mathrm{~m}$ and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of $30^{\circ}$ with the horizontal surface. What is the maximum height it reaches after the bounce, in metres? | 30 | 8 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A particle, of mass $10^{-3} \mathrm{~kg}$ and charge $1.0 \mathrm{C}$, is initially at rest. At time $t=0$, the particle comes under the influence of an electric field $\vec{E}(t)=E_{0} \sin \omega t \hat{i}$, where $E_{0}=1.0 \mathrm{~N}^{-1}$ and $\omega=10^{3} \mathrm{rad} \mathrm{s}^{-1}$. Consider the effect of only the electrical force on the particle. Then what is the maximum speed, in $m s^{-1}$, attained by the particle at subsequent times? | 2 | 9 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A moving coil galvanometer has 50 turns and each turn has an area $2 \times 10^{-4} \mathrm{~m}^{2}$. The magnetic field produced by the magnet inside the galvanometer is $0.02 T$. The torsional constant of the suspension wire is $10^{-4} \mathrm{~N} \mathrm{~m} \mathrm{rad}{ }^{-1}$. When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by $0.2 \mathrm{rad}$. The resistance of the coil of the galvanometer is $50 \Omega$. This galvanometer is to be converted into an ammeter capable of measuring current in the range $0-1.0 \mathrm{~A}$. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. What is the value of this shunt resistance, in ohms? | 5.56 | 10 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A steel wire of diameter $0.5 \mathrm{~mm}$ and Young's modulus $2 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$ carries a load of mass $M$. The length of the wire with the load is $1.0 \mathrm{~m}$. A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count $1.0 \mathrm{~mm}$, is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by $1.2 \mathrm{~kg}$, what is the vernier scale division which coincides with a main scale division? Take $g=10 \mathrm{~ms}^{-2}$ and $\pi=3.2$ | 3 | 11 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is $100 \mathrm{~K}$ and the universal gas constant $R=8.0 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$, what is the decrease in its internal energy, in Joule? | 900 | 12 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
In a photoelectric experiment a parallel beam of monochromatic light with power of $200 \mathrm{~W}$ is incident on a perfectly absorbing cathode of work function $6.25 \mathrm{eV}$. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is $100 \%$. A potential difference of $500 \mathrm{~V}$ is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force $F=n \times 10^{-4} N$ due to the impact of the electrons. What is the value of $n$?
Mass of the electron $m_{e}=9 \times 10^{-31} \mathrm{~kg}$ and $1.0 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J}$. | 24 | 13 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Consider a hydrogen-like ionized atom with atomic number $Z$ with a single electron. In the emission spectrum of this atom, the photon emitted in the $n=2$ to $n=1$ transition has energy $74.8 \mathrm{eV}$ higher than the photon emitted in the $n=3$ to $n=2$ transition. The ionization energy of the hydrogen atom is $13.6 \mathrm{eV}$. What is the value of $Z$? | 3 | 14 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the total number of compounds having at least one bridging oxo group among the molecules given below?
$\mathrm{N}_{2} \mathrm{O}_{3}, \mathrm{~N}_{2} \mathrm{O}_{5}, \mathrm{P}_{4} \mathrm{O}_{6}, \mathrm{P}_{4} \mathrm{O}_{7}, \mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{5}, \mathrm{H}_{5} \mathrm{P}_{3} \mathrm{O}_{10}, \mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}, \mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{5}$ | 6 | 25 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Galena (an ore) is partially oxidized by passing air through it at high temperature. After some time, the passage of air is stopped, but the heating is continued in a closed furnace such that the contents undergo self-reduction. What is the weight (in $\mathrm{kg}$ ) of $\mathrm{Pb}$ produced per $\mathrm{kg}$ of $\mathrm{O}_{2}$ consumed?
(Atomic weights in $\mathrm{g} \mathrm{mol}^{-1}: \mathrm{O}=16, \mathrm{~S}=32, \mathrm{~Pb}=207$ ) | 6.47 | 26 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
To measure the quantity of $\mathrm{MnCl}_{2}$ dissolved in an aqueous solution, it was completely converted to $\mathrm{KMnO}_{4}$ using the reaction, $\mathrm{MnCl}_{2}+\mathrm{K}_{2} \mathrm{~S}_{2} \mathrm{O}_{8}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{KMnO}_{4}+\mathrm{H}_{2} \mathrm{SO}_{4}+\mathrm{HCl}$ (equation not balanced).
Few drops of concentrated $\mathrm{HCl}$ were added to this solution and gently warmed. Further, oxalic acid (225 mg) was added in portions till the colour of the permanganate ion disappeared. The quantity of $\mathrm{MnCl}_{2}$ (in $\mathrm{mg}$ ) present in the initial solution is
(Atomic weights in $\mathrm{g} \mathrm{mol}^{-1}: \mathrm{Mn}=55, \mathrm{Cl}=35.5$ ) | 126 | 27 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
The surface of copper gets tarnished by the formation of copper oxide. $\mathrm{N}_{2}$ gas was passed to prevent the oxide formation during heating of copper at $1250 \mathrm{~K}$. However, the $\mathrm{N}_{2}$ gas contains 1 mole $\%$ of water vapour as impurity. The water vapour oxidises copper as per the reaction given below:
$2 \mathrm{Cu}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightarrow \mathrm{Cu}_{2} \mathrm{O}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{~g})$
$p_{\mathrm{H}_{2}}$ is the minimum partial pressure of $\mathrm{H}_{2}$ (in bar) needed to prevent the oxidation at $1250 \mathrm{~K}$. What is the value of $\ln \left(p_{\mathrm{H}_{2}}\right)$?
(Given: total pressure $=1$ bar, $R$ (universal gas constant $)=8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}, \ln (10)=2.3 \cdot \mathrm{Cu}(\mathrm{s})$ and $\mathrm{Cu}_{2} \mathrm{O}(\mathrm{s})$ are mutually immiscible.
At $1250 \mathrm{~K}: 2 \mathrm{Cu}(\mathrm{s})+1 / 2 \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{Cu}_{2} \mathrm{O}(\mathrm{s}) ; \Delta G^{\theta}=-78,000 \mathrm{~J} \mathrm{~mol}^{-1}$
\[
\mathrm{H}_{2}(\mathrm{~g})+1 / 2 \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) ; \quad \Delta G^{\theta}=-1,78,000 \mathrm{~J} \mathrm{~mol}^{-1} ; G \text { is the Gibbs energy) }
\] | -14.6 | 30 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Consider the following reversible reaction,
\[
\mathrm{A}(\mathrm{g})+\mathrm{B}(\mathrm{g}) \rightleftharpoons \mathrm{AB}(\mathrm{g})
\]
The activation energy of the backward reaction exceeds that of the forward reaction by $2 R T$ (in $\mathrm{J} \mathrm{mol}^{-1}$ ). If the pre-exponential factor of the forward reaction is 4 times that of the reverse reaction, what is the absolute value of $\Delta G^{\theta}$ (in $\mathrm{J} \mathrm{mol}^{-1}$ ) for the reaction at $300 \mathrm{~K}$?
(Given; $\ln (2)=0.7, R T=2500 \mathrm{~J} \mathrm{~mol}^{-1}$ at $300 \mathrm{~K}$ and $G$ is the Gibbs energy) | 8500 | 31 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Consider an electrochemical cell: $\mathrm{A}(\mathrm{s})\left|\mathrm{A}^{\mathrm{n}+}(\mathrm{aq}, 2 \mathrm{M}) \| \mathrm{B}^{2 \mathrm{n}+}(\mathrm{aq}, 1 \mathrm{M})\right| \mathrm{B}(\mathrm{s})$. The value of $\Delta H^{\theta}$ for the cell reaction is twice that of $\Delta G^{\theta}$ at $300 \mathrm{~K}$. If the emf of the cell is zero, what is the $\Delta S^{\ominus}$ (in $\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}$ ) of the cell reaction per mole of $\mathrm{B}$ formed at $300 \mathrm{~K}$?
(Given: $\ln (2)=0.7, R$ (universal gas constant) $=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} . H, S$ and $G$ are enthalpy, entropy and Gibbs energy, respectively.)
| -11.62 | 32 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the value of the integral
\[
\int_{0}^{\frac{1}{2}} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{\frac{1}{4}}} d x
\]? | 2 | 43 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{-1,0,1\}$. Then, what is the maximum possible value of the determinant of $P$? | 4 | 44 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $X$ be a set with exactly 5 elements and $Y$ be a set with exactly 7 elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$, then what is the value of $\frac{1}{5 !}(\beta-\alpha)$? | 119 | 45 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation
\[
\frac{d y}{d x}=(2+5 y)(5 y-2)
\]
then what is the value of $\lim _{x \rightarrow-\infty} f(x)$? | 0.4 | 46 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with $f(0)=1$ and satisfying the equation
\[
f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y) \text { for all } x, y \in \mathbb{R} .
\]
Then, the value of $\log _{e}(f(4))$ is | 2 | 47 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y=3$ (that is, the line segment $P Q$ is perpendicular to the plane $x+y=3$ and the mid-point of $P Q$ lies in the plane $x+y=3$ ) lies on the $z$-axis. Let the distance of $P$ from the $x$-axis be 5 . If $R$ is the image of $P$ in the $x y$-plane, then what is the length of $P R$? | 8 | 48 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Consider the cube in the first octant with sides $O P, O Q$ and $O R$ of length 1 , along the $x$-axis, $y$-axis and $z$-axis, respectively, where $O(0,0,0)$ is the origin. Let $S\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $O T$. If $\vec{p}=\overrightarrow{S P}, \vec{q}=\overrightarrow{S Q}, \vec{r}=\overrightarrow{S R}$ and $\vec{t}=\overrightarrow{S T}$, then what is the value of $|(\vec{p} \times \vec{q}) \times(\vec{r} \times \vec{t})|$? | 0.5 | 49 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let
\[
X=\left({ }^{10} C_{1}\right)^{2}+2\left({ }^{10} C_{2}\right)^{2}+3\left({ }^{10} C_{3}\right)^{2}+\cdots+10\left({ }^{10} C_{10}\right)^{2}
\]
where ${ }^{10} C_{r}, r \in\{1,2, \cdots, 10\}$ denote binomial coefficients. Then, what is the value of $\frac{1}{1430} X$? | 646 | 50 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A parallel plate capacitor of capacitance $C$ has spacing $d$ between two plates having area $A$. The region between the plates is filled with $N$ dielectric layers, parallel to its plates, each with thickness $\delta=\frac{d}{N}$. The dielectric constant of the $m^{t h}$ layer is $K_{m}=K\left(1+\frac{m}{N}\right)$. For a very large $N\left(>10^{3}\right)$, the capacitance $C$ is $\alpha\left(\frac{K \epsilon_{0} A}{d \ln 2}\right)$. What will be the value of $\alpha$? $\left[\epsilon_{0}\right.$ is the permittivity of free space] | 1 | 16 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Among $\mathrm{B}_{2} \mathrm{H}_{6}, \mathrm{~B}_{3} \mathrm{~N}_{3} \mathrm{H}_{6}, \mathrm{~N}_{2} \mathrm{O}, \mathrm{N}_{2} \mathrm{O}_{4}, \mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}$ and $\mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{8}$, what is the total number of molecules containing covalent bond between two atoms of the same kind? | 4 | 31 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
At $143 \mathrm{~K}$, the reaction of $\mathrm{XeF}_{4}$ with $\mathrm{O}_{2} \mathrm{~F}_{2}$ produces a xenon compound $\mathrm{Y}$. What is the total number of lone pair(s) of electrons present on the whole molecule of $\mathrm{Y}$? | 19 | 32 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
On dissolving $0.5 \mathrm{~g}$ of a non-volatile non-ionic solute to $39 \mathrm{~g}$ of benzene, its vapor pressure decreases from $650 \mathrm{~mm} \mathrm{Hg}$ to $640 \mathrm{~mm} \mathrm{Hg}$. What is the depression of freezing point of benzene (in $\mathrm{K}$ ) upon addition of the solute? (Given data: Molar mass and the molal freezing point depression constant of benzene are $78 \mathrm{~g}$ $\mathrm{mol}^{-1}$ and $5.12 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$, respectively) | 1.02 | 34 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Consider the kinetic data given in the following table for the reaction $\mathrm{A}+\mathrm{B}+\mathrm{C} \rightarrow$ Product.
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Experiment No. & $\begin{array}{c}{[\mathrm{A}]} \\ \left(\mathrm{mol} \mathrm{dm}^{-3}\right)\end{array}$ & $\begin{array}{c}{[\mathrm{B}]} \\ \left(\mathrm{mol} \mathrm{dm}^{-3}\right)\end{array}$ & $\begin{array}{c}{[\mathrm{C}]} \\ \left(\mathrm{mol} \mathrm{dm}^{-3}\right)\end{array}$ & $\begin{array}{c}\text { Rate of reaction } \\ \left(\mathrm{mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}\right)\end{array}$ \\
\hline
1 & 0.2 & 0.1 & 0.1 & $6.0 \times 10^{-5}$ \\
\hline
2 & 0.2 & 0.2 & 0.1 & $6.0 \times 10^{-5}$ \\
\hline
3 & 0.2 & 0.1 & 0.2 & $1.2 \times 10^{-4}$ \\
\hline
4 & 0.3 & 0.1 & 0.1 & $9.0 \times 10^{-5}$ \\
\hline
\end{tabular}
\end{center}
The rate of the reaction for $[\mathrm{A}]=0.15 \mathrm{~mol} \mathrm{dm}^{-3},[\mathrm{~B}]=0.25 \mathrm{~mol} \mathrm{dm}^{-3}$ and $[\mathrm{C}]=0.15 \mathrm{~mol} \mathrm{dm}^{-3}$ is found to be $\mathbf{Y} \times 10^{-5} \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}$. What is the value of $\mathbf{Y}$? | 6.75 | 35 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $\omega \neq 1$ be a cube root of unity. Then what is the minimum of the set
\[
\left\{\left|a+b \omega+c \omega^{2}\right|^{2}: a, b, c \text { distinct non-zero integers }\right\}
\] equal? | 3 | 49 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $A P(a ; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d>0$. If
\[
A P(1 ; 3) \cap A P(2 ; 5) \cap A P(3 ; 7)=A P(a ; d)
\]
then what does $a+d$ equal? | 157 | 50 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $S$ be the sample space of all $3 \times 3$ matrices with entries from the set $\{0,1\}$. Let the events $E_{1}$ and $E_{2}$ be given by
\[
\begin{aligned}
& E_{1}=\{A \in S: \operatorname{det} A=0\} \text { and } \\
& E_{2}=\{A \in S: \text { sum of entries of } A \text { is } 7\} .
\end{aligned}
\]
If a matrix is chosen at random from $S$, then what is the conditional probability $P\left(E_{1} \mid E_{2}\right)$? | 0.5 | 51 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8 x-6 y-23=0$. Let $\Gamma_{A}$ and $\Gamma_{B}$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_{A}$ and $\Gamma_{B}$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then what is the length of the line segment $A C$? | 10 | 52 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
If
\[
I=\frac{2}{\pi} \int_{-\pi / 4}^{\pi / 4} \frac{d x}{\left(1+e^{\sin x}\right)(2-\cos 2 x)}
\]
then what does $27 I^{2}$ equal? | 4 | 53 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Three lines are given by
\[
\vec{r} & =\lambda \hat{i}, \lambda \in \mathbb{R}
\]
\[\vec{r} & =\mu(\hat{i}+\hat{j}), \mu \in \mathbb{R}
\]
\[
\vec{r} =v(\hat{i}+\hat{j}+\hat{k}), v \in \mathbb{R}.
\]
Let the lines cut the plane $x+y+z=1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $A B C$ is $\triangle$ then what is the value of $(6 \Delta)^{2}$? | 0.75 | 54 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Suppose a ${ }_{88}^{226} R a$ nucleus at rest and in ground state undergoes $\alpha$-decay to a ${ }_{86}^{222} R n$ nucleus in its excited state. The kinetic energy of the emitted $\alpha$ particle is found to be $4.44 \mathrm{MeV}$. ${ }_{86}^{222} R n$ nucleus then goes to its ground state by $\gamma$-decay. What is the energy of the emitted $\gamma$ photon is $\mathrm{keV}$?
[Given: atomic mass of ${ }_{88}^{226} R a=226.005 \mathrm{u}$, atomic mass of ${ }_{86}^{222} R n=222.000 \mathrm{u}$, atomic mass of $\alpha$ particle $=4.000 \mathrm{u}, 1 \mathrm{u}=931 \mathrm{MeV} / \mathrm{c}^{2}, \mathrm{c}$ is speed of the light] | 135 | 13 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
An optical bench has $1.5 \mathrm{~m}$ long scale having four equal divisions in each $\mathrm{cm}$. While measuring the focal length of a convex lens, the lens is kept at $75 \mathrm{~cm}$ mark of the scale and the object pin is kept at $45 \mathrm{~cm}$ mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at $135 \mathrm{~cm}$ mark. In this experiment, what is the percentage error in the measurement of the focal length of the lens? | 0.69 | 14 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the amount of water produced (in g) in the oxidation of 1 mole of rhombic sulphur by conc. $\mathrm{HNO}_{3}$ to a compound with the highest oxidation state of sulphur?
(Given data: Molar mass of water $=18 \mathrm{~g} \mathrm{~mol}^{-1}$ ) | 288 | 27 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the total number of cis $\mathrm{N}-\mathrm{Mn}-\mathrm{Cl}$ bond angles (that is, $\mathrm{Mn}-\mathrm{N}$ and $\mathrm{Mn}-\mathrm{Cl}$ bonds in cis positions) present in a molecule of cis-[Mn(en $\left.)_{2} \mathrm{Cl}_{2}\right]$ complex?
(en $=\mathrm{NH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{2}$ ) | 6 | 28 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
The decomposition reaction $2 \mathrm{~N}_{2} \mathrm{O}_{5}(g) \stackrel{\Delta}{\rightarrow} 2 \mathrm{~N}_{2} \mathrm{O}_{4}(g)+\mathrm{O}_{2}(g)$ is started in a closed cylinder under isothermal isochoric condition at an initial pressure of $1 \mathrm{~atm}$. After $\mathrm{Y} \times 10^{3} \mathrm{~s}$, the pressure inside the cylinder is found to be $1.45 \mathrm{~atm}$. If the rate constant of the reaction is $5 \times 10^{-4} \mathrm{~s}^{-1}$, assuming ideal gas behavior, what is the value of $\mathrm{Y}$? | 2.3 | 29 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
The mole fraction of urea in an aqueous urea solution containing $900 \mathrm{~g}$ of water is 0.05 . If the density of the solution is $1.2 \mathrm{~g} \mathrm{~cm}^{-3}$, what is the molarity of urea solution? (Given data: Molar masses of urea and water are $60 \mathrm{~g} \mathrm{~mol}^{-1}$ and $18 \mathrm{~g} \mathrm{~mol}^{-1}$, respectively) | 2.98 | 30 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the molecular formula $\mathrm{C}_{4} \mathrm{H}_{8} \mathrm{O}$? | 10 | 32 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Suppose
\[
\operatorname{det}\left[\begin{array}{cc}
\sum_{k=0}^{n} k & \sum_{k=0}^{n}{ }^{n} C_{k} k^{2} \\
\sum_{k=0}^{n}{ }^{n} C_{k} k & \sum_{k=0}^{n}{ }^{n} C_{k} 3^{k}
\end{array}\right]=0
\]
holds for some positive integer $n$. Then what does $\sum_{k=0}^{n} \frac{{ }^{n} C_{k}}{k+1}$? | 6.2 | 45 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Five persons $A, B, C, D$ and $E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then what is the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats? | 30 | 46 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $|X|$ denote the number of elements in a set $X$. Let $S=\{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then what is the number of ordered pairs $(A, B)$ such that $1 \leq|B|<|A|$? | 422 | 47 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the value of
\[
\sec ^{-1}\left(\frac{1}{4} \sum_{k=0}^{10} \sec \left(\frac{7 \pi}{12}+\frac{k \pi}{2}\right) \sec \left(\frac{7 \pi}{12}+\frac{(k+1) \pi}{2}\right)\right)
\]
in the interval $\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]$? | 0 | 48 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
What is the value of the integral
\[
\int_{0}^{\pi / 2} \frac{3 \sqrt{\cos \theta}}{(\sqrt{\cos \theta}+\sqrt{\sin \theta})^{5}} d \theta
\]? | 0.5 | 49 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Put a uniform meter scale horizontally on your extended index fingers with the left one at $0.00 \mathrm{~cm}$ and the right one at $90.00 \mathrm{~cm}$. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_{R}$ from the center $(50.00 \mathrm{~cm})$ of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32 , respectively, what is the value of $x_{R}$ (in $\mathrm{cm})$? | 25.6 | 13 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Consider one mole of helium gas enclosed in a container at initial pressure $P_{1}$ and volume $V_{1}$. It expands isothermally to volume $4 V_{1}$. After this, the gas expands adiabatically and its volume becomes $32 V_{1}$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\text {iso }}$ and $W_{\text {adia }}$, respectively. If the ratio $\frac{W_{\text {iso }}}{W_{\text {adia }}}=f \ln 2$, then what is the value of $f$? | 1.77 | 16 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of $2 \mathrm{~ms}^{-1}$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is $320 \mathrm{~ms}^{-1}$, what is the smallest value of the percentage change required in the length of the pipe? | 0.62 | 17 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
A circular disc of radius $R$ carries surface charge density $\sigma(r)=\sigma_{0}\left(1-\frac{r}{R}\right)$, where $\sigma_{0}$ is a constant and $r$ is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is $\phi_{0}$. Electric flux through another spherical surface of radius $\frac{R}{4}$ and concentric with the disc is $\phi$. Then what is the ratio $\frac{\phi_{0}}{\phi}$? | 6.4 | 18 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
$5.00 \mathrm{~mL}$ of $0.10 \mathrm{M}$ oxalic acid solution taken in a conical flask is titrated against $\mathrm{NaOH}$ from a burette using phenolphthalein indicator. The volume of $\mathrm{NaOH}$ required for the appearance of permanent faint pink color is tabulated below for five experiments. What is the concentration, in molarity, of the $\mathrm{NaOH}$ solution?
\begin{center}
\begin{tabular}{|c|c|}
\hline
Exp. No. & Vol. of NaOH (mL) \\
\hline
$\mathbf{1}$ & 12.5 \\
\hline
$\mathbf{2}$ & 10.5 \\
\hline
$\mathbf{3}$ & 9.0 \\
\hline
$\mathbf{4}$ & 9.0 \\
\hline
$\mathbf{5}$ & 9.0 \\
\hline
\end{tabular}
\end{center} | 0.11 | 31 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Consider a 70\% efficient hydrogen-oxygen fuel cell working under standard conditions at 1 bar and $298 \mathrm{~K}$. Its cell reaction is
\[
\mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \rightarrow \mathrm{H}_{2} \mathrm{O}(l)
\]
The work derived from the cell on the consumption of $1.0 \times 10^{-3} \mathrm{~mol} \mathrm{of}_{2}(g)$ is used to compress $1.00 \mathrm{~mol}$ of a monoatomic ideal gas in a thermally insulated container. What is the change in the temperature (in K) of the ideal gas?
The standard reduction potentials for the two half-cells are given below.
\[
\begin{gathered}
\mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q)+4 e^{-} \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(l), \quad E^{0}=1.23 \mathrm{~V}, \\
2 \mathrm{H}^{+}(a q)+2 e^{-} \rightarrow \mathrm{H}_{2}(g), \quad E^{0}=0.00 \mathrm{~V}
\end{gathered}
\]
Use $F=96500 \mathrm{C} \mathrm{mol}^{-1}, R=8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. | 13.32 | 33 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Aluminium reacts with sulfuric acid to form aluminium sulfate and hydrogen. What is the volume of hydrogen gas in liters (L) produced at $300 \mathrm{~K}$ and $1.0 \mathrm{~atm}$ pressure, when $5.4 \mathrm{~g}$ of aluminium and $50.0 \mathrm{~mL}$ of $5.0 \mathrm{M}$ sulfuric acid are combined for the reaction?
(Use molar mass of aluminium as $27.0 \mathrm{~g} \mathrm{~mol}^{-1}, R=0.082 \mathrm{~atm} \mathrm{~L} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ ) | 6.15 | 34 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
${ }_{92}^{238} \mathrm{U}$ is known to undergo radioactive decay to form ${ }_{82}^{206} \mathrm{~Pb}$ by emitting alpha and beta particles. A rock initially contained $68 \times 10^{-6} \mathrm{~g}$ of ${ }_{92}^{238} \mathrm{U}$. If the number of alpha particles that it would emit during its radioactive decay of ${ }_{92}^{238} \mathrm{U}$ to ${ }_{82}^{206} \mathrm{~Pb}$ in three half-lives is $Z \times 10^{18}$, then what is the value of $Z$ ? | 1.2 | 35 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $m$ be the minimum possible value of $\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)$, where $y_{1}, y_{2}, y_{3}$ are real numbers for which $y_{1}+y_{2}+y_{3}=9$. Let $M$ be the maximum possible value of $\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)$, where $x_{1}, x_{2}, x_{3}$ are positive real numbers for which $x_{1}+x_{2}+x_{3}=9$. Then what is the value of $\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)$? | 8 | 49 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b_{1}, b_{2}, b_{3}, \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a_{1}=b_{1}=c$, then what is the number of all possible values of $c$, for which the equality
\[
2\left(a_{1}+a_{2}+\cdots+a_{n}\right)=b_{1}+b_{2}+\cdots+b_{n}
\]
holds for some positive integer $n$? | 1 | 50 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} | ||
Let $f:[0,2] \rightarrow \mathbb{R}$ be the function defined by
\[
f(x)=(3-\sin (2 \pi x)) \sin \left(\pi x-\frac{\pi}{4}\right)-\sin \left(3 \pi x+\frac{\pi}{4}\right)
\]
If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$, then what is the value of $\beta-\alpha$? | 1 | 51 | {
"question_source": "daman1209arora/jeebench"
} | null | {
"dummy_field": null
} |
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