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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 val... | Okay, so I've got this problem here about a metal being heated in a furnace, and there's a sensor above it measuring the power radiated, P. The sensor displays the logarithm base 2 of P over P0, where P0 is a constant. When the metal is at 487°C, the sensor shows a value of 1. Now, they want to know what the sensor dis... | 9 | 14 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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}^{1... | Okay, so I need to find the maximum kinetic energy of the beta-particle when Boron-12 undergoes beta decay to an excited state of Carbon-12. Hmm, let me think about this step by step.
First, I remember that in beta decay, a neutron in the nucleus converts into a proton, emitting a beta particle (electron) and an antin... | 9 | 15 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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?
Please reason step by step, and put your final ... | Okay, so I need to figure out how many emission lines are present when a hydrogen atom in its ground state is irradiated by light with a wavelength of 970 Angstroms. Hmm, let's break this down step by step.
First, I remember that when atoms absorb energy, they can get excited to higher energy levels, and when they dr... | 6 | 16 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to find the ratio of the terminal velocities of two spheres, P and Q. Both spheres have the same density, 8 gm/cm³, but different diameters. Sphere P has a diameter of 1 cm, and Sphere Q has a diameter of 0.5 cm. They are dropped into different liquids with different densities and viscosities. Sphere P ... | 3 | 17 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 th... | Okay, so I've got this problem here with two inductors and a resistor all connected in parallel across a 5V battery. The question is asking for the ratio of the maximum to the minimum current drawn from the battery, which is I_max over I_min. Hmm, alright, let me try to figure this out step by step.
First, I remember ... | 8 | 18 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 }}... | Okay, so I've got this problem here about mole fractions, molarity, molality, and density. Hmm, I need to find the ratio of the molecular weights of the solute and solvent. Let me try to break this down step by step.
First, let me recall what mole fraction is. The mole fraction of a solute is the number of moles of th... | 9 | 32 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $x$?
Please reason step by st... | Okay, so I need to figure out how the diffusion coefficient of an ideal gas changes when both its absolute temperature and pressure are altered. The problem states that the temperature is increased four times and the pressure is doubled. I remember that the diffusion coefficient depends on the mean free path and the me... | 4 | 33 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\mathbf{X}$?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I've got this chemistry problem here, and I need to figure out how many moles of a sulfur-containing product are produced when permanganate anions oxidize thiosulfate anions in a neutral or faintly alkaline solution. The problem states that 8 moles of permanganate anion are used, and they quantitatively oxidiz... | 6 | 34 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
What is the total number of distinct $x \in \mathbb{R}$ for which $\left|\begin{array}{ccc}x & x^{2} & 1+x^{3} \\ 2 x & 4 x^{2} & 1+8 x^{3} \\ 3 x & 9 x^{2} & 1+27 x^{3}\end{array}\right|=10$?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the number of distinct real solutions x for which the determinant of this 3x3 matrix is equal to 10. Let me write down the matrix first to get a clear picture.
The matrix is:
| x x² 1 + x³ |
| 2x 4x² 1 + 8x³ |
| 3x 9x² 1 + 27x³ |
So, determinant equals 10. Hmm, determina... | 2 | 50 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $n$?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the smallest positive integer \( m \) such that the coefficient of \( x^2 \) in the expansion of \( (1+x)^2 + (1+x)^3 + \cdots + (1+x)^{49} + (1+mx)^{50} \) is equal to \( (3n + 1) \binom{51}{3} \) for some positive integer \( n \). Then, I need to determine the value of \( n \).
First, let me ... | 5 | 51 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
What is the total number of distinct $x \in[0,1]$ for which $\int_{0}^{x} \frac{t^{2}}{1+t^{4}} d t=2 x-1$?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the number of distinct solutions \( x \) in the interval \([0,1]\) that satisfy the equation:
\[
\int_{0}^{x} \frac{t^{2}}{1+t^{4}} dt = 2x - 1
\]
Hmm, let's break this down step by step. First, I know that the integral on the left is a function of \( x \), so let me denote that as \( F(x) \).... | 1 | 52 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $6(\alpha+\beta)$?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the value of \(6(\alpha + \beta)\) given that the limit as \(x\) approaches 0 of \(\frac{x^{2} \sin (\beta x)}{\alpha x - \sin x}\) is equal to 1. Hmm, let me break this down step by step.
First, I remember that when dealing with limits involving trigonometric functions like sine, it's often us... | 7 | 53 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $P^{2}=-I$?
Please reason step by step, and... | Okay, so I need to figure out how many ordered pairs (r, s) where r and s are in {1, 2, 3} satisfy the equation P² = -I, where P is the given matrix. Let me break this down step by step.
First, let me understand what z is. It's given as z = (-1 + √3 i)/2. Hmm, that looks familiar. Isn't that one of the cube roots of 1... | 1 | 54 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\alpha$?
Pleas... | Okay, so I've got this physics problem here about surface tension and energy change when a liquid drop divides into smaller drops. Let me try to break it down step by step.
First, the problem states that a liquid drop with radius R = 10^{-2} meters has a surface tension S = 0.1/(4π) N/m. It divides into K identical dr... | 6 | 8 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $n_{f}$?
Please reason ... | Okay, so I have this problem here about an electron in a hydrogen atom transitioning between energy levels. The question is about finding the smallest possible final quantum number, \( n_f \), given the ratio of the initial potential energy to the final potential energy is 6.25. Hmm, let me try to break this down step ... | 5 | 9 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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? (Give... | Okay, so I need to find the beat frequency when a stationary source emits a sound, which is then reflected by a moving car and superposed with the original sound. Hmm, beats occur when two sound waves of slightly different frequencies interfere, right? So the beat frequency is the difference between these two frequenci... | 6 | 11 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
${ }^{131} \mathrm{I}$ 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... | Okay, so I need to figure out the total volume of blood in a person's body based on the given information about the radioactive isotope I-131. Let me try to break this down step by step.
First, let's understand the problem. We have a person who received an injection of serum labeled with I-131. The initial activity of... | 5 | 12 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $N$?
Pl... | Okay, so I need to find the number of atoms in 256 grams of a crystalline solid. The solid has a face-centered cubic (FCC) structure with a cell edge length of 400 pm, and the density is given as 8 g/cm³. The answer should be in terms of N × 10²⁴, so I need to find N.
First, let me recall what I know about FCC structu... | 2 | 26 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 fou... | Okay, so I need to find the limiting molar conductivity (Λm⁰) of a weak monobasic acid. Hmm, let's see what information I have.
First, the solution is 0.0015 M. That's the concentration, right? So, c = 0.0015 mol/dm³ or 0.0015 mol/L. Got that.
They used a conductivity cell with platinum electrodes. The distance betwe... | 6 | 27 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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, \mat... | Okay, so I need to find the sum of the number of lone pairs of electrons on each central atom in four different species: [TeBr6]^2-, [BrF2]^+, SNF3, and [XeF3]^-.
Hmm, let me start by recalling that the number of lone pairs on a central atom can be determined using the Lewis structure approach. The formula I remember ... | 6 | 28 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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}=... | Okay, so I need to figure out how many diamagnetic species are among the given molecules and ions. The list is H₂, He₂⁺, Li₂, Be₂, B₂, C₂, N₂, O₂⁻, and F₂. Hmm, diamagnetic species are those with all their electrons paired, meaning they have no unpaired electrons. So, I need to determine the number of unpaired electron... | 5 | 29 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to figure out for how many values of \( p \) the circle given by the equation \( x^{2} + y^{2} + 2x + 4y - p = 0 \) and the coordinate axes have exactly three common points. Hmm, let's break this down step by step.
First, I remember that the general equation of a circle is \( x^{2} + y^{2} + Dx + Ey + ... | 2 | 44 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to find the limit as x approaches 0 of g(x), where g(x) is defined as the integral from x to π/2 of [f’(t) cosec t - cot t cosec t f(t)] dt. Hmm, let me start by understanding the problem step by step.
First, let me write down what I know. The function f is differentiable from R to R, and we have some ... | 2 | 45 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 infinit... | Okay, so I have this system of linear equations given by a matrix multiplied by a vector [x, y, z] equals another vector [1, -1, 1]. The matrix is a 3x3 matrix with entries that are powers of alpha. The question is asking, if this system has infinitely many solutions, what is the value of 1 + alpha + alpha squared?
Hm... | 1 | 46 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\frac{y}{9 x}$?
Please reason ... | Okay, so I need to figure out the value of y/(9x), where x is the number of 10-letter words with no repeated letters, and y is the number of 10-letter words where exactly one letter is repeated twice and all others are unique. The letters we can use are A, B, C, D, E, F, G, H, I, J. That's 10 letters in total.
First,... | 5 | 47 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the length of the smallest side of a right-angled triangle where the sides are in arithmetic progression and the area is 24. Hmm, let me think about how to approach this.
First, let's recall what arithmetic progression means. It means that each side increases by a common difference. So, if I de... | 6 | 48 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 seco... | Okay, so I need to find the value of τ where the magnitude of vector A plus vector B equals the square root of 3 times the magnitude of vector A minus vector B. Both vectors A and B are given, with A being straightforward and B having some time dependence because of the cosine and sine terms. Let me write down what I k... | 2 | 7 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to figure out the frequency of beats heard by the stationary man when two other men are walking in the same direction, blowing whistles. Let me try to break this down step by step.
First, let me visualize the scenario. There's a horizontal straight line, and two men are walking along it in the same dir... | 5 | 8 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $(2-\sqrt{3}) / \sqrt{10} s$,... | Okay, so I have this physics problem here where a ring and a disc are rolling down an inclined plane without slipping. They start from rest, side by side, at the top of the incline which is at a 60-degree angle. The time difference between them reaching the ground is given as (2 - √3)/√10 seconds, and I need to find th... | 0.75 | 9 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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} \m... | Okay, so I need to figure out the average intensity of light after it passes through a convex lens. The sunlight has an intensity of 1.3 kW/m², and it's incident normally on the lens. The lens has a focal length of 20 cm, and we're looking at the intensity at a distance of 22 cm on the other side of the lens. They told... | 130 | 13 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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)_{... | Okay, so I need to figure out how many diamagnetic species are among the given list. Hmm, diamagnetic means that all the electrons are paired, right? So, no unpaired electrons. I remember that diamagnetic substances are weakly repelled by a magnetic field because their electrons all pair up, so there's no net magnetic ... | 1 | 25 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to solve this chemistry problem. Let me read it again and make sure I understand what's being asked.
We have ammonium sulfate treated with calcium hydroxide, and the ammonia produced is completely used by nickel(II) chloride hexahydrate to form a stable coordination compound. Both reactions are 100% co... | 2992 | 26 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 ... | Okay, so I need to figure out the ratio of anions to cations in this new structure Z that's made from modifying the unit cell of MX, which has an NaCl structure. Hmm, let's break this down step by step.
First, I should probably recall what the NaCl structure looks like. NaCl has a face-centered cubic (FCC) lattice fo... | 3 | 27 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 chan... | Okay, so I've got this electrochemistry problem here. Let me try to figure it out step by step. Hmm, the problem is about an electrochemical cell, and I need to find the new concentration of Mg²+ when the cell potential changes.
Alright, first, let me write down the given information. The cell is Mg(s) | Mg²+(aq, 1M)... | 10 | 28 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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_{... | Okay, so I've got this problem about two liquids, A and B, forming an ideal solution. Hmm, ideal solutions, right? They obey Raoult's Law, which says that the partial vapor pressure of each component is proportional to its mole fraction in the solution. So, the vapor pressure of the solution is the sum of the partial p... | 19 | 30 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to find the solubility of a salt AB in a solution with pH 3. The solubility is given as Y × 10^-3 mol L^-1, and I need to find Y. The solubility product (Ksp) of AB is 2 × 10^-10, and the ionization constant (Ka) of HB is 1 × 10^-8.
First, I remember that AB is a salt of a weak acid, HB. So, when AB d... | 4.47 | 31 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
What is the value of
\[
\left(\left(\log _{2} 9\right)^{2}\right)^{\frac{1}{\log _{2}\left(\log _{2} 9\right)}} \times(\sqrt{7})^{\frac{1}{\log _{4} 7}}
\]?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the value of this expression:
\[
\left(\left(\log _{2} 9\right)^{2}\right)^{\frac{1}{\log _{2}\left(\log _{2} 9\right)}} \times(\sqrt{7})^{\frac{1}{\log _{4} 7}}
\]
Hmm, that looks a bit complicated, but let's break it down step by step. Maybe if I handle each part separately, it will make mor... | 8 | 43 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to figure out how many 5-digit numbers there are using the digits {1,2,3,4,5}, where each digit can be repeated, and the number has to be divisible by 4. Hmm, let's start by recalling how to determine if a number is divisible by 4.
From what I remember, a number is divisible by 4 if the number formed b... | 625 | 44 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $X \cup Y$?
Please reason step by step, and put your final answer ... | Okay, so I need to find the number of elements in the union of sets X and Y. Both X and Y are sets containing the first 2018 terms of their respective arithmetic progressions. Let me try to break this down step by step.
First, let's recall what an arithmetic progression (AP) is. An AP is a sequence of numbers where ea... | 3748 | 45 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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}, \... | Okay, so I need to find the number of real solutions to this 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 \(\l... | 2 | 46 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $[L]$?
Please reason step by step, and put your final answer withi... | Okay, so I need to find the limit \( L \) of the sequence \( y_n \) as \( n \) approaches infinity, where
\[
y_{n} = \frac{1}{n} \left( (n+1)(n+2) \cdots (n+n) \right)^{\frac{1}{n}}
\]
And then find the greatest integer less than or equal to \( L \), which is denoted as \( [L] \).
First, let me try to understand wha... | 1 | 47 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 \co... | Okay, so I have this vector problem to solve. Let me try to break it down step by step. I'll start by writing down what's given and then figure out how to approach it.
We have two unit vectors, \(\vec{a}\) and \(\vec{b}\), which are orthogonal because their dot product is zero, \(\vec{a} \cdot \vec{b} = 0\). That's he... | 3 | 48 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\frac{b}{a}$?
Please reason step by step, and put y... | Okay, so I have this problem here where I need to find the value of \(\frac{b}{a}\) given some conditions about a trigonometric equation. Let me try to unpack this step by step.
First, the equation given is:
\[
\sqrt{3} a \cos x + 2 b \sin x = c
\]
where \(a\), \(b\), and \(c\) are non-zero real numbers, and \(x\) i... | 0.5 | 49 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 \... | Okay, so I need to find the value of \( n \) such that the area taken away by farmer \( F_2 \) is exactly 30% of the area of triangle \( PQR \). The triangle has vertices at \( P(0,0) \), \( Q(1,1) \), and \( R(2,0) \). The region taken away is between the side \( PQ \) and a curve of the form \( y = x^n \) where \( n ... | 4 | 50 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 co... | Okay, so I need to find the displacement of a block at time t equals tau, which is 4 seconds. The block is on a horizontal surface with a thin oil layer, so I think that means there's some friction involved because oil usually reduces friction, but maybe it's still there. The block has a mass of 0.4 kg and is initially... | 6.3 | 7 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 ... | Okay, so I have this physics problem here, and I need to figure out the maximum height the ball reaches after it bounces. Let me try to break it down step by step.
First, the problem says a ball is projected from the ground at a 45-degree angle. It reaches a maximum height of 120 meters and then returns to the ground.... | 30 | 8 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 o... | Okay, so I need to find the maximum speed attained by a particle under the influence of a time-varying electric field. Let me start by jotting down the given information.
The particle has a mass \( m = 10^{-3} \, \text{kg} \) and a charge \( q = 1.0 \, \text{C} \). It's initially at rest, so its initial velocity \( v(... | 2 | 9 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 g... | Okay, so I need to figure out the shunt resistance required to convert a galvanometer into an ammeter that can measure up to 1.0 A. Let me start by recalling what I know about galvanometers and ammeters.
First, a galvanometer is a device used to detect and measure small electric currents. To convert it into an ammeter... | 5.56 | 10 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 whic... | Okay, so I need to solve this problem about a steel wire, a vernier scale, and figuring out which division coincides after increasing the load. Hmm, let's see.
First, let me try to understand the setup. There's a steel wire with a diameter of 0.5 mm. Its Young's modulus is given as 2 x 10^11 N/m². It's carrying a load... | 3 | 11 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to find the decrease in internal energy when one mole of a monatomic ideal gas undergoes an adiabatic expansion, making its volume eight times the initial volume. The initial temperature is 100 K, and R is given as 8.0 J/mol·K. Hmm, let's break this down step by step.
First, I remember that for an adia... | 900 | 12 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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. Assum... | Okay, so I've got this physics problem here about the photoelectric effect. Hmm, let me try to break it down step by step. I remember that the photoelectric effect involves light hitting a metal surface and causing electrons to be emitted. The key points here are the power of the light, the work function, the frequency... | 24 | 13 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to find the atomic number \( Z \) of a hydrogen-like ionized atom. The problem says that the energy difference between the \( n=2 \) to \( n=1 \) transition is \( 74.8 \) eV higher than the \( n=3 \) to \( n=2 \) transition. Also, the ionization energy of hydrogen is given as \( 13.6 \) eV. Hmm, let's b... | 3 | 14 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to figure out how many compounds from the given list have at least one bridging oxo group. Hmm, I'm a bit rusty on my chemistry, but I remember that bridging oxo groups have to do with how oxygen atoms connect different atoms in a molecule. Let me try to recall.
First, what exactly is a bridging oxo gr... | 6 | 25 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\mat... | Okay, so I need to figure out the weight of lead (Pb) produced per kilogram of oxygen (O₂) consumed in this process. Hmm, let's see. The problem mentions that Galena, which is an ore, is partially oxidized by passing air through it at high temperatures. Then, after some time, the air supply is stopped, but heating cont... | 6.47 | 26 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
To measure the quantity of $\mathrm{MnCl}_{2}$ dissolved in an aqueous solution, it was completely converted to $\mathrm{KMnO}_{4}$ using the reaction, $\mathrm{MnCl}_{2}+\mathrm{K}_{2} \mathrm{~S}_{2} \mathrm{O}_{8}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{KMnO}_{4}+\mathrm{H}_{2} \mathrm{SO}_{4}+\mathrm{HCl}$ (e... | Okay, so I need to figure out how much MnCl₂ was originally dissolved in the solution. Let me go through the problem step by step.
First, the reaction given is:
MnCl₂ + K₂S₂O₈ + H₂O → KMnO₄ + H₂SO₄ + HCl
This equation isn't balanced, so I should probably start by balancing it. That way, I can understand the stoichio... | 126 | 27 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 g... | Okay, so I need to find the minimum partial pressure of hydrogen gas, \( p_{\mathrm{H}_2} \), required to prevent the oxidation of copper by water vapor at 1250 K. The given problem states that nitrogen gas with 1 mole% of water vapor is used, and the reaction involved is:
\[ 2 \mathrm{Cu}(\mathrm{s}) + \mathrm{H}_2 \... | -14.6 | 30 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 r... | Okay, so I've got this chemistry problem here about a reversible reaction, and I need to find the absolute value of the Gibbs free energy change, ΔG°, at 300 K. Let me try to break this down step by step.
First, the reaction given is:
A(g) + B(g) ⇌ AB(g)
They mention that the activation energy of the backward react... | 8500 | 31 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I've got this electrochemistry problem here. Let me try to understand what it's asking. It's about an electrochemical cell with specific electrodes and solutions. The cell is written as A(s)|A^n+(aq, 2M) || B^{2n+}(aq, 1M)|B(s). Hmm, so I need to figure out the entropy change, ΔS°, for the cell reaction per mo... | -11.62 | 32 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
What is the value of the integral
\[
\int_{0}^{\frac{1}{2}} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{\frac{1}{4}}} d x
\]?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to compute this integral:
\[
\int_{0}^{\frac{1}{2}} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{\frac{1}{4}}} dx
\]
Hmm, let me first simplify the integrand to see if I can make it more manageable. The denominator is \(\left((x+1)^2(1-x)^6\right)^{1/4}\). Let me break that down.
First, the exp... | 2 | 43 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $P$?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the maximum determinant of a 3x3 matrix where each entry is either -1, 0, or 1. Hmm, determinants can be tricky, but I remember that the determinant of a 3x3 matrix can be calculated using the rule of Sarrus or the general method of expansion by minors. Maybe I should start by recalling the form... | 4 | 44 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\frac{1}{5 !}(\beta-\alpha)$?
Please reason step by step, and put your final answer w... | Okay, so I need to figure out the value of (1/5!)(β - α), where α is the number of one-one functions from set X to Y, and β is the number of onto functions from Y to X. X has 5 elements and Y has 7 elements. Hmm, let's break this down step by step.
First, let's recall what one-one functions and onto functions are. A o... | 119 | 45 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\lim _{x \rightarrow-\infty} f(x)$?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the limit of f(x) as x approaches negative infinity, given that f(0) = 0 and the differential equation dy/dx = (2 + 5y)(5y - 2). Hmm, let's see. I remember that solving differential equations often involves separation of variables, so maybe I can try that here.
First, let me write down the diff... | 0.4 | 46 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to solve this problem where we have a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that's differentiable, satisfies \( f(0) = 1 \), and for all real numbers \( x \) and \( y \), it satisfies the equation:
\[
f(x + y) = f(x) f'(y) + f'(x) f(y)
\]
And we need to find \( \log_e(f(4)) \). Hmm, alri... | 2 | 47 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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,... | Okay, so I need to solve this geometry problem. Let me read it again and try to understand all the parts.
We have a point P in the first octant. First octant means all coordinates x, y, z are positive, right? So P has coordinates (a, b, c) where a, b, c > 0.
The image of P in the plane x + y = 3 is Q. Image here mean... | 8 | 48 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $O$ su... | Okay, so I need to find the value of |(p × q) × (r × t)| where p, q, r, t are vectors from the center S of the cube to the points P, Q, R, and T respectively. Let me try to visualize this cube first.
The cube is in the first octant with sides OP, OQ, and OR each of length 1 along the x, y, and z-axes. So, the cube has... | 0.5 | 49 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\frac{1}{1430} X$?
Please reason step by step, and put you... | Okay, so I need to find the value of \( \frac{1}{1430} X \) where \( X \) is given by the series:
\[
X = \left({ }^{10} C_{1}\right)^{2} + 2\left({ }^{10} C_{2}\right)^{2} + 3\left({ }^{10} C_{3}\right)^{2} + \cdots + 10\left({ }^{10} C_{10}\right)^{2}
\]
Hmm, okay. So, \( X \) is a sum where each term is \( k \times... | 646 | 50 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $K_{m}=K\left(1+\frac{m}{N}\right)$. F... | Okay, so I need to find the capacitance of a parallel plate capacitor that has multiple dielectric layers. Let me start by recalling the basic formula for capacitance. For a single dielectric, the capacitance is given by \( C = \frac{\epsilon_0 \epsilon_r A}{d} \), where \( \epsilon_r \) is the relative permittivity or... | 1 | 16 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 ... | Okay, so I need to figure out how many of the given molecules have covalent bonds between two atoms of the same kind. The molecules listed are: B₂H₆, B₃N₃H₆, N₂O, N₂O₄, H₂S₂O₃, and H₂S₂O₈.
First, let me recall what a covalent bond is. Covalent bonds are formed when two atoms share electrons. They can be between atoms... | 4 | 31 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\mathrm{Y}$?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to figure out the total number of lone pairs of electrons on the whole molecule of Y, which is produced when XeF4 reacts with O2F2 at 143 K. Hmm, I'm not entirely sure about the reaction, but I'll try to work it out step by step.
First, I remember that XeF4 is a common xenon compound. Xenon is in group... | 19 | 32 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 ma... | Okay, so I need to find the depression of the freezing point of benzene when a solute is added to it. Let me see, I have some data here: 0.5 grams of a non-volatile, non-ionic solute is dissolved in 39 grams of benzene. The vapor pressure of benzene decreases from 650 mmHg to 640 mmHg. Also, the molar mass of benzene i... | 1.02 | 34 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to figure out the rate of the reaction given the concentrations of A, B, and C. The reaction is A + B + C → Product, and I have some kinetic data from four experiments. The goal is to find the rate when [A] is 0.15 mol/dm³, [B] is 0.25 mol/dm³, and [C] is 0.15 mol/dm³. The rate will be Y × 10⁻⁵ mol/dm³ ... | 6.75 | 35 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the minimum of the set \(\left\{\left|a + b \omega + c \omega^{2}\right|^{2} : a, b, c \text{ are distinct non-zero integers}\right\}\), where \(\omega \neq 1\) is a cube root of unity. Hmm, cube roots of unity... I remember they satisfy the equation \(\omega^3 = 1\), so the cube roots are 1, \(... | 3 | 49 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the value of \( a + d \) where the intersection of three arithmetic progressions \( AP(1; 3) \), \( AP(2; 5) \), and \( AP(3; 7) \) results in another arithmetic progression \( AP(a; d) \). Hmm, let me break this down step by step.
First, let me recall what an arithmetic progression (AP) is. An... | 157 | 50 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 matr... | Okay, so I need to find the conditional probability \( P(E_1 \mid E_2) \), which is the probability that a randomly chosen 3x3 binary matrix (with entries 0 or 1) has a determinant of zero, given that the sum of its entries is 7.
First, I remember that conditional probability is defined as:
\[
P(E_1 \mid E_2) = \fra... | 0.5 | 51 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to find the length of the line segment AC, where C is the intersection point of a common tangent T to circles Γ_A and Γ_B, and the line passing through points A and B. The circles Γ_A and Γ_B have radii 2 and 1, respectively, with centers at A(2,3) and B, which is the reflection of A over the line 8x - ... | 10 | 52 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the value of \(27I^2\) where \(I\) is given by the integral:
\[
I = \frac{2}{\pi} \int_{-\pi/4}^{\pi/4} \frac{dx}{(1 + e^{\sin x})(2 - \cos 2x)}
\]
Hmm, let me start by understanding the integral. It's from \(-\pi/4\) to \(\pi/4\), which is a symmetric interval around zero. Maybe I can exploit... | 4 | 53 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $A B C$ ... | Okay, so I need to find the area of triangle ABC formed by the points where three given lines intersect the plane x + y + z = 1. Then, I have to compute (6Δ)², where Δ is the area of the triangle. Hmm, let's break this down step by step.
First, let me understand the given lines. They are given in vector form, so I nee... | 0.75 | 54 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 i... | Okay, so I need to find the energy of the gamma photon emitted when the Rn nucleus transitions from its excited state to the ground state. Hmm, let me think about this step by step.
First, I know that when Ra-226 undergoes alpha decay, it emits an alpha particle (which is a helium nucleus) and transforms into Rn-222. ... | 135 | 13 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 ... | Okay, so I need to figure out the percentage error in the measurement of the focal length of a convex lens using an optical bench setup. Let me go through the problem step by step.
First, the optical bench has a scale that's 1.5 meters long, and each centimeter is divided into four equal parts. That means each divisio... | 0.69 | 14 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $=18 \mathrm{~g} \mathrm{~mol}^{-1}$ )
Please reason step by step, and put your final answer within \boxed{}... | Okay, so I need to figure out how much water is produced when 1 mole of rhombic sulfur is oxidized by concentrated nitric acid to its highest oxidation state. Hmm, let me break this down step by step.
First, I remember that sulfur can have various oxidation states, and the highest one is +6. So, when sulfur is oxidize... | 288 | 27 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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}_{... | Okay, so I need to figure out the total number of cis N-Mn-Cl bond angles in the complex cis-[Mn(en)₂Cl₂]. Hmm, let me break this down step by step.
First, I remember that en is ethylenediamine, which is a bidentate ligand. That means each en molecule can bind to the central metal ion, manganese in this case, through ... | 6 | 28 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 pressu... | Okay, so I've got this chemistry problem here about the decomposition of dinitrogen pentoxide. Hmm, let me see. The reaction is 2 N₂O₅(g) → 2 N₂O₄(g) + O₂(g). It's happening in a closed cylinder under isothermal and isochoric conditions, which means the temperature and volume are constant. The initial pressure is 1 atm... | 2.3 | 29 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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} \mat... | Okay, so I need to find the molarity of a urea solution. Hmm, molarity is the number of moles of solute per liter of solution, right? The problem gives me the mole fraction of urea as 0.05, the mass of water as 900 grams, and the density of the solution as 1.2 g/cm³. They also provided the molar masses: urea is 60 g/mo... | 2.98 | 30 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
What is the total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the molecular formula $\mathrm{C}_{4} \mathrm{H}_{8} \mathrm{O}$?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to figure out the total number of isomers, both structural and stereoisomers, for cyclic ethers with the molecular formula C₄H₈O. Hmm, let me start by understanding what cyclic ethers are. Cyclic ethers are compounds where an oxygen atom is part of a ring structure, connected to two carbon atoms. The si... | 10 | 32 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\sum_{k=0}^{n} \frac{{ }^{n} C_{k}}{k+1}$?
Please reason step ... | Okay, so I've got this determinant problem here, and I need to find the value of this sum involving binomial coefficients. Let me start by writing down the determinant equation they gave me:
\[
\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}... | 6.2 | 45 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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?
Please reason step by step, and pu... | Okay, so I have this problem where five people, A, B, C, D, and E, are sitting in a circular arrangement. Each person is going to get a hat, and each hat can be one of three colors: red, blue, or green. The condition is that adjacent people must have different colored hats. I need to figure out how many ways there are ... | 30 | 46 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $1 \leq|B|<|A|$?
Please reason step by step, and put you... | Okay, so I need to figure out how many ordered pairs (A, B) there are such that A and B are independent events on the sample space S = {1,2,3,4,5,6}, and the size of B is less than the size of A, with both sizes being at least 1. Hmm, okay.
First, let me recall that two events A and B are independent if and only if P(... | 422 | 47 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]$?
Please reason step by step, and put your final answer within \boxed{}... | Okay, so I need to find the value of the inverse secant function of a certain sum. The expression is:
\[
\sec ^{-1}\left(\frac{1}{4} \sum_{k=0}^{10} \sec \left(\frac{7 \pi}{12}+\frac{k \pi}{2}\right) \sec \left(\frac{7 \pi}{12}+\frac{(k+1) \pi}{2}\right)\right)
\]
And I need to find this value in the interval \(\left... | 0 | 48 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
What is the value of the integral
\[
\int_{0}^{\pi / 2} \frac{3 \sqrt{\cos \theta}}{(\sqrt{\cos \theta}+\sqrt{\sin \theta})^{5}} d \theta
\]?
Please reason step by step, and put your final answer within \boxed{}. | Okay, so I need to find the value of the integral:
\[
\int_{0}^{\pi / 2} \frac{3 \sqrt{\cos \theta}}{(\sqrt{\cos \theta}+\sqrt{\sin \theta})^{5}} d \theta
\]
Hmm, this looks a bit complicated, but maybe I can simplify it somehow. Let me think about substitution or maybe some symmetry in the integral.
First, I notice... | 0.5 | 49 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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. Aft... | Okay, so I need to figure out the value of \( x_R \), which is the distance from the center of the scale where the right finger stops slipping. The setup is that there's a meter scale (100 cm) placed horizontally on two extended index fingers. Initially, the left finger is at 0.00 cm and the right finger is at 90.00 cm... | 25.6 | 13 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 {is... | Okay, so I've got this problem about helium gas expanding first isothermally and then adiabatically. I need to find the ratio of the work done during these two processes and express it as f times the natural log of 2. Hmm, let's break this down step by step.
First, I remember that for an isothermal process, the work d... | 1.77 | 16 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 ... | Okay, so I have this physics problem here, and I need to figure out the smallest percentage change in the length of a pipe required for resonance when a tuning fork is moved. Let me read it again:
"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 ms⁻¹... | 0.62 | 17 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 thro... | Okay, so I've got this problem about electric flux through two spherical surfaces enclosing a charged disc. Hmm, let me try to visualize it. There's a circular disc with radius R, and it has a surface charge density that varies with the radius. The charge density is given by σ(r) = σ₀(1 - r/R), where σ₀ is a constant a... | 6.4 | 18 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
$5.00 \mathrm{~mL}$ 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 concentr... | Okay, so I need to find the concentration of NaOH solution using the titration data provided. Hmm, let's see. I remember that titration involves neutralizing an acid with a base, and the point where they neutralize each other is called the equivalence point. The indicator, phenolphthalein, changes color when the soluti... | 0.11 | 31 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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{~mo... | Okay, so I need to figure out the change in temperature of a monoatomic ideal gas when it's compressed using the work derived from a hydrogen-oxygen fuel cell. Let me break this down step by step.
First, I should understand what the fuel cell does. It's a hydrogen-oxygen fuel cell with 70% efficiency. The reaction giv... | 13.32 | 33 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 ... | Okay, so I need to figure out the volume of hydrogen gas produced when aluminum reacts with sulfuric acid. Let me start by writing down the balanced chemical equation for this reaction. I remember that aluminum reacts with sulfuric acid to produce aluminum sulfate, water, and hydrogen gas.
The reaction is:
2 Al (s)... | 6.15 | 34 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
${ }_{92}^{238} \mathrm{U}$ 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 ${ ... | Okay, so I need to figure out how many alpha particles are emitted when a certain amount of uranium-238 decays into lead-206 over three half-lives. Let me break this down step by step.
First, I remember that uranium-238 undergoes radioactive decay by emitting alpha particles. Each time an alpha particle is emitted, th... | 1.2 | 35 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 ... | Okay, so I need to solve this problem where I have to find the minimum value of a logarithm expression and the maximum value of another logarithm expression, and then combine them using logarithms. Hmm, let me try to break it down step by step.
First, let's look at the problem again:
We have m defined as the minimum ... | 8 | 49 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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... | Okay, so I need to solve this problem where we have two sequences: an arithmetic progression (AP) and a geometric progression (GP). Both start with the same initial term, which is denoted as c. The AP has a common difference of 2, and the GP has a common ratio of 2. The problem states that for some positive integer n, ... | 1 | 50 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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 $\beta-\alpha$?
Please reason ... | Okay, so I need to find the value of β - α where the function f(x) is non-negative on the interval [α, β] within [0, 2]. The function is given by:
f(x) = (3 - sin(2πx)) sin(πx - π/4) - sin(3πx + π/4)
Hmm, that looks a bit complicated. Maybe I can simplify it using some trigonometric identities. Let me recall some pro... | 1 | 51 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Qwen-32B |
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