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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 have this problem where a metal is heated in a furnace, and there's a sensor above it that measures the power radiated by the metal. The sensor displays the value of log base 2 of (P/P₀), where P is the power radiated and P₀ is a constant. When the metal is at 487°C, the sensor shows 1. Now, I need to find t... | 9 | 14 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about beta decay, and I need to find the maximum kinetic energy of the beta-particle. Let me try to figure this out step by step.
First, the problem says that the isotope Boron-12, which is written as _{5}^{12}B, has a mass of 12.014 u. It undergoes beta decay to become Carbon-12 in an exc... | 9 | 15 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where a hydrogen atom in its ground state is irradiated by light with a wavelength of 970 Å, and I need to find the number of lines present in its emission spectrum. The given constants are \( h c / e = 1.237 \times 10^{-6} \) eV·m and the ground state energy of the hydrogen atom is -1... | 6 | 16 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this physics problem here about two solid spheres, P and Q, being dropped into different liquids, and I need to find the ratio of their terminal velocities. Hmm, terminal velocity is the maximum velocity a falling object reaches when the forces balance out, right? So, I remember that terminal velocity d... | 3 | 17 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem where two inductors, L₁ and L₂, with inductances 1 mH and 2 mH respectively, and internal resistances 3 Ω and 4 Ω, are connected in parallel with a resistor R of 12 Ω, all across a 5 V battery. The circuit is switched on at time t=0, and I need to find the ratio of the maximum to the minimu... | 8 | 18 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I need to find the ratio of the molecular weights of the solute and solvent. Let me try to figure this out step by step.
First, the problem says that the mole fraction of the solute is 0.1 at 298 K. Molarity and molality are given to be the same for this solution at that tempera... | 9 | 32 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about the diffusion coefficient of an ideal gas. Hmm, I remember that the diffusion coefficient depends on a few factors, but I need to recall exactly which ones. The problem says it's proportional to the mean free path and the mean speed. Let me jot that down.
So, if I denote the diffusio... | 4 | 33 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here: In a neutral or faintly alkaline solution, 8 moles of permanganate anion (MnO4^-) quantitatively oxidize thiosulphate anions (S2O3^2-) to produce X moles of a sulfur-containing product. I need to find the magnitude of X. Hmm, let's try to figure this out step by step.
First... | 6 | 34 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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{}. | Alright, so I have this determinant problem to solve. It says:
Find the total number of distinct real numbers \( x \) for which the determinant of the following matrix is equal to 10:
\[
\begin{vmatrix}
x & x^2 & 1 + x^3 \\
2x & 4x^2 & 1 + 8x^3 \\
3x & 9x^2 & 1 + 27x^3
\end{vmatrix} = 10
\]
Hmm, okay. So, I need to ... | 2 | 50 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I need to find the smallest positive integer \( m \) such that the coefficient of \( x^2 \) in the expansion of \( (1+x)^2 + (1+x)^3 + \cdots + (1+x)^{49} + (1 + m x)^{50} \) is equal to \( (3n + 1)^{51} \binom{51}{3} \) for some positive integer \( n \). Then, I have to find the... | 5 | 51 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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{}. | Alright, so I have this problem here: I need to find the total number of distinct x in the interval [0,1] for which the integral from 0 to x of (t²)/(1 + t⁴) dt equals 2x - 1. Hmm, okay. Let me try to figure this out step by step.
First, I know that the integral is equal to 2x - 1. So, I can write the equation as:
∫₀... | 1 | 52 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this limit problem here: the limit as x approaches 0 of (x² sin(βx)) divided by (αx - sin x), and it's given that this limit equals 1. I need to find the value of 6 times (α + β). Hmm, let me think about how to approach this.
First, I remember that when dealing with limits that result in indeterminate ... | 7 | 53 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem where I need to find the total number of ordered pairs (r, s) with r and s each being 1, 2, or 3. The matrix P is given as:
P = [ [(-z)^r, z^{2s}], [z^{2s}, z^r] ]
And I need to find when P squared equals -I, the identity matrix of order 2. The complex number z is given as (-1 + sqrt(3)i)... | 1 | 54 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here about a drop of liquid that's splitting into smaller drops. Let me try to understand what it's asking.
First, the liquid drop has a radius of R = 10^-2 meters, which is 0.01 meters. The surface tension is given as S = 0.1/(4π) N/m. The drop divides itself into K identical smaller dro... | 6 | 8 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 transitioning between energy levels in a hydrogen atom. The question is asking for the smallest possible final quantum number \( n_f \) given that the ratio of the initial potential energy \( V_i \) to the final potential energy \( V_f \) is 6.25. Hmm, let me try to b... | 5 | 9 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this physics problem here about sound reflection and beat frequencies. Hmm, let me try to figure it out step by step. I'm a bit rusty on this, but I'll take it slowly.
Alright, the problem says there's a stationary source emitting a sound with frequency \( f_0 = 492 \) Hz. A car is approaching this sou... | 6 | 11 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
${ }^{131} \mathrm{I}$ 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 have this problem about radioactive decay, specifically involving Iodine-131. Let me try to wrap my head around it step by step.
First, let me restate the problem to make sure I understand it. We have a serum with Iodine-131, which is radioactive. It has a half-life of 8 days. This serum is injected into a... | 5 | 12 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here about a crystalline solid with a face-centered cubic (FCC) structure. The cell edge is 400 pm, and the density is 8 g/cm³. I need to find the number of atoms in 256 grams of this crystal, and express it as N × 10²⁴. Hmm, okay, let me think about how to approach this.
First, I remember... | 2 | 26 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about finding the limiting molar conductivity of a weak monobasic acid using the given data. Let me try to figure this out step by step.
First, I know that conductance (σ) is related to the molar conductivity (Λm°) by the formula:
σ = Λm° × c
But wait, actually, isn't conductance also re... | 6 | 27 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here, and I need to find the sum of the number of lone pairs of electrons on each central atom in four different species. The species are [TeBr6]^2-, [BrF2]^+, SNF3, and [XeF3]^-. I'm a bit confused about how to approach this, but I'll try to work through each one step by step.
F... | 6 | 28 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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}=... | Alright, so I have this chemistry problem here, and I need to figure out how many diamagnetic species there are among a list of diatomic molecules. The species given are H₂, He₂⁺, Li₂, Be₂, B₂, C₂, N₂, O₂⁻, and F₂. The atomic numbers are provided: H=1, He=2, Li=3, Be=4, B=5, C=6, N=7, O=8, F=9.
First, I should rememb... | 5 | 29 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here: I need to find how many values of \( p \) make the circle defined by the equation \( x^{2} + y^{2} + 2x + 4y - p = 0 \) intersect the coordinate axes at exactly three points. Hmm, let's break this down step by step.
First, I remember that the coordinate axes are the x-axis and y-axis... | 2 | 44 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I need to find the limit of \( g(x) \) as \( x \) approaches 0. Let me try to figure this out step by step.
First, let's write down what's given. The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is differentiable, with \( f(0) = 0 \) and \( f\left(\frac{\pi}{2}\right) = ... | 2 | 45 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 linear system of equations given by a matrix and a vector, and I need to find the value of \(1 + \alpha + \alpha^2\) such that the system has infinitely many solutions. Hmm, I remember that for a system to have infinitely many solutions, the determinant of the coefficient matrix must be zero. That ... | 1 | 46 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I need to find the value of y divided by 9x, where x is the number of 10-letter words formed from the letters A, B, C, D, E, F, G, H, I, J without repeating any letters, and y is the number of such words where exactly one letter is repeated twice and no other letters are repeated... | 5 | 47 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem where the sides of a right-angled triangle are in arithmetic progression, and the area is 24. I need to find the length of the smallest side. Hmm, let me think about how to approach this.
First, let me recall what an arithmetic progression is. It's a sequence where each term after the firs... | 6 | 48 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem with two vectors, A and B. Vector A is given as a i hat, which I think means it's along the x-axis with magnitude a. Vector B is a bit more complicated: it's a times (cos ωt i hat + sin ωt j hat). Hmm, so vector B is also a vector with magnitude a, but it's rotating in the plane because it ... | 2 | 7 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this physics problem here, and I need to figure out the frequency of beats heard by a stationary man when two other men are walking and blowing whistles. Hmm, let me try to break this down step by step.
First, let me understand the problem. There are three men: two are walking in the same direction alo... | 5 | 8 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 about a ring and a disc rolling down an inclined plane. They start from rest at the top, side by side, and the plane is inclined at 60 degrees. The time difference between them reaching the ground is (2 - sqrt(3))/sqrt(10) seconds. I need to find the height of the top of the in... | 0.75 | 9 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this physics problem here about light intensity and a convex lens. Let me try to figure it out step by step. Hmm, the problem says that sunlight with an intensity of 1.3 kW per square meter is incident normally on a thin convex lens with a focal length of 20 cm. I need to find the average intensity of t... | 130 | 13 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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)_{... | Alright, so I have this chemistry problem here, and I need to figure out how many diamagnetic species are among the given list. Diamagnetic means that the species has all its electrons paired, right? So, their magnetic properties are such that they are not attracted to a magnetic field, unlike paramagnetic species whic... | 1 | 25 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here, and I need to figure out the combined weight of gypsum and a nickelammonia coordination compound produced from ammonium sulfate and nickel chloride hexahydrate. Hmm, let me try to break this down step by step.
First, let me write down the given data. I have 1584 grams of am... | 2992 | 26 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 ... | Alright, so I have this problem about ionic solids and unit cell structures. The compound in question is MX, which has the NaCl structure. I need to construct a new structure Z by following a series of steps. Then, I need to find the ratio of anions to cations in Z. Hmm, okay, let's break this down step by step.
First... | 3 | 27 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this electrochemical cell problem here. Let me try to figure it out step by step. The cell is Mg(s) | Mg²+ (aq, 1M) || Cu²+ (aq, 1M) | Cu(s), and the standard E° is 2.70 V at 300 K. Then, when the concentration of Mg²+ is changed to x M, the cell potential becomes 2.67 V at the same temperature. I need ... | 10 | 28 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about vapor pressure of solutions, and I need to find the ratio of mole fractions \( \frac{x_A}{x_B} \) in a solution. Let me try to figure this out step by step.
First, the problem says that liquids A and B form an ideal solution over the entire range of composition. That means I can use ... | 19 | 30 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here, and I need to find the value of Y, which is the solubility of a salt AB at pH 3. The given information is that the solubility product (Ksp) of AB is 2 × 10^-10, and the ionization constant (Ka) of HB is 1 × 10^-8. Hmm, let me try to figure this out step by step.
First, I kn... | 4.47 | 31 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this expression to evaluate:
\[
\left(\left(\log _{2} 9\right)^{2}\right)^{\frac{1}{\log _{2}\left(\log _{2} 9\right)}} \times(\sqrt{7})^{\frac{1}{\log _{4} 7}}
\]
Hmm, that looks a bit intimidating at first glance, but maybe if I break it down step by step, it'll make more sense. Let me write it down... | 8 | 43 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem: I need to find the number of 5-digit numbers that are divisible by 4. The digits can be from the set {1,2,3,4,5}, and repetition of digits is allowed. Hmm, okay. Let me try to figure this out step by step.
First, I remember that a number is divisible by 4 if the last two digits form a num... | 625 | 44 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem where I need to find the number of elements in the union of two sets, X and Y. Set X is the first 2018 terms of the arithmetic progression starting with 1, 6, 11, and so on. Set Y is the first 2018 terms of another arithmetic progression starting with 9, 16, 23, etc. I need to figure out ho... | 3748 | 45 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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}, \... | Alright, so I have this equation to solve:
\[
\sin ^{-1}\left(\sum_{i=1}^{\infty} x^{i+1}-x \sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^{i}\right)=\frac{\pi}{2}-\cos ^{-1}\left(\sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^{i}-\sum_{i=1}^{\infty}(-x)^{i}\right)
\]
And I need to find the number of real solutions in the... | 2 | 46 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I need to find the limit of a sequence \( y_n \) as \( n \) approaches infinity, and then take the floor of that limit. The sequence is defined as:
\[
y_{n} = \frac{1}{n} \left( (n+1)(n+2) \cdots (n + n) \right)^{\frac{1}{n}}
\]
Hmm, let me try to understand this. For each posi... | 1 | 47 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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... | Alright, so I have this vector problem here, and I need to find the value of 8 times the square of cosine alpha, given some conditions. Let me try to parse this step by step.
First, let me restate the problem in my own words to make sure I understand it. We have two unit vectors, a and b, and their dot product is zero... | 3 | 48 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 where I need to find the value of b/a given that the equation √3 a cos x + 2b sin x = c has two distinct real roots α and β in the interval [-π/2, π/2], and that α + β = π/3. Hmm, let's see how to approach this.
First, the equation is √3 a cos x + 2b sin x = c. This looks like a linear com... | 0.5 | 49 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem where a farmer F1 has a triangular piece of land with vertices at P(0,0), Q(1,1), and R(2,0). Farmer F2 takes away a region between the side PQ and a curve of the form y = x^n, where n > 1. The area taken away by F2 is 30% of the area of triangle PQR. I need to find the value of n.
First, ... | 4 | 50 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this physics problem here, and I need to figure out the displacement of a block after a certain time. Let me try to understand what's going on.
Alright, the problem says there's a solid horizontal surface covered with a thin layer of oil. There's a rectangular block of mass 0.4 kg resting on this surfa... | 6.3 | 7 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 about a ball being projected at a 45-degree angle, reaching a maximum height of 120 meters, and then bouncing back with some energy loss and changing direction. I need to find the maximum height after the bounce. Hmm, let's try to break this down step by step.
First, the ball ... | 30 | 8 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this physics problem here, and I need to figure out the maximum speed of a charged particle under an oscillating electric field. Let me try to break it down step by step.
First, let me read the problem again to make sure I understand it correctly. There's a particle with a mass of 10^-3 kg and a charge... | 2 | 9 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about converting a moving coil galvanometer into an ammeter by adding a shunt resistance. Hmm, I remember that galvanometers are used to measure current by the torque they experience when a current flows through them. The idea is that the coil in the galvanometer twists, causing a deflectio... | 5.56 | 10 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this physics problem here, and I need to figure out how to solve it. Let me read it again and understand what's given and what's being asked.
We have a steel wire with a diameter of 0.5 mm and a Young's modulus of 2 × 10^11 N/m². The wire is carrying a load of mass M, and its length is 1.0 m. There's a... | 3 | 11 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where one mole of a monatomic ideal gas undergoes an adiabatic expansion, and its volume becomes eight times its initial value. The initial temperature is 100 K, and the universal gas constant R is given as 8.0 J/mol·K. I need to find the decrease in internal energy. Hmm, let me think ... | 900 | 12 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this physics problem here about a photoelectric experiment. Let me try to figure it out step by step. I'm a bit rusty on some of these concepts, but I'll take it slowly.
First, the problem says that a parallel beam of monochromatic light with a power of 200 W is incident on a perfectly absorbing cathod... | 24 | 13 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about a hydrogen-like ionized atom with a single electron. The question is asking for the atomic number Z, given some information about the energy differences between photon emissions in two transitions. Let me try to break this down step by step.
First, I remember that in a hydrogen-like ... | 3 | 14 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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... | Alright, so I have this chemistry problem here, and I need to figure out how many of the given compounds have at least one bridging oxo group. The compounds listed are:
- N₂O₃
- N₂O₅
- P₄O₆
- P₄O₇
- H₄P₂O₅
- H₅P₃O₁₀
- H₂S₂O₃
- H₂S₂O₅
First, I should recall what a bridging oxo group is. From what I remember, a bridgin... | 6 | 25 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here about Galena, which is a type of ore. The question is about how much lead (Pb) is produced per kilogram of oxygen (O2) consumed when Galena is heated in a closed furnace after the air supply is stopped. Hmm, let me try to figure this out step by step.
First, I need to unders... | 6.47 | 26 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here, and I need to figure out how much MnCl₂ was dissolved in an aqueous solution. Let me try to understand the problem step by step.
First, the problem says that MnCl₂ was converted completely to KMnO₄ using a reaction. The given reaction is:
MnCl₂ + K₂S₂O₈ + H₂O → KMnO₄ + H₂S... | 126 | 27 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here about copper getting tarnished and how nitrogen gas with some water vapor is used to prevent it. Hmm, let me try to understand what's going on here.
First, the problem says that copper surfaces oxidize to form copper oxide, which causes tarnishing. To prevent this, they pass... | -14.6 | 30 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here about Gibbs free energy and reversible reactions. Let me try to figure it out step by step. I remember that Gibbs free energy change (ΔG°) is related to the equilibrium constant, right? The formula I recall is ΔG° = -RT ln K, where K is the equilibrium constant. But wait, I a... | 8500 | 31 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this electrochemical cell problem here. Let me try to figure it out step by step. The problem is about a cell notation and relating the enthalpy change (ΔH°) to the Gibbs free energy change (ΔG°) and then finding the entropy change (ΔS°). Hmm, okay.
The cell is written as:
A(s)|A^(n+)(aq, 2M) || B^(2n... | -11.62 | 32 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this integral to solve:
\[
\int_{0}^{\frac{1}{2}} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{\frac{1}{4}}} \, dx
\]
Hmm, that looks a bit intimidating at first glance, but let me try to break it down step by step. Maybe I can simplify the integrand first.
Looking at the denominator, it's rai... | 2 | 43 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem where I need to find the maximum possible determinant of a 3x3 matrix P where each entry is either -1, 0, or 1. Hmm, interesting. I remember that the determinant of a matrix can tell us things about the matrix's scaling factor and whether it's invertible or not. But I'm not exactly sure how... | 4 | 44 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I need to find the value of \(\frac{1}{5!}(\beta - \alpha)\), where \(\alpha\) is the number of one-one functions from a set \(X\) with 5 elements to a set \(Y\) with 7 elements, and \(\beta\) is the number of onto functions from \(Y\) to \(X\). Hmm, okay. Let me try to break thi... | 119 | 45 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this differential equation to solve: dy/dx = (2 + 5y)(5y - 2). And I need to find the limit as x approaches negative infinity of f(x), which is the function y. Hmm, let me think about how to approach this.
First, I remember that when dealing with differential equations, especially first-order ones, it ... | 0.4 | 46 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem where I need to find the value of log base e of f(4), given that f is a differentiable function from R to R, f(0) = 1, and it satisfies the equation f(x + y) = f(x)f’(y) + f’(x)f(y) for all real numbers x and y. Hmm, interesting. Let me try to unpack this step by step.
First, I know that t... | 2 | 47 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this geometry problem here, and I need to figure out the length of PR. Let me try to break it down step by step.
First, let's understand the problem:
We have a point P in the first octant. That means all its coordinates are positive, right? So, P is somewhere in the positive x, y, z space.
The image ... | 8 | 48 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about a cube in the first octant. The cube has sides OP, OQ, and OR, each of length 1, along the x, y, and z-axes respectively. The origin is O(0,0,0), and S is the center of the cube at (1/2, 1/2, 1/2). T is the vertex opposite to O, so that would be the far corner of the cube from the ori... | 0.5 | 49 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I need to find the value of (1/1430) times X, where X is defined as the sum from r=1 to r=10 of r times (10 choose r) squared. That is,
X = (10 choose 1)^2 + 2*(10 choose 2)^2 + 3*(10 choose 3)^2 + ... + 10*(10 choose 10)^2.
Hmm, okay. I need to compute this sum and then divide... | 646 | 50 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about a parallel plate capacitor with some dielectric layers, and I need to find the value of alpha (α) such that the capacitance C is equal to α times (K ε₀ A / (d ln 2)). Hmm, let's try to break this down step by step.
First, I remember that for a simple parallel plate capacitor without ... | 1 | 16 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 ... | Alright, so I have this chemistry problem here, and I need to figure out how many molecules among the given compounds have covalent bonds between two atoms of the same kind. The compounds listed are B₂H₆, B₃N₃H₆, N₂O, N₂O₄, H₂S₂O₃, and H₂S₂O₈. Hmm, okay. Let me break this down step by step.
First, I remember that a co... | 4 | 31 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here: At 143 K, the reaction of XeF4 with O2F2 produces a xenon compound Y. I need to find the total number of lone pair(s) of electrons present on the whole molecule of Y. Hmm, let me think about how to approach this.
First, I should figure out what kind of reaction this is. It ... | 19 | 32 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I need to find the depression of the freezing point of benzene after dissolving a certain amount of a solute in it. Hmm, let me see. The problem states that 0.5 grams of a non-volatile non-ionic solute is dissolved in 39 grams of benzene. The vapor pressure of benzene decreases f... | 1.02 | 34 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here about reaction rates, and I need to find the value of Y for a specific rate when given some concentrations. Hmm, let me see. The reaction is A + B + C → Product, and I have four experiments with different concentrations of A, B, C, and their corresponding reaction rates. Then... | 6.75 | 35 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem where I need to find the minimum of the set \(\left\{\left|a + b \omega + c \omega^{2}\right|^{2}: a, b, c \text { distinct non-zero integers}\right\}\), where \(\omega\) is a cube root of unity, not equal to 1. Hmm, cube roots of unity... I remember they are complex numbers that satisfy \(... | 3 | 49 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here about arithmetic progressions, and I need to find the value of \( a + d \) where the intersection of three different arithmetic progressions equals another arithmetic progression \( AP(a; d) \). Hmm, let me try to break this down step by step.
First, let me recall what an arithmetic p... | 157 | 50 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this probability problem here about 3x3 matrices with entries from {0,1}. The sample space S consists of all such matrices. We have two events, E1 and E2. E1 is the set of matrices where the determinant is zero, and E2 is the set of matrices where the sum of the entries is 7. The question is asking for ... | 0.5 | 51 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this geometry problem here, and I need to figure out the length of segment AC. Let me try to break it down step by step.
First, let me understand the problem again. We have point A at (2,3), and point B is the reflection of A over the line 8x - 6y - 23 = 0. Then, there are two circles: ΓA with radius 2... | 10 | 52 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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{}. | Alright, so I have this integral I need to compute:
\[
I = \frac{2}{\pi} \int_{-\pi / 4}^{\pi / 4} \frac{dx}{\left(1 + e^{\sin x}\right)\left(2 - \cos 2x\right)}
\]
And then I need to find 27 times I squared. Hmm, okay. Let me try to figure this out step by step.
First off, the integral looks a bit complicated becau... | 4 | 53 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem where I'm given three lines in vector form, and each of these lines intersects the plane x + y + z = 1 at points A, B, and C. I need to find the square of the area of triangle ABC, specifically (6Δ)^2. Hmm, let's break this down step by step.
First, let me write down the equations of the t... | 0.75 | 54 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about alpha decay and gamma decay in radium. Let me try to figure it out step by step. I'm a bit rusty on nuclear physics, but I remember that decay processes involve the conservation of energy and mass. Let me start by writing down what's given.
We have a radium-226 nucleus initially at r... | 135 | 13 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about measuring the focal length of a convex lens using an optical bench. Hmm, let me try to figure this out step by step.
First, the setup: there's an optical bench with a scale that's 1.5 meters long, divided into four equal parts in each centimeter. That means each small division is 0.... | 0.69 | 14 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here: I need to find the amount of water produced when 1 mole of rhombic sulfur is oxidized by concentrated nitric acid to a compound where sulfur is in its highest oxidation state. Hmm, let me think about how to approach this.
First, I should figure out the oxidation state of su... | 288 | 27 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here: I need to find the total number of cis Mn-N and Mn-Cl bonds in a molecule of cis-[Mn(en)₂Cl₂]. Hmm, let me think about how to approach this.
First, I should probably recall what the structure of the complex looks like. The complex is cis, which usually means that the substi... | 6 | 28 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here about a decomposition reaction in a closed cylinder. The reaction is N₂O₅(g) decomposing into N₂O₄(g) and O₂(g). It's under isothermal and isochoric conditions, which I think means the temperature and volume are constant. The initial pressure is 1 atm, and after Y × 10³ secon... | 2.3 | 29 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I need to find the molarity of an aqueous urea solution. Let me see what information I have. The problem states that the mole fraction of urea in the solution is 0.05, the solution has 900 grams of water, and the density of the solution is 1.2 g/cm³. The molar masses of urea and ... | 2.98 | 30 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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{}. | Alright, so I have to figure out the total number of isomers for cyclic ethers with the molecular formula C4H8O. Hmm, okay. Let me start by recalling what cyclic ethers are. They are compounds where an oxygen atom is part of a ring structure, right? So, the oxygen is connected to two carbon atoms in a ring, making it a... | 10 | 32 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 ... | Alright, so I have this determinant equation here, and I need to find the sum \(\sum_{k=0}^{n} \frac{{}^{n}C_{k}}{k+1}\). Hmm, okay, let me break this down step by step.
First, the determinant given is:
\[
\operatorname{det}\left[\begin{array}{cc}
\sum_{k=0}^{n} k \cdot {}^{n}C_{k} & \sum_{k=0}^{n} {}^{n}C_{k} k^{2}... | 6.2 | 45 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 seated in a circle. Each of them gets a hat that's either red, blue, or green. I need to figure out how many different ways the hats can be distributed so that adjacent people have different colored hats. Hmm, interesting. Let me try to break this d... | 30 | 46 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I'm given a sample space S = {1,2,3,4,5,6}, and each element is equally likely to occur. A and B are independent events associated with S, and I need to find the number of ordered pairs (A, B) such that the size of B is less than the size of A, specifically 1 ≤ |B| < |A|. Hmm, ok... | 422 | 47 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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{}... | Alright, so I have this problem here that I need to solve:
\[
\sec ^{-1}\left(\frac{1}{4} \sum_{k=0}^{10} \sec \left(\frac{7 \pi}{12}+\frac{k \pi}{2}\right) \sec \left(\frac{7 \pi}{12}+\frac{(k+1) \pi}{2}\right)\right)
\]
And I need to find its value in the interval \(\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]\). H... | 0 | 48 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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{}. | Alright, so I have this integral to solve:
\[
\int_{0}^{\pi / 2} \frac{3 \sqrt{\cos \theta}}{(\sqrt{\cos \theta}+\sqrt{\sin \theta})^{5}} d \theta
\]
Hmm, okay. Let me try to figure this out step by step. I'm not too confident about this, but I'll give it a shot.
First, I notice that the integral is from 0 to π/2, w... | 0.5 | 49 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this physics problem here about two fingers on a scale, and I need to figure out how far the right finger slides before it stops. Let me try to understand the problem first.
Alright, so there's a meter scale placed horizontally. The left finger is at 0.00 cm and the right finger is at 90.00 cm. Both fi... | 25.6 | 13 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about helium gas expanding in two different ways, first isothermally and then adiabatically. I need to find the ratio of the work done during each process, specifically W_iso over W_adia, which is given as f times the natural logarithm of 2, and I need to find f. Hmm, let me try to break th... | 1.77 | 16 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 about a tuning fork and a pipe. Let me try to understand what it's asking. A stationary tuning fork is in resonance with an air column in a pipe. That means the frequency of the tuning fork matches the frequency of the standing wave in the pipe, right? So, they vibrate at the s... | 0.62 | 17 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about electric flux through two different spherical surfaces enclosing a charged disc. Let me try to break it down step by step.
First, the setup: there's a circular disc with radius R, and it has a surface charge density given by σ(r) = σ₀(1 - r/R), where r is the distance from the center... | 6.4 | 18 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
$5.00 \mathrm{~mL}$ 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 have this chemistry problem here about titration. It involves oxalic acid and sodium hydroxide. Hmm, let me try to figure this out step by step.
First, the problem says that 5.00 mL of a 0.10 M oxalic acid solution is being titrated with NaOH using phenolphthalein as the indicator. They've given me the volu... | 0.11 | 31 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem about a hydrogen-oxygen fuel cell, and I need to find the change in temperature when the work from the cell is used to compress a monoatomic ideal gas. Hmm, let me break this down step by step.
First, the problem states that the fuel cell is 70% efficient. The cell reaction is given as:
\... | 13.32 | 33 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this chemistry problem here, and I need to figure out how much hydrogen gas is produced when aluminum reacts with sulfuric acid. Hmm, let me see. Alright, the problem says that 5.4 grams of aluminum reacts with 50.0 mL of 5.0 M sulfuric acid. I need to find the volume of hydrogen gas in liters at 300 K ... | 6.15 | 34 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
${ }_{92}^{238} \mathrm{U}$ 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 have this problem about radioactive decay, and I need to find the number of alpha particles emitted after three half-lives. Hmm, let me think step by step.
First, the problem says that Uranium-238 decays to Lead-206 by emitting alpha and beta particles. The initial amount of Uranium-238 is 68 × 10^-6 grams.... | 1.2 | 35 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here, and I need to find the value of log base 2 of (m cubed) plus log base 3 of (M squared). Hmm, that sounds a bit complicated, but let me break it down step by step.
First, let's understand what m and M represent. The problem says that m is the minimum possible value of log base 3 of (3... | 8 | 49 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this problem here where I have two sequences: one is an arithmetic progression with a common difference of 2, and the other is a geometric progression with a common ratio of 2. Both sequences start with the same positive integer c. The question is asking for the number of possible values of c such that ... | 1 | 50 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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 have this function f(x) defined on the interval [0,2], and I need to find the difference between α and β, where [α, β] is the set of x in [0,2] such that f(x) is greater than or equal to zero. Hmm, let me write down the function first to get a clear picture.
The function is:
f(x) = (3 - sin(2πx)) * sin(πx -... | 1 | 51 | {
"question_source": "daman1209arora/jeebench"
} | deepseek-ai/DeepSeek-R1-Distill-Llama-8B |
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