datavorous/entrance-exam-dataset huggingface dataset:
Consider the following relations:
$R=\{(x, y) \mid x, y$ are real numbers and $x=$ wy for some rational number w $\}$;
$S=\left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p\right.$ and $q$ are integers such that $n, q \neq 0$ and $\left.q m=p n\right\}$. Then

Options:
A
neither $\mathrm{R}$ nor $\mathrm{S}$ is an equivalence relation

B
$S$ is an equivalence relation but $R$ is not an equivalence relation

C
$R$ and $S$ both are equivalence relations

D
$\mathrm{R}$ is an equivalence relation but $\mathrm{S}$ is not an equivalence relation

The correct answer is:
$\mathrm{R}$ is an equivalence relation but $\mathrm{S}$ is not an equivalence relation

Solution:
xRy need not implies $y R x$
$\mathrm{S}: \frac{\mathrm{m}}{\mathrm{n}} \mathrm{s} \frac{\mathrm{p}}{\mathrm{q}} \Leftrightarrow \mathrm{qm}=\mathrm{pn}$ $\frac{m}{n} s \frac{m}{n}$ reflexive
$$
\begin{array}{ll}
\frac{m}{n} s \frac{p}{q} & \Rightarrow \frac{p}{q} s \frac{m}{n} \text { symmetric } \\
\frac{m}{n} s \frac{p}{q}, \frac{p}{q} s \frac{r}{s} & \Rightarrow q m=p n, p s=r q \quad \Rightarrow m s=r n \text { transitive. }
\end{array}
$$
$S$ is an equivalence relation.

Topic: Mathematics, Sets and Relations, JEE Main, JEE Main 2010
Consider the following reversible chemical reactions:
A2g+B2g⇌k12ABg .....(1)
6ABg⇌k23A2g+3B2g .....(2)
The relation between K1 and K2 is:

Options:
A K2=K1-3
B K1K2=13
C K2=K13
D K1K2=3

The correct answer is:
K2=K1-3

Solution:
A2+B2⇌2AB
k1=AB2A2B2
6AB⇌3A2+3B2
k2=A23B23AB6
So, 1k13=k2

Topic: Chemistry, Chemical Kinetics, JEE Main, JEE Main 2019 (09 Jan Shift 2)
Consider the following reversible reaction: \( \mathrm{NO}+\mathrm{NO}_{3} \rightleftharpoons 2 \mathrm{NO}_{2} \) If \( 1.0 \) mol of \( \mathrm{NO} \) is mixed with \( 3.0 \mathrm{~mol} \) of \( \mathrm{NO}_{3}, \mathrm{x}^{\prime} \) mole of \( \mathrm{NO}_{2} \) is produced at equilibrium. If \( 2.0 \mathrm{~mol} \) of \( \mathrm{NO} \) is added further, 'x' mol of \( \mathrm{NO}_{2} \) is further produced. What is the value of equilibrium constant?

Options:

The correct answer is:
4

Solution:
NO + NO 3 â 2 NO 2 ; K c 1 - x / 2 3 - x / 2 x
NO + NO 3 â 2 NO 2 ; K c 3 - x 3 - x 2 x
x 2 1 - x / 2 3 - x / 2 = 4 x 2 3 - x 2 = 4 x 2 2 - x 6 - x
3 - x 2 = 2 - x 6 - x
9 + x 2 - 6 x = 1 2 - 2 x - 6 x + x 2
9 - 6 x = 1 2 - 8 x
2 x = 3
x = 1 · 5
K c = 4 × 1 · 5 × 1 · 5 1 · 5 × 1 · 5 = 4

Topic: Chemistry, Chemical Equilibrium, JEE Main
Consider the following sequence for aspartic acid:
The pH (Isoelectric point) of aspartic acid is:

Options:
A 3.65
B 2.77
C 5.74
D 1.88

The correct answer is:
2.77

Solution:
pH=pK1+pK22
=1.88+3.652
=5.532
=2.765≈2.77
At isoelectric point of aspartic acid both carboxylic and amino from co-exist.

Topic: Chemistry, Biomolecules, JEE Main
Consider the following sequence for aspartic acid:
The pH (Isoelectric point) of aspartic acid is:

Options:
A 3.65
B 2.77
C 5.74
D 1.88

The correct answer is:
2.77

Solution:
pH=pK1+pK22
=1.88+3.652
=5.532
=2.765≈2.77
At isoelectric point of aspartic acid both carboxylic and amino from co-exist.

Topic: Chemistry, Biomolecules, JEE Main, JEE Main 2016 (09 Apr Online)
Consider the following sequence of reaction and identify the final product (Z).

CH 3 CH 2 CH 2 Br â Mgâdry ether (X) â CO 2 (Y) â H + (Z)

Options:
A CH3CH2COOH
B CH3CH2CH2CH2COOH
C CH3-C|H-CH3 COOH
D CH3CH2CH2COOH

The correct answer is:
CH3CH2CH2COOH

Solution:
The sequence of reaction is called carbonation of alkyl halide.
CH 3 CH 2 CH 2 Br → Mg/dry ether CH 3 CH 2 CH 2 MgBr X → CO 2 CH 3 CH 2 CH 2 CO 2 MgBr Y → H + CH 3 CH 2 CH 2 CO 2 H Z

Topic: Chemistry, Haloalkanes and Haloarenes, JEE Main
Consider the following sequence of reactions and identify the final product (Z).
C 6 H 6  +  CH 2  =  CHCH 2 Cl → AlCl 3 (X) → (ii)  H 2 O 2 ,  OH − (i)  BH 3 , THF (Y) → H + /Δ Z C 9 H 10

Options:
A PhCH2CH=CH2
B PhCH=CH-CH3
C
D PhC(CH3)=CH2

The correct answer is:

Solution:
(X) PhCH2CH=CH2
(Y) PhCH2CH2CH2OH and
(Z)

Topic: Chemistry, Alcohols Phenols and Ethers, JEE Main
Consider the following sequence of reactions
Compound ' $C$ ' is

Options:
A

B

C
$\mathrm{HOCH}_2-\mathrm{CH}=\mathrm{CH}_2$

D

The correct answer is:

Solution:

Topic: Chemistry, Hydrocarbons, JEE Main, JEE Main 2012 (26 May Online)
Consider the following sequence of reactions

The compound A is:

Options:
A HOCH2CH2CH2COOH
B CH3CHOH CH2COOH
C CH3CH2CHOH COOH
D CH32COH COOH

The correct answer is:
CH3CHOH CH2COOH

Solution:
The compound A is 3-hydroxy butanoic acid and the reaction is as follows

Topic: Chemistry, Aldehydes and Ketones, JEE Main
Consider the following sequence of reactions.

The final product C is

Options:
A
B
C
D

The correct answer is:

Solution:

Topic: Chemistry, Carboxylic Acid Derivatives, JEE Main
Consider the following sequence of reactions

The intermediate products (X) and (Y) are respectively

Options:
A
B
C
D

The correct answer is:

Solution:

Topic: Chemistry, Aldehydes and Ketones, JEE Main
Consider the following sequence of reactions

The product ‘B’ is

Options:
A

B

C

D

The correct answer is:

Solution:
The reaction of aniline (aromatic amine) with nitrous acid results in the diazonium salt formation which is benzene diazonium chloride.When benzene diazonium chloride reacts with N, N - dimethyl aniline at ice cold condition gives p - dimethyl aminoazobenzene. It is an example of coupling reaction.

Topic: Chemistry, Amines, JEE Main, JEE Main 2023 (15 Apr Shift 1)
Consider the following set of quantum numbers.

n	l	m1

A	3	3	-3

B	3	2	-2

C	2	1	+1

D	2	2	+2

The number of correct sets of quantum numbers is

Options:

The correct answer is:
2

Solution:
For a given 'n' value l will have 0 to n-1 values and for a given 'l' value m will have -m to +m values

For n=3, l=3, is not possible.Similarly for n=2, l=2 is also not possible, as n and l have same values.

So, set A, D are not possible.

If n=3, l can have 0 to 2 values and 'm' can have -2 to +2 values,

So, set B is possible.

If n=2, 'l' can have 0 and 1 values and 'm' can have -1 to +1 values.

So, set C is also possible.

Hence, set B and C are the correct set of quantum nmbers.

Topic: Chemistry, Structure of Atom, JEE Main, JEE Main 2022 (27 Jun Shift 2)
Consider the following single step reaction in gas phase at constant temperature.
\(2 \mathrm{~A}_{(\mathrm{g})}+\mathrm{B}_{(\mathrm{g})} \rightarrow \mathrm{C}_{(\mathrm{g})}\)
The initial rate of the reaction is recorded as \(\mathrm{r}_1\) when the reaction starts with \(1.5 \mathrm{~atm}\) pressure of \(\mathrm{A}\) and \(0.7 \mathrm{~atm}\) pressure of \(\mathrm{B}\). After some time, the rate \(\mathrm{r}_2\) is recorded when the pressure of \(\mathrm{C}\) becomes \(0.5 \mathrm{~atm}\). The ratio \(\mathrm{r}_1: \mathrm{r}_2\) is ______ \(\times 10^{-1}\). (Nearest integer)

Options:

The correct answer is:
315

Solution:
$2 \mathrm{~A}(\mathrm{~g})+\mathrm{B}(\mathrm{g}) \longrightarrow \mathrm{C}(\mathrm{g})$
$\begin{array}{ll}
\mathrm{r}_1 & 1.5 \mathrm{~atm} \quad 0.7 \mathrm{~atm} \\
\mathrm{r}_2 & 0.5 \mathrm{~atm} \quad 0.2 \mathrm{~atm} \quad 0.5 \mathrm{~atm} \\
& \because \mathrm{r}=\mathrm{K}\left[\mathrm{P}_{\mathrm{A}}\right]^2\left[\mathrm{P}_{\mathrm{B}}\right] \\
& \mathrm{r}_1=\mathrm{K}[1.5]^2[0.7] \\
& \mathrm{r}_2=\mathrm{K}[0.5]^2[0.2] \\
& \frac{\mathrm{r}_1}{\mathrm{r}_2}=9 \times \frac{7}{2}=31.5=315 \times 10^{-1}
\end{array}$
Ans. 315

Topic: Chemistry, Chemical Kinetics, JEE Main, JEE Main 2024 (05 Apr Shift 2)
Consider the following standard electrode potentials (Eo in volts) in aqueous solution:
ElementM3+/MM+/MAl-1.66+0.55Tl+1.26-0.34
Based on these data, which of the following statements is correct?

Options:
A Tl+ is more stable than Al3+
B Al+ is more stable than Al3+
C Tl3+ is more stable than Al3+
D Tl+ is more stable than Al+

The correct answer is:
Tl+ is more stable than Al+

Solution:
∆Go=-nFEo
if Eo= positive; then ∆Go=negative,
For those, conversions whose E o values are more positive are more favourable.
As we can see from then electrode potential data, that reduction of Al+ is spontaneous, but reduction of Tl+ is not. So, we can conclude from this that Tl+ is more stable than Al+

Topic: Chemistry, Electrochemistry, JEE Main, JEE Main 2017 (08 Apr Online)
Consider the following standard electrode potentials (Eo in volts) in aqueous solution:
ElementM3+/MM+/MAl-1.66+0.55Tl+1.26-0.34
Based on these data, which of the following statements is correct?

Options:
A Tl+ is more stable than Al3+
B Al+ is more stable than Al3+
C Tl3+ is more stable than Al3+
D Tl+ is more stable than Al+

The correct answer is:
Tl+ is more stable than Al+

Solution:
∆Go=-nFEo
if Eo= positive; then ∆Go=negative,
For those, conversions whose E o values are more positive are more favourable.
As we can see from then electrode potential data, that reduction of Al+ is spontaneous, but reduction of Tl+ is not. So, we can conclude from this that Tl+ is more stable than Al+

Topic: Chemistry, Electrochemistry, JEE Main
Consider the following statement

(A) NF3 molecules has a trigonal planar structure.

(B) Bond Length of N2 is shorter than O2.

(C) Isoelectronic molecules or ions have identical bond order.

(D) Dipole moment of H2S is higher than that of water molecule.

Choose the correct answer from the options given below:

Options:
A A and B are correct
B A and D are correct
C C and D are correct
D B and C are correct

The correct answer is:
B and C are correct

Solution:
In NF3, nitrogen is having three bond pairs and a lone pair, hence, it has pyramidal shape.

The bond order of N2=3, and The bond order of O2=2. Bond length is inversely related to bond order. Hence, bond length of N2 is shorter than O2

Isoelectronic species have identical bond order

Dipole moment of H2O is more than that of H2S due to higher electronegativity of O.

Topic: Chemistry, Chemical Bonding and Molecular Structure, JEE Main, JEE Main 2023 (15 Apr Shift 1)
Consider the following statement about electric field intensity and electric potential.
A. The electric field intensity due to a charged spherical shell is inversely proportional to square of its radius.
B. The electric potential due to a point charge is inversely proportional to the distance between charge and the point.

Options:
A Both A and B are incorrect
B Only B is correct
C Only A is correct
D Both A and B are correct

The correct answer is:
Only B is correct

Solution:
Only B is true. As for statement A, it is not mentioned whether the point in outside or inside the shell, further statement A is true only for the point on the surface of the spherical shell.

Topic: Physics, Electrostatics, JEE Main
Consider the following statements
(1) Mode can be computed from histogram
(2) Median is not independent of change of scale
(3) Variance is independent of change of origin and scale
Which of these is/are correct

Options:
A (1), (2) and (3)
B Only (2)
C Only (1) and (2)
D Only (1)

The correct answer is:
(1), (2) and (3)

Solution:
Mode can be computed from the histogram by drawing two lines diagonally in the inside of the modal class bar, starting from each upper corner of the bar to the upper corner of the adjacent bar. The foot of the perpendicular from the point of intersection of these two lines gives the mode. So, option (1) is correct.

Topic: Mathematics, Statistics, JEE Main
Consider the following statements
(1) Mode can be computed from histogram
(2) Median is not independent of change of scale
(3) Variance is independent of change of origin and scale
Which of these is/are correct?

Options:
A (1), (2) and (3)
B Only (2)
C Only (1) and (2)
D Only (1)

The correct answer is:
(1), (2) and (3)

Solution:
Mode can be computed from the histogram by drawing two lines diagonally in the inside of the modal class bar, starting from each upper corner of the bar to the upper corner of the adjacent bar. The foot of the perpendicular from the point of intersection of these two lines gives the mode. So, option (1) is correct.

Topic: Mathematics, Statistics, JEE Main
Consider the following statements:
1- The derivative where the function attains maxima or minima be zero.
2- If a function is differentiable at a point, then it must be continuous at that point. Which of the above statements is/are correct?

Options:
A 1 only
B 2 only
C Both 1 and 2
D Neither 1 nor 2

The correct answer is:
2 only

Solution:
$1.$ Not necessarily true
$2.$ True $(\because$ Differentiability $\Rightarrow$ Continuity)
#

Topic: Mathematics, Continuity and Differentiability, JEE Main
Consider the following statements:
1- The function $f(x)=|x|$ is not differentiable at $x=1$.
2- The function $f(x)=e^{x}$ is not differentiable at $x=0$. Which of the above statements is/are correct?

Options:
A 1 only
B 2 only
C Both 1 and 2
D Neither 1 nor 2

The correct answer is:
2 only

Solution:
Statement $1: \mathrm{f}(\mathrm{x})=\mid \mathrm{x}$
From the graph, the curve has sharp turn at $\mathrm{x}=0$. Therefore, the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}|$ is not differentiable only $\mathrm{x}=0$, it is differentiable at $\mathrm{x}=1$ Statement $2: \mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}$
$\mathrm{Rf}^{\prime}(0)=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(0+\mathrm{h})-\mathrm{f}(0)}{\mathrm{h}}$
$=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{(0+\mathrm{h})}-\mathrm{e}^{\circ}}{\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{\mathrm{h}}-1}{\mathrm{~h}}$
Use L'Hospital rule
$=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{\mathrm{h}}-0}{1}=\mathrm{e}^{\mathrm{o}}=1$
$f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{1-\mathrm{e}^{-\mathrm{h}}}{\mathrm{h}}$
Use L' Hospital rule
$=\lim _{h \rightarrow 0} \frac{\mathrm{e}^{-\mathrm{h}}}{1}=\mathrm{e}^{-0}=1$
Therefore $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}$ is differentiable at $\mathrm{x}=1$.

Topic: Mathematics, Continuity and Differentiability, JEE Main
Consider the following statements:
1- The function is continuous at every point in the interval $(1, \infty)$
2- The function is differentiable at $x=1.5$.
Which of the above statements is/are correct?
Let $f(x)$ be a function defined in $1 \leq x < \infty$ by $\quad[2014-I]$
$f(x)=\left\{\begin{array}{cc}2-x & \text { for } 1 \leq x \leq 2 \\ 3 x-x^{2} & \text { for } x>2\end{array}\right.$

Options:
A 1 only
B 2 only
C Both 1 and 2
D Neither 1 nor 2

The correct answer is:
2 only

Solution:
Statement $1:$ Given $f(x)=\left\{\begin{array}{ccc}2-x & \text { for } & 1 \leq x \leq 2 \\ 3 x-x^{2} & \text { for } & x>2\end{array}\right.$
function defined in $1 \leq \mathrm{x} < \infty$ the function is polynomial, so it is continous and differentiable in its domain $[1, \infty]-\{2\}$
LHLf $(2-0)=\lim _{h \rightarrow 0} f(2-h)$
$=\lim _{h \rightarrow 0} \mathrm{~h}=0$
$\operatorname{RHLf}(2+0)=\lim _{h \rightarrow 0}(2+h)$
$=6-4=2$
$\mathrm{f}(2)=2-2=0 \therefore \mathrm{LHL} \neq \mathrm{RHL}$
Statement 2 :
$\mathrm{Rf}^{\prime \prime}(1.5)=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(1.5+\mathrm{h})-\mathrm{f}(1.5)}{\mathrm{h}}$
$=\lim _{h \rightarrow 0} \frac{(2-1.5+h)-(2-1.5)}{h}=\lim _{h \rightarrow 0} \frac{-h}{h}=-1$
$\mathrm{Lf}^{\prime}(1.5)=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(1.5-\mathrm{h})-\mathrm{f}(1.5)}{-\mathrm{h}}$
$=\lim _{h \rightarrow 0} \frac{(2-1.5-h)-(2-1.5)}{-h}$
$=\lim _{h \rightarrow 0} \frac{h}{-h}=-1$
Therefore, the function is differentiable at $\mathrm{x}=1.5$

Topic: Mathematics, Continuity and Differentiability, NDA, NDA 2014 (Phase 1)
Consider the following statements:
$1.$ The probability that there are 53 Sundays in a leap year is twice the probability that there are 53 Sundays in a non-leap year.
$2.$ The probability that there are 5 Mondays in the month of March is thrice the probability that there are 5 Mondays in the month of April. Which of the statements given above is/are correct?

Options:
A 1 only
B 2 only
C Both 1 and 2
D Neither 1 nor 2

The correct answer is:
1 only

Solution:
Statement 1:
Anon leap year has 365 days. i.e., 52 weeks and 1 day. 1 day can be \{Sunday\}, \{Monday $\},$ Tuesday\},
\{Wednesday $\}$, \{Thursday\}, \{Friday\}, \{Saturday\}
In total, there are 7 possibilities and 1 possibility is Sunday.
Required probability $=\frac{1}{7}$
A leap year has 366 days. i.e., 52 weeks and 2 days. 2 days can be \{Sun, Mon\}, \{Mon, Tue\}, \{Tue, Wed $\}$,
\{Wed, Thu \}, \{Thu, Fri\}, \{Fri, Sat\}, \{Sat, Sun\}.
In total, there are 7 possibilities and 2 possibilities have Sundays.
Required probability $=\frac{2}{7}$
So, statement 1 is correct. Statement 2 :
March has 31 days. i.e., 4 complete weeks and 3 days. 3 days can be $\{\mathrm{S}, \mathrm{M}, \mathrm{T}\},\{\mathrm{M}, \mathrm{T}, \mathrm{W}\},\{\mathrm{T}, \mathrm{W}, \mathrm{Th}\},\{\mathrm{W}, \mathrm{Th}, \mathrm{F}\}$
$\{\mathrm{Th}, \mathrm{F}, \mathrm{Sa}\},\{\mathrm{F}, \mathrm{Sa}, \mathrm{S}\},\{\mathrm{Sa}, \mathrm{S}, \mathrm{M}\}$
In total 7 possibilities, Monday can come in 3 possibilities
$\therefore$ Required probabilities $=\frac{3}{7}$
April has 30 days. i.e., 4 complete weeks and 2 days. 2 days can be $\{\mathrm{S}, \mathrm{M}\},\{\mathrm{M}, \mathrm{T}\},\{\mathrm{T}, \mathrm{W}\},\{\mathrm{W}, \mathrm{Th}\},\{\mathrm{Th}, \mathrm{F}\}$
$\{\mathrm{F}, \mathrm{Sa}\},\{\mathrm{Sa}, \mathrm{S}\}$
In total 7 possibilities, Monday can come in 2 possibilities.
$\therefore$ Required probability $=\frac{2}{7}$
$\therefore$ Statement 2 is wrong.

Topic: Mathematics, Probability, JEE Main
Consider the following statements:
1. There exists no triangle $A B C$ for which $\sin A+\sin B=\sin C$.
2. If the angles of a triangle are in the ratio $1: 2: 3$, then its sides will be in the ratio $1: \sqrt{3} : 2$. Which of the above statements is/are correct?

Options:
A $1$ only
B $2$ only
C Both $1$ and $2$
D Neither $1$ nor $2$

The correct answer is:
Both $1$ and $2$

Solution:
Consider, $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are the sides of triangle and $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are the angles of the triangle.
We know that, $\frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}}=\frac{\mathrm{c}}{\sin \mathrm{C}}=\mathrm{k}$
$\Rightarrow \sin A=\frac{a}{\mathrm{k}}, \sin B=\frac{b}{\mathrm{k}}, \sin \mathrm{C}=\mathrm{ck}$
Given, $\sin A+\sin B=\sin C$
$\Rightarrow A+B=C$.
$\Rightarrow a+b=c$
This is not posible.
Hence, there exists no triangle $A B C$ for which $\sin A+\sin B=\sin C$.
(2) Given, the angles of a triangle are in the ratio $1: 2: 3$.
Consider the angles of a triangle are $\mathrm{A}=\mathrm{x}, \mathrm{B}=2 \mathrm{x}$ and $\mathrm{C}=3 \mathrm{x}$.
We know that, Sum of the angles of the triangle is $180^{\circ}$
Hence, $x+2 x+3 x=180$
$\begin{aligned} & 6 x=180^{\circ} \\ & x=30^{\circ}\end{aligned}$
Angles are $A=30^{\circ}, B=60^{\circ}$ and $C=90^{\circ}$
We know that, $\frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}}=\frac{\mathrm{c}}{\sin \mathrm{C}}=\mathrm{k}$
$\frac{\mathrm{a}}{\frac{1}{2}}=\frac{\mathrm{b}}{\frac{\sqrt{3}}{2}}=\frac{\mathrm{c}}{1}=\mathbf{k}$
$\frac{\mathrm{a}}{\mathrm{I}}=\frac{\mathrm{b}}{\sqrt{3}}=\frac{\mathrm{c}}{2}=\mathrm{k}$
Hence, the angles of a triangle are in the ratio $1: 2: 3$, then its sides will be in the ratio $1: \sqrt{3}: 2$..

Topic: Mathematics, Properties of Triangles, JEE Main
Consider the following statements $A$ and $B$ and identify the correct answer
A. In an elastic collision, if a body suffers a head on collision with another of same mass at rest, then the first body comes to rest while the other starts moving with the velocity of the first one.
B. Two bodies of equal mass suffering a head on elastic collision merely exchanges their velocities

Options:
A both $A$ and $B$ are true
B both $A$ and $B$ are false
C $A$ is true but $B$ is false
D $A$ is false but $B$ is true

The correct answer is:
both $A$ and $B$ are true

Solution:
Both statements $A$ and $B$ given in the system are true.

Topic: Physics, Center of Mass Momentum and Collision, JEE Main
Consider the following statements $A$ and $B$ and identify the correct answer
A. When a person walks on a rough surface the direction of frictional force exerted by the surface on the person is opposite to the direction of his motion.
B. When a cycle is in motion, the force of friction exerted by the ground on the front wheel is in the backward direction

Options:
A A and B are correct
B A is correct, B is wrong
C A and B are wrong
D A is wrong, B is correct

The correct answer is:
A and B are correct

Solution:
No Solution Available

Topic: Physics, Laws of Motion, JEE Main
Consider the following statements $A$ and $B$. Identify the correct choice in the given answer.
A. $p-n, p-p$ and $n-n$ forces between nucleons are not equal and charge dependent.
B. In nuclear reactor the fission reaction will be in accelerating state if the value of neutron reproduction factor $A>1$.

Options:
A Both A and B are correct
B Both A and B are wrong
C A is wrong and B is correct
D $\mathrm{A}$ is correct and $\mathrm{B}$ is wrong

The correct answer is:
A is wrong and B is correct

Solution:
The force acting between proton-neutron $(p-n)$ proton-proton $(p-p)$ and neutron-neutron $(n-n)$ are always equal and charge independent, so statement $A$ is wrong.
If reproduction factor of neutron $(A)>1$, then fission reaction will be in accelerating state, so statement $B$ is correct.

Topic: Physics, Nuclear Physics, JEE Main
Consider the following statements:
A. Atoms of each element emit characteristics spectrum.
B. According to Bohr's Postulate, an electron in a hydrogen atom revolves in a certain stationary orbit.

C. The density of nuclear matter depends on the size of the nucleus.
D. A free neutron is stable but a free proton decay is possible.
E. Radioactivity is an indication of the instability of nuclei. Choose the correct answer from the options given below.

Options:
A A, B, C, D and E
B A, B and E only.
C B and D only
D A,C and E only

The correct answer is:
A, B and E only.

Solution:
(A) True, an atom of each element emits a characteristic spectrum.

(B) True, according to Bohr's postulates mvr=nh2π and hence electron resides into orbits of specific radius called stationary orbits.

(C) False, the density of the nucleus is constant.

(D) False, A free neutron is unstable decays into proton and electron and antineutrino.

(E) True unstable nucleus show radioactivity.

Topic: Physics, Nuclear Physics, JEE Main, JEE Main 2021 (27 Jul Shift 2)