datavorous/entrance-exam-dataset huggingface dataset:
What is the relationship betwen Gibbs free energy of the cell reaction in a galvanic cell and the emf of the cell? When will the maximum work be obtained from a galvanic cell?

Options:

None

Solution:
When concentration of all reacting species is unity, then $E_{\text {cell }}=\mathrm{E}^{\circ}$ cell and
Gibbs free energy is given as $\Delta_r G^{\circ}=-n F E_{\text {cell }}^{\circ}$ where, $\Delta_r G^{\circ}=$ Standard Gibbs energy of the reaction $E_{\text {cell }}^{\circ}=$ Emf of the cell
$n F=$ Charge passed
To obtain maximum work from a galvanic cell, charge has to be passed reversibly.
The reversible work done by a galvanic cell is equal to decrease in its Gibbs energy.
$\Delta_r G=-n F E_{\text {cell }}$
As $E_{\text {cell }}$ is an intensive parameter but $\Delta_r G$ is an extensive thermodynamic property and the value depends on $n$.
For reaction,
$\mathrm{Zn}(s)+\mathrm{Cu}^{2+}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+\mathrm{Cu}(s)$ in a galvanic cell.
$\Delta_r G=-2 F E_{\text {cell }}[$ Here, $n=2]$

Topic: Chemistry, Electrochemistry
What is the result of competition between closely related species?

Options:
A Co:existence through resource partitioning
B Competitive exclusion of the inferior species
C Interference competition with abundant resources
D Competitive release through species removal

The correct answer is:
Competitive exclusion of the inferior species

Solution:
If two closely related species compete for the same resources, the competitively inferior one will be eliminated eventually, as stated by Gause's Competitive Exclusion Principle.

Topic: Biology, Organisms and Populations, NEET
What is the result of extinction of one species in a plant:pollinator mutualism?

Options:
A The other species in the mutualism becomes extinct
B The other species in the mutualism becomes more abundant
C The other species in the mutualism does not get impacted
D None of the above

The correct answer is:
The other species in the mutualism becomes extinct

Solution:
When a species becomes extinct in a plant:pollinator mutualism, other species in the mutualism become extinct.

Topic: Biology, Biodiversity and its Conservation, NEET
What is the result of the first meiotic division of a primary spermatocyte?

Options:
A Formation of two equal, haploid spermatids
B Formation of four equal, haploid spermatozoa
C Formation of two equal, diploid secondary spermatocytes
D Formation of two unequal, haploid spermatids

The correct answer is:
Formation of two equal, diploid secondary spermatocytes

Solution:
A primary spermatocyte undergoes the first meiotic division leading to the formation of two equal, diploid secondary spermatocytes.

Topic: Biology, Human Reproduction, NEET
What is the result of the first meiotic division of the primary oocyte?

Options:
A Formation of a large haploid secondary oocyte and a tiny first polar body
B The formation of a mature follicle
C The rupture of the Graafian follicle
D Ovulation

The correct answer is:
Formation of a large haploid secondary oocyte and a tiny first polar body

Solution:
The first meiotic division of the primary oocyte results in the formation of a large haploid secondary oocyte and a tiny first polar body.

Topic: Biology, Human Reproduction, NEET
What is the result of the reduction of the carbonyl group of aldehydes and ketones with zinc-amalgam and concentrated hydrochloric acid?

Options:
A Formation of alkanes
B Formation of primary alcohols
C Formation of secondary alcohols
D Formation of hydrazine

The correct answer is:
Formation of alkanes

Solution:
The reduction of the carbonyl group of aldehydes and ketones with zinc-amalgam and concentrated hydrochloric acid leads to the formation of alkanes.

Topic: Chemistry, Aldehydes, Ketones and Carboxylic Acids, NEET
What is the result of the second meiotic division of the secondary spermatocytes?

Options:
A Formation of two equal, haploid spermatids
B Formation of four equal, haploid spermatozoa
C Formation of two equal, diploid primary spermatocytes
D Formation of two unequal, haploid spermatids

The correct answer is:
Formation of four equal, haploid spermatozoa

Solution:
The secondary spermatocytes undergo the second meiotic division to produce four equal, haploid spermatids.

Topic: Biology, Human Reproduction, NEET
What is the result of the second meiotic division?

Options:
A The formation of a diploid zygote
B The fusion of a sperm with an ovum
C The formation of a second polar body and a haploid ovum
D The completion of the meiotic division of the secondary oocyte

The correct answer is:
The formation of a second polar body and a haploid ovum

Solution:
The second meiotic division is unequal and results in the formation of a second polar body and a haploid ovum.

Topic: Biology, Human Reproduction, NEET
What is the role of a promoter in a transcription unit?

Options:
A To prevent RNA from being translated into protein
B To define the boundaries of the region of DNA that would be transcribed
C To initiate the process of replication
D To catalyze the polymerization of RNA

The correct answer is:
To define the boundaries of the region of DNA that would be transcribed

Solution:
A promoter in a transcription unit helps define the boundaries of the region of DNA that would be transcribed.

Topic: Biology, Molecular Basis of Inheritance, NEET
What is the role of a terminator in a transcription unit?

Options:
A To initiate the process of transcription
B To define the boundaries of the region of DNA that would be transcribed
C To mark the end of the segment of DNA that would be transcribed
D To prevent RNA from being translated into protein

The correct answer is:
To mark the end of the segment of DNA that would be transcribed

Solution:
A terminator in a transcription unit marks the end of the segment of DNA that would be transcribed.

Topic: Biology, Molecular Basis of Inheritance, NEET
What is the role of all nations in conserving biodiversity?

Options:
A To conserve biodiversity in their political boundaries
B To conserve biodiversity in their geographical regions
C To conserve biodiversity in their local levels
D A collective responsibility of all nations

The correct answer is:
A collective responsibility of all nations

Solution:
Biodiversity knows no political boundaries and its conservation is therefore a collective responsibility of all nations.

Topic: Biology, Biodiversity and its Conservation, NEET
What is the role of NAD+ in cellular respiration?

Options:
A It is a nucleotide source for ATP synthesis.
B It functions as an electron carrier.
C It functions as an enzyme.
D It is the final electron acceptor for anaerobic respiration.

The correct answer is:
It functions as an electron carrier.

Solution:
Nicotinamide adenine dinucleotide is a coenzyme found in all living cells, and it's required for the fundamental biological processes that make life possible. It functions as an electron carrier.

Topic: Biology, Respiration in Plants, NEET, NEET 2018
What is the role of NAD+ in cellular respiration?

Options:
A It is a nucleotide source for ATP synthesis.
B It functions as an electron carrier.
C It functions as an enzyme.
D It is the final electron acceptor for anaerobic respiration.

The correct answer is:
It functions as an electron carrier.

Solution:
Nicotinamide adenine dinucleotide is a coenzyme found in all living cells, and it's required for the fundamental biological processes that make life possible. It functions as an electron carrier

Topic: Biology, Respiration in Plants, NEET
What is the role of the pollen tubes in pollination?

Options:
A To transport the male gametes to the egg cell
B To provide nourishment to the developing embryo
C To enter the embryo-sac and release the male gametes
D To transport the diploid secondary nucleus to the ovule

The correct answer is:
To enter the embryo-sac and release the male gametes

Solution:
Pollen tubes grow through the tissues of the stigma and style and reach the ovule, where they enter the embryo-sac and release the male gametes for fertilization.

Topic: Biology, Plant Kingdom, NEET
What is the role of the structural gene in a transcription unit?

Options:
A To initiate the process of transcription
B To define the boundaries of the region of DNA that would be transcribed
C To encode the information for the synthesis of a particular protein
D To prevent RNA from being translated into protein

The correct answer is:
To encode the information for the synthesis of a particular protein

Solution:
The structural gene in a transcription unit encodes the information for the synthesis of a particular protein.

Topic: Biology, Molecular Basis of Inheritance, NEET
What is the role of the zygote in pollination?

Options:
A To develop into an embryo
B ) To provide nourishment to the developing embryo
C To fuse with the diploid secondary nucleus
D To guide the pollen tubes to the ovules

The correct answer is:
To develop into an embryo

Solution:
The zygote, which is formed from the fusion of the male gamete and egg cell, develops into an embryo with one or two cotyledons.

Topic: Biology, Plant Kingdom, NEET
What is the second most abundant element in the earth's crust by mass?

Options:
A Carbon
B Germanium
C Silicon
D Tin

The correct answer is:
Silicon

Solution:
Silicon is the second (27.7% by mass) most abundant element in the earth's crust.

Topic: Chemistry, p Block Elements (Group 13 & 14), NEET
What is the secondary follicle characterised by?

Options:
A A fluid filled cavity
B The theca interna
C The zona pellucida
D The primary follicle

The correct answer is:
A fluid filled cavity

Solution:
The secondary follicle is characterised by a fluid filled cavity called antrum.

Topic: Biology, Human Reproduction, NEET
What is the secondary oocyte surrounded by?

Options:
A The zona pellucida
B The theca externa
C The theca interna
D A new membrane

The correct answer is:
The zona pellucida

Solution:
The secondary oocyte forms a new membrane called zona pellucida surrounding it.

Topic: Biology, Human Reproduction, NEET
What is the secondary structure of nucleic acids?

Options:
A The sequence of nucleotides in the chain
B Double-stranded helical structure
C Hydrogen bonds between pairs of bases
D Complementary base pairing

The correct answer is:
Double-stranded helical structure

Solution:
The secondary structure of nucleic acids is the double-stranded helical structure.

Topic: Chemistry, Biomolecules, NEET
What is the secondary structure of tRNA?

Options:
A A straight chain
B A circular loop
C A clover-leaf shape
D A double helix

The correct answer is:
A clover-leaf shape

Solution:
The secondary structure of tRNA is a clover-leaf shape.

Topic: Biology, Molecular Basis of Inheritance, NEET
What is the secondary valence of a metal ion according to Alfred Werner's concept?

Options:
A The valence involved in ionic bonding
B The valence that determines the metal's chemical reactivity
C The valence responsible for coordination with ligands
D The valence that determines the metal's atomic number

The correct answer is:
The valence responsible for coordination with ligands

Solution:
According to Alfred Werner's concept, the secondary valence of a metal ion is the valence responsible for coordination with ligands.

Topic: Chemistry, Coordination Compounds, NEET
What is the secondary valence of $\mathrm{Co}^{3+}$ ion according to Werner's theory in $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}_{2}\right]^{+}$ ?

Options:
A 5
B 4
C 3
D 6

The correct answer is:
6

Solution:
(C)
Secondary valence $=$ Coordination number $=6$

Topic: Chemistry, Coordination Compounds, MHT CET, MHT CET 2020 (16 Oct Shift 2)
What is the set of all points, where the function
$f(x)=\frac{x}{1+|x|}$ is differentiable?

Options:
A $(-\infty, \infty)$ only
B $(0, \infty)$ only
C $(-\infty, 0) \cup(0, \infty)$ only
D $(-\infty, 0)$ only

The correct answer is:
$(-\infty, \infty)$ only

Solution:
Given $f(x)=\frac{x}{1+|x|}$
$=\left\{\begin{array}{c}\frac{x}{1-x}, x < 0 \\ \frac{x}{1+x}, x \geq 0\end{array} \quad\left(\because|x|=\left\{\begin{array}{lll}x & \text { if } \quad x \geq 0 \\ -x & \text { if } \quad x < 0\end{array}\right)\right.\right.$
$\therefore \quad$ LHD $=f^{\prime}\left(0^{-}\right)=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{-h}$
$=\lim _{h \rightarrow 0} \frac{f(-h)-f(0)}{-h}$
and $\mathrm{RHD}=f^{\prime}\left(0^{+}\right)$
$=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$
$\quad$ Since, $\mathrm{LHD}=\mathrm{RHD}$
$\therefore \quad f(x)$ is differentiable at $x=0$
Hence, $f(x)$ is differentiable in $(-\infty, \infty)$.

Topic: Mathematics, Continuity and Differentiability, NDA, NDA 2009 (Phase 1)
What is the set of points $(x, y)$ satisfying the equations $x^{2}+y^{2}=4$ and $x+y=2 ?$

Options:
A $\{(2,0),(-2,0),(0,2)\}$
B $\{(0,2),(0,-2)\}$
C $\{(0,2),(2,0)\}$
D $\{(2,0),(-2,0),(0,2),(0,-2)\}$

The correct answer is:
$\{(0,2),(2,0)\}$

Solution:
The given equations are $x^{2}+v^{2}=4$
and $x+y=2$
$\quad x^{2}+(2-x)^{2}=4$
$\Rightarrow x^{2}+4+x^{2}-4 x=4$
$\Rightarrow 2 x^{2}=4 x$...(i)
$\Rightarrow x=2$
When $x=2$ then $y=2-2=0$.. (ii)
Similarly $y^{2}+(2-y)^{2}=4$
$\Rightarrow y^{2}+4+y^{2}-4 y=4$
$\Rightarrow y=2$ and $x=0$
These equations are satisfied by only $(2,0)$ and $(0,2)$. Hence, required set is $\{(0,2),(2,0)\}$

Topic: Mathematics, Quadratic Equation, NDA, NDA 2011 (Phase 1)
What is the seventeenth most abundant element by mass in the earth's crust?

Options:
A Silicon
B Germanium
C Carbon
D Tin

The correct answer is:
Carbon

Solution:
Carbon is the seventeenth most abundant element by mass in the earth's crust.

Topic: Chemistry, p Block Elements (Group 13 & 14), NEET
What is the seventh term of the sequence $0,3,8,15,24, \ldots$ ?

Options:
A 63
B 48
C 35
D 33

The correct answer is:
48

Solution:
$0+3=3,3+5=8,8+7=15,15+9=24,24+11=35$,
$35+13=48$
Sequence is $0,3,8,15,24,35,48$

Topic: Mathematics, Sequences and Series, NDA, NDA 2013 (Phase 2)
What is the significance of Alfred Werner's concept of primary and secondary valence?

Options:
A It explained the concept of atomic mass.
B It described the behavior of gases.
C It provided insights into the structures of coordination compounds.
D It explained the concept of atomic radius.

The correct answer is:
It provided insights into the structures of coordination compounds.

Solution:
Alfred Werner's concept of primary and secondary valence was instrumental in understanding the structures of coordination compounds.

Topic: Chemistry, Coordination Compounds, NEET
What is the significance of distinguishing between primary, secondary, and tertiary amines in these reactions?

Options:
A It helps in the estimation of protein content
B It is important for pharmaceutical applications
C It aids in the separation of amine mixtures
D It determines the colour of the final product

The correct answer is:
It aids in the separation of amine mixtures

Solution:
Distinguishing between primary, secondary, and tertiary amines is important for the separation of amine mixtures.

Topic: Chemistry, Amines, NEET
What is the significance of the intense activity at the boundary between ice and water in equilibrium?

Options:
A It signifies an unstable equilibrium.
B It is a static equilibrium.
C It indicates dynamic equilibrium with equal rates of molecule transfer.
D It signifies a change in pressure.

The correct answer is:
It indicates dynamic equilibrium with equal rates of molecule transfer.

Solution:
The intense activity at the boundary between ice and water is a characteristic of dynamic equilibrium with equal rates of molecule transfer.

Topic: Chemistry, Chemical Equilibrium, NEET
What is the significance of the terms "isolated gaseous atom" and "ground state" while defining the ionisation enthalpy and electron gain enthalpy?

Options:
A These terms indicate that the enthalpy values are measured in a vacuum.
B These terms indicate that the enthalpy values are measured at standard temperature and pressure.
C "Isolated gaseous atom" emphasises that the atom is in its most stable state, and "ground state" indicates that the electron configurations are in their lowest energy states.
D "Isolated gaseous atom" and "ground state" are technical terms without specific significance in the context of ionisation enthalpy and electron gain enthalpy

The correct answer is:
"Isolated gaseous atom" emphasises that the atom is in its most stable state, and "ground state" indicates that the electron configurations are in their lowest energy states.

Solution:
Correct Option is : (C)
"Isolated gaseous atom" emphasises that the atom is in its most stable state, and "ground state" indicates that the electron configurations are in their lowest energy states.

Topic: Chemistry, Classification of Elements and Periodicity in Properties, NEET
What is the sine of angle between the vectors $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$
and $-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} ?$

Options:
A $\sqrt{\frac{13}{7}}$
B $\frac{\sqrt{13}}{7}$
C $\frac{13}{\sqrt{7}}$
D None of these

The correct answer is:
$\frac{\sqrt{13}}{7}$

Solution:
We know that, the angle between the vectors
$a_{1} \hat{i}+b_{1} \hat{j}+c_{1} \hat{k}$ and $a_{2} \hat{i}+b_{2} \hat{j}+c_{2} \hat{k}$ is given by
$\cos \theta=\left[\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\right]$
$\therefore \quad$ Angle between the vectors $\hat{i}+2 \hat{j}+3 \hat{k}$ and
$-\hat{i}+2 \hat{j}+3 \hat{k}$ is given by
$\cos \theta=\left[\frac{1 \times(-1)+2 \times 2+3 \times 3}{\sqrt{1+4+9} \sqrt{1+4+9}}\right]$
$=\frac{-1+4+9}{14}=\frac{12}{14}=\frac{6}{7}$
Now, $\sin \theta=\sqrt{1-\cos ^{2} \theta}$
$=\sqrt{1-\frac{36}{49}}=\sqrt{\frac{49-36}{49}}=\sqrt{\frac{13}{49}}=\frac{\sqrt{13}}{7}$

Topic: Mathematics, Vector Algebra, NDA, NDA 2011 (Phase 1)
What is the slope of the line perpendicular to the line $\frac{x}{4}+\frac{y}{3}=1 ?$

Options:
A $\frac{3}{4}$
B $-\frac{3}{4}$
C $-\frac{4}{3}$
D $\frac{4}{3}$

The correct answer is:
$\frac{4}{3}$

Solution:
Given equation of the line $\frac{x}{4}+\frac{y}{3}=1$ can be written as $\begin{aligned} 3 x+4 y=12 & \Rightarrow & 4 y &=-3 x+12 \\ & \Rightarrow \quad & y &=\frac{-3}{4} x+\frac{12}{4} \end{aligned}$
The slope of the line $\frac{x}{y}+\frac{y}{3}=1$ is $\frac{-3}{4}$
$\therefore$ Slope of the line perpendicular to this line $=-\left(\frac{-1}{3 / 4}\right)=\frac{4}{3}$

Topic: Mathematics, Straight Lines, NDA, NDA 2010 (Phase 2)
What is the slope of the normal at the point (at $^{2}, 2$ at) of the parabola $y^{2}=4 a x$ ?

Options:
A $\frac{1}{t}$
B t
C $-t$
D $-\frac{1}{t} \quad$

The correct answer is:
$-t$

Solution:
Equation of parabola is $y^{2}=4 a x$
$2 y \frac{d y}{d x}=4 a$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2 \mathrm{a}}{\mathrm{y}}, \quad$ [slope of tangent]
So, slope of normal $=-\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)_{\left(\mathrm{at}^{2}, 2 \mathrm{at}\right)}$
$=-\left(\frac{\mathrm{y}}{2 \mathrm{a}}\right)=-\frac{2 \mathrm{at}}{2 \mathrm{a}}=-\mathrm{t}$

Topic: Mathematics, Application of Derivatives, NDA, NDA 2006 (Phase 1)
What is the slope of the tangent to the curve $\mathrm{x}=\mathrm{t}^{2}+3 \mathrm{t}-8, \mathrm{y}=2 \mathrm{t}^{2}-2 \mathrm{t}-5$ at $\mathrm{t}=2 ? \quad$

Options:
A $7 / 6$
B $6 / 7$
C 1
D $5 / 6$

The correct answer is:
$6 / 7$

Solution:
$\frac{\mathrm{dx}}{\mathrm{dt}}=2 \mathrm{t}+3, \frac{\mathrm{dy}}{\mathrm{dt}}=4 \mathrm{t}-2$
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy} / \mathrm{dt}}{\mathrm{d} \mathrm{x} / \mathrm{dt}}=\frac{4 \mathrm{t}-2}{2 \mathrm{t}+3}$
slope at $t=2=\frac{\mathrm{dy}}{\mathrm{dx}} / \mathrm{t}=2=\frac{4(2)-2}{2(2)+3}=\frac{6}{7}$

Topic: Mathematics, Application of Derivatives, NDA, NDA 2012 (Phase 1)
What is the slope of the tangent to the curve $y=\sin ^{1}\left(\sin ^{2} x\right)$ at $x=0 ?$

Options:
A 0
B 1
C 2
D None of these

The correct answer is:
0

Solution:
$\mathrm{y}=\sin ^{-1}\left(\sin ^{2} \mathrm{x}\right)$
$\frac{d y}{d x}=\frac{2 \sin x \cos x}{\sqrt{1-\sin ^{4} x}} \Rightarrow \frac{d y}{d x}=\frac{\sin 2 x}{\sqrt{1-\sin ^{4} x}}$
at $\mathrm{x}=0, \frac{\mathrm{dy}}{\mathrm{dx}}=0$

Topic: Mathematics, Application of Derivatives, NDA, NDA 2014 (Phase 1)
What is the smallest natural number $n$ such that $n !$ is divisible by $990 ?$

Options:
A 9
B 11
C 33
D 99

The correct answer is:
11

Solution:
Consider option 'a' Let us take $n=9$ Since, $9 !=9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1=362880$
which is not divisible by 990 .
Now assume, $n=11$ Since $11 !=39916800$
which is divisible by 990 . Thus, required smallest natural number 11

Topic: Mathematics, Permutation Combination, NDA, NDA 2009 (Phase 1)
What is the smallest value of $\mathrm{m}$ for which $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}+\mathrm{mx}+5$ is increasing in the interval $1 \leq \mathrm{x} \leq 2 ?$

Options:
A $\mathrm{m}=0$
B $\mathrm{m}=-1$
C $\mathrm{m}=-2$
D $\mathrm{m}=-3$

The correct answer is:
$\mathrm{m}=-2$

Solution:
$f(x)=x^{2}+m x+5$
Consider, option (c) $\mathrm{m}=-2$. $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-2 \mathrm{x}+5 \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{x}-2$
It is clear that $\mathrm{f}^{\prime}(\mathrm{x})>0$ in interval $1 \leq \mathrm{x} \leq 2$. $\therefore \quad \mathrm{m}=-2$

Topic: Mathematics, Application of Derivatives, NDA, NDA 2008 (Phase 1)
What is the solution of $(1+2 \mathrm{x}) \mathrm{dy}-(1-2 \mathrm{y}) \mathrm{dx}=0 ?$

Options:
A $x-y-2 x y=c$
B $\mathrm{y}-\mathrm{x}-2 \mathrm{xy}=\mathrm{c}$
C $y+x-2 x y=c$
D $x+y+2 x y=c$

The correct answer is:
$x-y-2 x y=c$

Solution:
$\begin{aligned} &(1+2 \mathrm{x}) \mathrm{dy}-(1-2 \mathrm{y}) \mathrm{dx}=0 \Rightarrow(1+2 \mathrm{x}) \mathrm{dy}=(1-2 \mathrm{y}) \mathrm{d} \mathrm{x} \\ & \Rightarrow \frac{\mathrm{dy}}{1-2 \mathrm{y}}=\frac{\mathrm{d} \mathrm{x}}{1+2 \mathrm{x}} \end{aligned}$
Integrating both sides, $\int \frac{\mathrm{dy}}{1-2 \mathrm{y}}=\int \frac{\mathrm{dx}}{1+2 \mathrm{x}}$
$\Rightarrow \frac{-1}{2} \log (1-2 \mathrm{y})=\frac{1}{2} \log (1+2 \mathrm{x})+\frac{1}{2} \log \mathrm{c}$
$\Rightarrow \log (1-2 y)+\log (1+2 x)=\log c$
$\Rightarrow(1+2 x)(1-2 y)=c$
$\Rightarrow 1+2 x-2 y-4 x y=c$
$\Rightarrow 2 x-2 y-4 x y=c$
$\Rightarrow x-y-2 x y=c$

Topic: Mathematics, Differential Equations, NDA, NDA 2018 (Phase 1)
What is the solution of $\frac{d y}{d x}+2 y=1$ satisfying $y(0)=0$ ?

Options:
A $y=\frac{1-e^{-2 x}}{2}$
B $y=\frac{1+e^{-2 x}}{2}$
C $y=1+e^{x}$
D $y=\frac{1+e^{x}}{2}$

The correct answer is:
$y=\frac{1-e^{-2 x}}{2}$

Solution:
$\frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{y}=1 \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=1-2 \mathrm{y}$
$\int \frac{\mathrm{dy}}{1-2 \mathrm{y}}=\int \mathrm{dx}$
$-\frac{1}{2} \log |1-2 y|=x+C$
at $\mathrm{x}=0, \mathrm{y}=0$
$-\frac{1}{2} \log 1=0+\mathrm{C} \Rightarrow \mathrm{C}=0$
$1-2 \mathrm{y}=\mathrm{e}^{-2 \mathrm{x}}$
$\mathrm{y}=\frac{1-\mathrm{e}^{-2 \mathrm{x}}}{2}$

Topic: Mathematics, Differential Equations, NDA, NDA 2014 (Phase 1)
What is the solution of the differential equation $3 e^{x} \tan y d x+\left(1+e^{x}\right) \sec ^{2} y d y=0 ? \quad$

Options:
A $\left(1+e^{x}\right) \tan y=c$
B $\left(1+e^{x}\right)^{3} \tan y=c$
C $\left(1+e^{x}\right)^{2} \tan y=c$
D $\left(1+e^{x}\right) \sec ^{2} y=c$
where $c$ is a constant of integration.

The correct answer is:
$\left(1+e^{x}\right)^{3} \tan y=c$

Solution:
$3 e^{x} \tan y d x+\left(1+e^{x}\right) \sec ^{2} y d y=0$
By separating the variables, we get
$$
\begin{array}{l}
3 e^{x} d x=\frac{-\left(1+e^{x}\right) \sec ^{2} y}{\tan y} d y \\
\Rightarrow \frac{3 e^{x}}{1+e^{x}} d x=-\frac{\sec ^{2} y}{\tan y} d y
\end{array}
$$
Integrate on both sides,
$\Rightarrow \int \frac{3 e^{x}}{1+e^{x}} d x+\int \frac{\sec ^{2} y}{\tan y} d y=0$
$\Rightarrow 3 \log \left(1+e^{x}\right)+\log \tan y=\log c$
$\Rightarrow \log \left(1+e^{x}\right)^{3} \tan y=\log c$
$(\because \log m+\log n=\log m n)$
$\Rightarrow\left(1+e^{x}\right)^{3} \tan y=c$

Topic: Mathematics, Differential Equations, NDA, NDA 2010 (Phase 1)