datavorous/entrance-exam-dataset huggingface dataset:
The condition that the equation \( l x+m y+n=0 \) represents the equation of a straight line in the normal form, is

Options:
A \( l^{2}+m^{2} \neq 0 ; n>0 \)
B \( l^{2}+m^{2} \neq 0 ; n < 0 \)
C \( l^{2}+m^{2}=1 ; n < 0 \)
D \( l^{2}+m^{2}=1 ; n>0 \)

The correct answer is:
\( l^{2}+m^{2}=1 ; n < 0 \)

Solution:
Given,

lx+my+n=0.......(i)

Standard normal form of an equation of a line is xcos⁡α+ysin⁡α=p, p>0......(ii)

(p is the perpendicular distance from the origin to the line, α is an angle made by its perpendicular with the positive direction of the x-axis)

Comparing the equation (ii) with the given equation (i), we get

∴l=cosα, m=sinα, n=-p.

And n<0.

As cos2α+sin2α=1⇒l2+m2=1, n<0

Topic: Mathematics, Straight Lines, JEE Main
The condition to be imposed on \( \beta \), so that \( (0, \beta) \) lies on or inside of the triangle having equation of sides as
\( y+3 x+2=0,3 y-2 x-5=0 \) and \( 4 y+x-14=0 \), is?

Options:
A \( 0 < \beta < \frac{5}{2} \)
B \( 0 < \beta < \frac{7}{2} \)
C \( \frac{5}{3} \leq \beta \leq \frac{7}{2} \)
D None of these

The correct answer is:
\( \frac{5}{3} \leq \beta \leq \frac{7}{2} \)

Solution:
Draw the lines y+3x+2=0, 3y-2x-5=0 and 4y+x-14=0

Given, 0, β lies inside the triangle formed by the lines y+3x+2=0, 3y-2x-5=0 & 4y+x-14=0. The above figure clearly shows that 0, β lies above the lines y+3x+2=0 & 3y-2x-5=0 and below the line 4y+x-14=0.

∴β+3×0+2>0, 3β-2×0-5≥0 & 4β+0-14≤0

⇒β>-2, β≥53 & β≤72

⇒β∈53, 72.

Topic: Mathematics, Straight Lines, JEE Main
The condition to be imposed on Î², so that (0, Î²) lies on or insideof the triangle having equation of sides as y+3x+2=0, 3y-2x-5=0 and 4y+x-14=0, is

Options:
A 0 < β < 5 2
B 0 < β < 7 2
C 5 3 â¤ Î² â¤ 7 2
D None of these

The correct answer is:
5 3 ≤ β ≤ 7 2

Solution:
We can find the position of Î² on y-axis by plotting the lines and their intersection point on the y-axis.

âµ Î² lie on the y-axis and inside the triangle formed by the lines.
âµ 5 3 â¤ Î² â¤ 7 2

Topic: Mathematics, Straight Lines, JEE Main
The conditional $(p \wedge q) \Rightarrow p$ is

Options:
A A tautology
B A fallacy i.e., contradiction
C Neither tautology nor fallacy
D None of these

The correct answer is:
A tautology

Solution:
$$
\begin{array}{|c|c|c|c|}
\hline p & q & p \wedge q & (p \wedge q) \Rightarrow p \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline
\end{array}
$$
$\therefore(p \wedge q) \Rightarrow p$ is a tautology.

Topic: Mathematics, Mathematical Reasoning, JEE Main
The conditions given below are in the context of observing Tyndall effect in colloidal solutions:
(A) The diameter of the colloidal particles is comparable to the wavelength of light used.
(B) The diameter of the colloidal particles is much smaller than the wavelength of light used.
(C) The diameter of the colloidal particles is much larger than the wavelength of light used.
(D) The refractive indices of the dispersed phase and the dispersion medium are comparable.
(E) The dispersed phase has a very different refractive index from the dispersion medium. Choose the most appropriate conditions from the options given below:

Options:
A (A) and (E) only
B (C) and (D) only
C (A) and (D) only
D (B) and (E) only

The correct answer is:
(A) and (E) only

Solution:
The phenomenon of scattering of light by colloidal particles as a result of which the path of the beam becomes visible is called a tyndall effect. Smaller the diameter and similar the magnitude of refractive indices, lesser is the scattering and hence the tyndall effect and viced-versa. The diameter of the dispersed phase particle should not be smaller than the wavelength of light used because they won't be able to scatter the light so, therefore, the diameter of the dispersed particles should be equal or not much smaller than the wavelength of the light used.
2. The refractive indices (i.e. the ratio of the velocity of light in vacuum to the velocity of light in any medium) of the dispersed phase and the dispersion medium should differ greatly in magnitude than only the particles will be able to scatter the light and tyndall effect will be observed. On the other hand, if the refractive indices of the dispersed phase and dispersion medium are almost similar in magnitude, then there will be no scattering of light and hence, therefore, no tyndall effect is observed.
Hence answer A and E are correct.

Topic: Chemistry, Surface Chemistry, JEE Main, JEE Main 2021 (20 Jul Shift 1)
The conductance of a 0.0015 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 cm with an area of cross-section of 1 cm2 . The conductance of this solution was found to be 5×10-7 S. The pH of the solution is 4. The value of limiting molar conductivity Λmo if this weak monobasic acid in aqueous solution is Z×102 S cm2mol-1 . The value of Z is

Options:

The correct answer is:
6

Solution:
[ H + ] =  10 −pH  =  10 −4  M
ÎM=kÃ1000M
=GÃlaÃ1000M
=5Ã10-7Ã1201Ã10000.0015
=40  Scm 2 mol −1
Now, H+=CÎ±=0.0015ÃÎmÎmâ
Î m â â=âZâÃâ 10 2 â=â 0.0015âÃâ40 10 â4
On solving Z = 6.

Topic: Chemistry, Electrochemistry, JEE Main
The conduction current is the same as displacement current when source is :-

Options:
A \( a c \) only
B \( d c \) only
C both \( a c \) and \( d c \)
D neither \( d c \) nor \( a c \)

The correct answer is:
both \( a c \) and \( d c \)

Solution:
Displacement current is, Id=ε0dϕdt ……∵gauss law ϕ=Qε0=ε0dQε0dt=dQdt

Conduction current is,

I=dQdt

Thus, conduction current is equal to displacement current irrespective of the voltage source.

Topic: Physics, Electromagnetic Waves, JEE Main
The conductivity of a semiconductor sample having electron concentration of 5Ã1018 e m-3, hole concentration of 5Ã1019 holes m-3, electron mobility of 2.0 m2 V-1 s-1 and hole mobility of 0.01 m2 V-1 s-1 is

(Take charge of an electron as 1.6Ã10-19 C )

Options:
A 1.83 Î© m-1
B 1.65 Î© m-1
C 1.20 Ω m-1
D 0.59 Ω m-1

The correct answer is:
1.65 Î© m-1

Solution:
The conductivity of a semiconductor is given by
Ï=eneÎ¼e+nhÎ¼h
=1.6Ã10-195Ã1018Ã2+5Ã1019Ã0.01
=1.6Ã10-191019+0.05Ã1019
=1.6+1.05
=1.65 (Î© m)â1

Topic: Physics, Semiconductors, JEE Main
The conductivity of a semiconductor sample having electron concentration of 5×1018 electrons m-3, hole concentration of 5×1019 holes m-3, electron mobility of 2.0 m2 V-1 s-1 and hole mobility of 0.01 m2 V-1 s-1 is
(Take charge of an electron as 1.6×10-19 C )

Options:
A 1.83 Î© m-1
B 1.65 Î© m-1
C 1.20 Ω m-1
D 0.59 Ω m-1

The correct answer is:
1.65 Î© m-1

Solution:
The conductivity of a semiconductor is given by
σ=eneμe+nhμh
=1.6×10-195×1018×2+5×1019×0.01
=1.6×10-191019+0.05×1019
=1.6+1.05
=1.65 (Ω m)−1

Topic: Physics, Semiconductors, JEE Main, JEE Main 2017 (08 Apr Online)
The conductivity of a solution of complex with formula CoCl3NH34 corresponds to 1:1 electrolyte, then the primary valency of central metal ion is

Options:

The correct answer is:
3

Solution:
Primary valencies are ionizable. If the complex ion exhibits positive charge, then primary valence corresponds to the number of charges present and is balanced by the same number of negative ions. Hence, primary valency can also be defined by the number of anions neutralizing the charge on the complex.

CoNH34Cl2Cl

The above complex can be represented as CoNH34Cl2+Cl-. Hence, the primary valency of above complex is 3.

Topic: Chemistry, Coordination Compounds, JEE Main, JEE Main 2022 (27 Jul Shift 1)
The conductivity of a weak acid HA of concentration $0.001 \mathrm{~mol} \mathrm{~L}^{-1}$ is $2.0 \times 10^{-5} \mathrm{~S} \mathrm{~cm}^{-1}$.
If $\Lambda_{\mathrm{m}}^{\circ}(\mathrm{HA})=190 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$, the ionization constant $\left(\mathrm{K}_{\mathrm{a}}\right)$ of $\mathrm{HA}$ is equal to _______ x $10^{-6}$.

Options:
A 24
B 48
C 12
D 45

The correct answer is:
12

Solution:
$\Lambda_{\mathrm{m}}=1000 \times \frac{\mathrm{K}}{\mathrm{M}}$
$=1000 \times \frac{2 \times 10^{-5}}{0.001}=20 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$
$\Rightarrow \alpha=\frac{\Lambda_{\mathrm{m}}}{\Lambda_{\mathrm{m}}^{\circ}}=\frac{20}{190}=\left(\frac{2}{19}\right)$
$\underset{0.001(1-\alpha)}{\mathrm{HA}} \rightleftharpoons \underset{0.001 \alpha}{\mathrm{H}^{+}}+\underset{0.001 \alpha}{\mathrm{A}^{-}}$
$K_{\mathrm{a}}=0.001\left(\frac{\alpha^2}{1-\alpha}\right)=\frac{0.001 \times\left(\frac{2}{19}\right)^2}{1-\left(\frac{2}{19}\right)}$
$=12.3 \times 10^{-6}$

Topic: Chemistry, Electrochemistry, JEE Main
The conductivity of strong electrolyte is

Options:
A Increase on dilution slightly
B Decrease on dilution
C Does not change with dilution
D Depend upon density of electrolytes itself

The correct answer is:
Increase on dilution slightly

Solution:
Generally strong electrolyte on dilution shows conductivity characters.

Topic: Chemistry, Electrochemistry, JEE Main
The configuration $1 s^2, 2 s^2 2 p^5, 3 s^1$ shows

Options:
A Excited state of $\mathrm{O}_2^{-}$
B Excited state of neon
C Excited state of fluorine
D Ground state of fluorine atom

The correct answer is:
Excited state of neon

Solution:
The ground state of neon is $1 s^2 2 s^2 2 p^6$ on excitation an electron from $2 p$ jumps to $3 s$ orbital. The excited neon configuration is $1 s^2 2 s^2 2 p^5 3 s^1$.

Topic: Chemistry, Structure of Atom, JEE Main
The conic represented by $x=2(\cos t+\sin t), y=5(\cos t-\sin t)$ is

Options:
A a circle
B a parabola
C an ellipse
D a hyperbola

The correct answer is:
an ellipse

Solution:
From given equations
$$
\begin{array}{l}
\frac{x}{2}=\cos t+\sin t ......(i)\\
\frac{y}{5}=\cos t-\sin t ......(ii)
\end{array}
$$
Eliminating t from (i) and (ii), we have $\frac{\mathrm{x}^{2}}{4}+\frac{\mathrm{y}^{2}}{25}=2 \Rightarrow \frac{\mathrm{x}^{2}}{8}+\frac{\mathrm{y}^{2}}{50}=1$, which is an ellipse.

Topic: Mathematics, Ellipse, JEE Main
The conjugate acid of $\mathrm{HPO}_3^{2-}$ is

Options:
A $\mathrm{H}_3 \mathrm{PO}_4$
B $\mathrm{H}_3 \mathrm{PO}_3$
C $\mathrm{H}_2 \mathrm{PO}_3^{-}$
D $\mathrm{PO}_4^{3-}$

The correct answer is:
$\mathrm{H}_2 \mathrm{PO}_3^{-}$

Solution:
$\mathrm{HPO}_3^{2-}+\mathrm{H}^{+} \rightarrow \mathrm{H}_2 \mathrm{PO}_3^{-}$

Topic: Chemistry, Ionic Equilibrium, JEE Main
The conjugate base of a strong acid is a

Options:
A Strong base
B Strong acid
C Weak acid
D Weak base

The correct answer is:
Weak base

Solution:
According to Bronsted-Lowry acid-base theory, the stronger of a pair of acids must form the weaker of a pair of conjugate bases. Therefore, the conjugate base of a strong acid is a weak base.

Topic: Chemistry, Ionic Equilibrium, JEE Main
The conjugate base of $\mathrm{H}_2 \mathrm{PO}_4^{-}$is

Options:
A
$\mathrm{PO}_4^{3-}$

B
$\mathrm{HPO}_4^{2-}$

C
$\mathrm{H}_3 \mathrm{PO}_4$

D
$\mathrm{P}_2 \mathrm{O}_5$

The correct answer is:
$\mathrm{HPO}_4^{2-}$

Solution:
$\because 6.02 \times 10^{+20}$ molecules of urea is present in $=\frac{0.0001 \times 1000}{100}=0.01 \mathrm{M}$

Topic: Chemistry, Ionic Equilibrium, JEE Main, JEE Main 2004
The conjugate base of hydrazoic acid is:

Options:
A
$\mathrm{N}^{-3}$

B
$\mathrm{N}_3^{-}$

C
$\mathrm{N}_2^{-}$

D
$\mathrm{HN}_3^{-}$

The correct answer is:
$\mathrm{N}_3^{-}$

Solution:
$\mathrm{N}_3 \mathrm{H} \rightleftharpoons \mathrm{N}_3^{-}+\mathrm{H}^{+}$ Hydrazoic acid i.e, conjugate base of hydrazoic acid is $\mathrm{N}_3^{-}$.

Topic: Chemistry, Ionic Equilibrium, JEE Main, JEE Main 2014 (12 Apr Online)
The conjugate base of $\mathrm{NH}_2^{-}$is

Options:
A $\mathrm{NH}_3$
B $\mathrm{NH}^{2-}$
C $\mathrm{NH}_4^{+}$
D $\mathrm{N}_3^{-}$

The correct answer is:
$\mathrm{NH}^{2-}$

Solution:
$N H_2^{-} \Leftrightarrow N H^{-2}+H^{+}$
Conjugate acid, base pair.

Topic: Chemistry, Ionic Equilibrium, JEE Main
The conjugate of $\frac{(2+i)^2}{3+i}$, in the form of $a+i b$, is

Options:
A $\frac{13}{2}+i\left(\frac{15}{2}\right)$
B $\frac{13}{10}+i\left(\frac{-15}{2}\right)$
C $\frac{13}{10}+i\left(\frac{-9}{10}\right)$
D $\frac{13}{10}+i\left(\frac{9}{10}\right)$

The correct answer is:
$\frac{13}{10}+i\left(\frac{-9}{10}\right)$

Solution:
$\begin{aligned} & z=\frac{(2+i)^2}{3+i}=\frac{3+4 i}{3+i} \times \frac{3-i}{3-i}=\frac{13}{10}+i \frac{9}{10} \\ & \text { Conjugate }=\frac{13}{10}-i \frac{9}{10} .\end{aligned}$

Topic: Mathematics, Complex Number, JEE Main
The conjugate of a complex number is $\frac{1}{i-1}$. Then the complex number is

Options:
A
$\frac{-1}{i-1}$

B
$\frac{1}{i+1}$

C
$\frac{-1}{i+1}$

D
$\frac{1}{i-1}$

The correct answer is:
$\frac{-1}{i+1}$

Solution:
Put $-i$ in place of $i$
Hence $\frac{-1}{i+1}$.

Topic: Mathematics, Complex Number, JEE Main, JEE Main 2008
The conjugate of the complex number $\frac{2+5 i}{4-3 i}$ is

Options:
A $\frac{7-26 i}{25}$
B $\frac{-7-26 i}{25}$
C $\frac{-7+26 i}{25}$
D $\frac{7+26 i}{25}$

The correct answer is:
$\frac{-7-26 i}{25}$

Solution:
$\frac{2+5 i}{4-3 i}=\frac{(2+5 i)(4+3 i)}{25}=\frac{-7+26 i}{25}$.
Therefore conjugate of the complex number is $\frac{-7-26 i}{25}$.

Topic: Mathematics, Complex Number, JEE Main
The consecutive odd integers whose sum is 452-212 are

Options:
A 43,45,.....,75
B 43,45,.....,79
C 43,45,.....,85
D 43,45,.....,89

The correct answer is:
43,45,.....,89

Solution:
Let n consecutive odd integers are
2m+1,2m+3,2m+5,.....,2m+2n-1
Given that,
452-212=2m+1+2m+3+2m+5+......+2m+2n-1
=2mn+1+3+5+.....+2n-1
=2mn+n2
=m2+2mn+n2-m2
⇒452-212=m+n2-m2
⇒m+n=45 and m=21
⇒n=24 and m=21
Hence, the numbers are
43,45,....,89

Topic: Mathematics, Sequences and Series, JEE Main
The consequence of global warming may be the following

Options:
A Decrease in average temperature of the earth
B Melting of Himalayan glaciers
C Eutrophication
D Increased biochemical oxygen demand

The correct answer is:
Melting of Himalayan glaciers

Solution:
Global warming refers to the steady temperature rise on our planet.Global warming is the unusually rapid increase in Earth's average surface temperature over th past century primarily due to the greenhouse gases released by people burning fossil fuels.

Global warming is the main reason of melting of Himalayan glaciers. Glacier ice loss linked to rising temperatures from climate change.

Topic: Chemistry, Environmental Chemistry, TS EAMCET, TS EAMCET 2020 (09 Sep Shift 1)
The continuous upward flow of water stream in tall trees is maintained due to ______________________.

Options:
A Guttation and transpiration
B Transpiration pull only
C Cohesive force between water molecules only
D Cohesive force between water molecules and transpiration pull

The correct answer is:
Cohesive force between water molecules and transpiration pull

Solution:
The continuous upward flow of water stream in a tall tree is maintained due to cohesive force between water molecules and transpiration pull.

Topic: Biology, Transport in Plants, NEET
The contrapositive of $(p \vee q) \Rightarrow r$ is

Options:
A $r \Rightarrow(p \vee q)$
B $\sim r \Rightarrow(p \vee q)$
C $\sim r \Rightarrow \sim p \wedge \sim q$
D $p \Rightarrow(q \vee r)$

The correct answer is:
$\sim r \Rightarrow \sim p \wedge \sim q$

Solution:
Contrapositive of $p \Rightarrow q$ is $\sim q \Rightarrow \sim p$
$\therefore$ contrapositive of $(p \vee q) \Rightarrow r$ is
$\sim r \Rightarrow \sim(p \vee q)$ i.e. $\sim r \Rightarrow(\sim p \wedge \sim q)$

Topic: Mathematics, Mathematical Reasoning, JEE Main
The contrapositive of (p∨q)⇒r is

Options:
A r⇒(p∨q)
B ~r⇒(p∨q)
C ~r⇒ ~p ∧~q
D p⇒(q∨r)

The correct answer is:
~r⇒ ~p ∧~q

Solution:
Contrapositive of p⇒q is ~q⇒ ~p
∴contrapositive of (p∨q)⇒r is ~r⇒ ~(p∨q) i.e. ~r⇒(~p ∧~q) .

Topic: Mathematics, Mathematical Reasoning, JEE Main
The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is

Options:
A if the area of a square increases four times, then its side is not doubled.
B if the area of a square increases four times, then its side is doubled.
C if the area of a square does not increase four times, then its side is not doubled.
D if the side of a square is not doubled, then its area does not increase four times.

The correct answer is:
if the area of a square does not increase four times, then its side is not doubled.

Solution:
Contrapositive of p→q is given by ~q→~p.
So, contrapositive of given statement is, "If the area of a square does not increase four times, then its side is not doubled".

Topic: Mathematics, Mathematical Reasoning, JEE Main, JEE Main 2016 (10 Apr Online)
The contrapositive of the statement "I go to school if it does not rain" is

Options:
A If it rains, I go to school.
B If it rains, I do not go to school.
C If I go to school, it rains.
D If I do not go to school, it rains.

The correct answer is:
If it rains, I do not go to school.

Solution:
Let p= It does not rain

q= I go to school

Contrapositive of if p  → q is if ∼q → ∼p

So, If it rains, I do not go to school.

Topic: Mathematics, Mathematical Reasoning, JEE Main, JEE Main 2014 (09 Apr Online)
The contrapositive of the statement ''if I am not feeling well, then I will go to the doctor" is

Options:
A if I will go to the doctor, then I am not feeling well.
B if I am feeling well, then I will not go to the doctor.
C if I will not go to the doctor, then I am feeling well.
D if I will go to the doctor, then I am feeling well.

The correct answer is:
if I will not go to the doctor, then I am feeling well.

Solution:
Statement: If I am not feeling well, then I will go to the doctor.

Let p: I am not feeling well and q: I will go to the doctor.

The given statement is p→q.

We know that the contrapositive of p→q is ~q→~p.

Contrapositive: If I will not go to the doctor, then I am feeling well.

Topic: Mathematics, Mathematical Reasoning, JEE Main, JEE Main 2014 (19 Apr Online)
The contrapositive of the statement "If I reach the station in time, then I will catch the train" is

Options:
A If I do not reach the station in time, then I will catch the train.
B If do not reach the station in time, then I will not catch the train.
C If I will catch the train, then I reach the station in time.
D If I will not catch the train, then I do not reach the station in time.

The correct answer is:
If I will not catch the train, then I do not reach the station in time.

Solution:
Contrapositive of p→q is ~q→~p.

⇒The statement will be "If I will not catch the train, then I do not reach the station in time".

Topic: Mathematics, Mathematical Reasoning, JEE Main, JEE Main 2020 (02 Sep Shift 1)
The contrapositive of the statement "If it is raining, then I will not come", is

Options:
A if I will come, then it is not raining.
B if I will come, then it is raining.
C if I will not come, then it is raining.
D if I will not come, then it is not raining.

The correct answer is:
if I will come, then it is not raining.

Solution:
Contrapositive of p⇒q is ~q⇒~ p.

So contrapositive of the statement "If it is raining, then I will not come", would be,
"If I will come, then it is not raining".

Topic: Mathematics, Mathematical Reasoning, JEE Main, JEE Main 2015 (10 Apr Online)
The contrapositive of the statement: "If the weather is fine then my friends will come and we go for a picnic." is

Options:
A The weather is fine but my friends will not come or we do not go for a picnic.
B If my friends do not come or we do not go for picnic then weather will not be fine.
C If the weather is not fine then my friends will not come or we do not go for a picnic.
D The weather is not fine but my friends will come and we go for a picnic.

The correct answer is:
If my friends do not come or we do not go for picnic then weather will not be fine.

Solution:
p→( q∧r )
Contrapositive ( ∽( q∧r ) )→∽p
∴( ∽q∨∽r )→∽p
If my friends do not come or we do not go for picnic then weather will not be fine.

Topic: Mathematics, Mathematical Reasoning, JEE Main
The contrapositive of the statement 'If two numbers are not equal, then their squares are not equal', is

Options:
A If the squares of two numbers are equal, then the numbers are not equal
B If the squares of two numbers are not equal, then the numbers are equal
C If the squares of two numbers are not equal, then the numbers are not equal
D If the squares of two numbers are equal, then the numbers are equal

The correct answer is:
If the squares of two numbers are equal, then the numbers are equal

Solution:
p→q

Contrapositive is ~ q→~p

So, answer is (d) - If the squares of two numbers are equal, then the numbers are equal.

Topic: Mathematics, Mathematical Reasoning, JEE Main, JEE Main 2017 (09 Apr Online)
The contrapositive of the statement "If two triangles are identical, then they are similar" is

Options:
A If two triangles are not similar, then they are not identical
B If two triangles are not identical, then they are not similar
C If two triangles are not identical, then they are similar
D If two triangles are not similar, then they are identical

The correct answer is:
If two triangles are not similar, then they are not identical

Solution:
Let, p: Two triangles are identical
q: Two triangles are similar

Clearly, the given statement in symbolic form is p→q.
Contrapositive of p→q is ∼q→∼p
i.e., If two triangles are not similar, then they are not identical.

Topic: Mathematics, Mathematical Reasoning, JEE Main
The contrapositive of the statement “If you are born in India, then you are a citizen of India”, is

Options:
A If you are not born in India, then you are not a citizen of India.
B If you are a citizen of India, then you are born in India.
C If you are born in India, then you are not a citizen of India.
D If you are not a citizen of India, then you are not born in India.

The correct answer is:
If you are not a citizen of India, then you are not born in India.

Solution:
As we know, contrapositive of p→q is ∼q→∼p
Hence, required contrapositive statement is “If you are not a citizen of India, then you are not born in India”.

Topic: Mathematics, Mathematical Reasoning, JEE Main, JEE Main 2019 (08 Apr Shift 1)
The contrapositive of the statement "If you will work, you will earn money" is:

Options:
A If you will not earn money, you will not work
B To earn money, you need to work
C You will earn money, if you will not work
D If you will earn money, you will work

The correct answer is:
If you will not earn money, you will not work

Solution:
Constrapositive of p→q is ~q→~p

⇒ If you will not earn money, you will not work.

option 1

Topic: Mathematics, Mathematical Reasoning, JEE Main, JEE Main 2021 (25 Feb Shift 2)
The contribution in the total current flowing through a semiconductor due to electrons and holes are 34 and 14 respectively. If the drift velocity of electrons is 52 times that of holes at this temperature, then the ratio of the concentration of electrons and holes is

Options:
A 6 : 5
B 5 : 6
C 3 : 2
D 2 : 3

The correct answer is:
6 : 5

Solution:
As we know, current density J=nqv
⇒ je=ne qve
and jh=nh qvh
jeJh=nenh×vevn
⇒3/41/4=nenh×52
⇒nenh=65

Topic: Physics, Semiconductors, JEE Main
The converse of ~p∧q⇒r is

Options:
A ~p∨q⇒r
B ~r⇒p∧q
C ~r⇒~p∧q
D p∨~q⇒~r

The correct answer is:
p∨~q⇒~r

Solution:
We know that,

Converse of A⇒B is B⇒A

So, using the above formula we get,

Converse of ~p∧q⇒r

≡r⇒(~p∧q)

≡~r∨~p∧q

≡~p∧q∨~r

≡~p∨~q∨~r as ~A∨B≡~A∧~B

≡p∨~q⇒~r

Topic: Mathematics, Mathematical Reasoning, JEE Main, JEE Main 2023 (11 Apr Shift 2)
The converse of p⇒q⇒r is

Options:
A q∧~r∨p
B ~q∨r∨p
C q∧~r∧~p
D q∧~r∧p

The correct answer is:
q∧~r∨p

Solution:
Converse of p⇒q⇒r is
q⇒r⇒p≡~q∨r⇒p
≡~~q∨r∨p
≡q∧~r∨p

Topic: Mathematics, Mathematical Reasoning, JEE Main