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If $p\rightarrow \left ( p\: \wedge \sim q \right )$ is false, then the truth values of p and q are respectively :
Option: 1 F, T
Option: 2 T, F
Option: 3 F, F
Option: 4 T, T

Relation Between Set Notation and Truth Table -

Sets can be used to identify basic logical structures of statements. Statements have two fundamental roles either it is true or false.

Let us understand with an example of two sets p{1,2} and q{2,3}.

$\begin{array}{|c|c|c|}\hline\quad p\vee q\quad & \quad p\cup q\quad&\quad 1,2,3\quad \\ \hline p\wedge q& p\cap q&2 \\ \hline p^c& \sim p & 3,4 \\ \hline q^c& \sim q&1,4 \\ \hline\end{array}$

Using this relation we get

$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline \text{Element } & \mathrm{\;\;\;\;\;}p\mathrm{\;\;\;\;\;}&\mathrm{\;\;\;}q\mathrm{\;\;\;}&\mathrm{\;\;\;\;\;}\sim p\mathrm{\;\;\;\;\;}&\mathrm{\;\;\;}\sim q\mathrm{\;\;\;} &\mathrm{\;\;\;}p\wedge q\mathrm{\;\;}&\mathrm{\;\;}p\vee q\mathrm{\;\;}&\sim\left (p\wedge q \right )\mathrm{\;\;}&\sim p\wedge\sim q\mathrm{\;\;} \\ \hline \hline 1& \mathrm{T}&\mathrm{F} & \mathrm{F} &\mathrm{T}&\mathrm{F}&\mathrm{T}&\mathrm{T}&\mathrm{F} \\ \hline2& \mathrm{T}&\mathrm{T} & \mathrm{F} &\mathrm{F}&\mathrm{T}&\mathrm{T}&\mathrm{F}&\mathrm{T} \\ \hline 3& \mathrm{F}&\mathrm{T} & \mathrm{T} &\mathrm{F}&\mathrm{F}&\mathrm{T}&\mathrm{T}&\mathrm{F} \\ \hline4& \mathrm{F}&\mathrm{F} & \mathrm{T} &\mathrm{T}&\mathrm{F}&\mathrm{F}&\mathrm{T}&\mathrm{F} \\ \hline\end{array}$

Practise Session - 2 -

Q1. Write the truth table for the following statement pattern:
$\\ \mathrm{1.\;\;\;\sim(p \vee((\sim q \Rightarrow q) \wedge q))}\\\mathrm{2.\;\;\;(p\wedge q)\vee(\sim r)}\\$

$\begin{array}{c|c|c}{\mathbf{p}} & {\mathbf{q}} & {(\mathbf{p} \rightarrow(\mathbf{p} \mathbf{\Lambda}-\mathbf{q}))} \\ \hline F & {F} & {\mathbf{T}} \\ \hline F & {T} & {\mathbf{T}} \\ \hline \mathbf{T} & {F} & {\mathbf{T}} \\ \hline \mathbf{T} & {T} & {F}\end{array}$

Correct Option (4)

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avinash.dongre

In a fluorescent lamp choke ( a small transformer) 100 V of reverse voltage is produced when the choke current changes uniformly from 0.25 A to 0 in a duration of 0.025 ms. The self-inductance of the choke (in mH) is estimated to be _____.
Option: 1 10
Option: 2 20
Option: 3 30
Option: 4 40

$V = L\frac{\text d i}{\text d t}\Rightarrow 100 = L\frac{0.25}{0.025} \\ \Rightarrow L = 10 \ \text{mH}$

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avinash.dongre

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The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the age (in years) of the newly appointed teacher is :
Option: 1 25
Option: 2 30
Option: 3 35
Option: 4 40

No concept add

mean age $= 40$ years

$\frac{\sum x_{i}}{25} = 40$ years

= sum of ages $\left ( s \right )$

new teacher be of age $T$ let the

now $\frac{S-60+T}{25}=39$

$1000-60+T=25\times 39$

$940+T=975$

$T=35\ years$

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Posted by

vishal kumar

Which one of the following is a tautology ?

Option: 1 $\left ( P\wedge \left ( P\rightarrow Q \right ) \right )\rightarrow Q$
Option: 2 $P\wedge (P\vee Q)$
Option: 3 $Q\rightarrow (P\wedge (P\rightarrow Q))$
Option: 4 $P\vee (P\wedge Q)$

Tautology And Contradiction -

Tautology

A compound statement is called tautology if it is always true for all possible truth values of its component statement.

For example, ( p ⇒ q ) ∨ ( q ⇒ p )

Contradiction (fallacy)

A compound statement is called a contradiction if it is always false for all possible truth values of its component statement.

For example, ∼( p ⇒ q ) ∨ ( q ⇒ p )

Truth Table

$\begin{array}{|c|c|c|c|c|c|c|c|}\hline \mathrm{\;\;\;\;\;}p\mathrm{\;\;\;\;\;}&\mathrm{\;\;\;}q\mathrm{\;\;\;}&\mathrm{\;\;\;\;\;}p\rightarrow q\mathrm{\;\;\;\;\;}&\mathrm{\;\;\;} q\rightarrow p\mathrm{\;\;\;} &\mathrm{\;\;\;}\left ( p\rightarrow q \right )\vee\left ( q\rightarrow p \right )\mathrm{\;\;}&\mathrm{\;\;\;}\sim\left ( \left ( p\rightarrow q \right )\vee\left ( q\rightarrow p \right ) \right ) \mathrm{\;\;} \\\hline \hline \mathrm{T}&\mathrm{T} & \mathrm{T} &\mathrm{T}&\mathrm{T}&\mathrm{F} \\ \hline \mathrm{T}&\mathrm{F} & \mathrm{F} &\mathrm{T}&\mathrm{T}&\mathrm{F} \\ \hline \mathrm{F}&\mathrm{T} & \mathrm{T} &\mathrm{F}&\mathrm{T}&\mathrm{F} \\ \hline \mathrm{F}&\mathrm{F} & \mathrm{T} &\mathrm{T}&\mathrm{T}&\mathrm{F} \\ \hline\end{array}$

Quantifiers

Quantifiers are phrases like ‘These exist’ and “for every”. We come across many mathematical statements containing these phrases.

For example –

p : For every prime number x, √x is an irrational number.

q : There exists a triangle whose all sides are equal.

Practise Session - 2 -

Q1. Write the truth table for the following statement pattern:
$\\ \mathrm{1.\;\;\;\sim(p \vee((\sim q \Rightarrow q) \wedge q))}\\\mathrm{2.\;\;\;(p\wedge q)\vee(\sim r)}\\$

$\begin{array}{c|c|c}{\mathbf{p}} & {\mathbf{q}} & {(\mathbf{p} \rightarrow \mathbf{q})} \\ \hline F & {F} & {\mathbf{T}} \\ \hline F & {T} & {\mathbf{T}} \\ \hline T & {F} & {F} \\ \hline T & {T} & {\mathbf{T}}\end{array}$ $\begin{array}{c|c|c}{\mathbf{p}} & {\mathbf{q}} & {(\mathbf{p} \wedge \mathbf{q})} \\ \hline F & {F} & {\mathbf{F}} \\ \hline F & {T} & {F} \\ \hline T & {F} & {F} \\ \hline T & {T} & {T}\end{array}$

$\begin{array}{c|c|c}{\mathbf{p}} & {\mathbf{q}} & {(p \wedge(p \rightarrow q)) \rightarrow q} \\ \hline F & {F} & {\mathbf{T}} \\ \hline F & {T} & {\mathbf{T}} \\ \hline T & {F} & {T} \\ \hline T & {T} & {\mathbf{T}}\end{array}$

Correct Option (1)

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Kuldeep Maurya

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At time $t=0$ magnetic field of $1000$ Gauss is passing perpendicularly through the area defined by the closed loop shown in the figure. If the magnetic field reduces linearly to $500\: Gauss$ , in the next $5s$ , then induced EMF (in $\mu V$ ) in the loop is :
Option: 1 56

Option: 3 48

Option: 5 36

Option: 7 28

Area of the given loop

$\\A=Area\ of\ rectangle-Area\ of \ 2\ triangles\ with\ height\ 2cm \\ A=16\times4-2\times4=56cm^2$

Magnitude of emf

$\\E=A\frac{dB}{dt}=56\times10^{-4}\times(\frac{(1000-500)\times10^{-4}}{5})\\ E=5600\times10^{-8}V=56\mu \ V$

So the correct graph is given in option 1.

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Posted by

vishal kumar

The logical statement $(p\Rightarrow q)\wedge (q\Rightarrow \sim p)$ is equivalent to :
Option: 1 $\sim p$
Option: 2 $p$
Option: 3 $q$
Option: 4 $\sim q$

Practise Session - 2 -

Q1. Write the truth table for the following statement pattern:
$\\ \mathrm{1.\;\;\;\sim(p \vee((\sim q \Rightarrow q) \wedge q))}\\\mathrm{2.\;\;\;(p\wedge q)\vee(\sim r)}\\$

$\begin{array}{c|c|ccc}{\mathbf{p}} & {\mathbf{q}} & {((\mathbf{p} \rightarrow \mathbf{q})} & {\mathbf{\wedge}(\mathbf{q} \rightarrow \neg \mathbf{p}))} & {} \\ \hline F & {F} & {} & {\mathbf{T}} \\ \hline F & {T} & {} & {\mathbf{T}} \\ \hline T & {F} & {} & {F} \\ \hline T & {T} & {} & {F}\end{array}$

which is equal to $\sim p$

Correct Option (1)

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Ritika Jonwal

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A loop ABCDEFA of straight edges has six corner points A(0,0,0), B(5,0,0),C(5,5,0), D(0,5,0), E(0,5,5) and F(0,0,5). The magnetic field in this region is $\vec B=(3\hat{i}+4\hat{j})T$ .The quantity of flux through the loop ABCDEFA (in Wb) is:-
Option: 1 175
Option: 2 60
Option: 3 25
Option: 4 170

Magnetic flux - - wherein

Magnetic flux-

The total number of magnetic lines of force passing normally through an area placed in a magnetic field is equal to
the magnetic flux linked with that area.

I.e for the below figure

Net magnetic flux through the surface is given by

$\phi_B = \oint \vec{B}\cdot \vec{dA}= BA\cos \Theta$

where

$\phi_B=$ Magnetic Flux

$B =$ Magnetic field

$\Theta =$ The angle between area vector and magnetic field vector

Magnetic flux is a scalar quantity.
Unit of magnetic flux -

It's S.I. unit is Weber (wb) or $Tesla\times m^2$ and its C.G.S. unit is maxwell(Mx).

and $1 \ w b=1 \ T m^{2}$ and $1 \ Mx = 10^{-8 } \ wb$

The dimension of magnetic flux is $ML^{2}T^{-2}A^{-1}$
if θ = 0 then $\phi = BA$ and Flux will be positive.
If $\theta =\frac{\pi }{2}$ then Flux will be zero (i.e $\phi = 0$ )

$\vec B=(3\hat{i}+4\hat{j})T$

Total area vectot=area of ABCD+area of DEFA= $5^2\hat{k}+5^2\hat{i}=25 (\hat{i}+\hat{k})$

Total Magnetic flux= $\vec{B}.\vec{A}=(3\hat{i}+4\hat{j}).(25 (\hat{i}+\hat{k}))=(75+100)wb=175 wb$

So option 3 will be correct answer.

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Ritika Jonwal

A LCR circuit behaves like a damped harmonic oscillator. Comparing it with a physical spring-mass damped oscillator having dampling constant 'b', the correct equivalence would be:-

Option: 1 $L\leftrightarrow k,C\leftrightarrow b,R\leftrightarrow m$

Option: 2 $L\leftrightarrow m,C\leftrightarrow k,R\leftrightarrow b$

Option: 3 $L\leftrightarrow m,C\leftrightarrow \frac{1}{k},R\leftrightarrow b$

Option: 4 $L\leftrightarrow \frac{1}{b},C\leftrightarrow \frac{1}{m},R\leftrightarrow \frac{1}{k}$

Series LCR circuit -

Series LCR circuit-

The Figure given above shows a circuit containing a capacitor ,resistor and inductor connected in series through an alternating / sinosoidal voltage source.

As they are in series so the same amount of current will flow in all the three circuit components and for the voltage, vector sum of potential drop across each component would be equal to the applied voltage.

Let 'i' be the amount of current in the circuit at any time and V_L,V_C and V_R the potential drop across L,C and R respectively then
$\begin{array}{l}{\mathrm{v}_{\mathrm{R}}=\mathrm{i} \mathrm{R} \rightarrow \text { Voltage is in phase with i }} \\ \\ {\mathrm{v}_{\mathrm{L}=\mathrm{i} \omega \mathrm{L}} \rightarrow \text { voltage is leading i by } 90^{\circ}} \\ \\ {\mathrm{v}_{\mathrm{c}}=\mathrm{i} / \mathrm{\omega} \mathrm{c} \rightarrow \text { voltage is lagging behind i by } 90^{\circ}}\end{array}\varepsilon$

By all these we can draw phasor diagram as shown below -

One thing should be noticed that we have assumed that V_L is greater than V_C which makes i lags behind V. If V_C > V_L then i lead V. So as per our assumption, there resultant will be (V_L -V_C). So, from the above phasor diagram V will represent resultant of vectors V_R and (V_L -V_C). So the equation become -

$\begin{aligned} V &=\sqrt{V_{R}^{2}+\left(V_{L}-V_{C}\right)^{2}} \\ \\ &=i \sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}} \\ \\ &=i \sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}} \\ \\ &=i Z \\ \text { where, } & \\ Z &=\sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}} \end{aligned}$

Here, Z is called Impedence of this circuit.

Now come to the phase angle. The phase angle for this case is given as -

$\tan \varphi=\frac{V_{L}-V_{C}}{V_{R}}=\frac{X_{L}-X_{C}}{R}=\frac{\omega L-\frac{1}{\omega C}}{R}$

Now from the equation of the phase angle three cases will arise. These three cases are -

(i) When, $\omega L \ > \ \frac{1}{\omega C}$

then, tanφ is positive i.e. φ is positive and voltage leads the current i.
(ii) When $\omega L \ < \ \frac{1}{\omega C}$

then, tanφ is negative i.e. φ is negative and voltage lags behind the current i.
(iii) When $\omega L \ = \ \frac{1}{\omega C}$ ,

then tanφ is zero i.e. φ is zero and voltage and current are in phase. This is called electrical resonance.

For the spring-mass damping system, the governing equation is given by

$m \frac{dx^2}{dt^2}+b\frac{dx}{dt}+kx=0...............(1)$

For an LCR damped oscillator, the equation is

$L \frac{dQ^2}{dt^2}+R\frac{dQ}{dt}+\frac{Q}{C}=0...............(2)$

Comparing 1 and 2

$\\L\leftrightarrow m\\C\leftrightarrow \frac{1}{k}\\R\leftrightarrow b$

So the answer is option (3)

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Ritika Jonwal

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Which of the following statements is a tautology ?
Option: 1 $\sim (p\wedge \sim q)\rightarrow p\vee q$
Option: 2 $\sim (p\vee \sim q)\rightarrow p\wedge q$
Option: 3 $p\vee (\sim q)\rightarrow p\wedge q$
Option: 4 $\sim (p\vee \sim q)\rightarrow p\vee q$

Tautology And Contradiction -

Tautology

A compound statement is called tautology if it is always true for all possible truth values of its component statement.

For example, ( p ⇒ q ) ∨ ( q ⇒ p )

Contradiction (fallacy)

A compound statement is called a contradiction if it is always false for all possible truth values of its component statement.

For example, ∼( p ⇒ q ) ∨ ( q ⇒ p )

Truth Table

Quantifiers

Quantifiers are phrases like ‘These exist’ and “for every”. We come across many mathematical statements containing these phrases.

For example –

p : For every prime number x, √x is an irrational number.

q : There exists a triangle whose all sides are equal.

Practise Session - 1 -

Q1. Write the negations of the following:
1. Ram is a doctor or peon.
2. Room is clean and big.
3. $\sqrt 5$ is a rational number.
4. If Ram is a doctor then he is smart

$\sim(p \vee \sim q) \rightarrow p \vee q$

$\begin{array}{l}{(\sim p \wedge q) \rightarrow(p \vee q)} \\ {\sim\{(\sim p \wedge q) \wedge(\sim p \wedge \sim q)\}} \\ {\sim\{\sim p \wedge f\}}\end{array}$

Correct Option (4)

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vishal kumar

A long solenoid of radius R carries a time(t) - dependent current $I(t)=I_ot(1-t)$ . A ring of radius 2R is placed coaxially near its middle. During the time interval $0\leq t\leq 1$ , the induced current (IR) and the induced EMF (VR) in the ring changes as:-
Option: 1 Direction of $I_{R}$ remains unchanged and $V_{R}$ is zero at $t = 0.25$
Option: 2 Direction of $I_{R}$ remains unchanged and $V_{R}$ is zero at $t = 0.5$
Option: 3 At $t = 0.25$ direction of $I_{R}$ reverses and $V_{R}$ is maximum
Option: 4 At $t = 0.5$ direction of $I_{R}$ reverses and $V_{R}$ is zero

Faraday's law of induction -

Faraday’s First Law-

Whenever the number of magnetic lines of force (Magnetic Flux) passing through a circuit changes an emf called induced emf is produced in the circuit. The induced emf persists only as long as there is a change of flux.

Faraday’s Second Law-

The induced emf is given by the rate of change of magnetic flux linked with the circuit.

i.e Rate of change of magnetic Flux= $\varepsilon = \frac{-d\phi }{dt}$

where $d\phi\rightarrow \phi _{2}-\phi _{1}$ = change in flux

So,

$B=\mu _0nI_0t(1-t)$

using $B_0=\mu _0nI_0$

SO $B=B_0t(1-t)$

Now considering solenoid as ideal solenoid extended up to infinite and ring as its centre

$\phi =BA_s=B_0t(1-t) A_s$

$e=-\frac{d \phi}{dt} =-B_0A_s (1-2t)$

Since induced emf is changing so current will also be changing

because $i=\frac{e}{R}$

So, since direction of emf is changing so direction of current is also changing.

And V_R will be zero at t=0.5 sec

So the option (4) is correct.

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Posted by

Ritika Jonwal

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Get Answers to all your Questions

All Questions

If is false, then the truth values of p and q are respectively : Option: 1 F, TOption: 2 T, F Option: 3 F, F Option: 4 T, T

Posted by

avinash.dongre

In a fluorescent lamp choke ( a small transformer) 100 V of reverse voltage is produced when the choke current changes uniformly from 0.25 A to 0 in a duration of 0.025 ms. The self-inductance of the choke (in mH) is estimated to be _____. Option: 1 10 Option: 2 20 Option: 3 30 Option: 4 40

Posted by

avinash.dongre

Crack CUET with india's "Best Teachers"

Posted by

vishal kumar

Which one of the following is a tautology ? Option: 1 Option: 2 Option: 3 Option: 4

Posted by

Kuldeep Maurya

JEE Main high-scoring chapters and topics

At time magnetic field of Gauss is passing perpendicularly through the area defined by the closed loop shown in the figure. If the magnetic field reduces linearly to , in the next , then induced EMF (in ) in the loop is : Option: 1 56 Option: 3 48 Option: 5 36 Option: 7 28

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If $p\rightarrow \left ( p\: \wedge \sim q \right )$ is false, then the truth values of p and q are respectively :
Option: 1 F, T
Option: 2 T, F
Option: 3 F, F
Option: 4 T, T

In a fluorescent lamp choke ( a small transformer) 100 V of reverse voltage is produced when the choke current changes uniformly from 0.25 A to 0 in a duration of 0.025 ms. The self-inductance of the choke (in mH) is estimated to be _____.
Option: 1 10
Option: 2 20
Option: 3 30
Option: 4 40

Which one of the following is a tautology ?

Option: 1 $\left ( P\wedge \left ( P\rightarrow Q \right ) \right )\rightarrow Q$
Option: 2 $P\wedge (P\vee Q)$
Option: 3 $Q\rightarrow (P\wedge (P\rightarrow Q))$
Option: 4 $P\vee (P\wedge Q)$

The logical statement $(p\Rightarrow q)\wedge (q\Rightarrow \sim p)$ is equivalent to :
Option: 1 $\sim p$
Option: 2 $p$
Option: 3 $q$
Option: 4 $\sim q$

A loop ABCDEFA of straight edges has six corner points A(0,0,0), B(5,0,0),C(5,5,0), D(0,5,0), E(0,5,5) and F(0,0,5). The magnetic field in this region is $\vec B=(3\hat{i}+4\hat{j})T$ .The quantity of flux through the loop ABCDEFA (in Wb) is:-
Option: 1 175
Option: 2 60
Option: 3 25
Option: 4 170

Which of the following statements is a tautology ?
Option: 1 $\sim (p\wedge \sim q)\rightarrow p\vee q$
Option: 2 $\sim (p\vee \sim q)\rightarrow p\wedge q$
Option: 3 $p\vee (\sim q)\rightarrow p\wedge q$
Option: 4 $\sim (p\vee \sim q)\rightarrow p\vee q$