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30 Jan 2023 Shift 2

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Physics30 questions

Q1JEE Main 2023MCQ4MLaws of Motion

A block of ${\sqrt{3}} {\text{kg}}$ is attached to a string whose other end is attached to the wall. An unknown force ${\text{F}}$ is applied so that the string makes an angle of $30^{\circ}$ with the wall. The tension ${\text{T}}$ is : $(.$ Given ${\text{g}}=10 {\text{ms}}^{-2})$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\theta=30^{\circ}$ $\;\cos \theta=\;{{\sqrt{3}} {\text{g}}}/{ {\text{T}}}$ $\;\Rightarrow \;{{\sqrt{3}}}/{2}=\;{{\sqrt{3}} {\text{g}}}/{ {\text{T}}}$ $\;\Rightarrow {\text{T}}=20 {\text{N}}$

Explanation Figure 2

Q2JEE Main 2023MCQ4MThermodynamics

A flask contains hydrogen and oxygen in the ratio of $2: 1$ by mass at temperature $27^{\circ} {\text{C}}$ . The ratio of average kinetic energy per molecule of hydrogen and oxygen respectively is : [30-Jan-2023 Shift 2]

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📖 Explanation

${\text{K}}_{ {\text{av}}}\;=\;\frac{5}{2} {\text{kT}}$ $\;\text{ Ratio }\;\;=1: 1$

Q3JEE Main 2023MCQ4MCurrent Electricity

The equivalent resistance between ${\text{A}}$ and ${\text{B}}$ is [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\;{1}/{ {\text{R}}_{ {\text{eq}}}}=\;\frac{1}{2}+\;\frac{1}{12}+\;\frac{1}{4}+\;\frac{1}{6}+\;\frac{1}{2}$ $\;\; =\;\frac{6+1+3+2+6}{12}=\;\frac{18}{12}=\;\frac{3}{2}$ $\;\Rightarrow {\text{R}}_{ {\text{eq}}}=\;\frac{2}{3} Ω$

Explanation Figure 2

Q4JEE Main 2023MCQ4MAtoms and Nuclei

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason $\text{bo} {R}$ . Assertion A : The nuclear density of nuclides ${ }_5^{10} {\text{B}},{ }_3^6 {\text{Li}},{ }_{26}^{56} {\text{Fe}},{ }_{10}^{20} {\text{Ne}}$ and ${ }_{83}^{209} {\text{Bi}}$ can be arranged as $\rho _{ {\text{Bi}}}^{ {\text{N}}} > \rho _{ {\text{Fe}}}^{ {\text{N}}} > \rho _{ {\text{Ne}}}^{ {\text{N}}} > \rho _{ {\text{B}}}^{ {\text{N}}} > \rho _{ {\text{Li}}}^{ {\text{N}}}$ .Reason $R$ : The radius $R$ of nucleus is related to its mass number $A$ as ${\text{R}}= {\text{R}}_0 {\text{A}}^{1 / 3}$ , where ${\text{R}}_0$ is a constant.In the light of the above statement, choose the correct answer from the options given below : [30-Jan-2023 Shift 2]

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Q5JEE Main 2023MCQ4MRay Optics and Optical Instruments

A thin prism ${\text{P}}_1$ with an angle $6^{\circ}$ and made of glass of refractive index $1.54$ is combined with another prism ${\text{P}}_2$ made from glass of refractive index $1.72$ to produce dispersion without average deviation. The angle of prism ${\text{P}}_2$ is : [30-Jan-2023 Shift 2]

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📖 Explanation

$\delta _1=\delta _2$ [for no average deviation] $\Rightarrow 6^{\circ}(1.54-1)= {\text{A}}(1.72-1)$ $\Rightarrow {\text{A}}=\;\frac{6^{\circ} \times 0.54}{0.72}$ $=\;\frac{18^{\circ}}{4}=4.5^{\circ}$

Q6JEE Main 2023MCQ4MSemiconductor Electronics

The output ${\text{Y}}$ for the inputs ${\text{A}}$ and ${\text{B}}$ of circuit is given by Truth table of the shown circuit is : [30-Jan-2023 Shift 2]

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Q7JEE Main 2023MCQ4MMotion in a Straight Line

A vehicle travels $4 {\text{km}}$ with speed of $3 {\text{km}} / {\text{h}}$ and another $4 {\text{km}}$ with speed of $5 {\text{km}} / {\text{h}}$ , then its average speed is: [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\;{2}/{ {\text{V}}_{ {\text{av}}}}=\;\frac{1}{3}+\;\frac{1}{5}=\;\frac{8}{15}$ $\;\Rightarrow \;\; {\text{V}}_{ {\text{av}}}=\;\frac{15}{4}=3.75 {\text{km}} / {\text{h}}$

Q8JEE Main 2023MCQ4MElectrostatics

As shown in the figure, a point charge $Q$ is placed at the centre of conducting spherical shell of inner radius $a$ and outer radius $b$ . The electric field due to charge Q in three different regions I, II and III is given by : $( {\text{I}}: {\text{r}} < {\text{a}}$ , II : ${\text{a}} < {\text{r}} < {\text{b}}$ , III : $r > b$ ) [30-Jan-2023 Shift 2]

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Q9JEE Main 2023MCQ4MMagnetic Effects of Current and Magnetism

As shown in the figure, a current of $2 {\text{A}}$ flowing in an equilateral triangle of side $4 {\sqrt{3}} {\text{cm}}$ . The magnetic field at the centroid ${\text{O}}$ of the triangle is:(Neglect the effect of earth's magnetic field.) [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{d}} \tan 60^{\circ}=2 {\sqrt{3}}$ $\; {\text{d}}=2 {\text{cm}}$ $\; {\text{B}}=3 \times \;{\mu _0 {\text{i}}}/{2 \pi {\text{d}}} sin \;60^{\circ}$ $\;=3 \times \;\frac{2 \times 10^{-7} \times 2}{2 \times 10^{-2}} \times \;{{\sqrt{3}}}/{2}$ $\;=3 {\sqrt{3}} \times 10^{-5}$

Q10JEE Main 2023MCQ4MAlternating Current

In the given circuit, rms value of current $( {\text{I}}_{ {\text{rms}}})$ through the resistor ${\text{R}}$ is : [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{z}}={\sqrt{100^{2+}(200-100)^2}}$ $\;=100 {\sqrt{2}} Ω$ $\; {\text{i}}_{ {\text{rms}}}=\;{ {\text{V}}_{ {\text{rms}}}}/{ {\text{z}}}=\;{200 {\sqrt{2}}}/{100 {\sqrt{2}}}$ $\;=2 {\text{A}}$

Q11JEE Main 2023MCQ4MCentre of Mass and Collision

A machine gun of mass $10 {\text{kg}}$ fires $20 {\text{g}}$ bullets at the rate of 180 bullets per minute with a speed of $100 {\text{m}} {\ s}^{-1}$ each. The recoil velocity of the gun is : [30-Jan-2023 Shift 2]

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📖 Explanation

$\;20 \times 10^{-3} \times \;\frac{180}{60} \times 100=10 {\text{V}}$ $\;\Rightarrow {\text{v}}=0.6 {\text{m}} / {\ s}$

Q12JEE Main 2023MCQ4MThermodynamics

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : Efficiency of a reversible heat engine will be highest at $-273^{\circ} {\text{C}}$ temperature of cold reservoir.Reason R : The efficiency of Carnot's engine depends not only on temperature of cold reservoir but it depends on the temperature of hot reservoir too and is given as $η= (1-\;{ {\text{T}}_2}/{ {\text{T}}_1})$ .In the light of the above statements, choose the correct answer from the options given below : [30-Jan-2023 Shift 2]

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📖 Explanation

Both ${\text{A}}$ and ${\text{R}}$ are true and ${\text{R}}$ is the correct explanation of A

Q13JEE Main 2023MCQ4MUnits and Measurements

Match List I with List II. List I List II A. Torque I. ${\text{kg}} {\text{m}}^{-1} {\ s}^{-2}$ B. Energy density II. ${\text{kg}} \;{ms } ^ { - 1 }$ C. Pressure gradient III. ${\text{kg}} {\text{m}}^{-2} {\ s}^{-2}$ D. Impulse IV. ${\text{kg}} {\text{m}}^2 {\ s}^{-2}$ Choose the correct answer from the options given below : [30-Jan-2023 Shift 2]

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📖 Explanation

To match the physical quantities with their SI units, we derive the units for each term:
A. Torque is defined as force multiplied by perpendicular distance. Its unit is $({\text{kg}} \cdot {\text{m/s}}^2) \times {\text{m}} = {\text{kg}} {\text{m}}^2 {\text{s}}^{-2}$ , which corresponds to IV.

B. Energy density is energy per unit volume. Its unit is $\frac{\text{Energy}}{\text{Volume}} = \frac{{\text{kg}} {\text{m}}^2 {\text{s}}^{-2}}{{\text{m}}^3} = {\text{kg}} {\text{m}}^{-1} {\text{s}}^{-2}$ , which corresponds to I.

C. Pressure gradient is pressure per unit length. Its unit is $\frac{\text{Pressure}}{\text{Length}} = \frac{{\text{kg}} {\text{m}}^{-1} {\text{s}}^{-2}}{{\text{m}}} = {\text{kg}} {\text{m}}^{-2} {\text{s}}^{-2}$ , which corresponds to III.

D. Impulse is the change in momentum, or force multiplied by time. Its unit is $({\text{kg}} \cdot {\text{m/s}}^2) \times {\text{s}} = {\text{kg}} {\text{m}} {\text{s}}^{-1}$ , which corresponds to II.

The correct matching sequence is A-IV, B-I, C-III, D-II.

Q14JEE Main 2023MCQ4MOscillations

For a simple harmonic motion in a mass spring system shown, the surface is frictionless. When the mass of the block is $1 {\text{kg}}$ , the angular frequency is $\omega _1$ . When the mass block is $2 {\text{kg}}$ the angular frequency is $\omega _2$ . The ratio $\omega _2 / \omega _1$ is : [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\omega =\sqrt{\;{ {\text{k}}}/{ {\text{m}}}}$ $\;\;\frac{\omega _2}{\omega _1}=\sqrt{\;{ {\text{m}}_1}/{ {\text{m}}_2}}=\sqrt{\;\frac{1}{2}}$

Q15JEE Main 2023MCQ4MDual Nature of Matter and Radiation

An electron accelerated through a potential difference $V_1$ has a de-Broglie wavelength of $\lambda$ . When the potential is changed to ${\text{V}}_2$ , its de-Broglie wavelength increases by $50 \%$ . The value of $(\;{ {\text{V}}_1}/{ {\text{V}}_2})$ is equal to : [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{KE}}=\;{ {\text{P}}^2}/{2 {\text{m}}}, \;\; {\text{P}}=\;{ {\text{h}}}/{\lambda }$ $\; {\text{eV}}_1=\;{ ({ {\text{h}}}/{\lambda }) ^2}/{2 {\text{m}}}$ $\; {\text{eV}}_2=\;{ ({ {\text{h}}}/{1.5 \lambda }) ^2}/{2 {\text{m}}}$ $\;\;{ {\text{V}}_1}/{ {\text{V}}_2}=(1.5)^2=\;\frac{9}{4}$

Q16JEE Main 2023MCQ4MElectromagnetic Waves

Match List I with List II : List I List II A. Attenuation I. Combination of a receiver and transmitter. B. Transducer II. Process of retrieval of information from the carrier wave at received C. Demodulation III. Converts one form of energy into another D. Repeater IV. Loss of strength of a signal while propagating through a mediumChoose the correct answer from the options given below : [30-Jan-2023 Shift 2]

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Q17JEE Main 2023MCQ4MMagnetic Effects of Current and Magnetism

A current carrying rectangular loop PQRS is made of uniform wire. The length ${\text{PR}}= {\text{QS}}=5 {\text{cm}}$ and ${\text{PQ}}= {\text{RS}}=100 {\text{cm}}$ . If ammeter current reading changes from I to 2I, the ratio of magnetic forces per unit length on the wire ${\text{PQ}}$ due to wire ${\text{RS}}$ in the two cases respectively $f_{ {\text{PQ}}}^{ {\text{I}}}: f_{ {\text{PQ}}}^{2 {\text{I}}}$ is : [30-Jan-2023 Shift 2]

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📖 Explanation

$\;F \propto I_1 I_2$ $\;F_1: F_{21}=1: 4$

Q18JEE Main 2023MCQ4MProperties of Matter

A force is applied to a steel wire ' ${\text{A}}$ ', rigidly clamped at one end. As a result elongation in the wire is $0.2 {\text{mm}}$ . If same force is applied to another steel wire ' ${\text{B}}$ ' of double the length and a diameter $2.4$ times that of the wire ' $A$ ', the elongation in the wire ' $B$ ' will be (wires having uniform circular cross sections) [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{Y}}=\;{ {\text{F}} / {\text{A}}}/{\frac{\Delta ℓ}{ℓ}}$ $\;\Rightarrow {\text{F}}=\;{ {\text{YA}}}/{ℓ} \Delta ℓ$ $\; (\;{ {\text{A}} \Delta ℓ}/{ℓ}) _1= (\;{ {\text{A}} \Delta ℓ}/{ℓ}) _2$ $\;\Rightarrow \;\frac{\Delta ℓ_2}{\Delta ℓ_1}=\;{ {\text{A}}_1}/{ {\text{A}}_2} \times \;\frac{ℓ_2}{ℓ_1}$ $\;\Rightarrow \;\frac{\Delta ℓ_2}{0.2}=\;\frac{1}{2.4 \times 2.4} \times \;\frac{2}{1}$ $\;\Rightarrow \Delta ℓ_2=6.9 \times 10^{-2} {\text{mm}}$

Q19JEE Main 2023MCQ4MGravitation

An object is allowed to fall from a height ${\text{R}}$ above the earth, where ${\text{R}}$ is the radius of earth. Its velocity when it strikes the earth's surface, ignoring air resistance, will be : [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\;\text{ Loss \in PE = Gain \in KE }\;$ $\; (-\;{ {\text{GMm}}}/{2 {\text{R}}}) - (-\;{ {\text{GMm}}}/{ {\text{R}}}) =\;\frac{1}{2} {\ mv}^2$ $\;\Rightarrow {\text{v}}^2=\;{ {\text{GM}}}/{ {\text{R}}}= {\text{gR}}$ $\;\Rightarrow {\text{v}}=\sqrt{ {\text{gR}}}$

Q20JEE Main 2023MCQ4MElectromagnetic Waves

A point source of $100 {\text{W}}$ emits light with $5 \%$ efficiency. At a distance of $5 {\text{m}}$ from the source, the intensity produced by the electric field component is : [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{I}}_{ {\text{EF}}}=\;\frac{1}{2} \times \;\frac{5}{4 \pi \times 5^2}$ $\;=\;\frac{1}{40 \pi} {\text{W}} / {\text{m}}^2$

Q21JEE Main 2023NAT4MThermodynamics

A faulty thermometer reads $5^{\circ} {\text{C}}$ in melting ice and $95^{\circ} {\text{C}}$ in steam. The correct temperature on absolute scale will be ________ . ${\text{K}}$ when the faulty thermometer reads $41^{\circ} {\text{C}}$ . [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\;\frac{41^{\circ}-5^{\circ}}{95^{\circ}-5^{\circ}}=\;{ {\text{C}}-0^{\circ}}/{100^{\circ}-0^{\circ}}$ $\;\Rightarrow \;\; {\text{C}}=\;\frac{36}{90} \times 100=40^{\circ} {\text{C}}=313 {\text{K}}$

Q22JEE Main 2023NAT4MCurrent Electricity

If the potential difference between ${\text{B}}$ and ${\text{D}}$ is zero, the value of $x$ is $\;\frac{1}{n} Ω$ . The value of $n$ is _______ [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\;\frac{2}{3}=\;\frac{\;\frac{x}{x+1}}{x}$ $\;\Rightarrow \;\frac{2}{3}=\;\frac{1}{x+1}$ $\;\Rightarrow \;\; x=0.5=\;\frac{1}{2}$ $\;n=2$

Q23JEE Main 2023NAT4MOscillations

The velocity of a particle executing SHM varies with displacement $( {\text{x}})$ as $4 {\text{v}}^2=50- {\text{x}}^2$ . The time period of oscillations is $\;\frac{x}{7} {\ s}$ . The value of $x$ is _______. $(\;\text{ Take }\; \pi=\;\frac{22}{7})$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\;4 {\text{v}}^2=50- {\text{x}}^2$ $\;\Rightarrow \;\; {\text{v}}=\;\frac{1}{2} \sqrt{50- {\text{x}}^2}$ $\;\omega =\;\frac{1}{2}$ $\; {\text{T}}=\;\frac{2 \pi}{\omega }=4 \pi=\;\frac{88}{7}$ $\; {\text{x}}=88$

Q24JEE Main 2023NAT4MWave Optics

In a Young's double slit experiment, the intensities at two points, for the path difference $\;\frac{\lambda }{4}$ and $\;\frac{\lambda }{3}$ ( $\lambda$ being the wavelength of light used) are $I_1$ and $I_2$ respectively. If $I_0$ denotes the intensity produced by each one of the individual slits, then $\;{ {\text{I}}_1+ {\text{I}}_2}/{ {\text{I}}_0}=. . . . . .$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{I}}=4 {\text{I}}_0 \cos ^2 (\;\frac{\Delta \varphi }{2})$ $\; {\text{I}}_1=4 {\text{I}}_0 \cos ^2 (\;\frac{\pi}{4}) =2 {\text{I}}_0$ $\; {\text{I}}_2=4 {\text{I}}_0 \cos ^2 (\;\frac{2 \pi}{3}) = {\text{I}}_0$ $\;\Rightarrow \;{ {\text{I}}_1+ {\text{I}}_2}/{ {\text{I}}_0}=3$

Q25JEE Main 2023NAT4MAtoms and Nuclei

A radioactive nucleus decays by two different process. The half life of the first process is 5 minutes and that of the second process is $30 {\ s}$ . The effective half-life of the nucleus is calculated to be $\;\frac{\alpha}{11}$ s. The value of $\alpha$ is _______. [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\;{ {\text{dN}}_1}/{ {\text{dt}}}=-\lambda _1 {\text{N}} \;\; \;{ {\text{dN}}_2}/{ {\text{dt}}}=-\lambda _2 {\text{N}}$ $\;\;{ {\text{dN}}}/{ {\text{dt}}}=- (\lambda _1+\lambda _2) {\text{N}}$ $\;\Rightarrow \lambda _{ {\text{eq}}}=\lambda _1+\lambda _2$ $\;\Rightarrow \;{1}/{ {\text{t}}_{1 / 2}}=\;\frac{1}{300}+\;\frac{1}{30}=\;\frac{11}{300}$ $\;\Rightarrow {\text{t}}_{1 / 2}=\;\frac{300}{11}$

Q26JEE Main 2023NAT4MWork, Power and Energy

A body of mass $2 {\text{kg}}$ is initially at rest. It starts moving unidirectionally under the influence of a source of constant power P. Its displacement in $4 {\ s}$ is $\;\frac{1}{3} \alpha^2 \sqrt{ {\text{P}}} {\text{m}}$ . The value of $\alpha$ will be _______ [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\;\frac{1}{2} {\text{mV}}^2= {\text{Pt}}$ $\; {\text{V}}=\sqrt{\;{2 {\text{Pt}}}/{ {\text{m}}}}$ $\;\;{ {\text{dx}}}/{ {\text{dt}}}=\sqrt{\;{2 {\text{Pt}}}/{ {\text{m}}}}$ $\; {\text{x}}=\sqrt{\;{2 {\text{P}}}/{ {\text{m}}} \;\frac{2}{3}} \$ [ {\text{t}}^{3 / 2}]$ _0^4; {\text{x}}=;{16 \sqrt{ {\text{P}}}}/{3}=;\frac{1}{3} \times 16 \sqrt{ {\text{P}}};\alpha=4$

Q27JEE Main 2023NAT4MElectrostatics

As shown in figure, a cuboid lies in a region with electric field ${\text{E}}=2 {\text{x}}^2 \;{ {\text{i}}}^{\^}-4 {\text{y}} \;{ {\text{j}}}^{\^}+6 \;{ {\text{k}}}^{\^} \;\; {\text{N}} / {\text{C}}$ . The magnitude of charge within the cuboid is $n \in _0 {\text{C}}$ . The value of ${\text{n}}$ is ________ (if dimension of cuboid is $1 \times 2 \times 3 {\text{m}}^3$ ) [30-Jan-2023 Shift 2]

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📖 Explanation

${ \vec{\text{E}}=2 {\text{x}}^2 \;{ {\text{i}}}^{\^}-4 {\text{y}} \;{ {\text{j}}}^{\^}+6 \;{ {\text{k}}}^{\^}$ $\;\varphi _{\;\text{net }\;}=-8 \times 3+2 \times 6=-12$ $\;-12=\;{ {\text{q}}}/{E_0}$ $\;| {\text{q}}|=12 E_0$

Explanation Figure 2

Q28JEE Main 2023NAT4MElectromagnetic Induction

In an ac generator, a rectangular coil of 100 turns each having area $14 \times 10^{-2} {\text{m}}^2$ is rotated at $360 {\text{rev}} / {\min}$ about an axis perpendicular to a uniform magnetic field of magnitude $3.0 {\text{T}}$ . The maximum value of the emf produced will be V _______. $(.$ Take $\pi=\;\frac{22}{7})$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\;ξ_{\max }= {\text{NAB}} \omega$ $\;=100 \times 14 \times 10^{-2} \times 3 \times \;\frac{360 \times 2 \pi}{60}$ $\;=1584 {\text{V}}$

Q29JEE Main 2023NAT4MCircular Motion

A stone tied to $180 {\text{cm}}$ long string at its end is making 28 revolutions in horizontal circle in every minute. The magnitude of acceleration of stone is $\;{1936}/{ {\text{x}}} {\text{ms}}^{-2}$ . The value of ${\text{x}}$ ________. $(\;\text{ Take }\; \pi=\;\frac{22}{7})$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\;a=\omega ^2 {\text{R}}= (\;\frac{28 \times 2 \pi}{60}) ^2 \times 1.8$ $\;= (\;\frac{56}{60} \times \;\frac{22}{7}) ^2 \times 1.8$ $\;=\;\frac{(44)^2}{225} \times 1.8$ $\;=\;\frac{1936 \times 1.8}{225}$ $\;x=125$

Q30JEE Main 2023NAT4MRotational Motion

A uniform disc of mass $0.5 {\text{kg}}$ and radius ${\text{r}}$ is projected with velocity $18 {\text{m}} / {\ s}$ at ${\text{t}}=0 {\ s}$ on a rough horizontal surface. It starts off with a purely sliding motion at ${\text{t}}=0 {\ s}$ . After $2 {\ s}$ it acquires a purely rolling motion (see figure). The total kinetic energy of the disc after $2 {\ s}$ will be _______ J. (given, coefficient of friction is $0.3$ and ${\text{g}}=10 {\text{m}} / {\ s}^2)$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{a}}=-\mu _{ {\text{k}}} {\text{g}}=-3$ $\; {\text{V}}=18-3 \times 2$ $\; {\text{V}}=12 {\text{m}} / {\ s}$ $\; {\text{KE}}=\;\frac{1}{2} {\ mv}^{2+}\;\frac{1}{2} \;{ {\text{mr}}^2}/{2} \;{ {\text{v}}^2}/{ {\text{r}}^2}$ $\; {\text{KE}}=\;\frac{3}{4} {\ mv}^2$ $\; {\text{KE}}=3 \times 18=54 {\text{J}}$

Chemistry30 questions

Q31JEE Main 2023MCQ4Mp Block Elements

Which of the following reaction is correct? [30-Jan-2023 Shift 2]

🚀 Solve in Practice Mode

📖 Explanation

$4 {\text{LiNO}}_3 {\to }^{\Delta } 2 {\text{Li}}_2 {\text{O}}+4 {\text{NO}}_2+ {\text{O}}_2$

Q32JEE Main 2023MCQ4MOrganic Chemistry - Some Basic Principles

The most stable carbocation for the following is: [30-Jan-2023 Shift 2]

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📖 Explanation

The $+ {\text{M}}$ effect of ${\text{NH}}_2$ is stabilizing the carbocation.

Explanation Figure 2

Q33JEE Main 2023MCQ4MAlcohols, Phenols and Ethers

The correct order of ${\text{pK}}_{ {\text{a}}}$ values for the following compounds is: [30-Jan-2023 Shift 2]

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📖 Explanation

Due to $- {\text{M}}$ effect of $- {\text{NO}}_2$ group, it increases acidity $+ {\text{M}}$ effect of ${\text{N}} ( {\text{CH}}_3) _2$ decreases acidity.Hyperconjugation of isopropyl decrease acidity $\therefore$ order of acidic strength(c) $>$ (a) $>$ (d) $>$ (b)

Q34JEE Main 2023MCQ4MHaloalkanes and Haloarenes

Decreasing order towards $S_N 1$ reaction for the following compounds is: [30-Jan-2023 Shift 2]

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📖 Explanation

The rate of $S_N 1$ reaction depends upon stability of carbocation which follows the order $\therefore$ Reactivity order(b) $>$ (d) $>$ (c) $>$ (a)

Q35JEE Main 2023MCQ4MAmines

In the above conversion of compound $( {\text{X}})$ to product $( {\text{Y}})$ , the sequence of reagents to be used will be: [30-Jan-2023 Shift 2]

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Q36JEE Main 2023MCQ4MStructure of Atom

Maximum number of electrons that can be accommodated in shell with ${\text{n}}=4$ are: [30-Jan-2023 Shift 2]

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📖 Explanation

The number of electrons in the orbitals of sub-shell of $n=4$ are 4s 2 4p 6 4d 10 4f 14 (Total) 32

Q37JEE Main 2023MCQ4MCoordination Compounds

Match List I with List II: List I (Complexes) List II (Hybridis ation) (A) $[ {\text{Ni}}( {\text{CO}})_4]$ I. ${\ sp}^3$ (B) $[ {\text{Cu}} ( {\text{NH}}_3) _4] ^{2+}$ II. ${\text{dsp}}^2$ (C) $[ {\text{Fe}} ( {\text{NH}}_3) _6] ^{2+}$ III. ${\ sp}^3 {\text{d}}^2$ (D) $[ {\text{Fe}} ( {\text{H}}_2 {\text{O}}) _6] ^{2+}$ IV. ${\text{d}}^2 {\ sp}^3$ [30-Jan-2023 Shift 2]

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📖 Explanation

For $[ {\text{Fe}} ( {\text{NH}}_3) _6] ^{+2}, \Delta _0 < {\text{P}}$ , hence the pairing of electrons does not occur in $t_{2 g}$ . Therefore complex is outer orbital and its hybridisation is ${\ sp}^3 {\text{d}}^2$ . List I (Complexes) List II (Hybridis ation) $[ {\text{Ni}}( {\text{CO}})_4]$ ${\ sp}^3$ $[ {\text{Cu}} ( {\text{NH}}_3) _4] ^{2+}$ ${\text{dsp}}^2$ $[ {\text{Fe}} ( {\text{NH}}_3) _6] ^{2+}$ ${\ sp}^3 {\text{d}}^2$ $[ {\text{Fe}} ( {\text{H}}_2 {\text{O}}) _6] ^{2+}$ ${\ sp}^3 {\text{d}}^2$

Q38JEE Main 2023MCQ4MCoordination Compounds

The ${\text{Cl}}- {\text{Co}}- {\text{Cl}}$ bond angle values in a fac $[ {\text{Co}} ( {\text{NH}}_3) _3 {\text{Cl}}_3]$ complex is/are: [30-Jan-2023 Shift 2]

🚀 Solve in Practice Mode

📖 Explanation

The ${\text{Cl}}- {\text{Co}}- {\text{Cl}}$ bond angle in above octahedral complex is $90^{\circ}$

Explanation Figure 1

Q39JEE Main 2023MCQ4MAldehydes, Ketones and Carboxylic Acids

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.Assertion A : can be easily reduced using ${\text{Zn}}- {\text{Hg}} / {\text{HCl}}$ to Reason R : ${\text{Zn}}- {\text{Hg}} / {\text{HCl}}$ is used to reduce carbonyl group to $- {\text{CH}}_2-$ group.In the light of the above statements, choose the correct answer from the options given below: [30-Jan-2023 Shift 2]

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📖 Explanation

The acid sensitive alcohol group reacts with ${\text{HCl}}$ , hence Clemmenson reduction is not suitable for above conversion.

Explanation Figure 2

Q40JEE Main 2023MCQ4MChemical Bonding and Molecular Structure

Chlorides of which metal are soluble in organic solvents: [30-Jan-2023 Shift 2]

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📖 Explanation

${\text{BeCl}}_2$ having covalent nature is soluble in organic solvent.

Q41JEE Main 2023MCQ4MBiomolecules

Given below are two statements: One is labelled as Assertion A and the other labelled as Reason R.Assertion A: Antihistamines do not affect the secretion of acid in stomach.Reason $\text{bo} {R}$ : Antiallergic and antacid drugs work on different receptors.In the light of the above statements, choose the correct answer from the options given below: [30-Jan-2023 Shift 2]

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📖 Explanation

Antiallergic and antacid drugs work on different receptorsNCERT(XII) vol. 2 page no. $451-452$

Q42JEE Main 2023MCQ4MStructure of Atom

The wave function $(Ψ)$ of $2 {\ s}$ is given by $Ψ_{2 {\ s}}=\;{1}/{2 {\sqrt{2 \pi}}} (\;{1}/{ {\text{a}}_0}) ^{1 / 2} (2-\;{ {\text{r}}}/{ {\text{a}}_0}) {\text{e}}^{- {\text{r}} / 2 {\text{a}}_0}$ At ${\text{r}}= {\text{r}}_0$ , radial node is formed. Thus, ${\text{r}}_0$ in terms of ${\text{a}}_0$ [30-Jan-2023 Shift 2]

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📖 Explanation

At node $Ψ_{2 {\ s}}=0$ $\; \therefore 2-\;{ {\text{r}}_0}/{ {\text{a}}_0}=0$ $\; \therefore {\text{r}}_0=2 {\text{a}}_0$

Q43JEE Main 2023MCQ4MRedox Reactions

${\text{KMnO}}_4$ oxidises ${\text{I}}^{-}$ in acidic and neutral/faintly alkaline solution, respectively to [30-Jan-2023 Shift 2]

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📖 Explanation

In acidic medium $2 {\text{MnO}}_4^{-}+10 {\text{I}}^{-}+16 {\text{H}}^{+} \to 2 {\text{Mn}}^{2+}+5 {\text{I}}_2+8 {\text{H}}_2 {\text{O}}$ In neutral/faintly alkaline solution $2 {\text{MnO}}_4^{-}+ {\text{I}}^{-}+ {\text{H}}_2 {\text{O}} \to 2 {\text{MnO}}_2+2 {\text{OH}}^{-}+ {\text{IO}}_3^{-}$

Q44JEE Main 2023MCQ4Mp Block Elements

Bond dissociation energy of ${\text{E}}- {\text{H}}$ bond of the " ${\text{H}}_2 {\text{E}}$ " hydrides of group 16 elements (given below), follows order.(A) ${\text{O}}$ (B) ${\text{S}}$ (C) ${\text{Se}}$ (D) ${\text{Te}}$ [30-Jan-2023 Shift 2]

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📖 Explanation

Bond dissociation energy of E-H bond in hydrides of group 16 follows the order ${\text{H}}_2 {\text{O}} > {\text{H}}_2 {\text{S}} > {\text{H}}_2 {\text{Se}} > {\text{H}}_2 {\text{Te}}$

Q45JEE Main 2023MCQ4MBiomolecules

The water quality of a pond was analysed and its BOD was found to be 4 . The pond has [30-Jan-2023 Shift 2]

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📖 Explanation

Clean water as BOD value of $< 5$ while polluted water has BOD of 15 or more.

Q46JEE Main 2023MCQ4MPractical Organic Chemistry

Match List I with List II: List I (Mixture) List II (SeparationTechnique) (A) ${\text{CHCl}}_3+ {\text{C}}_6 {\text{H}}_5 {\text{NH}}_2$ I. Steam distillation (B) ${\text{C}}_6 {\text{H}}_{14}+ {\text{C}}_5 {\text{H}}_{12}$ II. Differential extraction (C) ${\text{C}}_6 {\text{H}}_5 {\text{NH}}_2+ {\text{H}}_2 {\text{O}}$ III. Distillation (D) Organic compound $\;\text{ \in }\; {\text{H}}_2 {\text{O}}$ IV. Fractionaldistillation [30-Jan-2023 Shift 2]

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📖 Explanation

List I (Mixture) List II (SeparationTechnique) ${\text{CHCl}}_3+ {\text{C}}_6 {\text{H}}_5 {\text{NH}}_2$ Distillation ${\text{C}}_6 {\text{H}}_{14}+ {\text{C}}_5 {\text{H}}_{12}$ Fractionaldistillation ${\text{C}}_6 {\text{H}}_5 {\text{NH}}_2+ {\text{H}}_2 {\text{O}}$ Steam distillation Organic compound $\;\text{ \in }\; {\text{H}}_2 {\text{O}}$ Differential extractionNCERT (XI) Vol. 2 Page No. 359, 360.

Q47JEE Main 2023MCQ4MChemical Bonding and Molecular Structure

Boric acid in solid, whereas ${\text{BF}}_3$ is gas at room temperature because of [30-Jan-2023 Shift 2]

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📖 Explanation

Boric acid has strong hydrogen bonding while ${\text{BF}}_3$ does not. Therefore boric acid is solid.

Q48JEE Main 2023MCQ4MElectrochemistry

Given below are two statements: Statement I: During Electrolytic refining, the pure metal is made to act as anode and its impure metallic form is used as cathode.Statement II: During the Hall-Heroult electrolysis process, purified ${\text{Al}}_2 {\text{O}}_3$ is mixed with ${\text{Na}}_3 {\text{AlF}}_6$ to lower the melting point of the mixture.In the light of the above statements, choose the most appropriate answer from the options given below: [30-Jan-2023 Shift 2]

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📖 Explanation

In Electrolytic refining, the pure metal is used as cathode and impure metal is used as anode. ${\text{Na}}_3 {\text{AlF}}_6$ is added during electrolysis of ${\text{Al}}_2 {\text{O}}_3$ to lower the melting point and increase conductivity.

Q49JEE Main 2023MCQ4MSalt Analysis

Formulae for Nessler's reagent is: [30-Jan-2023 Shift 2]

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📖 Explanation

Nessler's reagent is ${\text{K}}_2 {\text{HgI}}_4$ .

Q50JEE Main 2023MCQ4MCoordination Compounds

$1 {\text{L}}, 0.02 {\text{M}}$ solution of $[ {\text{Co}} ( {\text{NH}}_3) _5 {\text{SO}}_4] {\text{Br}}$ is mixed with $1 {\text{L}}, 0.02 {\text{M}}$ solution of $[ {\text{Co}} ( {\text{NH}}_3) _5 {\text{Br}}] {\text{SO}}_4$ . The resulting solution is divided into two equal parts $( {\text{X}})$ and treated with excess ${\text{AgNO}}_3$ solution and ${\text{BaCl}}_2$ solution respectively as shown below: $1 {\text{L}} {\text{Solution}}( {\text{X}})+ {\text{AgNO}}_3$ solution (excess) $\to {\text{Y}}$ $1 {\text{L}}$ Solution $( {\text{X}})+ {\text{BaCl}}_2$ solution (excess) $\to {\text{Z}}$ The number of moles of ${\text{Y}}$ and ${\text{Z}}$ respectively are [30-Jan-2023 Shift 2]

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Q51JEE Main 2023NAT4MChemical Thermodynamics

1 mole of ideal gas is allowed to expand reversibly and adiabatically from a temperature of $27^{\circ} {\text{C}}$ . The work done is $3 {\text{kJ}} {\text{mol}}^{-1}$ . The final temperature of the gas is _______ ${\text{K}}$ (Nearest integer). Given ${\text{C}}_{ {\text{v}}}=20 {\text{J}} {\text{mol}}^{-1} {\text{K}}^{-1}$ . [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{q}}=0$ $\;\Delta {\text{U}}= {\text{W}}$ $\;1 \times 20 \times [ {\text{T}}_2-300] =-3000$ $\; {\text{T}}_2-300=-150$ $\; {\text{T}}_2=150 {\text{K}}$

Q52JEE Main 2023NAT4MSolid State

Iron oxide ${\text{FeO}}$ , crystallises in a cubic lattice with a unit cell edge length of $5.0 Å$ . If density of the ${\text{FeO}}$ in the crystal is $4.0 {\text{g}} {\text{cm}}^{-3}$ , then the number of ${\text{FeO}}$ units present per unit cell is _______ (Nearest integer)Given : Molar mass of ${\text{Fe}}$ and ${\text{O}}$ is 56 and $16 {\text{g}}$ ${\text{mol}}^{-1}$ respectively. ${\text{N}}_{ {\text{A}}}=6.0 \times 10^{23} {\text{mol}}^{-1}$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{d}}=\;{ {\text{Z}} \times {\text{M}}}/{ {\text{N}}_0 \times {\text{a}}^3}$ $\;4=\;{ {\text{Z}} \times 72}/{6 \times 10^{23} \times 125 \times 10^{-24}}$ $\; {\text{Z}}=4.166 ≃ 4$

Q53JEE Main 2023NAT4MChemical Kinetics

An organic compound undergoes first order decomposition. If the time taken for the $60 \%$ decomposition is $540 {\ s}$ , then the time required for $90 \%$ decomposition will be is ______ S. (Nearest integer).Given : $\ln 10=2.3 ; \log 2=0.3$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\;{ {\text{t}}_1}/{ {\text{t}}_2}=\;{\;{1}/{ {\text{K}}} \ln { {\text{a}}_0}/{0.4 {\text{a}}_0}}/{{1}/{ {\text{K}}} \ln \;{ {\text{a}}_0}/{0.1 {\text{a}}_0}}$ $\;\;{540}/{ {\text{t}}_2}=\;\frac{\ln \frac{10}{4}}{\ln 10}$ $\;\;{540}/{ {\text{t}}_2}=\;\frac{\log 10-\log 4}{\log 10}$ $\;\;{540}/{ {\text{t}}_2}=\;\frac{1-0.6}{1}$ $\;\Rightarrow \;{540}/{ {\text{t}}_2}=0.4$ $\;\Rightarrow {\text{t}}_2=\;\frac{540}{0.4}=1350 {\ sec}$

Q54JEE Main 2023NAT4MSolutions

Lead storage battery contains $38 \%$ by weight solution of ${\text{H}}_2 {\text{SO}}_4$ . The van't Hoff factor is $2.67$ at this concentration. The temperature in Kelvin at which the solution in the battery will freeze is _______ (Nearest integer).Given ${\text{K}}_{ {\text{f}}}=1.8 {\text{K}} {\text{kg}} {\text{mol}}^{-1}$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\Delta {\text{T}}_{ {\text{f}}}= {\text{i}} . {\text{K}}_{ {\text{f}}} \cdot {\text{m}}$ $\;\Rightarrow \Delta {\text{T}}_{ {\text{f}}}=2.67 \times 1.8 \times \;\frac{38}{98} \times \;\frac{1000}{62}$ $\;\Rightarrow \Delta {\text{T}}_{ {\text{f}}}=30.05$ $\; \therefore \;\text{ F.P. }\;=243 {\text{K}}$

Q55JEE Main 2023NAT4MChemical Equilibrium

Consider the following equation: $2 {\text{SO}}_2( {\text{g}})+ {\text{O}}_2( {\text{g}}) ⇌ 2 {\text{SO}}_3( {\text{g}}), \Delta {\text{H}}=-190 {\text{kJ}}$ The number of factors which will increase the yield of ${\text{SO}}_3$ at equilibrium from the following is _______A. Increasing temperatureB. Increasing pressureC. Adding more ${\text{SO}}_2$ D. Adding more ${\text{O}}_2$ E. Addition of catalyst [30-Jan-2023 Shift 2]

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📖 Explanation

The yield of ${\text{SO}}_3$ at equilibrium will be due to:B. Increasing pressureC. Adding more ${\text{SO}}_2$ D. Adding more ${\text{O}}_2$

Q56JEE Main 2023NAT4MStates of Matter

The graph of $\log \;{ {\text{x}}}/{ {\text{m}}}$ vs $\log {\text{p}}$ for an adsorption process is a straight line inclined at an angle of $45^{\circ}$ with intercept equal to $0.6020$ . The mass of gas adsorbed per unit mass of adsorbent at the pressure of $0.4 {\text{atm}}$ is _______ $\times 10^{-1}$ (Nearest integer)Given : $\log 2=0.3010$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\log \;{ {\text{x}}}/{ {\text{m}}}=\log {\text{k}}+\;{1}/{ {\text{n}}} \log P$ $\;\;{1}/{ {\text{n}}}=\tan 45^{\circ}=1$ $\;\log {\text{k}}=0.6020=\log 4$ $\;\Rightarrow {\text{K}}=4$ $\; \therefore \;{ {\text{x}}}/{ {\text{m}}}=K \cdot {\text{P}}^{1 / {\text{n}}}$ $\;\;{ {\text{x}}}/{ {\text{m}}}=4(0.4)=1.6$ $\;\;{ {\text{x}}}/{ {\text{m}}}=1.6=16 \times 10^{-1}$

Explanation Figure 1

Q57JEE Main 2023NAT4MPractical Organic Chemistry

Number of compounds from the following which will not dissolve in cold ${\text{NaHCO}}_3$ and ${\text{NaOH}}$ solutions but will dissolve in hot ${\text{NaOH}}$ solution is _______. [30-Jan-2023 Shift 2]

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📖 Explanation

Compound $2,3,7$

Q58JEE Main 2023NAT4MBiomolecules

A short peptide on complete hydrolysis produces 3 moles of glycine $( {\text{G}})$ , two moles of leucine $( {\text{L}})$ and two moles of valine (V) per mole of peptide. The number of peptide linkages in it are _______. [30-Jan-2023 Shift 2]

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📖 Explanation

Number of peptide linkage $=($ amino acid $-1)$ $=7-1=6$

Q59JEE Main 2023NAT4MSolutions

The strength of 50 volume solution of hydrogen peroxide is ________ ${\text{g}} / {\text{L}}$ (Nearest integer).Given:Molar mass of ${\text{H}}_2 {\text{O}}_2$ is $34 {\text{g}} {\text{mol}}^{-1}$ Molar volume of gas at ${\text{STP}}=22.7 {\text{L}}$ . [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\text{ Molarity }\;=\;\frac{50}{11.35}$ $\therefore$ Strength in ${\text{gm}} / {\text{L}}=\;\frac{50}{11.35} \times 34$

Q60JEE Main 2023NAT4MElectrochemistry

The electrode potential of the following half cell at $\;298 {\text{K}} \;\text{. }\;$ $\; {\text{X}} | {\text{X}}^{2+}(0.001 {\text{M}}) \| {\text{Y}}^{2+}(0.01 {\text{M}})| {\text{Y}}$ is _______ $\times 10^{-2} {\text{V}}$ (Nearest integer).Given: ${\text{E}}_{ {\text{x}}^{2+} \mid {\text{x}}}^0=-2.36 {\text{V}}$ ${\text{E}}_{ {\text{Y}}^{2+}{ }_{ {\text{Y}}}}^0=+0.36 {\text{V}}$ $\;{2.303 {\text{RT}}}/{ {\text{F}}}=0.06 {\text{V}}$ [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{X}}+ {\text{Y}}^{2+} \to {\text{Y}}+ {\text{X}}^{2+}$ $\; {\text{E}}_{ {\text{Cell}}}^0=0.36-(-2.36)=2.72 {\text{V}}$ $\; {\text{E}}_{\;\text{Cell }\;}=2.72-\;\frac{0.06}{2} \log \;\frac{0.001}{0.01}$ $\;=2.72+0.03=2.75 {\text{V}}$ $\;=275 \times 10^{-2} {\text{V}}$ $\;$

Mathematics30 questions

Q61JEE Main 2023MCQ4MSets and Relations

Consider the following statements: ${\text{P}}$ : I have fever ${\text{Q}}$ : I will not take medicine ${\text{R}}$ : I will take restThe statement "If I have fever, then I will take medicine and I will take rest" is equivalent to: [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{P}} \to (∼ {\text{Q}} ∧ {\text{R}})$ $\;∼ {\text{P}} ∨(∼ {\text{Q}} ∧ {\text{R}})$ $\;(∼ {\text{P}} ∨ ∼ {\text{Q}}) ∧(∼ {\text{P}} ∨ {\text{R}})$

Q62JEE Main 2023MCQ4MStraight Lines and Pair of Straight Lines

Let ${\text{A}}$ be a point on the ${\text{x}}$ -axis. Common tangents are drawn from $A$ to the curves $x^{2+}y^2=8$ and $y^2=$ 16x. If one of these tangents touches the two curves at ${\text{Q}}$ and ${\text{R}}$ , then $( {\text{QR}})^2$ is equal to [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{y}}= {\text{mx}}+\;{4}/{ {\text{m}}}$ $\;\;{ |{4}/{ {\text{m}}}|}/{\sqrt{1+ {\text{m}}^2}}=2 {\sqrt{2}} \therefore {\text{m}}=\pm 1$ Let ${\text{m}}=1, (\;{ {\text{a}}}/{ {\text{m}}^2}, \;{2 {\text{a}}}/{ {\text{m}}})$ ${\text{R}}(4,8)$ Point of contact on circle $Q(-2,2)$ $\therefore ( {\text{QR}})^2=36+36=72$

Q63JEE Main 2023MCQ4MArea Under The Curves

Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x^{2-}p x+\;\frac{5}{4} p=0$ are rational. Then the area of the region $\{(x, y): 0 \le y$ $\le ( {\text{x}}- {\text{q}})^2, 0 \le {\text{x}} \le {\text{q}}\}$ is [30-Jan-2023 Shift 2]

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📖 Explanation

$\;x^{2-}p x+\;\frac{5 p}{4}=0$ $\;D=p^{2-}5 p=p(p-5)$ $\; \therefore q=9$ $\;0 \le y \le (x-9)^2$ Area $=int_{0}^{9}(x-9)^2 \text{dx}=243$

Explanation Figure 1

Q64JEE Main 2023MCQ4MApplication of Derivatives

If the functions ${\text{f}}( {\text{x}})=\;{ {\text{x}}^3}/{3}+2 b {\text{bx}}+\;\frac{a x^2}{2}$ and $g(x)=\;\frac{x^3}{3}+a x+b x^2, a \ne 2 b$ have a common extreme point, then ${\text{a}}+2 {\text{b}}+7$ is equal to [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{f}}^{'}( {\text{x}})= {\text{x}}^{2+}2 b+a x$ $\; {\text{g}}^{'}( {\text{x}})= {\text{x}}^{2+} {\text{a}}+2 {\text{bx}}$ $\;(2 {\text{b}}- {\text{a}})- {\text{x}}(2 {\text{b}}- {\text{a}})=0$ $\; \therefore {\text{x}}=1 \;\text{ is the common root }\;$ $\;\;\text{ Put }\; {\text{x}}=1 \;\text{ \in }\; {\text{f}}^{'}( {\text{x}})=0 \;\text{ or }\; {\text{g}}^{'}( {\text{x}})=0$ $\;1+2 b+a=0$ $\;7+2 b+a=6$

Q65JEE Main 2023MCQ4MApplication of Derivatives

The range of the function $f(x)={\sqrt{3-x}}+{\sqrt{2+x}}$ is [30-Jan-2023 Shift 2]

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📖 Explanation

$\;y^2=3-x+2+x+2 {\sqrt{(3-x)(2+x)}}$ $\;=5+2 {\sqrt{6+x-x^2}}$ $\;y^2=5+2 \sqrt{\;\frac{25}{4}- (x-\;\frac{1}{2}) ^2}$ $\;y_{\max }={\sqrt{5+5}}={\sqrt{10}}$ $\;y_{\min }={\sqrt{5}}$

Q66JEE Main 2023MCQ4MDifferential Equations

The solution of the differential equation $\;\frac{d y}{d x}=- (\;\frac{x^{2+}3 y^2}{3 x^{2+}y^2}) , y(1)=0 \;\text{ is }\;$ [30-Jan-2023 Shift 2]

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📖 Explanation

Put $y=v x$ $\;v+x \;\frac{d v}{d x}=- (\;\frac{1+3 v^2}{3+v^2})$ $\;x \;\frac{d v}{d x}=-\;\frac{(v+1)^3}{3+v^2}$ $\;\;\frac{ (3+v^2) d v}{(v+1)^3}+\;\frac{d x}{x}=0$ $\;\int \;\frac{4 d v}{(v+1)^3}+\int \;\frac{d v}{v+1}-\int \;\frac{2 d v}{(v+1)^2}+\int \;\frac{d x}{x}=0$ $\;\;{-2}/{( {\text{v}}+1)^2}+\ln ( {\text{v}}+1)+\;{2}/{ {\text{v}}+1}+\ln {\text{x}}= {\text{c}}$ $\;\;\frac{-2 x^2}{(x+y)^2}+\ln (\;\frac{x+y}{x}) +\;\frac{2 x}{x+y}+\ln x=c$ $\;\;\frac{2 x y}{(x+y)^2}+\ln (x+y)=c$ $\; \therefore {\text{c}}=0 \;\text{, as }\; {\text{x}}=1, {\text{y}}=0$ $\; \therefore \;\frac{2 x y}{(x+y)^2}+\ln (x+y)=0$ $\;$

Q67JEE Main 2023MCQ4MBinomial Theorem

Let ${\text{x}}=(8 {\sqrt{3}}+13)^{13}$ and ${\text{y}}=(7 {\sqrt{2}}+9)^9$ . If $[ {\text{t}}]$ denotes the greatest integer $\le {\text{t}}$ , then [30-Jan-2023 Shift 2]

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📖 Explanation

${\text{x}}=(8 {\sqrt{3}}+13)={ }^{13} {\text{C}}_0 \cdot (8 {\sqrt{3}})^{13}+{ }^{13} {\text{C}}_1(8 {\sqrt{3}})^{12}(13)^{1+}. . .$ $\; {\text{x}}^{'}=(8 {\sqrt{3}}-13)^{13}={ }^{13} {\text{C}}_0(8 {\sqrt{3}})^{13}-{ }^{13} {\text{C}}_1(8 {\sqrt{3}})^{12}(13)^{1+}. . .$ $\; {\text{x}}- {\text{x}}^{'}=2 [{ }^{13} {\text{C}}_1 \cdot (8 {\sqrt{3}})^{12}(13)^{1+}{ }^{13} {\text{C}}_3(8 {\sqrt{3}})^{10} \cdot (13)^3 . . .]$ therefore, $x-x^{'}$ is even integer, hence $[x]$ is evenNow, ${\text{y}}=(7 {\sqrt{2}}+9)^9={ }^9 {\text{C}}_0(7 {\sqrt{2}})^{9+}{ }^9 {\text{C}}_1(7 {\sqrt{2}})^8(9)^1$ $\;+{ }^9 {\text{C}}_2(7 {\sqrt{2}})^7(9)^2 . . . . . . .$ $\; {\text{y}}^{'}=(7 {\sqrt{2}}-9)^9={ }^9 {\text{C}}_0(7 {\sqrt{2}})^{9-}{ }^9 {\text{C}}_1(7 {\sqrt{2}})^8(9)^1$ $\;+{ }^9 {\text{C}}_2(7 {\sqrt{2}})^7(9)^2 . . . . . . .$ $\; {\text{y}}- {\text{y}}^{'}=2 [{ }^9 {\text{C}}_1(7 {\sqrt{2}})^8(9)^{1+}{ }^9 {\text{C}}_3(7 {\sqrt{2}})^6(9)^{3+}. . .]$ $y-y^{'}=$ Even integer, hence $[y]$ is even

Q68JEE Main 2023MCQ4MThree Dimensional Geometry

A vector ${ \vec{\text{v}}$ in the first octant is inclined to the ${\text{x}}$ axis at $60^{\circ}$ , to the $y$ -axis at $45^{\circ}$ and to the z-axis at an acute angle. If a plane passing through the points $({\sqrt{2}},-1,1)$ and $( {\text{a}}, {\text{b}}, {\text{c}})$ , is normal to ${ \vec{\text{v}}$ , then [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\;{ {\text{v}}}^{\^}=\cos 60^{\circ} \;{ {\text{i}}}^{\^}+\cos 45^{\circ} \;{ {\text{j}}}^{\^}+\cos \gamma \;{ {\text{k}}}^{\^}$ $\;\Rightarrow \;\frac{1}{4}+\;\frac{1}{2}+\cos ^2 \gamma =1 \;\; (\gamma \to \;\text{ Acute }\;)$ $\;\Rightarrow \cos \gamma =\;\frac{1}{2}$ $\;\Rightarrow \gamma =60^{\circ}$ Equation of plane is $\;\;\frac{1}{2}( {\text{x}}-{\sqrt{2}})+\;{1}/{{\sqrt{2}}}( {\text{y}}+1)+\;\frac{1}{2}( {\text{z}}-1)=0$ $\;\Rightarrow {\text{x}}+{\sqrt{2}} {\text{y}}+ {\text{z}}=1$ $( {\text{a}}, {\text{b}}, {\text{c}})$ lies on it. $\Rightarrow {\text{a}}+{\sqrt{2}} {\text{b}}+ {\text{c}}=1$

Q69JEE Main 2023MCQ4MLimits, Continuity and Differentiability

Let ${\text{f}}, {\text{g}}$ and ${\text{h}}$ be the real valued functions defined on $ℝ$ as $f(x)={\begin{cases}\\ \\ \frac{x}{{|x|}} & x \ne 0 \\ 1 & x=0}\end{cases}$ , $g(x)={\begin{cases}\\ \\ \frac{ sin \\ \\ (x+1)}{(x+1)}, , x \ne -1 \\ 1 & x=-1}\end{cases}$ and $h(x)=2[x]-f(x)$ , where $[x]$ is the greatest integer $\le x$ . Then the value of $\lim _{x \to 1} g(h(x-1))$ is [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\text{ LHL }\;=\lim _{ {\text{k}} \to 0} {\text{g}}( {\text{h}}(- {\text{k}}))\;, {\text{k}} > 0$ $=\lim _{ {\text{k}} \to 0} {\text{g}}(-2+1)\; \because {\text{f}}( {\text{x}})=-1 ∀ {\text{x}} < 0$ $= {\text{g}}(-1)=1\;$ ${\text{RHL}}=\lim _{ {\text{k}} \to 0} {\text{g}}( {\text{h}}( {\text{k}}))\;, {\text{k}} > 0$ $=\lim _{ {\text{k}} \to 0} {\text{g}}(-1)\;, \because {\text{f}}( {\text{x}})=1, ∀ {\text{x}} > 0$ $=1\;$

Q70JEE Main 2023MCQ4MPermutations and Combinations

The number of ways of selecting two numbers a and b, ${\text{a}} \in \{2,4,6, . . . ., 100\} \;\;$ and $\;\; {\text{b}} \in \{1,3,5, . . . ., 99\}$ such that 2 is the remainder when ${\text{a}}+ {\text{b}}$ is divided by 23 is [30-Jan-2023 Shift 2]

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📖 Explanation

$\;a \in \{2,4,6,8,10, . . . ., 100\}$ $\;b \in \{1,3,5,7,9, . . . ., 99\}$ Now, $a+b \in \{25,71,117,163\}$ (i) $a+b=25$ , no. of ordered pairs (a, b) is 12(ii) $a+b=71$ , no. of ordered pairs (a, b) is 35(iii) ${\text{a}}+ {\text{b}}=117$ , no. of ordered pairs (a, b) is 42(iv) $a+b=163$ , no. of ordered pairs (a, b) is 19 $\therefore$ total $=108$ pairs

Q71JEE Main 2023MCQ4MMatrices and Determinants

If ${\text{P}}$ is a $3 \times 3$ real matrix such that ${\text{P}}^{ {\text{T}}}= {\text{aP}}+( {\text{a}}-1) {\text{I}}$ , where ${\text{a}}>1$ , then [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{P}}^{ {\text{T}}}= {\text{aP}}+( {\text{a}}-1) {\text{I}}$ $\;\Rightarrow {\text{P}}= {\text{aP}}^{ {\text{T}}}+( {\text{a}}-1) {\text{I}}$ $\;\Rightarrow {\text{P}}^{ {\text{T}}}- {\text{P}}= {\text{a}} ( {\text{P}}- {\text{P}}^{ {\text{T}}})$ $\;\Rightarrow {\text{P}}= {\text{P}}^{ {\text{T}}}, \;\text{ as }\; {\text{a}} \ne -1$ $\;\;\text{ Now, }\; {\text{P}}= {\text{aP}}+( {\text{a}}-1) {\text{I}}$ $\;\Rightarrow {\text{P}}=- {\text{I}} \Rightarrow | {\text{P}}|=1$ $\;\Rightarrow |{\text{Adj}} {\text{P}}|=1$

Q72JEE Main 2023MCQ4MVector Algebra

Let $\lambda \in ℝ, \vec{a}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}, \vec{b}=\hat{i}-\lambda \hat{j}+2 \hat{k}$ .If $((\vec{a}+\vec{b}) \times (\vec{a} \times \vec{b})) \times (\vec{a}-\vec{b})=8 \hat{i}-40 \hat{j}-24 \hat{k}$ , then $|\lambda (\vec{a}+\vec{b}) \times (\vec{a}-\vec{b})|^2$ is equal to [30-Jan-2023 Shift 2]

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📖 Explanation

$\;{ \vec{\text{a}}=\lambda \;{ {\text{i}}}^{\^}+2 \;{ {\text{j}}}^{\^}-3 \;{ {\text{k}}}^{\^}$ $\;{ \vec{\text{b}}=\;{ {\text{i}}}^{\^}-\lambda \;{ {\text{j}}}^{\^}+2 \;{ {\text{k}}}^{\^}$ $\;\Rightarrow ({ \vec{\text{b}}-{ \vec{\text{a}}) \times (({ \vec{\text{a}}+{ \vec{\text{b}}) \times ({ \vec{\text{a}} \times { \vec{\text{b}}))=8 \;{ {\text{i}}}^{\^}-40 \;{ {\text{j}}}^{\^}-24 \;{ {\text{k}}}^{\^}$ $\;\Rightarrow (({ \vec{\text{a}}-{ \vec{\text{b}}) \cdot ({ \vec{\text{a}}+{ \vec{\text{b}}))({ \vec{\text{a}} \times { \vec{\text{b}})=8 \;{ {\text{i}}}^{\^}-40 {\text{j}}-24 \;{ {\text{k}}}^{\^}$ $\;\Rightarrow 8({ \vec{\text{a}} \times { \vec{\text{b}})=8 \;{ {\text{i}}}^{\^}-40 \;{ {\text{j}}}^{\^}-24 \;{ {\text{k}}}^{\^}$ Now, $\vec{a} \times \vec{b}={\begin{vmatrix}\\ \\ { {\text{i}}}^{\^} & \\ \\ { {\text{j}}}^{\^} & \\ \\ { {\text{k}}}^{\^} \\ \lambda & 2 & -3 \\ 1 & -\lambda & 2\end{vmatrix}}$ $\;=(4-3 \lambda ) \;{ {\text{i}}}^{\^}-(2 \lambda +3) \;{ {\text{j}}}^{\^}+ (-\lambda ^{2-}2) \;{ {\text{k}}}^{\^}$ $\;\Rightarrow \lambda =1$ $\; \therefore { \vec{\text{a}}=\;{ {\text{i}}}^{\^}+2 \;{ {\text{j}}}^{\^}-3 \;{ {\text{k}}}^{\^}$ $\;{ \vec{\text{b}}=\;{ {\text{i}}}^{\^}-\;{ {\text{j}}}^{\^}+2 \;{ {\text{k}}}^{\^}$ $\;\Rightarrow { \vec{\text{a}}+{ \vec{\text{b}}=2 \;{ {\text{i}}}^{\^}+\;{ {\text{j}}}^{\^}-\;{ {\text{k}}}^{\^}, { \vec{\text{a}}-{ \vec{\text{b}}=3 \;{ {\text{j}}}^{\^}-5 \;{ {\text{k}}}^{\^}$ $\;\Rightarrow (\vec{\text{a}}+\vec{\text{b}} ) \times ( \vec{\text{a}}- \vec{\text{b}})={\begin{vmatrix}\\ \\ { {\text{i}}}^{\^} & \\ \\ { {\text{j}}}^{\^} & \\ \\ { {\text{k}}}^{\^} \\ 2 & 1 & -1 \\ 0 & 3 & -5\end{vmatrix}}=2 \;{ {\text{i}}}^{\^}+10 \;{ {\text{j}}}^{\^}+6 \;{ {\text{k}}}^{\^}$ $\; \therefore \;\text{ required answer }\;=4+100+36=140$

Q73JEE Main 2023MCQ4MVector Algebra

Let $\vec{a}$ and $\vec{b}$ be two vectors. Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$ . If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$ , then the value of $\vec{b} \cdot \vec{c}$ is [30-Jan-2023 Shift 2]

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📖 Explanation

$\;{ \vec{\text{c}}=(2 { \vec{\text{a}} \times { \vec{\text{b}})-3 { \vec{\text{b}}$ $\;{ \vec{\text{b}} \cdot { \vec{\text{c}}={ \vec{\text{b}} \cdot (2 { \vec{\text{a}} \times { \vec{\text{b}})-3 { \vec{\text{b}} \cdot { \vec{\text{b}}$ $\;=-3| {\text{b}}|^2$ $\;=-48$

Q74JEE Main 2023MCQ4MSequences and Series

Let ${\text{a}}_1=1, {\text{a}}_2, {\text{a}}_3, {\text{a}}_4, . . .$ . be consecutive natural numbers. Then $\tan ^{-1} (\;{1}/{1+ {\text{a}}_1 {\text{a}}_2}) +\tan ^{-1} (\;{1}/{1+ {\text{a}}_2 {\text{a}}_3})$ $+. . . . .+\tan ^{-1} (\;{1}/{1+ {\text{a}}_{2021} {\text{a}}_{2022}})$ is equal to [30-Jan-2023 Shift 2]

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📖 Explanation

Sol. ${\text{a}}_2- {\text{a}}_1= {\text{a}}_3- {\text{a}}_2=. . . . .= {\text{a}}_{2022}- {\text{a}}_{2021}=1$ . $\; \therefore \tan ^{-1} (\;{ {\text{a}}_2- {\text{a}}_1}/{1+ {\text{a}}_1 {\text{a}}_2}) +\tan ^{-1} (\;{ {\text{a}}_3- {\text{a}}_2}/{1+ {\text{a}}_2 {\text{a}}_3}) +. . . . .+\tan ^{-1} (\;{ {\text{a}}_{2022}- {\text{a}}_{2021}}/{1+ {\text{a}}_{2021} {\text{a}}_{2022}})$ $\;= \$ [ (\tan ^{-1} a_2) -\tan ^{-1} a_1]$ + $[\tan ^{-1} a_3-\tan ^{-1} a_2]$ +. . . . .;+ [\tan ^{-1} {\text{a}}{2022}-\tan ^{-1} {\text{a}}{2021}];=\tan ^{-1} {\text{a}}_{2022}-\tan ^{-1} {\text{a}}_1;=\tan ^{-1}(2022)-\tan ^{-1} 1=\tan ^{-1} 2022-;\frac{\pi}{4} ;\text{ (option 3) };;= (;\frac{\pi}{2}-\cot ^{-1}(2022)) -;\frac{\pi}{4};=;\frac{\pi}{4}-\cot ^{-1}(2022) ;\text{ (option 1) };;$

Q75JEE Main 2023MCQ4MParabola

The parabolas : $a x^{2+}2 b x+c y=0$ and $d x^{2+}2 e x+f y=0$ intersect on the line $y=1$ . If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then [30-Jan-2023 Shift 2]

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📖 Explanation

$\;a x^{2+}2 b x+c=0$ $\;\Rightarrow a x^{2+}2 {\sqrt{a c}} x+c=0 ( \because b^2=a c)$ $\;\Rightarrow (x {\sqrt{a}}+{\sqrt{c}})^2=0$ $\;x^{2-}\;{{\sqrt{c}}}/{{\sqrt{a}}} \;\; . . . . . .$ $\;\;\text{ Now, }\; \text{dx}^{2+}2 e x+f=0$ $\;\Rightarrow d (\;\frac{c}{a}) +2 e [-\;{{\sqrt{c}}}/{{\sqrt{a}}}] +f=0$ $\;\Rightarrow \;\frac{d c}{a}+f=2 e \sqrt{\;\frac{c}{a}}$ $\;\Rightarrow \;{ {\text{d}}}/{ {\text{a}}}+\;{ {\text{f}}}/{ {\text{c}}}=2 {\text{e}} \sqrt{\;{1}/{ {\text{ac}}}}$ $\;\Rightarrow \;{ {\text{d}}}/{ {\text{a}}}+\;{ {\text{f}}}/{ {\text{c}}}=\;{2 {\text{e}}}/{ {\text{b}}}[\;\text{ as }\; {\text{b}}=\sqrt{ {\text{ae}}}]$ $\; \therefore \;{ {\text{d}}}/{ {\text{a}}}, \;{ {\text{e}}}/{ {\text{b}}}, \;{ {\text{f}}}/{ {\text{c}}} \;\text{ are \in A.P. }\;$

Q76JEE Main 2023MCQ4MThree Dimensional Geometry

If a plane passes through the points $(-1, k, 0),(2, k,-1)$ , $(1,1,2)$ and is parallel to the line $\;\frac{x-1}{1}=\;\frac{2 y+1}{2}$ $=\;{ {\text{z}}+1}/{-1}$ , then the value of $\;{ {\text{k}}^{2+}1}/{( {\text{k}}-1)( {\text{k}}-2)}$ is [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\;{ {\text{x}}-1}/{1}=\;{2 {\text{y}}+1}/{2}=\;{ {\text{z}}+1}/{-1}$ $\;\;{ {\text{x}}-1}/{1}=\;{ {\text{y}}+\frac{1}{2}}/{1}=\;{ {\text{z}}+1}/{-1}$ $\;\;\text{ Points : A }\;(-1, {\text{k}}, 0), {\text{B}}(2, {\text{k}},-1), {\text{C}}(1,1,2)$ $\;{ \vec{\text{CA}}=-2 \;{ {\text{i}}}^{\^}+( {\text{k}}-1) \;{ {\text{j}}}^{\^}-2 \;{ {\text{k}}}^{\^}$ $\;{ \vec{\text{CB}}=\;{ {\text{i}}}^{\^}+( {\text{k}}-1) \;{ {\text{j}}}^{\^}-3 \;{ {\text{k}}}^{\^}$ ${ \vec{\text{CA}} \times { \vec{\text{CB}}={\begin{vmatrix}\\ \\ { {\text{i}}}^{\^} & \\ \\ { {\text{j}}}^{\^} & \\ \\ { {\text{k}}}^{\^} \\ -2 & {\text{k}}-1 & -2 \\ 1 & {\text{k}}-1 & -3\end{vmatrix}}$ $\;=\;{ {\text{i}}}^{\^}(-3 {\text{k}}+3+2 {\text{k}}-2)-\;{ {\text{j}}}^{\^}(6+2)+\;{ {\text{k}}}^{\^}(-2 {\text{k}}+2- {\text{k}}+1)$ $\; \;\; =(1- {\text{k}}) \;{ {\text{i}}}^{\^}-8 \;{ {\text{j}}}^{\^}+(3-3 {\text{k}}) \;{ {\text{k}}}^{\^}$ The line $\;\frac{x-1}{1}=\;\frac{y+\frac{1}{2}}{1}=\;\frac{z+1}{-1}$ is perpendicular to normal vector. $\; \therefore 1 \cdot (1- {\text{k}})+1(-8)+(-1)(3-3 {\text{k}})=0$ $\;\Rightarrow 1- {\text{k}}-8-3+3 {\text{k}}=0$ $\;\Rightarrow 2 {\text{k}}=10 \Rightarrow {\text{k}}=5$ $\; \therefore \;{ {\text{k}}^{2+}1}/{( {\text{k}}-1)( {\text{k}}-2)}=\;\frac{26}{4 \cdot 3}=\;\frac{13}{6}$

Q77JEE Main 2023MCQ4MSequences and Series

Let $a, b, c>1, a^3, b^3$ and $c^3$ be in A.P., and $\log _a b$ , $\log _c a$ and $\log _b c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\;\frac{a+4 b+c}{3}$ and the common difference is $\;{ {\text{a}}-8 {\text{b}}+ {\text{c}}}/{10}$ is $-444$ , then abc is equal to [30-Jan-2023 Shift 2]

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📖 Explanation

As $a^3, b^3, c^3$ be in A.P. $\to a^{3+}c^3=2 b^3$ . . . (1) $\log _a^b, \log _c^a, \log _b^c$ are in G.P. $\; \therefore \;\frac{\log b}{\log a} \cdot \;\frac{\log c}{\log b}= (\;\frac{\log a}{\log c}) ^2$ $\; \therefore (\log a)^3=(\log c)^3 \Rightarrow a=c$ . . . (2)From (1) and (2) $\; {\text{a}}= {\text{b}}= {\text{c}}$ $\; {\text{T}}_1=\;{ {\text{a}}+4 {\text{b}}+ {\text{c}}}/{3}=2 {\text{a}} ; {\text{d}}=\;{ {\text{a}}-8 {\text{b}}+ {\text{c}}}/{10}=\;{-6 {\text{a}}}/{10}=\;\frac{-3}{5} {\text{a}}$ $\; \therefore {\text{S}}_{20}=\;\frac{20}{2} [4 {\text{a}}+19 (-\;\frac{3}{5} {\text{a}}) ]$ $\;=10 [\;{20 {\text{a}}-57 {\text{a}}}/{5}]$ $\;=-74 {\text{a}}$ $\; \therefore -74 {\text{a}}=-444 \Rightarrow {\text{a}}=6$ $\; \therefore {\text{abc}}=6^3=216$

Q78JEE Main 2023MCQ4MStatistics

Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of 100 consecutive positive integers ${\text{a}}_1, {\text{a}}_2, {\text{a}}_3, . . . ., {\text{a}}_{100}$ is 25 . Then ${\text{S}}$ is [30-Jan-2023 Shift 2]

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📖 Explanation

let $a_1$ be any natural number $\; {\text{a}}_1, {\text{a}}_1+1, {\text{a}}_1+2, . . . ., {\text{a}}_1+99 \;\text{ are values of }\; {\text{a}}_{ {\text{i}}}{ }^{'} {\text{S}}$ $\;\overline{ {\text{x}}}=\;{ {\text{a}}_1+ ( {\text{a}}_1+1) + ( {\text{a}}_1+2) +. . . .+ {\text{a}}_1+99}/{100}$ $\;=\;{100 {\text{a}}_1+(1+2+. . . .+99)}/{100}= {\text{a}}_1+\;\frac{99 \times 100}{2 \times 100}$ $\;= {\text{a}}_1+\;\frac{99}{2}$ $\;\;\text{ Mean deviation about mean }\;=\;{sum_{1=1}^{100} |x_{ {\text{i}}}-\overline{ {\text{x}}}|}/{100}$ $\;=\;\frac{2 (\frac{99}{2}+\frac{97}{2}+\frac{95}{2}+. . . .+\frac{1}{2}) }{100}$ $\;=\;\frac{1+3+. . . .+99}{100}$ $\;=\;\frac{\;\frac{50}{2}[1+99]}{100}$ $\;=25$ So, it is true for every natural no. ' $a_1{ }^{'}$

Q79JEE Main 2023MCQ4MDefinite Integrals

$\lim _{n \to \infty } \;\frac{3}{n} \{4+ (2+\;\frac{1}{n}) ^{2+} (2+\;\frac{2}{n}) ^{2+}. . .+ (3-\;\frac{1}{n}) ^2\}$ is equal to [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\lim _{n \to \infty } \;\frac{3}{n} sum_{r=0}^{n-1} (2+\;\frac{r}{n}) ^2$ $\;=3 int_{0}^{1}(2+x)^2 \text{dx}=27-8=19$

Q80JEE Main 2023MCQ4MMatrices and Determinants

For $\alpha, \beta \in ℝ$ , suppose the system of linear equations $\;x-y+z=5$ $\;2 x+2 y+\alpha z=8$ $\;3 x-y+4 z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of [30-Jan-2023 Shift 2]

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📖 Explanation

$\;{\begin{vmatrix}1 & -1 & 1 \\ 2 & 2 & \alpha \\ 3 & -1 & 4\end{vmatrix}}=0 ; 8+\alpha-2(-4+1)+3(-\alpha-2)=0$ $\;8+\alpha+6-3 \alpha-6=0$ $\;\alpha=4$

Q81JEE Main 2023NAT4MSequences and Series

$50^{\;\text{th }\;}$ root of a number $x$ is 12 and $50^{\;\text{th }\;}$ root of another number ${\text{y}}$ is 18 . Then the remainder obtained on dividing $(x+y)$ by 25 is _______. [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{x}}+ {\text{y}}=12^{50}+18^{50}=(150-6)^{25}+(325-1)^{25}$ $\;=25 {\text{K}}- (6^{25}+1) =25 {\text{K}}- ((5+1)^{25}+1)$ $\;=25 {\text{K}}_1-2 \;\; \;\text{ Remainder }\;=23$

Q82JEE Main 2023NAT4MFunctions

Let ${\text{A}}=\{1,2,3,5,8,9\}$ . Then the number of possible functions ${\text{f}}: {\text{A}} \to {\text{A}}$ such that ${\text{f}}( {\text{m}} \cdot {\text{n}})= {\text{f}}( {\text{m}}) \cdot {\text{f}}( {\text{n}})$ for every ${\text{m}}, {\text{n}} \in {\text{A}}$ with ${\text{m}} \cdot {\text{n}} \in {\text{A}}$ is equal to _______. [30-Jan-2023 Shift 2]

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📖 Explanation

${\text{f}}(1)=1 ; {\text{f}}(9)= {\text{f}}(3) \times {\text{f}}(3)$ i.e., $f(3)=1$ or 3Total function $=1 \times 6 \times 2 \times 6 \times 6 \times 1=432$

Q83JEE Main 2023NAT4MCircles

Let ${\text{P}} ( {\text{a}}_1, {\text{b}}_1)$ and ${\text{Q}} ( {\text{a}}_2, {\text{b}}_2)$ be two distinct points on a circle with center ${\text{C}}({\sqrt{2}}, {\sqrt{3}})$ . Let ${\text{O}}$ be the origin and ${\text{OC}}$ be perpendicular to both ${\text{CP}}$ and CQ. If the area of the triangle OCP is $\;{{\sqrt{35}}}/{2}$ , then ${\text{a}}_1^{2+} {\text{a}}_2^{2+} {\text{b}}_1^{2+} {\text{b}}_2^2$ is equal to _______. [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\frac{1}{2} \times {\text{PC}} \times {\sqrt{5}}=\;{{\sqrt{35}}}/{2} ; {\text{PC}}={\sqrt{7}}$ $\; {\text{a}}_1^{2+} {\text{b}}_1^{2+} {\text{a}}_2^{2+} {\text{b}}_2^2= {\text{OP}}^{2+} {\text{OQ}}^2$ $\;=2(5+7)=24$

Explanation Figure 1

Q84JEE Main 2023NAT4MSequences and Series

The $8^{\;\text{th }\;}$ common term of the series $\; {\text{S}}_1=3+7+11+15+19+. . . .$ $\; {\text{S}}_2=1+6+11+16+21+. . .$ is _______. [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{T}}_8=11+(8-1) \times 20$ $\;=11+140=151$

Q85JEE Main 2023NAT4MThree Dimensional Geometry

Let a line ${\text{L}}$ pass through the point ${\text{P}}(2,3,1)$ and be parallel to the line $x+3 y-2 z-2=0=x-y+2 z$ . If the distance of ${\text{L}}$ from the point $(5,3,8)$ is $\alpha$ , then $3 \alpha^2$ is equal to _______. [30-Jan-2023 Shift 2]

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📖 Explanation

${\begin{vmatrix}\\ \\ { {\text{i}}}^{\^} & \\ \\ { {\text{j}}}^{\^} & \\ \\ { {\text{k}}}^{\^} \\ 1 & 3 & -2 \\ 1 & -1 & 2\end{vmatrix}}=4 \;{ {\text{i}}}^{\^}-4 \;{ {\text{j}}}^{\^}-4 \;{ {\text{k}}}^{\^}$ $\therefore$ Equation of line is $\;{ {\text{x}}-2}/{1}=\;{ {\text{y}}-3}/{-1}=\;{ {\text{z}}-1}/{-1}$ Let Q be $(5,3,8)$ and foot of $⟂$ from Q on this line be ${\text{R}}$ .Now, ${\text{R}} \equiv ( {\text{k}}+2,- {\text{k}}+3,- {\text{k}}+1)$ DR of ${\text{QR}}$ are $( {\text{k}}-3,- {\text{k}},- {\text{k}}-7)$ $\; \therefore (1)( {\text{k}}-3)+(-1)(- {\text{k}})+(-1)(- {\text{k}}-7)=0$ $\;\Rightarrow {\text{k}}=-\;\frac{4}{3}$ $\; \therefore \alpha^2= (\;\frac{13}{3}) ^{2+} (\;\frac{4}{3}) ^{2+} (\;\frac{17}{3}) ^2=\;\frac{474}{9}$ $\; \therefore 3 \alpha^2=158$

Q86JEE Main 2023NAT4MIndefinite Integrals

If $\int {\sqrt{\ sec 2 x-1}} \text{dx}=\alpha \log _e |\cos 2 x+\beta+\sqrt{\cos 2 x (1+\cos \;\frac{1}{\beta} x) }|$ $+$ constant, then $\beta-\alpha$ is equal to _______. [30-Jan-2023 Shift 2]

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📖 Explanation

$\;\int {\sqrt{\ sec 2 x-1}} \text{dx}=\int \sqrt{\;\frac{1-\cos 2 x}{\cos 2 x}} \text{dx}$ $\;={\sqrt{2}} \int \;{ sin \;x}/{{\sqrt{2 \cos ^2 x-1}}} \text{dx}$ $\;\;\text{ put }\; \cos x=t \Rightarrow - sin \;x \text{dx}=d t$ $\;=-{\sqrt{2}} \int \;{ {\text{dt}}}/{{\sqrt{2 t^{2-}1}}}$ $\;=-\ln |{\sqrt{2}} \cos x+{\sqrt{\cos 2 x}}|+c$ $\;=-\;\frac{1}{2} \ln |2 \cos ^2 x+\cos 2 x+2 {\sqrt{\cos 2 x}} \cdot {\sqrt{2}} \cos x|+c$ $\;=-\;\frac{1}{2} \ln |\cos ^{2 x}+\;\frac{1}{2}+{\sqrt{\cos 2 x}} \cdot {\sqrt{1+\cos 2 x}}|+c$ $\; \because \beta=\;\frac{1}{2}, \alpha=-\;\frac{1}{2} \Rightarrow \beta-\alpha=1$

Q87JEE Main 2023NAT4MQuadratic Equation and Inequalities

If the value of real number $a > 0$ for which $x^{2-}5 a x$ $+1=0$ and ${\text{x}}^{2-} {\text{ax}}-5=0$ have a common real roots is $\;{3}/{{\sqrt{2 \beta}}}$ then $\beta$ is equal to ________. [30-Jan-2023 Shift 2]

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📖 Explanation

Two equations have common root $\; \therefore (4 a)(26 a)=(-6)^2=36$ $\;\Rightarrow a^2=\;\frac{9}{26} \;\; \therefore a=\;{3}/{{\sqrt{26}}} \Rightarrow \beta=13$

Q88JEE Main 2023NAT4MPermutations and Combinations

The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2,3,3,5 is _______. [30-Jan-2023 Shift 2]

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📖 Explanation

Digits are $1,2,2,2,3,3,5$ If unit digit 5 , then total numbers $=\;\frac{6 !}{3 ! 2 !}$ If unit digit 3 , then total numbers $=\;\frac{6 !}{3 !}$ If unit digit 1 , then total numbers $=\;\frac{6 !}{3 ! 2 !}$ $\therefore$ total numbers $=60+60+120=240$

Q89JEE Main 2023NAT4MProbability

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $q$ . If ${\text{p}}: {\text{q}}= {\text{m}}$ $: {\text{n}}$ , where ${\text{m}}$ and ${\text{n}}$ are coprime, then ${\text{m}}+ {\text{n}}$ is equal to _______. [30-Jan-2023 Shift 2]

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📖 Explanation

$\; {\text{p}}=\;{{ }^6 {\text{C}}_1}/{6 \times 6}=\;\frac{1}{6}$ $\; {\text{q}}=\;{{ }^6 {\text{C}}_1 \times { }^5 {\text{C}}_1 \times 4}/{6 \times 6 \times 6 \times 6}=\;\frac{5}{54}$ $\; \therefore {\text{p}}: {\text{q}}=9: 5 \Rightarrow {\text{m}}+ {\text{n}}=14$

Q90JEE Main 2023NAT4MArea Under The Curves

Let ${\text{A}}$ be the area of the region $\{(x, y): y \ge x^2, y \ge (1-x)^2, y \le 2 x(1-x)\} \;\text{. }\;$ Then $540 {\text{A}}$ is equal to [30-Jan-2023 Shift 2]

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📖 Explanation

$A=2 int_{{\;\frac{1}{3}}}^{\;\frac{1}{2}} (2 x-2 x^{2-}(1-x)^2) \text{dx}$ $\;=2 \$ [2 {\text{x}}^{2-} {\text{x}}^{3-} {\text{x}}]$ _{1 / 3}^{1 / 2}; \therefore {\text{A}}=;\frac{5}{108} \Rightarrow 540 {\text{A}}=;\frac{5}{108} \times 540=25$

Explanation Figure 1

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