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JEE Main 2021 (17 Mar Shift 1)

2021 · 90 questions · First 10 Solutions Free
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Q1 mcq
If an electron is moving in the $n^{th\,}$ orbit of the hydrogen atom, then its velocity $(v_{n})$ for the $n^{th\,}$ orbit is given as:
a.
$v_{n} \propto n$
b.
$v_{n} \propto \frac{1}{n}$ Correct Answer
c.
$v_{n} \propto n^{2}$
d.
$v_{n} \propto \frac{1}{n^{2}}$

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Q2 mcq
Two ideal polyatomic gases at temperatures $T_{1}$ and $T_{2}$ are mixed so that there is no loss of energy. If $F_{1}$ and $F_{2}, \,m_{1}$ and $m_{2}, \,n_{1}$ and $n_{2}$ be the degrees of freedom, masses, number of molecules of the first and second gas respectively, the temperature of mixture of these two gases is:
a.
$\frac{n_{1}T_{1}+n_{2}T_{2}}{n_{1}+n_{2}}$
b.
$\frac{n_{1}F_{1}T_{1}+n_{2}F_{2}T_{2}}{n_{1}F_{1}+n_{2}F_{2}}$ Correct Answer
c.
$\frac{n_{1}F_{1}T_{1}+n_{2}F_{2}T_{2}}{F_{1}+F_{2}}$
d.
$\frac{n_{1}F_{1}T_{1}+n_{2}F_{2}T_{2}}{n_{1}+n_{2}}$

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Q3 nat
A parallel plate capacitor whose capacitance $C$ is $14\, pF$ is charged by a battery to a potential difference $V = 12\,V$ between its plates. The charging battery is now disconnected and a porcelain plate with $k= 7$ is inserted between the plates, then the plate would oscillate back and forth between the plates with a constant mechanical energy of _______ $pJ.$ (Assume no friction)
✓ Accepted Answer: 864

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Q4 mcq
A triangular plate is shown. A force $\vec{F} = 4ˆi -3ˆj$ is applied at point $P.$ The torque at point $P$ with respect to point $O$ and $Q$ are:
Question 4 image
a.
$-15 -20, 15 -20\sqrt{3} \sqrt{3}$ Correct Answer
b.
$15 + 20, 15 -20\sqrt{3} \sqrt{3}$
c.
$15 -20, 15 + 20\sqrt{3} \sqrt{3}$
d.
$-15 + 20, 15 + 20\sqrt{3} \sqrt{3}$

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Q5 mcq
The thickness at the centre of a plano convex lens is $3\, mm$ and the diameter is $6\, cm.$ If the speed of light in the material of the lens is $2 \times 10^{8}\,m\,s^{-1}.$ The focal length of the lens is ___________.
a.
$0. 30\, cm$
b.
$15\, cm$
c.
$1. 5\, cm$
d.
$30\, cm$ Correct Answer

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Q6 mcq
The vernier scale used for measurement has a positive zero error of $0. 2\, mm.$ If while taking a measurement it was noted that $0$ on the vernier scale lies between $8. 5\, cm$ and $8. 6\, cm$ . Vernier coincidence is $6,$ then the correct value of measurement is $cm.$ (least count $= 0. 01\, cm$ )
a.
$8. 36\, cm$
b.
$8. 54\, cm$ Correct Answer
c.
$8. 58\, cm$
d.
$8. 56\, cm$

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Q7 mcq
An $AC$ current is given by $I =I_{1}sin\omega t+I_{2}cos\omega t.$ A hot wire ammeter will give a reading:
a.
$\sqrt{\,} \frac{I^{2}_{1}-I^{2}_{2}}{2}$
b.
$\sqrt{\,} \frac{I^{2}_{1}+I^{2}_{2}}{2}$ Correct Answer
c.
$\frac{I_{1}+I_{2}}{\sqrt{2}}$
d.
$\frac{I_{1}+I_{2}}{2\sqrt{2}}$

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Q8 nat
If $2. 5 \times 10^{-6}\,N$ average force is exerted by a light wave on a non-reflecting surface of $30\, cm^{2}$ area during $40\, min$ of time span, the energy flux of light just before it falls on the surface is _______ $W\, cm^{-2}.$ (Round off to the Nearest Integer), (Assume complete absorption and normal incidence conditions are there)
✓ Accepted Answer: 25

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Q9 nat
The radius in kilometer to which the present radius of earth $(R= 6400\, km)$ to be compressed so that the escape velocity is increased $10$ time is _______.
✓ Accepted Answer: 64

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Q10 nat
The angular speed of truck wheel is increased from $900\, rpm$ to $2460\, rpm$ in $26$ seconds. The number of revolutions by the truck engine during this time is ______. (Assuming the acceleration to be uniform).
✓ Accepted Answer: 728

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Q11 mcq
Which level of the single ionized carbon has the same energy as the ground state energy of hydrogen atom?
a.
$1$
b.
$6$
c.
$4$
d.
$8$
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Q12 mcq
A current of $10\,A$ exists in a wire of cross-sectional area of $5\, mm^{2}$ with a drift velocity of $2 \times 10^{-3}\,m\,s^{-1}.$ The number of free electrons in each cubic meter of the wire is
a.
$2 \times 10^{6}$
b.
$625 \times 10^{25}$
c.
$2 \times 10^{25}$
d.
$1 \times 10^{23}$
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Q13 nat
Two blocks ( $m= 0. 5\, kg$ and $M = 4. 5\, kg$ ) are arranged on a horizontal frictionless table as shown in the figure. The coefficient of static friction between the two blocks is $\frac{3}{7}.$ Then the maximum horizontal force that can be applied on the larger block so that the blocks move together is $N.$ (Round off to the Nearest Integer) [Take $g$ as $9. 8\,m\,s^{-2}$ ]
Question 13 image
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Q14 mcq
Two identical metal wires of thermal conductivities $K_{1}$ and $K_{2}$ respectively are connected in series. The effective thermal conductivity of the combination is:
a.
$\frac{2\,K_{1}K_{2}}{K_{1}+K_{2}}$
b.
$\frac{K_{1}+K_{2}}{2\,K_{1}K_{2}}$
c.
$\frac{K_{1}+K_{2}}{K_{1}K_{2}}$
d.
$\frac{K_{1}K_{2}}{K_{1}+K_{2}}$
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Q15 mcq
A polyatomic ideal gas has $24$ vibrational modes. What is the value of $\gamma ?$
a.
$1. 03$
b.
$1. 30$
c.
$1. 37$
d.
$10. 3$
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Q16 mcq
An electron of mass $m$ and a photon have same energy $E.$ The ratio of wavelength of electron to that of photon is : ( $c$ being the velocity of light)
a.
$\frac{1}{c}(\frac{2m}{E} ) \frac{1}{2}$
b.
$\frac{1}{c}( \frac{E}{2m})\frac{1}{2}$
c.
$(\frac{E}{2\,m})\frac{1}{2}$
d.
$c(2mE) \frac{1}{2}$
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Q17 mcq
A boy is rolling a $0. 5\, kg$ ball on the frictionless floor with the speed of $20\,m\,s^{-1}.$ The ball gets deflected by an obstacle on the way. After deflection it moves with $5%$ of its initial kinetic energy. What is the speed of the ball now?
a.
$19. 0\,m\,s^{-1}$
b.
$4. 4\,m\,s^{-1}$
c.
$14. 41\,m\,s^{-1}$
d.
$1. 00\,m\,s^{-1}$
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Q18 nat
The equivalent resistance of series combination of two resistors is $s.$ When they are connected in parallel, the equivalent resistance is $p.$ If $s=np,$ then the minimum value for $n$ is ______. (Round off to the Nearest Integer)
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Q19 mcq
The output of the given combination gates represents:
Question 19 image
a.
XOR Gate
b.
NAND Gate
c.
AND Gate
d.
NOR Gate
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Q20 mcq
When two soap bubbles of radii $a$ and $b(b>a)$ coalesce, the radius of curvature of common surface is:
a.
$\frac{ab}{b-a}$
b.
$\frac{a+b}{ab}$
c.
$\frac{b-a}{ab}$
d.
$\frac{ab}{a+b}$
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Q21 mcq
A modern grand-prix racing car of mass $m$ is travelling on a flat track in a circular arc of radius $R$ with a speed $v.$ If the coefficient of static friction between the tyres and the track is $\mu _{s},$ then the magnitude of negative lift $F_{L}$ acting downwards on the car is:
Question 21 image
a.
$m(\frac{v^{2}}{\mu _{s}R} +g)$
b.
$m(\frac{v^{2}}{\mu _{s}R} - g)$
c.
$m(g- \frac{v^{2}}{\mu _{s}R})$
d.
$-m(g+ \frac{v^{2}}{\mu _{s}R})$
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Q22 mcq
For what value of displacement the kinetic energy and potential energy of a simple harmonic oscillation become equal?
a.
$x= 0$
b.
$x= \pm A$
c.
$x= \pm \frac{A}{\sqrt{2}}$
d.
$x= \frac{A}{2}$
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Q23 nat
For VHF signal broadcasting, _______ $km^{2}$ of maximum service area will be covered by an antenna tower of height $30\,m,$ if the receiving antenna is placed at ground. Let radius of the earth be $6400\, km.$ (Round off to the Nearest Integer) (Take $\pi$ as $3. 14$ )
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Q24 mcq
A solenoid of $1000$ turns per metre has a core with relative permeability $500.$ Insulated windings of the solenoid carry an electric current of $5\,A.$ The magnetic flux density produced by the solenoid is: (Permeability of free space $= 4\pi \times 10^{-7}\,H\,m^{-1}$ )
a.
$(\pi )\,T$
b.
$(2 \times 10^{-3}\pi )\,T$
c.
$(\frac{\pi}{5})\,T$
d.
$(10^{-4}\pi )\,T$
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Q25 nat
The following bodies, $(1)$ a ring $(2)$ a disc $(3)$ a solid cylinder $(4)$ a solid sphere, of same mass $m$ and radius $R$ are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is [Mark the body as per their respective numbering given in the question]
Question 25 image
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Q26 mcq
A Carnot's engine working between $400\,K$ and $800\,K$ has a work output of $1200\,J$ per cycle. The amount of heat energy supplied to the engine from the source in each cycle is:
a.
$3200\,J$
b.
$1800\,J$
c.
$1600\,J$
d.
$2400\,J$
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Q27 nat
Consider two identical springs each of spring constant $k$ and negligible mass compared to the mass $M$ as shown. Fig. $1$ shows one of them and Fig. $2$ shows their series combination. The ratios of time period of oscillation of the two SHM is $\frac{T_{b}}{T_{a}} = \sqrt{x},$ where value of $x$ is ______. (Round off to the Nearest Integer)
Question 27 image
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Q28 mcq
A mass $M$ hangs on a massless rod of length $l$ which rotates at a constant angular frequency. The mass $M$ moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $\omega .$ The angular momentum of $M$ about point $A$ is $L_{A}$ which lies in the positive $z$ direction and the angular momentum of $M$ about $B$ is $L_{B}.$ The correct statement for this system is:
Question 28 image
a.
$L_{A}$ and $L_{B}$ are both constant in magnitude and direction
b.
$L_{B}$ is constant in direction with varying magnitude
c.
$L_{B}$ is constant, both in magnitude and direction
d.
$L_{A}$ is constant, both in magnitude and direction
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Q29 mcq
A car accelerates from rest at a constant rate $\alpha$ for some time after which it decelerates at a constant rate $\beta$ to come to rest. If the total time elapsed is t seconds, the total distance travelled is:
a.
$\frac{4\alpha \beta}{(\alpha +\beta )}t^{2}$
b.
$\frac{2\alpha \beta}{(\alpha +\beta )}t^{2}$
c.
$\frac{\alpha \beta}{2(\alpha +\beta )}t^{2}$
d.
$\frac{\alpha \beta}{4(\alpha +\beta )} t^{2}$
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Q30 nat
Four identical rectangular plates with length, $l= 2\, cm$ and breadth, $b= \frac{3}{2}\, cm$ are arranged as shown in figure. The equivalent capacitance between $A$ and $C$ is $\frac{x\epsilon _{0}}{d}.$ The value of $x$ is _______. (Round off to the Nearest Integer)
Question 30 image
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Q31 nat
For a certain first order reaction $32%$ of the reactant is left after $570\,s.$ The rate constant of this reaction is $\times 10^{-3}\,s^{-1}.$ (Round off to the Nearest Integer). [Given: $log_{10}\,2 = 0. 301, \, ln 10 = 2. 303$ ]
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Q32 nat
The standard enthalpies of formation of $Al_{2}O_{3}$ and $CaO$ are $-1675\, kJ \, mol^{-1}$ and $-635\, kJ \, mol^{-1}$ respectively. For the reaction $3CaO + 2Al \rightarrow 3Ca + Al\rightarrow_{2}O_{3}$ the standard reaction enthalpy $\Delta _{r}H^{0} =$ ________ $kJ.$ (Round off to the Nearest Integer).
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Q33 mcq
With respect to drug-enzyme interaction, identify the wrong statement:
a.
Non-Competitive inhibitor binds to the allosteric site
b.
Allosteric inhibitor changes the enzyme's active site
c.
Allosteric inhibitor competes with the enzyme's active site
d.
Competitive inhibitor binds to the enzyme's active site
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Q34 mcq
Mesityl oxide is a common name of:
a.
$2, \,4$ -Dimethylpentan- $3$ -one
b.
$3$ -Methylcyclohexanecarbaldehyde
c.
$2$ -Methylcyclohexanone
d.
$4$ -Methylpent- $3$ -en- $2$ -one
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Q35 mcq
The INCORRECT statement(s) about heavy water is (are) (A) used as a moderator in nuclear reactor (B) obtained as a by-product in fertilizer industry. (C) used for the study of reaction mechanism (D) has a higher dielectric constant than water Choose the correct answer from the options given below:
a.
(B) only
b.
(C) only
c.
(D) only
d.
(B) and (D) only
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Q36 nat
$0. 01$ moles of a weak acid $HA \,(K_{a} = 2. 0 \times 10^{-6})$ is dissolved in $1. 0\,L$ of $0. 1M\, HCl$ solution. The degree of dissociation of $HA$ is __________ $\times 10^{-5}$ (Round off to the Nearest Integer). [Neglect volume change on adding $HA$ and assume degree of dissociation $<< 1$ ]
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Q37 mcq
Which of the following compound CANNOT act as a Lewis base?
a.
$NF_{3}$
b.
$PCl_{5}$
c.
$SF_{4}$
d.
$ClF_{3}$
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Q38 mcq
Given below are two statements: Statement-I: Retardation factor $(R_{f})$ can be measured in meter/centimetre. Statement-II: $R_{f}$ value of a compound remains constant in all solvents. Choose the most appropriate answer from the options given below:
a.
Statement-I is true but statement-II is false
b.
Both statement-I and statement-II are true
c.
Both statement-I and statement-II are false
d.
Statement-I is false but statement-II is true
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Q39 nat
In the above reaction, $3. 9\,g$ of benzene on nitration gives $4. 92\,g$ of nitrobenzene. The percentage yield of nitrobenzene in the above reaction is _________ $%$ . (Round off to the Nearest Integer). (Given atomic mass : $C\, : 12. 0\,u, \,H\, : 1. 0\,u\,O\, : 16. 0\,u, N\, : 14. 0\,u$ )
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Q40 nat
A certain orbital has $n = 4$ and $m_{l} = -3.$ The number of radial nodes in this orbital is _______. (Round off to the Nearest Integer).
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Q41 mcq
The above reaction requires which of the following reaction conditions?
a.
$573\,K, \, Cu, \,300\, atm$
b.
$623\,K, \, Cu, \,300\, atm$
c.
$573\,K, \,300\, atm$
d.
$623\,K, \,300\, atm$
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Q42 mcq
Question 42 image
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Q43 nat
The mole fraction of a solute in a $100$ molal aqueous solution ________ $\times 10^{-2}.$ (Round off to the Nearest Integer). [Given : Atomic masses: $H\, : 1. 0\,u, \,O\, : 16. 0\,u$ ]
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Q44 mcq
The absolute value of the electron gain enthalpy of halogens satisfies:
a.
$I > Br > Cl > F$
b.
$Cl > Br > F > I$
c.
$Cl > F > Br > I$
d.
$F > Cl > Br > I$
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Q45 nat
The reaction of white phosphorus on boiling with alkali in inert atmosphere resulted in the formation of product $A.$ The reaction $1\, mol$ of $A$ with excess of $AgNO_{3}$ in aqueous medium gives ________ mole(s) of $Ag.$ (Round off to the Nearest Integer).
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Q46 mcq
What is the spin-only magnetic moment value (B.M.) of a divalent metal ion with atomic number $25,$ in its aqueous solution?
a.
$5. 92$
b.
$5. 0$
c.
zero
d.
$5. 26$
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Q47 mcq
Which of the following reaction is an example of ammonolysis?
a.
$C_{6}H_{5}COCl + C_{6}H_{5}NH_{2} \longrightarrow C_{6}H_{5}CONHC_{6}H$
b.
$^{5} C_{6}H_{5}CH_{2}CN \rightarrow C\rightarrow^{[H]}_{6}H_{5}CH_{2}CH_{2}NH_{2}$
c.
$C_{6}H_{5}NH_{2}^{HCl}\rightarrow \rightarrow C_{6}H_{5}NH_{3}Cl^{-}$
d.
$C H CH Cl + NH \longrightarrow C H CH NH$ $-^{6}^{5}^{2}^{3}^{6}^{5}^{2}^{2}$
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Q48 mcq
A central atom in a molecule has two lone pairs of electrons and forms three single bonds. The shape of this molecule is
a.
see-saw
b.
planar triangular
c.
T-shaped
d.
trigonal pyramidal
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Q49 nat
$15\, mL$ of aqueous solution of $Fe^{2+}$ in acidic medium completely reacted with $20\, mL$ of $0. 03\,M$ aqueous $Cr_{2}O_{7}^{2-}.$ The molarity of the $Fe^{2+}$ solution is _______ $\times 10^{-2}M$ (Round off to the Nearest Integer).
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Q50 mcq
The product "A" in the above reaction is:
Question 50 image
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Q51 mcq
Given below are two statements: Statement I: Potassium permanganate on heating at $573\,K$ forms potassium manganate. Statement II: Both potassium permanganate and potassium manganate are tetrahedral and paramagnetic in nature. In the light of the above statements, choose the most appropriate answer from the options given below:
a.
Statement I is true but statement II is false
b.
Both statement I and statement II are true
c.
Statement I is false but statement II is true
d.
Both statement I and statement II are false
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Q52 mcq
A colloidal system consisting of a gas dispersed in a solid is called a/an:
a.
solid sol
b.
gel
c.
aerosol
d.
foam
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Q53 mcq
Hoffmann bromamide degradation of benzamide gives product $A,$ which upon heating with $CHCl_{3}$ and $NaOH$ gives product $B.$ The structures of $A$ and $B$ are:
Question 53 image
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Q54 nat
The pressure exerted by a non-reactive gaseous mixture of $6. 4\,g$ of methane and $8. 8\,g$ of carbon dioxide in a $10\,L$ vessel at $27^{\circ}C$ is _____ $kPa.$ (Round off to the Nearest Integer). [Assume gases are ideal, $R = 8. 314\,J\, mol^{-1}\,K^{-1}$ Atomic masses: $C\, : 12. 0\,u, \,H\, : 1. 0\,u, \,\,O\, : 16. 0\,u$ ]
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Q55 mcq
The point of intersection and sudden increase in the slope, in the diagram given below, respectively, indicates:
Question 55 image
a.
$\Delta G = 0$ and melting or boiling point of the metal oxide
b.
$\Delta G > 0$ and decomposition of the metal oxide
c.
$\Delta G < 0$ and decomposition of the metal oxide
d.
$\Delta G = 0$ and reduction of the metal oxide
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Q56 mcq
Which of the following is an aromatic compound?
Question 56 image
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Q57 mcq
The correct order of conductivity of ions in water is:
a.
$Na^{+} > K^{+} > Rb^{+} > Cs^{+}$
b.
$Cs^{+} > Rb^{+} > K^{+} > Na^{+}$
c.
$K^{+} > Na^{+} > Cs^{+} > Rb^{+}$
d.
$Rb^{+} > Na^{+} > K^{+} > Li^{+}$
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Q58 nat
The oxygen dissolved in water exerts a partial pressure of $20\, kPa$ in the vapour above water. The molar solubility of oxygen in water is ______ $\times 10^{-5}\, mol \, dm^{-3}.$ (Round off to the Nearest Integer). [Given : Henry's law constant $= K_{H} = 8. 0 \times 10^{4}\, kPa$ for $O_{2}.$ Density of water with dissolved oxygen $= 1. 0\, kg \, dm^{-3}$ ]
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Q59 mcq
Reducing smog is a mixture of:
a.
Smoke, fog and $O_{3}$
b.
Smoke, fog and $SO_{2}$
c.
Smoke, fog and $CH_{2} = CH -CHO$
d.
Smoke, fog and $N_{2}O_{3}$
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Q60 mcq
Which of the following is correct structure of tyrosine?
Question 60 image
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Q61 nat
If the function $f(x) = \frac{cos(sinx)-cosx}{x^{4}}$ is continuous at each point in its domain and $f(0) = \frac{1}{k},$ then $k$ is _________.
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Q62 nat
$2 3$ If $A=[ ],$ then the value of $det(A^{4})+ det(A^{10} - (Adj(2\,A))^{10})$ is equal to ________. $0 -1$
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Q63 mcq
Let $\vec{a}= 2ˆi-3ˆj+ 4ˆk$ and $\vec{b}= 7ˆi+ ˆj-6ˆk$ If $\vec{r}\times \vec{a}= \vec{r}\times \vec{b}, \vec{r}\cdot (ˆi+ 2ˆj+ ˆk)= -3,$ then $\vec{r}\cdot (2ˆi-3ˆj+ˆk)$ is equal to:
a.
$12$
b.
$8$
c.
$13$
d.
$10$
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Q64 nat
If $(2021)^{3762}$ is divided by $17,$ then the remainder is _______.
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Q65 mcq
The area of the triangle with vertices $P(z), \,Q(iz)$ and $R(z+iz)$ is
a.
$1$
b.
$\frac{1}{2} z^{2}$
c.
$\frac{1}{2}$
d.
$\frac{1}{2} z+iz^{2}$
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Q66 mcq
The value of $_{x}lim_{\rightarrow 0\rightarrow }_{+} \frac{cos^{-1}(x-[x]^{2})\cdot sin^{-1}(x-[x]^{2})}{x-x^{3}},$ where $[x]$ denotes the greatest integer $\leq x$ is:
a.
$\pi$
b.
$0$
c.
$\frac{\pi}{4}$
d.
$\frac{\pi}{2}$
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Q67 nat
If $\vec{a}=\alpha ˆi+\beta ˆj+ 3ˆk, \,\vec{b}= -\beta ˆi- \alpha ˆj- ˆk$ and $\vec{c}= ˆi-2ˆj- ˆk$ such that $\vec{a}\cdot \vec{b}= 1$ and $\vec{b}\cdot \vec{c}= -3,$ then $\frac{1}{3}((\vec{a}\times \vec{b})\cdot \vec{c})$ is equal to _______.
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Q68 mcq
If the fourth term in the expansion of $(x+x^{log_{2}x})^{7}$ is $4480,$ then the value of $x$ where $x\in N$ is equal to:
a.
$2$
b.
$4$
c.
$3$
d.
$1$
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Q69 mcq
$0 sin\alpha$ If $A=[ ]$ and $det(A^{2} - \frac{1}{2} I)= 0,$ then a possible value of $\alpha$ is $sin\alpha 0$
a.
$\frac{\pi}{2}$
b.
$\frac{\pi}{3}$
c.
$\frac{\pi}{4}$
d.
$\frac{\pi}{6}$
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Q70 mcq
In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?
Question 70 image
a.
P and Q
b.
P and R
c.
Q and R
d.
None of these
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Q71 nat
The maximum value of $z$ in the following equation $z= 6xy+y^{2},$ where $3x+ 4y\leq 100$ and $4x+ 3y\leq 75$ for $x\geq 0$ and $y\geq 0$ is
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Q72 mcq
The value of $4 + \frac{1}{5+^{1}_{1}}$ is: $4+5+\frac{1}{4+……\infty}$
a.
$2 + \frac{2}{5}\sqrt{30}$
b.
$2 + \frac{4}{\sqrt{5}}\sqrt{30}$
c.
$4 + \frac{4}{\sqrt{5}}\sqrt{30}$
d.
$5 + \frac{2}{5}\sqrt{30}$
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Q73 mcq
Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{dy}{dx} =xy-1 +x- y; \,y(0) = 0$
a.
$y(1) = e^{-}\frac{1}{2} -1$
b.
$y(1) =e\frac{1}{2} - e^{-}\frac{1}{2}$
c.
$y(1) = 1$
d.
$y(1) =e\frac{1}{2} -1$
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Q74 nat
The minimum distance between any two points $P_{1}$ and $P_{2}$ while considering point $P_{1}$ on one circle and point $P_{2}$ on the other circle for the given circles' equations $x^{2} +y^{2} -10x-10y+ 41 = 0$ $x^{2} +y^{2} -24x-10y+ 160 = 0$ is ________
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Q75 mcq
The line $2x- y+ 1 = 0$ is a tangent to the circle at the point $(2, \,5)$ and the centre of the circle lies on $x-2y= 4.$ Then, the radius of the circle is:
a.
$3\sqrt{5}$
b.
$5\sqrt{3}$
c.
$5\sqrt{4}$
d.
$4\sqrt{5}$
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Q76 mcq
The system of equations $kx+y+z= 1, \,x+ky+z=k$ and $x+y+zk=k^{2}$ has no solution if $k$ is equal to:
a.
$0$
b.
$1$
c.
$-1$
d.
$-2$
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Q77 mcq
The equation of the plane which contains the $y$ -axis and passes through the point $(1, \,2, \,3)$ is:
a.
$x+ 3z= 10$
b.
$x+ 3z= 0$
c.
$3x+z= 6$
d.
$3x- z= 0$
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Q78 mcq
$|$ $||$ In a triangle $PQR,$ the co-ordinates of the points $P$ and $Q$ are $(-2, 4)$ and $(4, -2)$ respectively. If the equation of the perpendicular bisector of $PR$ is $2x- y+ 2 = 0,$ then the centre of the circumcircle of the $\Delta PQR$ is:
a.
$(-1, \,0)$
b.
$(-2, \,-2)$
c.
$(0, \,2)$
d.
$(1, \,4)$
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Q79 nat
$||$ $||$ Let there be three independent events $E_{1},E_{2}$ and $E_{3}.$ The probability that only $E_{1}$ occurs is $\alpha$ only $E_{2}$ occurs is $\beta$ and only $E_{3}$ occurs is $\gamma .$ Let $'p'$ denote the probability of none of events occurs that satisfies the equations $(\alpha -2\beta )p =\alpha \beta$ and $(\beta -3\gamma )p = 2\beta \gamma .$ All the given probabilities are assumed to lie in the interval $(0, \,1).$ Then, $\frac{\,Probability ofoccurrence of\,E_{1}}{\,Probability ofoccurrence of\,E_{3}}$ is equal to ________.
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Q80 mcq
Which of the following statement is correct for the function $g(\alpha )$ for $\alpha \in R$ such that $g(\alpha ) =\int _{6}\frac{\pi}{\pi 3} \frac{sin^{\alpha }x}{cos^{\alpha }x+sin^{\alpha }x}dx$
a.
$g(\alpha )$ is a strictly increasing function
b.
$g(\alpha )$ has an inflection point at $\alpha = -\frac{1}{2}$
c.
$g(\alpha )$ is a strictly decreasing function
d.
$g(\alpha )$ is an even function
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Q81 mcq
Two dices are rolled. If both dices have six faces numbered $1, \,2, \,3, \,5, \,7$ and $11,$ then the probability that the sum of the numbers on the top faces is less than or equal to $8$ is:
a.
$\frac{4}{9}$
b.
$\frac{17}{36}$
c.
$\frac{5}{12}$
d.
$\frac{1}{2}$
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Q82 mcq
If $cot^{-1}(\alpha )= cot^{-1} 2 + cot^{-1} 8 + cot^{-1} 18 + cot^{-1} 32 + … .$ upto $100$ terms, then $\alpha$ is:
a.
$1. 01$
b.
$1. 00$
c.
$1. 02$
d.
$1. 03$
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Q83 mcq
The inverse of $y= 5^{logx}$ is:
a.
$x= 5^{logy}$
b.
$x=y^{log 5}$
c.
$y=x\frac{1}{log 5}$
d.
$x= 5 \frac{1}{logy}$
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Q84 nat
If $[\cdot ]$ represents the greatest integer function, then the value of $\int _{0}^{\sqrt{\,} }\frac{\pi}{2} [[x^{2}]-cosx]dx$ is ___________.
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Q85 nat
If the equation of the plane passing through the line of intersection of the planes $2x-7y+ 4z-3 = 0, \,3x-5y+ 4z+ 11 = 0$ and the point $(-2, \,1, \,3)$ is $ax+by+cz-7 = 0,$ then the value of $2a+b+c-7$ is _________.
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Q86 mcq
The sum of possible values of $x$ for $tan^{-1}(x+ 1) + cot^{-1}(\frac{1}{x-1} )= tan^{-1}(\frac{8}{31})$ is:
a.
$-\frac{32}{4}$
b.
$- \frac{31}{4}$
c.
$-\frac{30}{4}$
d.
$- \frac{33}{4}$
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Q87 nat
If $f(x) = sin(cos^{-1}(\frac{1-2^{2}^{x}}{1+2^{2}^{x}}))$ and its first derivative with respect to $x$ is $-\frac{b}{a}log_{e}2$ when $x= 1,$ where $a$ and $b$ are integers, then the minimum value of $a^{2} - b^{2}$ is _______.
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Q88 mcq
If the Boolean expression $(p\Rightarrow q) \Leftrightarrow (q*(~p))$ is a tautology, then the Boolean expression $p*(~q)$ is equivalent to:
a.
$q\Rightarrow p$
b.
$~q\Rightarrow p$
c.
$p\Rightarrow ~q$
d.
$p\Rightarrow q$
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Q89 mcq
Team $'A'$ consists of $7$ boys and $n$ girls and Team $'B'$ has $4$ boys and $6$ girls. If a total of $52$ single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $n$ is equal to:
a.
$5$
b.
$2$
c.
$4$
d.
$6$
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Q90 mcq
Choose the incorrect statement about the two circles whose equations are given below: $x^{2} +y^{2} -10x-10y+ 41 = 0$ and $x^{2} +y^{2} -16x-10y+ 80 = 0$
a.
Distance between two centres is the average of radii of both the circles.
b.
Both circles' centres lie inside region of one another.
c.
Both circles pass through the centre of each other.
d.
Circles have two intersection points.
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