JEE Main 2020 · 09 Jan 2020 · Shift 2

Q1.A small spherical droplet of density $d$ is floating exactly half immersed in a liquid of density $\rho$ and surface tension T . The radius of the droplet is (take note that the surface tension applies an upward force on the droplet):

Q2.There is a small source of light at some depth below the surface of water (refractive index $=\frac{4}{3}$ ) in a tank of large cross sectional surface area. Neglecting any reflection from the bottom and absorption by water, percentage of light that emerges out of surface is (n …

Q3.For the four sets of three measured physical quantifies as given below. Which of the following options is correct?
(i) ${\mathrm{A}}_1=24.36,{\mathrm{B}}_1=0.0724,{\mathrm{C}}_1=256.2$
(ii) ${\mathrm{A}}_2=24.44,{\mathrm{B}}_2=16.082,{\mathrm{C}}_2=240.2$
(iii) …

Q4.A plane electromagnetic wave is propagating along the direction $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$, with its polarization along the direction $\hat{\mathrm{k}}$. The correct form of the magnetic field of the wave would be (here ${\mathrm{B}}_0$ is an appropriate constant)

Q5.A wire of length L and mass per unit length $6.0\times 10^{-3}\mathrm{kg}{\mathrm{m}}^{-1}$ is put under tension of 540 N. Two consecutive frequencies that it resonates at are: 420 Hz and 490 Hz . Then L in meters is

Q6.A particle of mass $m$ is projected with a speed $u$ form the ground at an angle $\theta =\frac{\pi }{3}$ w.r.t. horizontal (x-axis). When it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity uî. The horiz …

Q7.In LC circuit the inductance $\mathrm{L}=40\mathrm{mH}$ and capacitance $\mathrm{C}=100\mu \mathrm{F}$. If a voltage $\mathrm{V}(\mathrm{t})=10\sin (314\mathrm{t})$ is applied to the circuit, the current in the circuit is given as

Q8.The current i in the network is:

Q9.A rod of length $L$ has non-uniform linear mass density given by $\rho (x)=a+b{\left(\frac{x}{L}\right)}^2$, where $a$ and $b$ are constants and $0\le x\le L$. The value of $x$ for the centre of mass of the rod is at:

Q10.Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volume is $1:4$, the ratio of their diameters is

Q11.Planet $A$ has mass $M$ and radius $R$. Plant $B$ has half the mass and half the radius of Planet $A$. If the escape velocities from the planets $A$ and $B$ are $V_A$ and $V_B$, respectively, then $\frac{V_A}{V_B}=\frac{n}{4}$. The value of ' $n$ ' i …

Q12.A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses $m_1$ and $m_2\left(m_1>m_2\right)$ are attached to the ends of the string. The syste …

Q13.The energy required to ionize a hydrogen like ion in its ground state is 9 Rydbergs. What is the wavelength of the radiation emitted when the electron in this ion jumps from the second excited state to the ground state?

Q14.An electron of mass $m$ and magnitude of charge $|e|$ initially at rest gets accelerated by a constant electric field E . The rate of change of de-Broglie wavelength of this electron at time $t$ ignoring relativistic effects is

Q15.A particle starts from the origin at $t=0$ with an initial velocity of $3.0\hat{i}\mathrm{m}/\mathrm{s}$ and moves in the $x$ - $y$ plane with a constant acceleration $(6.0\hat{i}+4.0\hat{j})\mathrm{m}/{\mathrm{s}}^2$. The $x$-coordinate of the particle at t …

Q16.A small circular loop of conducting wire has radius a and carries current I. It is placed in a uniform magnetic field B perpendicular to its plane such that when rotated slightly about its diameter and released, it starts performing simple harmonic motion of time period T . If th …

Q17.Two identical capacitors A and B, charged to the same potential 5 V are connected in two different circuits as shown below at time $t=0$. If the charge on capacitors $A$ and $B$ at time $t=CR$ is $Q_A$ and $Q_B$ respectively, then (Here e is the base of natural logarithm)

Q18.An electron gun is placed inside a long solenoid of radius $R$ on its axis. The solenoid has $n$ turns/length and carries a current I. The electron gun shoots an electron along the radius of the solenoid with speed v. If the electron does not hit the surface of the solenoid, maxi …

Q19.A spring mass system (mass m, spring constant k and natural length $\ell$) rests in equilibrium on a horizontal disc. The free end of the spring is fixed at the centre of the disc. If the disc together with spring mass system, rotates about its axis with an angular velocity …

Q20.In a meter bridge experiment S is a standard resistance. $R$ is a resistance wire. It is found that balancing length is $\ell =25$ cm. If $R$ is replaced by a wire of half length and half diameter that of R of same material, then the balancing distance ${\ell }^{\prime }$ (in cm) will now be …

Q21.Starting at temperature 300 K, one mole of an ideal diatomic gas ($\gamma =1.4$) is first compressed adiabatically from volume $V_1$ to $V_2=\frac{V_1}{16}$. It is then allowed to expand isobarically to volume $2V_2$. If all the processes are the quasi-static then the fina …

Q22.An electric field $\vec{E}=4x\hat{i}-(y^2+1)\hat{j}$ N/C passes through the box shown in figure. The flux of the electric field through surfaces ABCD and BCGF are marked as ${\varphi }_I$ and ${\varphi }_{II}$ respectively. The difference between $({\varphi }_I-{\varphi }_{II})$ is (in …

Q23.In a Young's double slit experiment 15 fringes are observed on a small portion of the screen when light of wavelength 500 nm is used. Ten fringes are observed on the same section of the screen when another light source of wavelength $\lambda$ is used. Then the value of $\lambda$ …

Q24.The circuit as shown in figure is working as a 8 V dc regulated voltage source. When 12 V is used as input, the power dissipated (in mW) in each diode is (considering both zener diodes are identical) ____.

Q1.Amongst the following, the form of water with the lowest ionic conductance at 298 K is :

Q2.The correct order of the spin-only magnetic moments of the following complexes is:
(I) $\left[\mathrm{Cr}{\left({\mathrm{H}}_2\mathrm{O}\right)}_6\right]{\mathrm{Br}}_2$
(II) ${\mathrm{Na}}_4\left[\mathrm{Fe}(\mathrm{CN}{)}_6\right]$
(III) …

Q3.Which polymer has 'chiral' monomer(s)?

Q4.The first and second ionisation enthalpies of a metal are 496 and $4560\mathrm{kJ}{\mathrm{mol}}^{-1}$, respectively. How many moles of HCl and ${\mathrm{H}}_2{\mathrm{SO}}_4$, respectively, will be needed to react completely with 1 mole of the metal hydroxide ?

Q5.Among the statements (a) - (d), the correct ones are:
(a)Lithium has the highest hydration enthalpy among the alkali metals.
(b) Lithium chloride is insoluble in pyridine.
(c) Lithium cannot form ethynide upon its reaction with ethyne
(d) Both lithium and magnesium react slowly w …

Q6.The solubility product of $\mathrm{Cr}(\mathrm{OH}{)}_3$ at 298 K is $6.0\times 10^{-31}$. The concentration of hydroxide ions in a saturated solution of $\mathrm{Cr}(\mathrm{OH}{)}_3$ will be :

Q7.5 g of zinc is treated separately with an excess of
(a) dilute hydrochloric acid and
(b) aqueous sodium hydroxide.

The ratio of the volumes of ${\mathrm{H}}_2$ evolved in these two reactions is : -

Q8.If the figure shown below reactant A (represented by square) is in equilibrium with product B (represented by circle). The equilibrium constant is:

Q9.A mixture of gases ${\mathrm{O}}_2,{\mathrm{H}}_2$ and CO are taken in a closed vessel containing charcoal. The graph that represents the correct behaviour of pressure with time is:

Q10.The number of $s{p}^2$ hybrid orbitals in a molecule of benzene is

Q11.Biochemical Oxygen Demand (BOD) is the amount of oxygen required (in ppm):

Q12.The true statement amongst the following is:

Q13.A, B and C are three biomolecules. The results of the tests performed on them are given below:
| (C) | Negative | Negative | Posi …

Q14.Consider the following reactions:

The compound [P] is

Q15.Which of the following reactions will not produce a racemic product?

Q16.Which of the following has the shortest $\mathrm{C}-\mathrm{Cl}$ bond?

Q17.The decreasing order of basicity of the following amines is

(I) (see structure)
(II) (see structure)
(III) (see structure)
(IV) (see structure)

Q18.The reaction of ${\mathrm{H}}_3{\mathrm{N}}_3{\mathrm{B}}_3{\mathrm{Cl}}_3(\mathrm{A})$ with ${\mathrm{LiBH}}_4$ in tetrahydrofuran gives inorganic benzene (B). Further, the reaction of (A) with (C) leads to ${\mathrm{H}}_3{\mathrm{N}}_3{\mathrm{B}}_3(\mathrm{Me}{)}_3$. …

Q19.In the following reaction A is :

(i) ${\mathrm{Br}}_2,\mathrm{h}\nu$
(ii) KOH (alc.)
(iii) ${\mathrm{O}}_3$
(iv) ${\left({\mathrm{CH}}_3\right)}_2\mathrm{S}$
(v) $\mathrm{NaOH}(\mathrm{aq})+\Delta$

(see product structure)

Q20.The isomer (s) of $\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_4{\mathrm{Cl}}_2\right]$ that has/have a $\mathrm{Cl}-\mathrm{Co}-\mathrm{Cl}$ angle of $90^{\circ }$, is/are :

Q21.The sum of the total number of bonds between chromium and oxygen atoms in chromate and dichromate ions is ____

Q22.A sample of milk splits after 60 min. at 300 K and after 40 min. at 400 K when the population of lactobacillus acidophilus in it doubles. The activation energy (in $\mathrm{kJ}/\mathrm{mol}$) for this process is closest to ____

(Given, …

Q23.A Cylinder containing an ideal gas (0.1 mol of $1.0{\mathrm{dm}}^3$) is in thermal equilibrium with a large volume of 0.5 molal aqueous solution of ethylene glycol at its freezing point. If the stoppers ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ (as shown in the figure) are suddenly …

Q24.Consider the following reactions


The mass percentage of carbon in A is ____.

Q25.10.30 mg of ${\mathrm{O}}_2$ is dissolved into a liter of sea water of density $1.03\mathrm{g}/\mathrm{mL}$. The concentration of ${\mathrm{O}}_2$ in ppm is ____.

Q1.If $\mathrm{x}={\sum }_{\mathrm{n}=0}^{\infty }(-1{)}^{\mathrm{n}}{\tan }^{2\mathrm{n}}\theta$ and $\mathrm{y}={\sum }_{\mathrm{n}=0}^{\infty }{\cos }^{2\mathrm{n}}\theta$, for $0<\theta <\frac{\pi }{4}$, then:

Q2.If $p\to (p\wedge \sim q)$ is false, then the truth values of $p$ and $q$ are respectively:

Q3.Let $a,b\in R,a\ne 0$ be such that the equation $a{x}^2-2bx+5=0$ has a repeated root $\alpha$, which is also a root of the equation, $x^2-2bx-10=0$. If $\beta$ is the other root of this equation, then ${\alpha }^2+{\beta }^2$ is equal to:

Q4.Let $[t]$ denote the greatest integer $\le t$ and ${\lim }_{x\to 0}x\left[\frac{4}{x}\right]=A$. Then the function, $f(x)=\left[x^2\right]\sin (\pi x)$ is discontinuous, when $x$ is equal to:

Q5.Let a function $f:[0,5]\to R$ be continuous, $f(1)=3$ and $F$ be defined as:
$F(x)={\int }_1^x{t}^{2}g(t)dt$, where $g(t)={\int }_1^{t}f(u)du$. Then for the function $F$, the point $x=1$ is:

Q6.A random variable $X$ has the following probability distribution:

\begin{tabular}{ccccccc}
$X$ & $:$ & 1 & 2 & 3 & 4 & 5 \\
$P(X)$ & $:$ & $K^2$ & $2K$ & $K$ & $2K$ & $5K^2$
\end{tabular}

Then $\mathrm{P}(\mathrm{X}>2)$ is equal to:

Q7.If $\int \frac{d\theta }{{\cos }^2\theta (\tan 2\theta +\sec 2\theta )}=\lambda \tan \theta +2{\log }_e|f(\theta )|+C$ where $C$ is a constant of integration, then the ordered pair $(\lambda ,f(\theta ))$ is equal to:

Q8.If $\frac{dy}{dx}=\frac{xy}{x^2+y^2};y(1)=1$; then a value of $x$ satisfying $y(x)=e$ is:

Q9.Let ${\mathrm{a}}_{\mathrm{n}}$ be the ${\mathrm{n}}^{\text{th }}$ term of a G.P. of positive terms. If ${\sum }_{\mathrm{n}=1}^{100}{\mathrm{a}}_{2\mathrm{n}+1}=200$ and ${\sum }_{\mathrm{n}=1}^{100}{\mathrm{a}}_{2\mathrm{n}}=100$, then ${\sum }_{n=1}^{200}{a}_n$ is equal to:

Q10.If z be a complex number satisfying $|\mathrm{Re}(\mathrm{z})|+|\mathrm{Im}(\mathrm{z})|=4$, then $|\mathrm{z}|$ cannot be:

Q11.Given : $f(x)=\{\begin{array}{cc}x,& 0\le x<\frac{1}{2}\\ \frac{1}{2},& x=\frac{1}{2}\\ 1-x,& \frac{1}{2}<x\le 1\end{array}$ and $g(x)={\left(x-\frac{1}{2}\right)}^2,x\in R$. Then the area (in sq. units) of the region bounded by the curves, $y=f(x)$ a …

Q12.In the expansion of ${\left(\frac{x}{\cos \theta }+\frac{1}{x\sin \theta }\right)}^{16}$, if $I_1$ is the least value of the term independent of $x$ when $\frac{\pi }{16}\le \theta \le \frac{\pi }{8}$, and $I_2$ is the least value of the term independent of $x$ when …

Q13.If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is:

Q14.If $A=\{x\in R:|x|<2\}$ and $B=\{x\in R:|x-2|\ge 3\}$; then:

Q15.Let $a-2b+c=1$. If $f(x)=\left|\begin{array}{ccc}x+a& x+2& x+1\\ x+b& x+3& x+2\\ x+c& x+4& x+3\end{array}\right|$, then:

Q16.Let $f$ and $g$ be differentiable functions on $R$ such that fog is the identity function. If for some $\mathrm{a},\mathrm{b}\in \mathrm{R},{\mathrm{g}}^{\prime }(\mathrm{a})=5$ and $\mathrm{g}(\mathrm{a})=\mathrm{b}$, then ${\mathrm{f}}^{\prime }(\mathrm{b})$ is equal to:

Q17.If one end of a focal chord $AB$ of the parabola $y^2=8x$ is at $A\left(\frac{1}{2},-2\right)$, then the equation of the tangent to it at $B$ is:

Q18.The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line, $x+6y=8$; then its eccentricity is:

Q19.The following system of linear equation
$7x+6y-2z=0$
$3x+4y+2z=0$ has
$x-2y-6z=0$,

Q20.If the curves, $x^2-6x+y^2+8=0$ and $x^2-8y+y^2+16-k=0,(k>0)$ touch each other at a point, then the largest value of $k$ is ______.

Q21.If ${\mathrm{C}}_{\mathrm{r}}={}^{25}{\mathrm{C}}_{\mathrm{r}}$ and ${\mathrm{C}}_0+5\cdot {\mathrm{C}}_1+9\cdot {\mathrm{C}}_2+\dots \dots \dots +(101)\cdot {\mathrm{C}}_{25}=2{}^{25}\cdot \mathrm{k}$, then k is equal to ______.

Q22.Let $\vec{a},\vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{3},|\vec{b}|=5,\vec{b}\cdot \vec{c}=10$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi }{3}$. If $\vec{a}$ is perpendicular to the vector $\vec{b}\times \vec{c}$, then …

Q23.If the distance between the plane, $23x-10y-2z+48=0$ and the plane containing the lines $\frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3}$ and $\frac{x+3}{2}=\frac{y+2}{6}=\frac{z-1}{\lambda }(\lambda \in R)$ is equal to $\frac{k}{\sqrt{633}}$, then $k$ is equal to ______.

Q24.The number of terms common to the two A.P.'s $3,7,11,\dots \dots \dots \dots ..407$ and $2,9,16$, ..... 709 is ______