JEE Main 2019 · 10 Apr 2019 · Shift 1

Q1.A particle of mass $m$ is moving along a trajectory given by

The torque, acting on the particle about the origin, at $t=0$ is:

Q2.A stationary source emits sound waves of frequency 500 Hz. Two observers moving along a line passing through the source detect sound to be of frequencies 480 Hz and 530 Hz. Their respective speeds are, in ${\mathrm{ms}}^{-1}$ (Given speed of sound $=300\mathrm{m}/\mathrm{s}$)

Q3.Two particles, of masses M and 2M, moving as shown, with speeds of $10\mathrm{m}/\mathrm{s}$ and $5\mathrm{m}/\mathrm{s}$, collide elastically at the origin. After the collision, they move along the indicated directions with speeds $v_1$ and $v_2$ respectively. The value …

Q4.Given below in the left column are different modes of communication using the kinds of waves given in the right column.

From the options given below, find the most appropriate match between entries in the left and the right column.

Q5.In an experiment, the resistance of a material is plotted as a function of temperature (in some range). As shown in the figure, it is a straight line. One may conclude that:

Q6.A cylinder with fixed capacity of 67.2 lit contains helium gas at STP. The amount of heat needed to raise the temperature of the gas by $20^{\circ }\mathrm{C}$ is: [Given that $\mathrm{R}=8.31\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$]

Q7.A ball is thrown upward with an initial velocity ${\mathrm{V}}_0$ from the surface of the earth. The motion of the ball is affected by a drag force equal to $\mathrm{m}\gamma v^2$ (where m is mass of the ball, $v$ is its Instantneous velocity and $\gamma$ is a constant). Time …

Q8.A thin disc of mass $M$ and radius $R$ has mass per unit area $\sigma (r)=k{r}^2$ where $r$ is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is:

Q9.Two coaxial discs, having moments of inertia ${\mathrm{I}}_1$ and $\frac{{\mathrm{I}}_1}{2}$, area rotating with respectively angular velocities ${\omega }_1$ and $\frac{{\omega }_1}{2}$, about their common axes. They are brought in contact with each other and thereafter they rotat …

Q10.The ratio of surface tensions of mercury and water is given to be 7.5 while the ratio of their densities is 13.6 . Their contact angles, with glass, are close to $135^{\circ }$ and $0^{\circ }$, respectively. It is observed that mercury gets depressed by an amount $h$ in a capillar …

Q11.Figure shows charge (q) versus voltage (V) graph for series and parallel combination of to given capacitors. The capacitances are:

Q12.Two wires $A$ and $B$ are carrying currents $I_1$ and $I_2$ as shown in the figure. The separation between them is d . A third wire C carrying a current I is to be kept parallel to them at a distance x from A such that the net force acting on it is zero. The possible values o …

Q13.A proton, an electron, and a Helium nucleus, have the same energy. They are in circular orbitals in a plane due to magnetic field perpendicular to the plane. Let $r_p,r_e$ and $r_{He}$ be their respective radii, then,

Q14.The value of acceleration due to gravity at Earth's surface is $9.8{\mathrm{ms}}^{-2}$. The altitude above its surface at which the acceleration due to gravity decreases to $4.9{\mathrm{ms}}^{-2}$, is close to: (Radius of earth $=6.4\times 10^6\mathrm{m}$)

Q15.In the given circuit, an ideal voltmeter connected across the $10\Omega$ resistance reads 2 V . The internal resistance r, of each cell is:

Q16.The electric field of a plane electromagnetic wave is given by

The corresponding magnetic field $\vec{B}$ is then given by:

Q17.An npn transistor operates as a common emitter amplifier, with a power gain of 60 dB . The input circuit resistance is $100\Omega$ and the output load resistance is $10\mathrm{k}\Omega$. the common emitter current gain $\beta$ is:

Q18.In a meter bridge experiment, the circuit diagram and the corresponding observation table are shown in figure.

\begin{tabular}{ccc}
Sl. No. & $\mathrm{R}(\Omega )$ & $l(\mathrm{cm})$ \\

1. & 1000 & 60 \\
2. & 100 & 13 \\
3. & 10 & 1.5 \\
4. & 1 & 1.0

\end{tabular}

Which of …

Q19.A transformer consisting of 300 turns in the primary and 150 turns in the secondary gives output power of 2.2 kW. If the current in the secondary coils is 10 A, then the input voltage and current in the primary coil are:

Q20.A uniformly charged ring of radius $3a$ and total charge $q$ is placed in $xy$-plane centered at origin. A point charge q is moving towards the ring along the z-axis and has speed v at $z=4a$. The minimum value of $v$ such that it crosses the origin is:

Q21.A current of 5 A passes through a copper conductor (resistivity $=1.7\times 10^{-8}\Omega \mathrm{m}$) of radius of cross-section 5 mm. Find the mobility of the charges if their drift velocity is 1.1 $\times 10^{-3}\mathrm{m}/\mathrm{s}$.

Q22.n-moles of an ideal gas with constant volume heat capacity ${\mathrm{C}}_{\mathrm{V}}$ undergo an isobaric expansion by certain volume. The ratio of the work done in the process, to the heat supplied is

Q23.A message signal of frequency 100 MHz and peak voltage 100 V is used to execute amplitude modulation on a carrier wave of frequency 300 GHz and peak voltage 400 V. The modulation index and difference between the two side band frequencies are:

Q24.A ray of light AO in vacuum is incident on a glass slab at angle $60^{\circ }$ and refracted at angle $30^{\circ }$ along OB as shown in the figure. The optical path length of light ray from $A$ to $B$ is:

Q25.The displacement of a damped harmonic oscillator is given by


The time taken for its amplitude of vibration to drop to half of its initial value is close to:

Q26.Two radioactive materials A and B have decay constants $10\lambda$ and $\lambda$, respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of a to that of $B$ will be $1/e$ after a time:

Q27.A $25\times 10^{-3}{\mathrm{m}}^3$ volume cylinder is filled with 1 mol of ${\mathrm{O}}_2$ gas at room temperature ( 300 K ). The molecular diameter of ${\mathrm{O}}_2$, and its root mean square speed, are found to be 0.3 nm and $200\mathrm{m}/\mathrm{s}$, respect …

Q28.In a photoelectric effect experiment the threshold wavelength of the metal is 380 nm. If the wavelength of the incident light is 260 nm, the maximum kinetic energy of the emitted electrons will be: Given E (in eV) $=\frac{1237}{\lambda (\mathrm{in}\mathrm{nm})}$

Q29.One plano-convex and one plano-concave lens of same radius of curvature ' R ' but of different materials are joined side by side as shown in the figure. If the refractive index of the material of 1 is ${\mu }_1$ and that of 2 is ${\mu }_2$, then the focal length of the combinat …

Q1.The species that can have a trans-isomer is:
(en = ethane-1,2-diamine, ox = oxalate)

Q2.The synonym for water gas when used in the production of methanol is

Q3.The regions of the atmosphere, where clouds form and where we live, respectively, are:

Q4.Consider the hydrated ions of ${\mathrm{Ti}}^{2+},{\mathrm{V}}^{2+},{\mathrm{Ti}}^{3+}$ and ${\mathrm{Sc}}^{3+}$. The correct order of their spin-only magnetic moments is:

Q5.Amylopectin is composed of

Q6.A gas undergoes physical adsorption on a surface and follows the given Freundlich adsorption isotherm equation


Adsorption of the gas increases with:

Q7.A bacterial infection in an internal would grows as ${\mathrm{N}}^{\prime }(\mathrm{t})={\mathrm{N}}_0\exp (\mathrm{t})$, where the time t is in hours. A dose of antibiotic, taken orally, needs 1 hour to reach the wound. Once it reaches there, the bacterial population goes down a …

Q8.The graph between $|\psi {|}^2$ and r(radial distance) is shown below. This represents:

Q9.Consider the following table:

\begin{tabular}{lll}
Gas & $\mathrm{a}/(\mathrm{k}\mathrm{Pa}{\mathrm{dm}}^6{\mathrm{mol}}^{-1})$ & $\mathrm{b}/({\mathrm{dm}}^3{\mathrm{mol}}^{-1})$ \\
A & 642.32 & 0.05196 \\
B & 155.21 & 0.04136 \\
C & 431.91 & 0.05196 \\
D & 155.21 & 0.4 …

Q10.During the change of ${\mathrm{O}}_2$ to ${\mathrm{O}}_2^{-}$, the incoming electron goes to the orbital:

Q11.The principle of column chromatography is

Q12.At 300 K and 1 atmospheric pressure, 10 mL of a hydrocarbon required 55 mL of ${\mathrm{O}}_2$ for complete combustion, and 40 mL of ${\mathrm{CO}}_2$ is formed. The formula of the hydrocarbon is:

Q13.Three complexes, …

Q14.Major products of the following reaction are:

Q15.Match the refining methods (Column I) with metals (Column II).

Column I
(Refining methods)
(I) Liquation
(II) zone Refining
(III) Mond process
(IV) Van Arkel Method

Column II
(Metals)
(a) Zr
(b) Ni
(C) Sn
(d) Ga

Q16.The increasing order of the reactivity of the following compounds towards electrophilic aromatic substitution reaction is:

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

Q17.A process will be spontaneous at all temperatures if:

Q18.Consider the following statements
(a) The pH of a mixture containing 400 mL of $0.1\mathrm{M}{\mathrm{H}}_2{\mathrm{SO}}_4$ and 400 mL of 0.1 M NaOH will be approximately 1.3 .
(b) Ionic product of water is temperature dependent.
(c) A monobasic acid with …

Q19.Ethylamine $\left({\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_2\right)$ can be obtained from N -ethylphthalimide on treatment with:

Q20.The correct order of catenation is:

Q21.Which of the following is a condensation polymer?

Q22.Consider the statements S1 and S2:
S1: Conductivity always increases with decrease in the concentration of electrolyte.
S2: Molar conductivity always increases with decrease in the concentration of electrolyte.
The correct option among the following is:

Q23.Increasing rate of ${\mathrm{S}}_{\mathrm{N}}1$ reaction in the following compounds is

Q24.The oxoacid of sulphur that does not contain bond between sulphur atoms is:

Q25.The isoelectronic set of ions is

Q26.The alloy used in the construction of aircrafts is

Q27.At room temperature, a dilute solution of urea is prepared by dissolving 0.60 g of urea in 360 g of water. If the vapour pressure of pure water at this temperature is 35 mm Hg, lowering of vapour pressure will be: (molar mass of urea $=60\mathrm{g}{\mathrm{mol}}^{-1}$)

Q1.If $Q(0,-1,-3)$ is the image of the point $P$ in the plane $3x-y+4z=2$ and $R$ is the point $(3,-1,-2)$, then the area (in square units) of $△\mathrm{PQR}$ is:

Q2.If the coefficients of $x^2$ and $x^3$ are both zero, in the expansion of the expression $(1+ax+b{x}^2)(1-3x{)}^{15}$ in powers of $x$, then the ordered pair $(a,b)$ is equal to

Q3.The sum

Upto $10^{\text{th }}$ term, is

Q4.If the length of the perpendicular from the point $(\beta ,0,\beta )$ ($\beta \ne 0$) to the line, $\frac{x}{1}=\frac{y-1}{0}=\frac{z+1}{-1}$ is $\sqrt{\frac{3}{2}}$, then $\beta$ is equal to:

Q5.If $\mathrm{f}(\mathrm{x})=\{\begin{array}{cc}\frac{\sin (\mathrm{p}+1)\mathrm{x}+\sin \mathrm{x}}{\mathrm{x}}& ,x<0\\ \mathrm{q}& ,x=0\\ \frac{\sqrt{\mathrm{x}+{\mathrm{x}}^2}-\sqrt{\mathrm{x}}}{{\mathrm{x}}^{3/2}}& ,x>0\end{array}$

Is continuous at $x=0$ …

Q6.Which one of the following Boolean expressions is a tautology?

Q7.If $\alpha$ and $\beta$ are the roots of the quadratic equation, $x^2+x\sin \theta -2\sin \theta =0,\theta \in \left(0,\frac{\pi }{2}\right)$ then $\frac{{\alpha }^{12}+{\beta }^{12}}{\left({\alpha }^{-12}+{\beta }^{-12}\right)(\alpha -\beta {)}^{24}}$ is equal to:

Q8.ABC is a triangular park with $\mathrm{AB}=\mathrm{AC}=100$ metres. A vertical tower is situated at the mid-point of $BC$. If the angles of elevation of the top of the tower at $A$ and $B$ are ${\cot }^{-1}(3\sqrt{2})$ and ${\mathrm{cosec}}^{-1}(2\sqrt{2})$ respec …

Q9.If $a>0$ and $z=\frac{(1+i{)}^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$, then $\bar{z}$ is equal to:

Q10.If ${\Delta }_1=\left|\begin{array}{ccc}\mathrm{x}& \sin \theta & \cos \theta \\ \sin \theta & -\mathrm{x}& 1\\ \cos \theta & 1& \mathrm{x}\end{array}\right|$ and …

Q11.If $y=f(x)$ is the solution of the differential equation $\frac{dy}{dx}=(\tan x-y){\sec }^{2}x$, $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, such that $y(0)=0$, then $y\left(-\frac{\pi }{4}\right)$ is equal to

Q12.Let $f:R\to R$ be differentiable at $c\in R$ and $f(c)=0$. If $g(x)=|f(x)|$, then at $x=c,g$ is:

Q13.Let $f(x)=x^2,x\in R$. For any $A\subseteq R$, define $g(A)=\{x\in R:f(x)\in A\}$. If $S=[0,4]$, then which one of the following statements is not true?

Q14.If the system of linear equations

$x+3y+\lambda z=\mu$ ($\lambda ,\mu \in R$), has infinitely many solutions then the value of $\lambda +\mu$ is:

Q15.The number of 6 digit numbers that can be formed using the digits $0,1,2,5,7$ and $9$ which are divisible by 11 and no digit is repeated, is

Q16.The line $x=y$ touches a circle at the point $(1,1)$. If the circle also passes through the point $(1,-3)$, then its radius is

Q17.Let $f(x)=e^x-x$ and $g(x)=x^2-x,\forall x\in R$. Then the set of all $x\in R$, where the function $\mathrm{h}(\mathrm{x})=(\mathrm{fog})(\mathrm{x})$ is increasing is:

Q18.The value of ${\int }_0^{2\pi }[\sin 2x(1+\cos 3x)]dx$ where $[t]$ denotes the greatest integer function, is:

Q19.If the circles $x^2+y^2+5Kx+2y+K=0$ and $2\left(x^2+y^2\right)+2Kx+3y-1=0,(K\in R)$, intersect at the points $P$ and $Q$, then the line $4x+5y-K=0$ passes through $P$ and $Q$ for:

Q20.If the line $x-2y=12$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $\left(3,\frac{-9}{2}\right)$, then the length of the latus rectum of the ellipse is

Q21.If ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots ,{\mathrm{a}}_{\mathrm{n}}$ are in A.P and ${\mathrm{a}}_1+{\mathrm{a}}_4+{\mathrm{a}}_7+\dots +{\mathrm{a}}_{16}=114$, then ${\mathrm{a}}_1+{\mathrm{a}}_6+{\mathrm{a}}_{11}+{\mathrm{a}}_{16}$ is equal to

Q22.If for some $\mathrm{x}\in \mathrm{R}$, the frequency distribution of the marks obtained by 20 students in a test is

\begin{tabular}{ccccc}
Marks & 2 & 3 & 5 & 7 \\
Frequency & $(x+1{)}^2$ & $2x-5$ & $x^2-3x$ & $x$
\end{tabular}

Then the mean of the marks is:

Q23.If ${\lim }_{x\to 1}\frac{x^4-1}{x-1}={\lim }_{x\to k}\frac{x^3-k^3}{x^2-k^2}$, then $k$ is

Q24.All the pairs $(\mathrm{x},\mathrm{y})$ that satisfy the inequality
$2^{\sqrt{{\sin }^{2}x-2\sin x+5}}\cdot \frac{1}{4^{{\sin }^{2}y}}\le 1$ also
satisfy the equation:

Q25.
is equal to:

Q26.If a directrix of a hyperbola centered at the origin and passing through the point $(4,-2\sqrt{3})$ is $5x=4\sqrt{5}$ and its eccentricity is $e$, then

Q27.The region represented by $|x-y|\le 2$ and $|x+y|\le 2$ is bounded by a:

Q28.Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is:

Q29.Let $A(3,0,-1),B(2,10,6)$ and $C(1,2,1)$ be the vertices of a triangle and $M$ be the midpoint of AC. If G divides BM in the ratio, $2:1$, then $\cos (\angle \mathrm{GOA})$ (O being the origin) is equal to:

Q30.If $\int \frac{dx}{{\left(x^2-2x+10\right)}^2}$
$=A\left({\tan }^{-1}\left(\frac{x-1}{3}\right)+\frac{f(x)}{x^2-2x+10}\right)+C$
Where $C$ is a constant of integration, then: