JEE Main 2020 B.E./B.Tech Question Paper with Answer Keys (January 9, 2020- Shift 1)

If the number of five digit numbers with distinct digits and 2 at the 10^th place is 336 k, then k is equal to :

• (1) \(6\)
• (2) \(8\)
• (3) \(4\)
• (4) \(7\)

The value of
\(\cos^3\left(\frac{\pi}{8}\right)\cos\left(\frac{3\pi}{8}\right)+\sin^3\left(\frac{\pi}{8}\right)\sin\left(\frac{3\pi}{8}\right)\)
is :

• (1) \(\frac{1}{\sqrt{2}}\)
• (2) \(\frac{1}{2}\)
• (3) \(\frac{1}{4}\)
• (4) \(\frac{1}{2\sqrt{2}}\)

If for all real triplets (a, b, c), \(f(x)=a+bx+cx^2\); then
\(\int_0^1 f(x)\,dx\)
is equal to :

• (1) \(\frac{1}{6}\left\{f(0)+f(1)+4f\left(\frac{1}{2}\right)\right\}\)
• (2) \(2\left\{3f(1)+2f\left(\frac{1}{2}\right)\right\}\)
• (3) \(\frac{1}{3}\left\{f(0)+f\left(\frac{1}{2}\right)\right\}\)
• (4) \(\frac{1}{2}\left\{f(1)+3f\left(\frac{1}{2}\right)\right\}\)

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4\pi} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is:

• (1) \( 225\pi \)
• (2) \( 128\pi \)
• (3) \( 196\pi \)
• (4) \( 256\pi \)

Let z be complex number such that
\(\left|\frac{z-i}{z+2i}\right|=1\)
and \(|z|=\frac{5}{2}\).
Then the value of \(|z+3i|\) is :

• (1) \(2\sqrt{3}\)
• (2) \(\sqrt{10}\)
• (3) \(\frac{15}{4}\)
• (4) \(\frac{7}{2}\)

The value of
\(\int_0^{2\pi}\frac{x\sin^8x}{\sin^8x+\cos^8x}\,dx\)
is equal to :

• (1) \(4\pi\)
• (2) \(2\pi\)
• (3) \(\pi^2\)
• (4) \(2\pi^2\)

The integral
\(\int\frac{dx}{(x+4)^{\frac87}(x-3)^{\frac67}}\)
is equal to :
(where C is a constant of integration)

• (1) \(\frac{1}{2}\left(\frac{x-3}{x+4}\right)^{\frac37}+C\)
• (2) \(\left(\frac{x-3}{x+4}\right)^{\frac17}+C\)
• (3) \(-\frac{1}{13}\left(\frac{x-3}{x+4}\right)^{\frac{13}{7}}+C\)
• (4) \(-\left(\frac{x-3}{x+4}\right)^{-\frac17}+C\)

Let \(f\) be any function continuous on [a, b] and twice differentiable on (a, b). If for all \(x\in(a,b)\), \(f'(x)>0\) and \(f''(x)<0\), then for any \(c\in(a,b)\),
\(\frac{f(c)-f(a)}{f(b)-f(c)}\)
is greater than :

• (1) \(1\)
• (2) \(\frac{b-c}{c-a}\)
• (3) \(\frac{b+a}{b-a}\)
• (4) \(\frac{c-a}{b-c}\)

In a box, there are 20 cards, out of which 10 are lebelled as A and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained. The probability that the second A-card appears before the third B-card is :

• (1) \(\frac{13}{16}\)
• (2) \(\frac{11}{16}\)
• (3) \(\frac{15}{16}\)
• (4) \(\frac{9}{16}\)

If
\(f(x)=\begin{cases}\dfrac{\sin(a+2)x+\sin x}{x}; & x<0\\[2mm] b\ ; & x=0\\[2mm] \dfrac{(x+3x^2)^{\frac13}-x^{\frac13}}{x^{\frac43}}; & x>0\end{cases}\)
is continuous at x = 0, then a + 2b is equal to :

• (1) \(0\)
• (2) \(-1\)
• (3) \(-2\)
• (4) \(1\)

Let C be the centroid of the triangle with vertices (3, \(-1\)), (1, 3) and (2, 4). Let P be the point of intersection of the lines \(x+3y-1=0\) and \(3x-y+1=0\). Then the line passing through the points C and P also passes through the point :

• (1) \((-9,-7)\)
• (2) \((9,7)\)
• (3) \((7,6)\)
• (4) \((-9,-6)\)

The number of real roots of the equation,
\(e^{4x}+e^{3x}-4e^{2x}+e^x+1=0\) is :

• (1) \(1\)
• (2) \(2\)
• (3) \(3\)
• (4) \(4\)

Let the observations \(x_i\ (1\le i\le10)\) satisfy the equations,
\(\sum_{i=1}^{10}(x_1-5)=10\) and \(\sum_{i=1}^{10}(x_1-5)^2=40\).
If \(\mu\) and \(\lambda\) are the mean and the variance of the observations, \(x_1-3,x_2-3,\dots,x_{10}-3\), then the ordered pair \((\mu,\lambda)\) is equal to :

• (1) \((6,6)\)
• (2) \((3,3)\)
• (3) \((3,6)\)
• (4) \((6,3)\)

If \(f'(x) = \tan^{-1}(\sec x + \tan x)\), \(-\frac{\pi}{2} \lt x \lt \frac{\pi}{2}\), and \(f(0) = 0\), then \(f(1)\) is equal to :

• (1) \(\frac{1}{4}\)
• (2) \(\frac{\pi-1}{4}\)
• (3) \(\frac{\pi+1}{4}\)
• (4) \(\frac{\pi+2}{4}\)

If for some \(\alpha\) and \(\beta\) in R, the intersection of the following three places
\(x + 4y - 2z = 1\)
\(x + 7y - 5z = b\)
\(x + 5y + \alpha z = 5\)
is a line in \(\mathbb{R}^3\), then \(\alpha + \beta\) is equal to :

• (1) -10
• (2) 0
• (3) 10
• (4) 2

Let \( e_1 \) and \( e_2 \) be the eccentricities of the ellipse \frac{x^2}{b^2} + \frac{y^2}{25} = 1 and the hyperbola \frac{x^2}{16} - \frac{y^2}{b^2} = 1, respectively. If \( b<5 \) and \( e_1 e_2 = 1 \), then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:

• (1) \( \frac{4}{5} \)
• (2) \( \frac{3}{5} \)
• (3) \( \frac{\sqrt{7}}{4} \)
• (4) \( \frac{\sqrt{3}}{2} \)

A circle touches the y-axis at the point \((0, 4)\) and passes through the point \((2, 0)\). Which of the following lines is not a tangent to this circle?

• (1) \(3x - 4y - 24 = 0\)
• (2) \(4x + 3y - 8 = 0\)
• (3) \(3x + 4y - 6 = 0\)
• (4) \(4x - 3y + 17 = 0\)

If the matrices \(A = \begin{bmatrix}1&1&2\\1&3&4\\1&-1&3\end{bmatrix}\), \(B = \mathrm{adj}A\) and \(C = 3A\), then \(\frac{|\mathrm{adj}B|}{|C|}\) is equal to :

• (1) 8
• (2) 2
• (3) 72
• (4) 16

Negation of the statement :
\(\sqrt{5}\) is an integer or 5 is an irrational is :

• (1) \(\sqrt{5}\) is not an integer and 5 is not irrational.
• (2) \(\sqrt{5}\) is irrational or 5 is an integer.
• (3) \(\sqrt{5}\) is an integer and 5 is irrational.
• (4) \(\sqrt{5}\) is not an integer or 5 is not irrational.

The product \(2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdots\) to \(\infty\) is equal to :

• (1) \(2^{\frac{1}{4}}\)
• (2) \(2^{\frac{1}{2}}\)
• (3) 1
• (4) 2

The number of distinct solutions of the equation \(\log_{\frac{1}{2}}|\sin x| = 2 - \log_{\frac{1}{2}}|\cos x|\) in the interval \([0, 2\pi]\), is ____.

The projection of the line segment joining the points \((1, -1, 3)\) and \((2, -4, 11)\) on the line joining the points \((-1, 2, 3)\) and \((3, -2, 10)\) is ____________.

The coefficient of \(x^4\) is the expansion of \((1 + x + x^2)^{10}\) is _____.

If for \(x \ge 0\), \(y = y(x)\) is the solution of the differential equation \((x+1)\,dy = \left((x+1)^2 + y - 3\right)dx\), \(y(2) = 0\), then \(y(3)\) is equal to ______.

If the vectors, \(\vec{p} = (a+1)\hat{i} + a\hat{j} + a\hat{k}\), \(\vec{q} = a\hat{i} + (a+1)\hat{j} + a\hat{k}\) and \(\vec{r} = a\hat{i} + a\hat{j} + (a+1)\hat{k}\) \((a \in R)\) are coplanar and \(3(\vec{p}\cdot\vec{q})^2 - \lambda|\vec{r}\times\vec{q}|^2 = 0\), then the value of \(\lambda\) is ______.

Water flows in a horizontal tube (see figure). The pressure of water changes by \(700\, Nm^{-2}\) between \(A\) and \(B\) where the area of cross section are \(40\, cm^2\) and \(20\, cm^2\), respectively. Find the rate of flow of water through the tube. (density of water = \(1000\, kgm^{-3}\))

• (1) \(3020\, cm^3/s\)
• (2) \(2420\, cm^3/s\)
• (3) \(2720\, cm^3/s\)
• (4) \(1810\,cm^3/s\)

An electric dipole of dipole moment is 6.0 x 10^-6 Cm placed in a uniform electric field of 1.5 x 10³ NC^-1 in such a way that dipole moment is along electric field. The work done in rotating dipole by 180º in this field will be _____mJ.

In the given circuit diagram, a wire is joining points B and D. The current in this wire is :

• (1) 4 A
• (2) 2 A
• (3) zero
• (4) 0.4 A

Two particles of equal mass m have respective initial velocities \(u\hat{i}\) and \(u\left(\frac{\hat{i}+\hat{j}}{2}\right).\) They collide completely inelastically. The energy lost in the process is :

• (1) \(\frac{3}{4} mu^{2}\)
• (2) \(\sqrt{\frac{2}{3}} mu^{2}\)
• (3) \(\frac{1}{3} mu^{2}\)
• (4) \(\frac{1}{8} mu^{2}\)

Three solid spheres each of mass \(m\) and diameter \(d\) are stuck together such that the lines connecting the centres form an equilateral triangle of side of length \(d\). The ratio \(I_0/I_A\) of moment of inertia \(I_0\) of the system about an axis passing the centroid and about center of any of the spheres \(I_A\) and perpendicular to the plane of the triangle is :

• (1) \(\frac{15}{13}\)
• (2) \(\frac{13}{15}\)
• (3) \(\frac{13}{23}\)
• (4) \(\frac{23}{13}\)

Three harmonic waves having equal frequency \(\nu\) and same intensity \(I_0\), have phase angles 0, \(\frac{\pi}{4}\) and \(-\frac{\pi}{4}\) respectively. When they are superimposed the intensity of the resultant wave is close to :

• (1) 5.8 \(I_0\)
• (2) 3 \(I_0\)
• (3) 0.2 \(I_0\)
• (4) \(I_0\)

A charged particle of mass 'm' and charge 'q' moving under the influence of uniform electric field \(E\hat{i}\) and a uniform magnetic field \(B\hat{k}\) follows a trajectory from point P to Q as shown in figure. The velocities at P and Q are respectively, \(v\hat{i}\) and \(-2v\hat{j}\). Then which of the following statements (A, B, C, D) are the correct ? (Trajectory shown is schematic and not to scale) :

(A) \(E=\frac{3}{4}\left(\frac{mv^2}{qa}\right)\)
(B) Rate of work done by the electric field at P is \(\frac{3}{4}\left(\frac{mv^3}{a}\right)\)
(C) Rate of work done by both the fields at Q is zero
(D) The difference between the magnitude of angular momentum of the particle at P and Q is \(2mav\).

• (1) (A), (B), (C), (D)
• (2) (A), (B), (C)
• (3) (A), (C), (D)
• (4) (B), (C), (D)

A vessel of depth \(2\,h\) is half filled with a liquid of refractive index \(2\sqrt{2}\) and the upper half with another liquid of refractive index \(\sqrt{2}\) The liquids are immiscible. The apparent depth of the inner surface of the bottom of vessel will be :

• (1) \(\frac{h}{\sqrt{2}}\)
• (2) \(\frac{3}{4}h\sqrt{2}\)
• (3) \(\frac{h}{3\sqrt{2}}\)
• (4) \(\frac{h}{2\left(\sqrt{2}+1\right)}\)

Consider two ideal diatomic gases \(A\) and \(B\) at some temperature \(T\). Molecules of the gas A are rigid , and have an mass \(m\). Molecules of the gas \(B\) have an additional vibrational mode, and have a mass \(\frac{m}{4}.\) The ratio of the specific heats \((C^{A}_{V}\) and \(C^{B}_{V})\) of gas \(A\) and \(B\), respectively is :

• (1) 5:09
• (2) 7:09
• (3) 3:05
• (4) 5:07

The electric fields of two plane electromagnetic plane waves in vacuum are given by
\(\vec{E_1}=E_0\hat{j}\cos(\omega t-kx)\) and
\(\vec{E_2}=E_0\hat{k}\cos(\omega t-ky)\)
At \(t=0\), a particle of charge \(q\) is at origin with a velocity \(\vec{v}=0.8c\,\hat{j}\) (\(c\) is the speed of light in vacuum). The instantaneous force experienced by the particle is :

• (1) \(E_0q\left(0.8\hat{i}-\hat{j}+0.4\hat{k}\right)\)
• (2) \(E_0q\left(-0.8\hat{i}+\hat{j}+\hat{k}\right)\)
• (3) \(E_0q\left(0.8\hat{i}+\hat{j}+0.2\hat{k}\right)\)
• (4) \(E_0q\left(0.4\hat{i}-3\hat{j}+0.8\hat{k}\right)\)

Radiation, with wavelength \(6561\,\mathrm{\AA}\) falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of \(3\times10^{-4}\) T. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :

• (1) 1.8 eV
• (2) 0.8 eV
• (3) 1.1 eV
• (4) 1.6 eV

If de-Broglie wavelength is \(\lambda\) when energy is E. Find the wavelength at \(\frac{E}{4}\) (Kinetic Energy).

• (1) \(2\lambda \)
• (2) \(\sqrt{2}\lambda \)
• (3) \(\lambda \)
• (4) \(\frac{\lambda }{\sqrt{2}}\)

The aperture diameter of a telescope is 5 m. The separation between the moon and the ear this \(4 \times 10^5\) km. With light of wavelength of 5500 \(?\), the minimum separation between objects on the surface of moon, so that they are just resolved, is close to :

• (1) 600 m
• (2) 20 m
• (3) 60 m
• (4) 200 m

A quantity \(f\) is given by \(f=\sqrt{\frac{hc^5}{G}}\) where \(c\) is speed of light, \(G\) universal gravitational constant and \(h\) is the Planck's constant. Dimension of \(f\) is that of :

• (1) Energy
• (2) Momentum
• (3) Area
• (4) Volume

In a screw gauge, when the circular scale is given five complete rotations, it moves linearly by \(2.5\,mm\lt /strong\gt \). If the circular scale has 100 divisions, the least count of the screw gauge is:}