JEE Main 2021 — 17 Mar Shift 1 (Previous Year Paper)

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:

1. A.$\frac{2\alpha\beta}{(\alpha+\beta)}t^2$
2. B.$\frac{\alpha\beta}{2(\alpha+\beta)}t^2$✓
3. C.$\frac{4\alpha\beta}{(\alpha+\beta)}t^2$
4. D.$\frac{\alpha\beta}{4(\alpha+\beta)}t^2$

Correct answer: B

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: (Assume forces on the four tyres are identical and g = acceleration due to gravity)

1. A.$m\left(\frac{v^2}{\mu_sR}+g\right)$
2. B.$m\left(g-\frac{v^2}{\mu_sR}\right)$
3. C.$m\left(\frac{v^2}{\mu_sR}-g\right)$✓
4. D.$-m\left(g+\frac{v^2}{\mu_sR}\right)$

Correct answer: C

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:

1. A.2400 J✓
2. B.1600 J
3. C.3200 J
4. D.1800 J

Correct answer: A

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 $=4\pi\times10^{-7}$ H/m)

1. A.$10^{-4}\pi$ T
2. B.$2\times10^{-3}\pi$ T
3. C.$\frac\pi5$ T
4. D.$\pi$ T✓

Correct answer: D

A current of 10 A exists in a wire of crosssectional area of 5 mm$^2$ with a drift velocity of $2\times10^{-3}$ ms$^{-1}$. The number of free electrons in each cubic meter of the wire is ________.

1. A.$2\times10^6$
2. B.$1\times10^{23}$
3. C.$625\times10^{25}$✓
4. D.$2\times10^{25}$

Correct answer: C

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 point B is $L_B$. The correct statement for this system is:

1. A.$L_A$ and $L_B$ are both constant in magnitude and direction
2. B.$L_A$ is constant, both in magnitude and direction✓
3. C.$L_B$ is constant, both in magnitude and direction
4. D.$L_B$ is constant in direction with varying magnitude

Correct answer: B

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:

1. A.$\frac{I_1+I_2}{\sqrt2}$
2. B.$\frac{I_1+I_2}{2\sqrt2}$
3. C.$\sqrt{\frac{I_1^2+I_2^2}{2}}$✓
4. D.$\sqrt{\frac{I_1^2-I_2^2}{2}}$

Correct answer: C

A polyatomic ideal gas has 24 vibrational modes. What is the value of $\gamma$?

1. A.10.3
2. B.1.30
3. C.1.03✓
4. D.1.37

Correct answer: C

When two soap bubbles of radii a and b (b > a) coalesce, the radius of curvature of common surface is:

1. A.$\frac{ab}{b-a}$✓
2. B.$\frac{ab}{a+b}$
3. C.$\frac{b-a}{ab}$
4. D.$\frac{a+b}{ab}$

Correct answer: A

A triangular plate is shown. A force $\vec F=4\hat i-3\hat j$ is applied at point P. The torque at point P with respect to point 'O' and 'Q' are:

1. A.$-15-20\sqrt3$, $15-20\sqrt3$✓
2. B.$15+20\sqrt3$, $15-20\sqrt3$
3. C.$15-20\sqrt3$, $15+20\sqrt3$
4. D.$-15+20\sqrt3$, $15+20\sqrt3$

Correct answer: A

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:

1. A.$\frac{K_1K_2}{K_1+K_2}$
2. B.$\frac{2K_1K_2}{K_1+K_2}$✓
3. C.$\frac{K_1+K_2}{2K_1K_2}$
4. D.$\frac{K_1+K_2}{K_1K_2}$

Correct answer: B

A boy is rolling a 0.5 kg ball on the frictionless floor with the speed of 20 ms$^{-1}$. 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?

1. A.1.00 ms$^{-1}$
2. B.4.47 ms$^{-1}$✓
3. C.14.41 ms$^{-1}$
4. D.19.0 ms$^{-1}$

Correct answer: B

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 $2\times10^8$ ms$^{-1}$. The focal length of the lens is ________.

1. A.30 cm✓
2. B.15 cm
3. C.1.5 cm
4. D.0.30 cm

Correct answer: A

Which level of the single ionized carbon has the same energy as the ground state energy of hydrogen atom?

1. A.1
2. B.4
3. C.6✓
4. D.8

Correct answer: C

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:

1. A.$\frac{n_1F_1T_1+n_2F_2T_2}{n_1+n_2}$
2. B.$\frac{n_1F_1T_1+n_2F_2T_2}{F_1+F_2}$
3. C.$\frac{n_1F_1T_1+n_2F_2T_2}{n_1F_1+n_2F_2}$✓
4. D.$\frac{n_1T_1+n_2T_2}{n_1+n_2}$

Correct answer: C

For what value of displacement the kinetic energy and potential energy of a simple harmonic oscillation become equal?

1. A.$x=\pm A$
2. B.$x=0$
3. C.$x=\pm\frac{A}{\sqrt2}$✓
4. D.$x=\frac A2$

Correct answer: C

The output of the given combination gates represents:

1. A.AND Gate
2. B.NOR Gate
3. C.NAND Gate✓
4. D.XOR Gate

Correct answer: C

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)

1. A.$\left(\frac{E}{2m}\right)^{1/2}$
2. B.$c(2mE)^{1/2}$
3. C.$\frac1c\left(\frac{2m}{E}\right)^{1/2}$
4. D.$\frac1c\left(\frac{E}{2m}\right)^{1/2}$✓

Correct answer: D

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)

1. A.8.58 cm
2. B.8.56 cm
3. C.8.54 cm✓
4. D.8.36 cm

Correct answer: C

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:

1. A.$v_n\propto n^2$
2. B.$v_n\propto n$
3. C.$v_n\propto\frac1{n^2}$
4. D.$v_n\propto\frac1n$✓

Correct answer: D

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)

Correct answer: 1206

If $2.5\times10^{-6}$ N average force is exerted by a light wave on a non-reflecting surface of 30 cm$^2$ area during 40 minutes 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)

Correct answer: 25

Four identical rectangular plates with length, $l=2$ cm and breadth, $b=\frac32$ cm are arranged as shown in figure. The equivalent capacitance between A and C is $\frac{x\varepsilon_0}{d}$. The value of x is ________. (Round off to the Nearest Integer)

Correct answer: 2

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)

Correct answer: 4

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 porcelin 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)

Correct answer: 864

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 times is ________.

Correct answer: 64

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).

Correct answer: 728

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)

Correct answer: 2

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]

Correct answer: 4

Two blocks (m = 0.5 kg and M = 4.5 kg) are arranged on a horizontal frictionless table as shown in figure. The coefficient of static friction between the two blocks is $\frac37$. 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 ms$^{-2}$]

Correct answer: 21

A central atom in a molecule has two lone pairs of electrons and forms three single bonds. The shape of this molecule is:

1. A.planar triangular
2. B.T-shaped✓
3. C.see-saw
4. D.trigonal pyramidal

Correct answer: B

A colloidal system consisting of a gas dispersed in a solid is called a/an:

1. A.aerosol
2. B.solid sol✓
3. C.foam
4. D.gel

Correct answer: B

The absolute value of the electron gain enthalpy of halogens satisfies:

1. A.F > Cl > Br > I
2. B.Cl > F > Br > I✓
3. C.Cl > Br > F > I
4. D.I > Br > Cl > F

Correct answer: B

The point of intersection and sudden increase in the slope, in the diagram given below, respectively, indicates:

1. A.$\Delta G<0$ and decomposition of the metal oxide
2. B.$\Delta G>0$ and decomposition of the metal oxide
3. C.$\Delta G=0$ and melting or boiling point of the metal oxide✓
4. D.$\Delta G=0$ and reduction of the metal oxide

Correct answer: C

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:

1. A.(C) only
2. B.(B) only
3. C.(D) only✓
4. D.(B) and (D) only

Correct answer: C

The correct order of conductivity of ions in water is:

1. A.Cs$^+$ > Rb$^+$ > K$^+$ > Na$^+$✓
2. B.Na$^+$ > K$^+$ > Rb$^+$ > Cs$^+$
3. C.K$^+$ > Na$^+$ > Cs$^+$ > Rb$^+$
4. D.Rb$^+$ > Na$^+$ > K$^+$ > Li$^+$

Correct answer: A

Which of the following compound CANNOT act as a Lewis base?

1. A.ClF$_3$
2. B.PCl$_5$✓
3. C.NF$_3$
4. D.SF$_4$

Correct answer: B

What is the spin-only magnetic moment value (BM) of a divalent metal ion with atomic number 25, in it's aqueous solution?

1. A.5.0
2. B.5.26
3. C.5.92✓
4. D.zero

Correct answer: C

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:

1. A.Both statement I and statement II are false
2. B.Both statement I and statement II are true
3. C.Statement I is true but statement II is false✓
4. D.Statement I is false but statement II is true

Correct answer: C

Reducing smog is a mixture of:

1. A.Smoke, fog and O$_3$
2. B.Smoke, fog and SO$_2$✓
3. C.Smoke, fog and N$_2$O$_3$
4. D.Smoke, fog and CH$_2$=CH–CHO

Correct answer: B

Given below are two statements: Statement I: Retardation factor ($R_f$) can be measured in meter/centimeter. Statement II: $R_f$ value of a compound remains constant in all solvents. Choose the most appropriate answer from the options given below:

1. A.Both statement I and statement II are true
2. B.Both statement I and statement II are false✓
3. C.Statement I is true but statement II is false
4. D.Statement I is false but statement II is true

Correct answer: B

Which of the following is an aromatic compound?

1. A.cyclopentadiene (neutral)
2. B.cyclopentadienyl anion✓
3. C.cyclopentadienyl cation
4. D.a cyclopentadiene drawn with two negative charges (dianion)

Correct answer: B

1-methyl-1-vinylcyclohexane $\xrightarrow{\text{HBr, CCl}_4}$ A (Major Product). Product "A" in the above chemical reaction is:

1. A.1-(2-bromoethyl)-1-methylcyclohexane
2. B.1-bromo-1,2-dimethylcyclohexane
3. C.1-bromo-1-isopropylcyclohexane✓
4. D.1-(1-bromoethyl)-2-methylcyclohexane type (Br on side-chain CH, CH3 on adjacent ring carbon)

Correct answer: C

Chlorobenzene + NaOH $\rightarrow$ sodium phenoxide. The above reaction requires which of the following reaction conditions?

1. A.623 K, Cu, 300 atm
2. B.573 K, Cu, 300 atm
3. C.623 K, 300 atm✓
4. D.573 K, 300 atm

Correct answer: C

Mesityl oxide is a common name of:

1. A.4-Methyl pent-3-en-2-one✓
2. B.2,4-Dimethyl pentan-3-one
3. C.2-Methyl cyclohexanone
4. D.3-Methyl cyclohexane carbaldehyde

Correct answer: A

Hoffmann bromomide degradation of benzamide gives product A, which upon heating with CHCl$_3$ and NaOH gives product B. The structures of A and B are:

1. A.A = 4-bromobenzamide; B = 4-bromo-2-formylbenzamide
2. B.A = aniline; B = phenyl isocyanide (C6H5NC)✓
3. C.A = 4-bromoaniline; B = 2-amino-5-bromobenzaldehyde
4. D.A = aniline; B = 2-aminobenzaldehyde

Correct answer: B

Ethyl acetoacetate $\mathrm{CH_3COCH_2COOC_2H_5}\xrightarrow{\text{Ethylene Glycol, H}^+}$ A (Major Product). The product "A" in the above reaction is:

1. A.hemiacetal: CH3C(OH)(OCH2CH2OH)CH2COOC2H5
2. B.2-methyl-1,3-dioxolane-2-acetic acid (ketal with free COOH)
3. C.3-ethoxy-3-hydroxycyclobutanone
4. D.ethyl 2-methyl-1,3-dioxolane-2-acetate (ketal, ester retained)✓

Correct answer: D

Which of the following reaction is an example of ammonolysis?

1. A.$\mathrm{C_6H_5COCl+C_6H_5NH_2\rightarrow C_6H_5CONHC_6H_5}$
2. B.$\mathrm{C_6H_5CH_2CN\xrightarrow{[H]}C_6H_5CH_2CH_2NH_2}$
3. C.$\mathrm{C_6H_5CH_2Cl+NH_3\rightarrow C_6H_5CH_2NH_2}$✓
4. D.$\mathrm{C_6H_5NH_2\xrightarrow{HCl}C_6H_5\overset{+}{N}H_3Cl^-}$

Correct answer: C

With respect to drug-enzyme interaction, identify the wrong statement.

1. A.Competitive inhibitor binds to the enzyme's active site
2. B.Allosteric inhibitor changes the enzyme's active site
3. C.Allosteric inhibitor competes with the enzyme's active site✓
4. D.Non-Competitive inhibitor binds to the allosteric site

Correct answer: C

Which of the following is correct structure of tyrosine?

1. A.H2N-CH(COOH)-CH2-C6H4-OH with OH meta
2. B.H2N-CH(COOH)-CH2CH2-C6H4-OH with OH meta
3. C.H2N-CH(COOH)-CH2-C6H4-OH with OH para✓
4. D.H2N-CH(COOH)-CH2CH2-C6H4-OH with OH para

Correct answer: C

The mole fraction of a solute in a 100 molal aqueous solution is ________ $\times10^{-2}$. (Round off to the Nearest Integer). [Given: Atomic masses: H: 1.0 u, O: 16.0 u]

Correct answer: 64

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°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]

Correct answer: 150

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).

Correct answer: 0

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 $\mathrm{3CaO+2Al\rightarrow3Ca+Al_2O_3}$ the standard reaction enthalpy $\Delta_rH^0$ = ________ kJ. (Round off to the Nearest Integer).

Correct answer: 230

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 ________ $\times10^{-5}$ mol dm$^{-3}$. (Round off to the Nearest Integer). [Given: Henry\'s law constant = $K_H=8.0\times10^4$ kPa for O$_2$. Density of water with dissolved oxygen = 1.0 kg dm$^{-3}$]

Correct answer: 25

0.01 moles of a weak acid HA ($K_a=2.0\times10^{-6}$) is dissolved in 1.0 L of 0.1 M HCl solution. The degree of dissociation of HA is ________ $\times10^{-5}$ (Round off to the Nearest Integer). [Neglect volume change on adding HA. Assume degree of dissociation <<1]

Correct answer: 2

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 ________ $\times10^{-2}$ M. (Round off to the Nearest Integer).

Correct answer: 24

For a certain first order reaction 32% of the reactant is left after 570 s. The rate constant of this reaction is ________ $\times10^{-3}$ s$^{-1}$. (Round off to the Nearest Integer). [Given: $\log_{10}2=0.301$, ln 10 = 2.303]

Correct answer: 2

The reaction of white phosphorus on boiling with alkali in inert atmosphere resulted in the formation of product 'A'. The reaction of 1 mol of 'A' with excess of AgNO$_3$ in aqueous medium gives ________ mol(s) of Ag. (Round off to the Nearest Integer).

Correct answer: 4

Benzene $\xrightarrow{\text{HNO}_3/\text{H}_2\text{SO}_4}$ nitrobenzene. 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)

Correct answer: 80

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?

1. A.P and Q
2. B.P and R
3. C.Q and R
4. D.None of these✓

Correct answer: D

The area of the triangle with vertices $A(z)$, $B(iz)$ and $C(z+iz)$ is:

1. A.$\frac12|z+iz|^2$
2. B.$\frac12|z|^2$✓
3. C.$\frac12$
4. D.1

Correct answer: B

If $A=\begin{pmatrix}0&\sin\alpha\\\sin\alpha&0\end{pmatrix}$ and $\det\left(A^2-\frac12I\right)=0$, then a possible value of $\alpha$ is:

1. A.$\frac\pi4$✓
2. B.$\frac\pi3$
3. C.$\frac\pi6$
4. D.$\frac\pi2$

Correct answer: A

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:

1. A.0
2. B.1
3. C.−1
4. D.−2✓

Correct answer: D

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:

1. A.2
2. B.4✓
3. C.5
4. D.6

Correct answer: B

If the fourth term in the expansion of $(x+x^{\log_2x})^7$ is 4480, then the value of x where $x\in\mathbb{N}$ is equal to:

1. A.1
2. B.2✓
3. C.3
4. D.4

Correct answer: B

The value of $4+\cfrac{1}{5+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{4+\dots\infty}}}}$ is:

1. A.$2+\frac25\sqrt{30}$✓
2. B.$2+\frac4{\sqrt5}\sqrt{30}$
3. C.$5+\frac25\sqrt{30}$
4. D.$4+\frac4{\sqrt5}\sqrt{30}$

Correct answer: A

If $\cot^{-1}(\alpha)=\cot^{-1}2+\cot^{-1}8+\cot^{-1}18+\cot^{-1}32+\dots$ upto 100 terms, then $\alpha$ is:

1. A.1.00
2. B.1.01✓
3. C.1.02
4. D.1.03

Correct answer: B

The inverse of $y=5^{\log x}$ is:

1. A.$x=5^{\frac1{\log y}}$
2. B.$x=y^{\frac1{\log5}}$✓
3. C.$x=5^{\log y}$
4. D.$x=y^{\log5}$

Correct answer: B

The value of $\lim_{x\to0^+}\frac{\cos^{-1}(x-[x]^2)\cdot\sin^{-1}(x-[x]^2)}{x-x^3}$, where [x] denotes the greatest integer $\le x$ is:

1. A.$\frac\pi4$
2. B.0
3. C.$\frac\pi2$✓
4. D.$\pi$

Correct answer: C

Which of the following statements is incorrect for the function $g(\alpha)$ for $\alpha\in\mathbb{R}$ such that $g(\alpha)=\int_{\pi/6}^{\pi/3}\frac{\sin^\alpha x}{\cos^\alpha x+\sin^\alpha x}dx$

1. A.$g(\alpha)$ is a strictly increasing function
2. B.$g(\alpha)$ is a strictly decreasing function
3. C.$g(\alpha)$ has an inflection point at $\alpha=-\frac12$
4. D.$g(\alpha)$ is an even function✓

Correct answer: D

Which of the following is true for y(x) that satisfies the differential equation $\frac{dy}{dx}=xy-1+x-y$; $y(0)=0$:

1. A.$y(1)=e^{-\frac12}-1$✓
2. B.$y(1)=e^{\frac12}-1$
3. C.$y(1)=e^{\frac12}-e^{-\frac12}$
4. D.$y(1)=1$

Correct answer: A

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:

1. A.(1, 4)
2. B.(0, 2)
3. C.(−1, 0)
4. D.(−2, −2)✓

Correct answer: D

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:

1. A.$5\sqrt3$
2. B.$5\sqrt4$
3. C.$4\sqrt5$
4. D.$3\sqrt5$✓

Correct answer: D

Choose the incorrect statement about the two circles whose equations are given below: $x^2+y^2-10x-10y+41=0$ and $x^2+y^2-16x-10y+80=0$

1. A.Circles have two intersection points.
2. B.Both circles pass through the centre of each other.
3. C.Both circles' centres lie inside region of one another.✓
4. D.Distance between two centres is the average of radii of both the circles.

Correct answer: C

The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is:

1. A.$3x+z=6$
2. B.$x+3z=10$
3. C.$x+3z=0$
4. D.$3x-z=0$✓

Correct answer: D

If the Boolean expression $(p\Rightarrow q)\Leftrightarrow(q*(\sim p))$ is a tautology, then the Boolean expression $p*(\sim q)$ is equivalent to:

1. A.$p\Rightarrow q$
2. B.$p\Rightarrow\sim q$
3. C.$q\Rightarrow p$✓
4. D.$\sim q\Rightarrow p$

Correct answer: C

Let $\vec a=2\hat i-3\hat j+4\hat k$ and $\vec b=7\hat i+\hat j-6\hat k$. If $\vec r\times\vec a=\vec r\times\vec b$, $\vec r\cdot(\hat i+2\hat j+\hat k)=-3$, then $\vec r\cdot(2\hat i-3\hat j+\hat k)$ is equal to:

1. A.8
2. B.10
3. C.12✓
4. D.13

Correct answer: C

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:

1. A.$\frac5{12}$
2. B.$\frac49$
3. C.$\frac{17}{36}$✓
4. D.$\frac12$

Correct answer: C

The sum of possible values of x for $\tan^{-1}(x+1)+\cot^{-1}\left(\frac1{x-1}\right)=\tan^{-1}\left(\frac8{31}\right)$ is:

1. A.$-\frac{33}4$
2. B.$-\frac{32}4$✓
3. C.$-\frac{31}4$
4. D.$-\frac{30}4$

Correct answer: B

The maximum value of z in the following equation $z=6xy+y^2$, where $3x+4y\le100$ and $4x+3y\le75$ for $x\ge0$ and $y\ge0$ is ________.

Correct answer: 904

If $A=\begin{bmatrix}2&3\\0&-1\end{bmatrix}$, then the value of $\det(A^4)+\det(A^{10}-(\mathrm{Adj}(2A))^{10})$ is equal to ________.

Correct answer: 16

If $(2021)^{3762}$ is divided by 17, then the remainder is ________.

Correct answer: 4

If the function $f(x)=\frac{\cos(\sin x)-\cos x}{x^4}$ is continuous at each point in its domain and $f(0)=\frac1k$, then k is ________.

Correct answer: 6

If [·] represents the greatest integer function, then the value of $\left|\int_0^{\sqrt{\pi/2}}\left[[x^2]-\cos x\right]dx\right|$ is ________.

Correct answer: 1

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 ________.

Correct answer: 1

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 ________.

Correct answer: 4

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, $\frac{\text{Probability of occurrence of }E_1}{\text{Probability of occurrence of }E_3}$ is equal to ________.

Correct answer: 6

If $f(x)=\sin\left(\cos^{-1}\left(\frac{1-2^{2x}}{1+2^{2x}}\right)\right)$ and its first derivative with respect to x is $-\frac ba\log_e2$ when $x=1$, where a and b are integers, then the minimum value of $|a^2-b^2|$ is ________.

Correct answer: 481

If $\vec a=\alpha\hat i+\beta\hat j+3\hat k$, $\vec b=-\beta\hat i-\alpha\hat j-\hat k$ and $\vec c=\hat i-2\hat j-\hat k$ such that $\vec a\cdot\vec b=1$ and $\vec b\cdot\vec c=-3$, then $\frac13((\vec a\times\vec b)\cdot\vec c)$ is equal to ________.

Correct answer: 2