JEE Main 2024 — 6 Apr Shift 1 (Previous Year Paper)

To find the spring constant (k) of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of k is :

1. A.5%✓
2. B.1%
3. C.3%
4. D.4%

Correct answer: A

Match List I with List II. List I: A. Torque; B. Magnetic field; C. Magnetic moment; D. Permeability of free space. List II: I. $[M^1L^1T^{-2}A^{-2}]$; II. $[L^2A^1]$; III. $[M^1T^{-2}A^{-1}]$; IV. $[M^1L^2T^{-2}]$. Choose the correct answer from the options given below:

1. A.A-III, B-I, C-II, D-IV
2. B.A-IV, B-II, C-III, D-I
3. C.A-IV, B-III, C-II, D-I✓
4. D.A-I, B-III, C-II, D-IV

Correct answer: C

A train starting from rest first accelerates uniformly up to a speed of 80 km/h for time t, then it moves with a constant speed for time 3t. The average speed of the train for this duration of journey will be (in km/h) :

1. A.40
2. B.80
3. C.30
4. D.70✓

Correct answer: D

A light string passing over a smooth light pulley connects two blocks of masses $m_1$ and $m_2$ (where $m_2>m_1$). If the acceleration of the system is $\frac{g}{\sqrt2}$, then the ratio of the masses $\frac{m_1}{m_2}$ is:

1. A.$\frac{1+\sqrt5}{\sqrt5-1}$
2. B.$\frac{\sqrt2-1}{\sqrt2+1}$✓
3. C.$\frac{1+\sqrt5}{\sqrt2-1}$
4. D.$\frac{\sqrt3+1}{\sqrt2-1}$

Correct answer: B

A bullet of mass 50 g is fired with a speed 100 m/s on a plywood and emerges with 40 m/s. The percentage loss of kinetic energy is :

1. A.84%✓
2. B.16%
3. C.32%
4. D.44%

Correct answer: A

Four particles A, B, C, D of mass $\frac m2$, m, 2m, 4m, have same momentum, respectively. The particle with maximum kinetic energy is :

1. A.B
2. B.A✓
3. C.D
4. D.C

Correct answer: B

To project a body of mass m from earth's surface to infinity, the required kinetic energy is (assume, the radius of earth is $R_E$, g = acceleration due to gravity on the surface of earth):

1. A.$2mgR_E$
2. B.$4mgR_E$
3. C.$mgR_E$✓
4. D.$\frac12mgR_E$

Correct answer: C

A small ball of mass m and density ρ is dropped in a viscous liquid of density $\rho_0$. After sometime, the ball falls with constant velocity. The viscous force on the ball is :

1. A.$mg(1-\rho\rho_0)$
2. B.$mg\left(1+\frac{\rho}{\rho_0}\right)$
3. C.$mg\left(\frac{\rho_0}{\rho}-1\right)$
4. D.$mg\left(1-\frac{\rho_0}{\rho}\right)$✓

Correct answer: D

A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample, is:

1. A.$\frac1{32}$
2. B.$\frac{2\sqrt2}{1}$✓
3. C.$\frac{1}{2\sqrt2}$
4. D.$\frac14$

Correct answer: B

The specific heat at constant pressure of a real gas obeying $PV^2=RT$ equation is:

1. A.$\frac R3+C_V$
2. B.$C_V+R$
3. C.$C_V+\frac{R}{2V}$✓
4. D.R

Correct answer: C

An element $\Delta l=\Delta x\hat i$ is placed at the origin and carries a large current I = 10 A. The magnetic field on the y-axis at a distance of 0.5 m from the elements Δx of 1 cm length is:

1. A.$4\times10^{-8}$ T✓
2. B.$10\times10^{-8}$ T
3. C.$8\times10^{-8}$ T
4. D.$12\times10^{-8}$ T

Correct answer: A

Given below are two statements: Statement I: In an LCR series circuit, current is maximum at resonance. Statement II: Current in a purely resistive circuit can never be less than that in a series LCR circuit when connected to same voltage source. In the light of the above statements, choose the correct from the options given below:

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

Correct answer: C

Electromagnetic waves travel in a medium with speed of $1.5\times10^8$ m s$^{-1}$. The relative permeability of the medium is 2.0. The relative permittivity will be:

1. A.2✓
2. B.4
3. C.5
4. D.1

Correct answer: A

In photoelectric experiment energy of 2.48 eV irradiates a photo sensitive material. The stopping potential was measured to be 0.5 V. Work function of the photo sensitive material is :

1. A.1.68 eV
2. B.2.48 eV
3. C.1.98 eV✓
4. D.0.5 eV

Correct answer: C

Which of the following phenomena does not explain by wave nature of light. A. reflection B. diffraction C. photoelectric effect D. interference E. polarization Choose the most appropriate answer from the options given below:

1. A.E only
2. B.B, D only
3. C.C only✓
4. D.A, C only

Correct answer: C

The ratio of the shortest wavelength of Balmer series to the shortest wavelength of Lyman series for hydrogen atom is :

1. A.4 : 1✓
2. B.1 : 4
3. C.2 : 1
4. D.1 : 2

Correct answer: A

The correct truth table for the following logic circuit is :

1. A.A B Y: 0 0 1; 0 1 1; 1 0 0; 1 1 1✓
2. B.A B Y: 0 0 0; 0 1 1; 1 0 0; 1 1 1
3. C.A B Y: 0 0 1; 0 1 1; 1 0 0; 1 1 0
4. D.A B Y: 0 0 0; 0 1 0; 1 0 0; 1 1 1

Correct answer: A

While measuring diameter of wire using screw gauge the following readings were noted. Main scale reading is 1 mm and circular scale reading is equal to 42 divisions. Pitch of screw gauge is 1 mm and it has 100 divisions on circular scale. The diameter of the wire is $\frac{x}{50}$ mm. The value of x is :

1. A.21
2. B.142
3. C.71✓
4. D.42

Correct answer: C

For three vectors $\vec A=(-x\hat i-6\hat j-2\hat k)$, $\vec B=(-\hat i+4\hat j+3\hat k)$ and $\vec C=(-8\hat i-\hat j+3\hat k)$, if $\vec A\cdot(\vec B\times\vec C)=0$, then value of x is ______

Correct answer: 4

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be ______ hours 30 minutes.

Correct answer: 13

A big drop is formed by coalescing 1000 small droplets of water. The ratio of surface energy of 1000 droplets to that of energy of big drop is $\frac{10}{x}$. The value of x is ______

Correct answer: 1

A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s. The maximum velocity of the particle is ______ cm/s.

Correct answer: 12

Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point P is $\frac{x\sigma}{\epsilon_0}$. The value of x is ______ (all quantities are measured in SI units).

Correct answer: 2

A wire of resistance R and radius r is stretched till its radius became r/2. If new resistance of the stretched wire is xR, then value of x is ______

Correct answer: 16

A circular coil having 200 turns, $2.5\times10^{-4}$ m² area and carrying 100 μA current is placed in a uniform magnetic field of 1T. Initially the magnetic dipole moment $(\vec M)$ was directed along $\vec B$. Amount of work, required to rotate the coil through 90° from its initial orientation such that $\vec M$ becomes perpendicular to $\vec B$, is ______ μJ.

Correct answer: 5

When a dc voltage of 100 V is applied to an inductor, a dc current of 5 A flows through it. When an ac voltage of 200 V peak value is connected to inductor, its inductive reactance is found to be $20\sqrt3$ Ω. The power dissipated in the circuit is ______ W.

Correct answer: 250

The refractive index of prism is $\mu=\sqrt3$ and the ratio of the angle of minimum deviation to the angle of prism is one. The value of angle of prism is ______.

Correct answer: 60

Radius of a certain orbit of hydrogen atom is 8.48 Å. If energy of electron in this orbit is E/x, then x = ______ (Given $a_0=0.529$ Å, E = energy of electron in ground state).

Correct answer: 16

The density of 'x' M solution ('x' molar) of NaOH is 1.12 g mL$^{-1}$, while in molality, the concentration of the solution is 3 m (3 molal). Then x is (Given : Molar mass of NaOH is 40 g/mol)

1. A.3.5
2. B.3.8
3. C.2.8
4. D.3.0✓

Correct answer: D

The electron affinity value are negative for A. Be → Be$^-$ B. N → N$^-$ C. O → O$^{2-}$ D. Na → Na$^-$ E. Al → Al$^-$ Choose the most appropriate answer from the options given below :

1. A.D and E only
2. B.A, B and C only✓
3. C.A and D only
4. D.A, B, D and E only

Correct answer: B

Which of the following material is not a semiconductor.

1. A.Silicon
2. B.Copper oxide
3. C.Germanium
4. D.Graphite✓

Correct answer: D

Match List I with List II. List-I (Hybridization): A. $sp^3$; B. $dsp^2$; C. $sp^3d$; D. $sp^3d^2$. List-II (Orientation in Space): I. Trigonal bipyramidal; II. Octahedral; III. Tetrahedral; IV. Square planar.

1. A.A-IV, B-III, C-I, D-II
2. B.A-III, B-IV, C-I, D-II✓
3. C.A-III, B-I, C-IV, D-II
4. D.A-II, B-I, C-IV, D-III

Correct answer: B

Match List I with List II. List-I (Compound/Species): A. $\text{SF}_4$; B. $\text{BrF}_3$; C. $\text{BrO}_3^-$; D. $\text{NH}_4^+$. List-II (Shape/Geometry): I. Tetrahedral; II. Pyramidal; III. See saw; IV. Bent T-Shape.

1. A.A-II, B-III, C-I, D-IV
2. B.A-II, B-IV, C-III, D-I
3. C.A-III, B-IV, C-II, D-I✓
4. D.A-III, B-II, C-IV, D-I

Correct answer: C

Match List I with List II. List-I (Molecule/Species): A. $\text{SO}_2\text{Cl}_2$; B. NO; C. $\text{NO}_2^-$; D. $\text{I}_3^-$. List-II (Property/Shape): I. Paramagnetic; II. Diamagnetic; III. Tetrahedral; IV. Linear.

1. A.A-II, B-III, C-I, D-IV
2. B.A-III, B-I, C-II, D-IV✓
3. C.A-IV, B-I, C-III, D-II
4. D.A-III, B-IV, C-II, D-I

Correct answer: B

At −20°C and 1 atm pressure, a cylinder is filled with equal number of $\text{H}_2$, $\text{I}_2$ and HI molecules for the reaction $\text{H}_2(g)+\text{I}_2(g)\rightleftharpoons2\text{HI}(g)$, the $K_p$ for the process is $x\times10^{-1}$. x = ______ [Given : R = 0.082 L atm K$^{-1}$ mol$^{-1}$]

1. A.0.01
2. B.10✓
3. C.2
4. D.1

Correct answer: B

Functional group present in sulphonic acids is :

1. A.$-\text{SO}_4\text{H}$
2. B.$-\text{SO}_3\text{H}$✓
3. C.$-\text{S(=O)-OH}$
4. D.$-\text{SO}_2$

Correct answer: B

Which of the following statements are correct? A. Glycerol is purified by vacuum distillation because it decomposes at its normal boiling point. B. Aniline can be purified by steam distillation as aniline is miscible in water. C. Ethanol can be separated from ethanol water mixture by azeotropic distillation because it forms azeotrope. D. An organic compound is pure, if mixed M.P. is remained same. Choose the most appropriate answer from the options given below :

1. A.A, B, C only
2. B.A, C, D only✓
3. C.A, B, D only
4. D.B, C, D only

Correct answer: B

Which of the following is metamer of the given compound (X)? (X) = N-phenylcyclohexanecarboxamide ($\text{C}_6\text{H}_5\text{-NH-CO-C}_6\text{H}_{11}$)

1. A.N-cyclohexylbenzamide ($\text{C}_6\text{H}_{11}\text{-NH-CO-C}_6\text{H}_5$)✓
2. B.N-phenylbenzamide
3. C.N-cyclohexylcyclohexanecarboxamide
4. D.4-(cyclohexylamino)benzaldehyde

Correct answer: A

Given below are two statements: Statement I : Gallium is used in the manufacturing of thermometers. Statement II : A thermometer containing gallium is useful for measuring the freezing point (256 K) of brine solution. In the light of the above statements, choose the correct answer from the options given below :

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

Correct answer: D

A conductivity cell with two electrodes (dark side) are half filled with infinitely dilute aqueous solution of a weak electrolyte. If volume is doubled by adding more water at constant temperature, the molar conductivity of the cell will -

1. A.decrease sharply
2. B.increase sharply
3. C.remain same or can not be measured accurately✓
4. D.depend upon type of electrolyte

Correct answer: C

The number of element from the following that do not belong to lanthanoids is Eu, Cm, Er, Tb, Yb and Lu

1. A.3
2. B.4
3. C.1✓
4. D.5

Correct answer: C

Match List I with List II. List-I (Compound): A. Iodoform; B. Carbon tetrachloride; C. CFC; D. DDT. List-II (Uses): I. Fire extinguisher; II. Insecticide; III. Antiseptic; IV. Refrigerants.

1. A.A-I, B-II, C-III, D-IV
2. B.A-III, B-I, C-IV, D-II✓
3. C.A-II, B-IV, C-I, D-III
4. D.A-III, B-II, C-IV, D-I

Correct answer: B

The following complexes $[\text{CoCl(NH}_3)_5]^{2+}$ (A), $[\text{Co(CN)}_6]^{3-}$ (B), $[\text{Co(NH}_3)_5(\text{H}_2\text{O})]^{3+}$ (C), $[\text{Cu(H}_2\text{O)}_4]^{2+}$ (D). The correct order of A, B, C and D in terms of wavenumber of light absorbed is :

1. A.C < D < A < B
2. B.B < C < A < D
3. C.A < C < B < D
4. D.D < A < C < B✓

Correct answer: D

Given below are two statements : Statement I : Picric acid is 2,4,6 - trinitrotoluene. Statement II : Phenol - 2,4 - disulphonic acid is treated with Conc. $\text{HNO}_3$ to get picric acid. In the light of the above statements, choose the most appropriate answer from the options given below :

1. A.Statement I is incorrect but Statement II is correct✓
2. B.Both Statement I and Statement II are incorrect
3. C.Statement I is correct but Statement II is incorrect
4. D.Both Statement I and Statement II are correct

Correct answer: A

In Reimer - Tiemann reaction, phenol is converted into salicylaldehyde through an intermediate. The structure of intermediate is ______

1. A.Sodium 2-(dichloromethyl)phenoxide✓
2. B.Sodium 2-formylphenoxide
3. C.Sodium 2-methylphenoxide
4. D.2-(Dichloromethyl)phenol

Correct answer: A

Which among the following aldehydes is most reactive towards nucleophilic addition reactions?

1. A.$\text{C}_3\text{H}_7\text{-CHO}$
2. B.$\text{C}_2\text{H}_5\text{-CHO}$
3. C.$\text{CH}_3\text{-CHO}$
4. D.$\text{H-CHO}$✓

Correct answer: D

Match List I with List II. List-I (Precipitating reagent and conditions): A. $\text{NH}_4\text{Cl}+\text{NH}_4\text{OH}$; B. $\text{NH}_4\text{OH}+\text{Na}_2\text{CO}_3$; C. $\text{NH}_4\text{OH}+\text{NH}_4\text{Cl}+\text{H}_2\text{S}$ gas; D. dilute HCl. List-II (Cation): I. $\text{Mn}^{2+}$; II. $\text{Pb}^{2+}$; III. $\text{Al}^{3+}$; IV. $\text{Sr}^{2+}$.

1. A.A-III, B-IV, C-I, D-II✓
2. B.A-III, B-IV, C-II, D-I
3. C.A-IV, B-III, C-II, D-I
4. D.A-IV, B-III, C-I, D-II

Correct answer: A

DNA molecule contains 4 bases whose structure are shown below. One of the structures is not correct, identify the incorrect base structure.

1. A.Thymine-type pyrimidine (5-methyluracil) as drawn
2. B.Guanine-type purine (drawn with a methyl group on the imidazole carbon)✓
3. C.Cytosine as drawn
4. D.Adenine as drawn

Correct answer: B

Frequency of the de-Broglie wave of electron in Bohr's first orbit of hydrogen atom is ______ $\times10^{13}$ Hz (nearest integer). [Given : $R_H$ (Rydberg constant) = $2.18\times10^{-18}$ J, h (Plank's constant) = $6.6\times10^{-34}$ J s.]

Correct answer: 661

Number of molecules from the following which can exhibit hydrogen bonding is ______ (nearest integer). $\text{CH}_3\text{OH}$, $\text{H}_2\text{O}$, $\text{C}_2\text{H}_6$, $\text{C}_6\text{H}_6$, o-nitrophenol, HF, $\text{NH}_3$

Correct answer: 5

An ideal gas, $\bar C_v=\frac52R$, is expanded adiabatically against a constant pressure of 1 atm untill it doubles in volume. If the initial temperature and pressure is 298 K and 5 atm, respectively then the final temperature is ______ K (nearest integer). [$\bar C_v$ is the molar heat capacity at constant volume]

Correct answer: 274

The major product of the following reaction is P. $\text{CH}_3\text{C}\equiv\text{C-CH}_3\xrightarrow{\text{(i) Na/liq. NH}_3\ \text{(ii) dil. KMnO}_4,\ 273\text{ K}}$ P. Number of oxygen atoms present in product 'P' is ______ (nearest integer)

Correct answer: 2

Consider the dissociation of the weak acid HX as given below $\text{HX(aq)}\rightleftharpoons\text{H}^+(aq)+\text{X}^-(aq)$, $K_a=1.2\times10^{-5}$ [$K_a$ : dissociation constant]. The osmotic pressure of 0.03M aqueous solution of HX at 300 K is ______ $\times10^{-2}$ bar (nearest integer). [Given : R = 0.083 L bar mol$^{-1}$ K$^{-1}$]

Correct answer: 76

Time required for 99.9% completion of a first order reaction is ______ times the time required for completion of 90% reaction. (nearest integer)

Correct answer: 3

Among CrO, $\text{Cr}_2\text{O}_3$ and $\text{CrO}_3$, the sum of spin-only magnetic moment values of basic and amphoteric oxides is ______ $10^{-2}$ BM (nearest integer). (Given atomic number of Cr is 24)

Correct answer: 877

The difference in the 'spin-only' magnetic moment values of $\text{KMnO}_4$ and the manganese product formed during titration of $\text{KMnO}_4$ against oxalic acid in acidic medium is ______ BM. (nearest integer)

Correct answer: 6

The major products from the following reaction sequence are product A and product B. Cyclohexene $\xrightarrow{\text{(i) Br}_2\ \text{(ii) CH}_2\text{=CH-CH}_2\text{O}^-\text{Na}^+\text{ (1.0 eq.)}}$ A; Cyclohexene $\xrightarrow{\text{(i) Br}_2\ \text{(ii) alc. KOH (3 eq.)}}$ B. The total sum of π electrons in product A and product B are ______ (nearest integer)

Correct answer: 8

9.3 g of pure aniline upon diazotisation followed by coupling with phenol gives an orange dye. The mass of orange dye produced (assume 100% yield/conversion) is ______ g. (nearest integer)

Correct answer: 20

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

1. A.48
2. B.56
3. C.24
4. D.16✓

Correct answer: D

Let $A=\{n\in[100,700]\cap\mathbb{N}:n$ is neither a multiple of 3 nor a multiple of 4$\}$. Then the number of elements in A is

1. A.290
2. B.280
3. C.300✓
4. D.310

Correct answer: C

Let a variable line of slope m > 0 passing through the point (4, −9) intersect the coordinate axes at the points A and B. The minimum value of the sum of the distances of A and B from the origin is

1. A.30
2. B.25✓
3. C.15
4. D.10

Correct answer: B

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are m and n, respectively, then $m+n^2$ is equal to

1. A.408✓
2. B.414
3. C.396
4. D.312

Correct answer: A

The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

1. A.1.8
2. B.1.94
3. C.$\sqrt{3.96}$✓
4. D.$\sqrt{3.86}$

Correct answer: C

Let the relations $R_1$ and $R_2$ on the set X = {1, 2, 3, ..., 20} be given by $R_1=\{(x,y):2x-3y=2\}$ and $R_2=\{(x,y):-5x+4y=0\}$. If M and N be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then M + N equals

1. A.12
2. B.16
3. C.8
4. D.10✓

Correct answer: D

The function $f:\mathbb{R}\to\mathbb{R}$, $f(x)=\frac{x^2+2x-15}{x^2-4x+9}$, $x\in\mathbb{R}$ is

1. A.one-one but not onto.
2. B.both one-one and onto.
3. C.onto but not one-one.
4. D.neither one-one nor onto.✓

Correct answer: D

If $f(x)=\begin{cases}x^3\sin\left(\frac1x\right), & x\ne0\\ 0, & x=0\end{cases}$ then

1. A.$f''\left(\frac2\pi\right)=\frac{24-\pi^2}{2\pi}$✓
2. B.$f''\left(\frac2\pi\right)=\frac{12-\pi^2}{2\pi}$
3. C.$f''(0)=1$
4. D.$f''(0)=0$

Correct answer: A

The interval in which the function $f(x)=x^x$, x > 0, is strictly increasing is

1. A.$\left(0,\frac1e\right]$
2. B.$(0,\infty)$
3. C.$\left[\frac1e,\infty\right)$✓
4. D.$\left[\frac1{e^2},1\right)$

Correct answer: C

$\int_0^{\pi/4}\frac{\cos^2x\sin^2x}{(\cos^3x+\sin^3x)^2}dx$ is equal to

1. A.1/6✓
2. B.1/3
3. C.1/12
4. D.1/9

Correct answer: A

Let y = y(x) be the solution of the differential equation $(1+x^2)\frac{dy}{dx}+y=e^{\tan^{-1}x}$, y(1) = 0. Then y(0) is

1. A.$\frac12\left(e^{\pi/2}-1\right)$
2. B.$\frac12\left(1-e^{\pi/2}\right)$✓
3. C.$\frac14\left(1-e^{\pi/2}\right)$
4. D.$\frac14\left(e^{\pi/2}-1\right)$

Correct answer: B

Let y = y(x) be the solution of the differential equation $(2x\log_ex)\frac{dy}{dx}+2y=\frac3x\log_ex$, x > 0 and $y(e^{-1})=0$. Then, y(e) is equal to

1. A.$-\frac3e$✓
2. B.$-\frac3{2e}$
3. C.$-\frac2{3e}$
4. D.$-\frac2e$

Correct answer: A

The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is

1. A.$8\sqrt3$
2. B.$4\sqrt3$✓
3. C.$5\sqrt3$
4. D.$6\sqrt3$

Correct answer: B

A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the remaining are manufactured at plant B. 80% of the motorcycles manufactured at plant A are rated of the standard quality, while 90% of the motorcycles manufactured at plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If p is the probability that it was manufactured at plant B, then 126p is

1. A.54✓
2. B.66
3. C.64
4. D.56

Correct answer: A

Let $x_1,x_2,x_3,x_4$ be the solution of the equation $4x^4+8x^3-17x^2-12x+9=0$ and $(4+x_1^2)(4+x_2^2)(4+x_3^2)(4+x_4^2)=\frac{125}{16}m$. Then the value of m is

Correct answer: 221

Let the first term of a series be $T_1=6$ and its $r^{th}$ term $T_r=3T_{r-1}+6^r$, r = 2, 3, ..., n. If the sum of the first n terms of this series is $\frac15(n^2-12n+39)(4\cdot6^n-5\cdot3^n+1)$, then n is equal to ______

Correct answer: 6

If the second, third and fourth terms in the expansion of $(x+y)^n$ are 135, 30 and $\frac{10}{3}$, respectively, then $6(n^3+x^2+y)$ is equal to ______

Correct answer: 806

Let a conic C pass through the point (4, −2) and P(x, y), x ≥ 3, be any point on C. Let the slope of the line touching the conic C only at a single point P be half the slope of the line joining the points P and (3, −5). If the focal distance of the point (7, 1) on C is d, then 12d equals ______

Correct answer: 75

Let $L_1$, $L_2$ be the lines passing through the point P(0, 1) and touching the parabola $9x^2+12x+18y-14=0$. Let Q and R be the points on the lines $L_1$ and $L_2$ such that the △PQR is an isosceles triangle with base QR. If the slopes of the lines QR are $m_1$ and $m_2$, then $16(m_1^2+m_2^2)$ is equal to ______

Correct answer: 68

Let αβγ = 45; α, β, γ ∈ R. If x(α, 1, 2) + y(1, β, 2) + z(2, 3, γ) = (0, 0, 0) for some x, y, z ∈ R, xyz ≠ 0, then 6α + 4β + γ is equal to ______

Correct answer: 55

For n ∈ N, if $\cot^{-1}3+\cot^{-1}4+\cot^{-1}5+\cot^{-1}n=\frac\pi4$, then n is equal to ______

Correct answer: 47

Let $r_k=\frac{\int_0^1(1-x^7)^kdx}{\int_0^1(1-x^7)^{k+1}dx}$, k ∈ N. Then the value of $\sum_{k=1}^{10}\frac{1}{7(r_k-1)}$ is equal to ______

Correct answer: 65

Let $\vec a=2\hat i-3\hat j+4\hat k$, $\vec b=3\hat i+4\hat j-5\hat k$ and a vector $\vec c$ be such that $\vec a\times(\vec b+\vec c)+\vec b\times\vec c=\hat i+8\hat j+13\hat k$. If $\vec a\cdot\vec c=13$, then $(24-\vec b\cdot\vec c)$ is equal to ______

Correct answer: 46

Let P be the point (10, −2, −1) and Q be the foot of the perpendicular drawn from the point R(1, 7, 6) on the line passing through the points (2, −5, 11) and (−6, 7, −5). Then the length of the line segment PQ is equal to ______

Correct answer: 13