JEE Main 2024 — 5 Apr Shift 2 (Previous Year Paper)

A particle moves in x – y plane under the influence of a force $\vec F$ such that its linear momentum is $\vec p(t)=\hat i\cos(kt)-\hat j\sin(kt)$. If k is constant, the angle between $\vec F$ and $\vec p$ will be :

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

Correct answer: C

What is the dimensional formula of $ab^{-1}$ in the equation $\left(P+\frac{a}{V^2}\right)(V-b)=RT$, where letters have their usual meaning.

1. A.$[M^{-1}L^5T^3]$
2. B.$[M^6L^7T^4]$
3. C.$[ML^2T^{-2}]$✓
4. D.$[M^0L^3T^{-2}]$

Correct answer: C

A man carrying a monkey on his shoulder does cycling smoothly on a circular track of radius 9 m and completes 120 revolutions in 3 minutes. The magnitude of centripetal acceleration of monkey is (in m/s²) :

1. A.$57600\pi^2$ m s$^{-2}$
2. B.Zero
3. C.$4\pi^2$ m s$^{-2}$
4. D.$16\pi^2$ m s$^{-2}$✓

Correct answer: D

A heavy box of mass 50 kg is moving on a horizontal surface. If co-efficient of kinetic friction between the box and horizontal surface is 0.3 then force of kinetic friction is :

1. A.1.47 N
2. B.147 N✓
3. C.14.7 N
4. D.1470 N

Correct answer: B

A body is moving unidirectionally under the influence of a constant power source. Its displacement in time t is proportional to :

1. A.$t$
2. B.$t^{3/2}$✓
3. C.$t^2$
4. D.$t^{2/3}$

Correct answer: B

A satellite revolving around a planet in stationary orbit has time period 6 hours. The mass of planet is one-fourth the mass of earth. The radius orbit of planet is : (Given = Radius of geo-stationary orbit for earth is $4.2\times10^4$ km)

1. A.$1.4\times10^4$ km
2. B.$1.05\times10^4$ km✓
3. C.$8.4\times10^4$ km
4. D.$1.68\times10^5$ km

Correct answer: B

Match List-I with List-II : List-I: (A) A force that restores an elastic body of unit area to its original state; (B) Two equal and opposite forces parallel to opposite faces; (C) Forces perpendicular everywhere to the surface per unit area same everywhere; (D) Two equal and opposite forces perpendicular to opposite faces List-II: (I) Bulk modulus; (II) Young's modulus; (III) Stress; (IV) Shear modulus Choose the correct answer from the options given below :

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

Correct answer: B

During an adiabatic process, if the pressure of a gas is found to be proportional to the cube of its absolute temperature, then the ratio of $\frac{C_P}{C_V}$ for the gas is :

1. A.$\frac53$
2. B.$\frac97$
3. C.$\frac32$✓
4. D.$\frac75$

Correct answer: C

If n is the number density and d is the diameter of the molecule, then the average distance covered by a molecule between two successive collisions (i.e. mean free path) is represented by :

1. A.$\sqrt2n\pi d^2$
2. B.$\frac{1}{\sqrt{2n\pi d^2}}$
3. C.$\frac{1}{\sqrt2n\pi d^2}$✓
4. D.$\frac{1}{\sqrt2n^2\pi^2d^2}$

Correct answer: C

The vehicles carrying inflammable fluids usually have metallic chains touching the ground :

1. A.To protect tyres from catching dirt from ground
2. B.To alert other vehicles
3. C.It is a custom
4. D.To conduct excess charge due to air friction to ground and prevent sparking✓

Correct answer: D

A galvanometer of resistance 100Ω when connected in series with 400Ω measures a voltage of upto 10 V. The value of resistance required to convert the galvanometer into ammeter to read upto 10 A is $x\times10^{-2}$ Ω. The value of x is :

1. A.2
2. B.800
3. C.20✓
4. D.200

Correct answer: C

The ratio of heat dissipated per second through the resistance 5Ω and 10Ω in the circuit given below is :

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

Correct answer: B

The electrostatic force $(\vec F_1)$ and magnetic force $(\vec F_2)$ acting on a charge q moving with velocity v can be written :

1. A.$\vec F_1=q\vec E$, $\vec F_2=q(\vec V\times\vec B)$✓
2. B.$\vec F_1=q\vec B$, $\vec F_2=q(\vec B\times\vec V)$
3. C.$\vec F_1=q\vec E$, $\vec F_2=q(\vec B\times\vec V)$
4. D.$\vec F_1=q\vec V\cdot\vec E$, $\vec F_2=q(\vec B\cdot\vec V)$

Correct answer: A

A series LCR circuit is subjected to an ac signal of 200 V, 50 Hz. If the voltage across the inductor (L = 10 mH) is 31.4 V, then the current in this circuit is ______.

1. A.68 A
2. B.63 A
3. C.10 A✓
4. D.10 mA

Correct answer: C

Match List-I with List-II : List-I (EM-Wave): (A) Infra-red; (B) Ultraviolet; (C) X-rays; (D) Gamma rays. List-II (Wavelength Range): (I) $<10^{-3}$ nm; (II) 400 nm to 1 nm; (III) 1 mm to 700 nm; (IV) 1 nm to $10^{-3}$ nm

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

Correct answer: A

Given below are two statements : Statement I : When the white light passed through a prism, the red light bends lesser than yellow and violet. Statement II : The refractive indices are different for different wavelengths in dispersive medium. In the light of the above statements, chose the correct answer 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

Which of the following statement is not true about stopping potential $(V_0)$?

1. A.It is 1/e times the maximum kinetic energy of electrons emitted.
2. B.It increases with increase in intensity of the incident light.✓
3. C.It depends on the nature of emitter material.
4. D.It depends upon frequency of the incident light.

Correct answer: B

The angular momentum of an electron in a hydrogen atom is proportional to : (Where r is the radius of orbit of electron)

1. A.r
2. B.$\sqrt r$✓
3. C.$\frac1{\sqrt r}$
4. D.$\frac1r$

Correct answer: B

The output (Y) of logic circuit given below is 0 only when :

1. A.A = 1, B = 0
2. B.A = 0, B = 1
3. C.A = 0, B = 0✓
4. D.A = 1, B = 1

Correct answer: C

A vernier callipers has 20 divisions on the vernier scale, which coincides with $19^{th}$ division on the main scale. The least count of the instrument is 0.1 mm. One main scale division is equal to ______ mm.

1. A.0.5
2. B.2✓
3. C.5
4. D.1

Correct answer: B

The maximum height reached by a projectile is 64 m. If the initial velocity is halved, the new maximum height of the projectile is ______ m.

Correct answer: 16

A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is $\frac x5$. The value of x is ______.

Correct answer: 2

A hydraulic press containing water has two arms with diameters as mentioned in the figure. A force of 10 N is applied on the surface of water in the thinner arm. The force required to be applied on the surface of water in the thicker arm to maintain equilibrium of water is ______ N.

Correct answer: 1000

A sonometer wire of resonating length 90 cm has a fundamental frequency of 400 Hz when kept under some tension. The resonating length of the wire with fundamental frequency of 600 Hz under same tension ______ cm.

Correct answer: 60

The electric field at point p due to an electric dipole is E. The electric field at point R on equitorial line will be $\frac Ex$. The value of x :

Correct answer: 16

A wire of resistance 20Ω is divided into 10 equal parts, resulting pairs. A combination of two parts are connected in parallel and so on. Now resulting pairs of parallel combination are connected in series. The equivalent resistance of final combination is ______ Ω.

Correct answer: 5

A solenoid of length 0.5 m has a radius of 1 cm and is made up of 'm' number of turns. It carries a current of 5 A. If the magnitude of the magnetic field inside the solenoid is $6.28\times10^{-3}$ T then the value of m is ______.

Correct answer: 500

The current in an inductor is given by I = (3t + 8) where t is in second. The magnitude of induced emf produced in the inductor is 12 mV. The self-inductance of the inductor ______ mH.

Correct answer: 4

In a single slit experiment, a parallel beam of green light of wavelength 550 nm passes through a slit of width 0.20 mm. The transmitted light is collected on a screen 100 cm away. The distance of first order minima from the central maximum will be $x\times10^{-5}$ m. The value of x is :

Correct answer: 275

The shortest wavelength of the spectral lines in the Lyman series of hydrogen spectrum is 915 Å. The longest wavelength of spectral lines in the Balmer series will be ______ Å.

Correct answer: 6588

The number of moles of methane required to produce 11 g $\text{CO}_2$(g) after complete combustion is : (Given molar mass of methane in g mol$^{-1}$ : 16)

1. A.0.35
2. B.0.5
3. C.0.75
4. D.0.25✓

Correct answer: D

Given below are two statements : Statement I : The metallic radius of Na is 1.86 Å and the ionic radius of Na$^+$ is lesser than 1.86 Å. Statement II : Ions are always smaller in size than the corresponding elements. 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 false
2. B.Statement I is incorrect but Statement II is true
3. C.Both Statement I and Statement II are true
4. D.Statement I is correct but Statement II is false✓

Correct answer: D

Match List I with List II : List I: (A) ICl (B) $\text{ICl}_3$ (C) $\text{ClF}_5$ (D) $\text{IF}_7$. List II: (I) T - shape (II) pyramidal (III) Pentagonal bipyramidal (IV) Linear. Choose the correct answer from the options given below :

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

Correct answer: A

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : $\text{NH}_3$ and $\text{NF}_3$ molecule have pyramidal shape with a lone pair of electrons on nitrogen atom. The resultant dipole moment of $\text{NH}_3$ is greater than that of $\text{NF}_3$. Reason (R) : In $\text{NH}_3$, the orbital dipole due to lone pair is in the same direction as the resultant dipole moment of the N – H bonds. F is the most electronegative element. In the light of the above statements, choose the correct answer from the options given below :

1. A.Both (A) and (R) are true and (R) is the correct explanation of (A)✓
2. B.(A) is false but (R) is true
3. C.Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
4. D.(A) is true but (R) is false

Correct answer: A

The correct statements from the following are : (A) The decreasing order of atomic radii of group 13 elements is Tl > In > Ga > Al > B. (B) Down the group 13 electronegativity decreases from top to bottom. (C) Al dissolves in dil. HCl and liberates $\text{H}_2$ but conc. $\text{HNO}_3$ renders Al passive by forming a protective oxide layer on the surface. (D) All elements of group 13 exhibits highly stable +1 oxidation state. (E) Hybridisation of Al in $[\text{Al(H}_2\text{O)}_6]^{3+}$ ion is $sp^3d^2$. Choose the correct answer from the options given below :

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

Correct answer: C

The correct nomenclature for the following compound is : $\text{CH}_2\text{=CH-CH}_2\text{-CH(OH)-CH}_2\text{-CH(CHO)-COOH}$

1. A.2-formyl-4-hydroxyhept-7-enoic acid
2. B.2-formyl-4-hydroxyhept-6-enoic acid✓
3. C.2-carboxy-4-hydroxyhept-7-enal
4. D.2-carboxy-4-hydroxyhept-6-enal

Correct answer: B

Match List I with List II. List I (Pair of compounds): (A) n-propanol and Isopropanol (B) Methoxypropane and ethoxyethane (C) Propanone and propanal (D) Neopentane and Isopentane. List II (Isomerism): (I) Metamerism (II) Chain Isomerism (III) Position Isomerism (IV) Functional Isomerism

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

Correct answer: D

Identify A and B in the given chemical reaction sequence : Benzene + succinic anhydride $\xrightarrow{\text{AlCl}_3}$ A $\xrightarrow{\text{Zn-Hg/HCl}}$ B $\xrightarrow{\text{H}^+}$ α-tetralone

1. A.A = naphthalene-1,4-dione type; B = tetralin-1,4-diol
2. B.A = 3-phenylpropanoic acid; B = 3-phenylpropan-1-ol
3. C.A = 4-oxo-4-phenylbutanoic acid; B = 4-phenylbutanoic acid✓
4. D.A = 1,4-naphthoquinone; B = naphthalene

Correct answer: C

Consider the given chemical reaction : Cyclohexene $\xrightarrow{\text{KMnO}_4-\text{H}_2\text{SO}_4,\ \text{Heat}}$ Product "A". Product "A" is :

1. A.picric acid
2. B.acetic acid
3. C.adipic acid✓
4. D.oxalic acid

Correct answer: C

The quantity of silver deposited when one coulomb charge is passed through $\text{AgNO}_3$ solution :

1. A.1 g of silver
2. B.1 electrochemical equivalent of silver✓
3. C.1 chemical equivalent of silver
4. D.0.1 g atom of silver

Correct answer: B

For the electro chemical cell, If $E^0_{(M^{2+}/M)}=0.46$ V and $E^0_{(X/X^{2-})}=0.34$ V. Which of the following is correct?

1. A.$M+X\to M^{2+}+X^{2-}$ is a spontaneous reaction
2. B.$E_{cell}=0.80$ V
3. C.$E_{cell}=-0.80$ V
4. D.$M^{2+}+X^{2-}\to M+X$ is a spontaneous reaction✓

Correct answer: D

The number of ions from the following that have the ability to liberate hydrogen from a dilute acid is ______. $\text{Ti}^{2+}$, $\text{Cr}^{2+}$ and $\text{V}^{2+}$

1. A.2
2. B.3✓
3. C.1
4. D.0

Correct answer: B

While preparing crystals of Mohr's salt, dil $\text{H}_2\text{SO}_4$ is added to a mixture of ferrous sulphate and ammonium sulphate, before dissolving this mixture in water, dil $\text{H}_2\text{SO}_4$ is added here to :

1. A.prevent the hydrolysis of ferrous sulphate✓
2. B.prevent the hydrolysis of ammonium sulphate
3. C.make the medium strongly acidic
4. D.increase the rate of formation of crystals

Correct answer: A

The number of complexes from the following with no electrons in the $t_2$ orbital is $\text{TiCl}_4$, $[\text{MnO}_4]^-$, $[\text{FeO}_4]^{2-}$, $[\text{FeCl}_4]^-$, $[\text{CoCl}_4]^{2-}$

1. A.1
2. B.4
3. C.3✓
4. D.2

Correct answer: C

The metal atom present in the complex MABXL (where A, B, X and L are unidentate ligands and M is metal) involves $sp^3$ hybridization. The number of geometrical isomers exhibited by the complex is:

1. A.2
2. B.0✓
3. C.4
4. D.3

Correct answer: B

Identify the major product in the following reaction. 1-Bromo-1-methylcyclopentane $\xrightarrow{\overline{\text{O}}\text{H},\ \text{C}_2\text{H}_5\text{OH}}$ Major Product

1. A.Methylenecyclopentane
2. B.Methylcyclopentane
3. C.1-Methylcyclopentene✓
4. D.1-Bromocyclopentene

Correct answer: C

$\text{CH}_3\text{CH}_2\text{-OH}\xrightarrow{\text{(i) Jone's Reagent (ii) KMnO}_4\ \text{(iii) NaOH, CaO, }\Delta}$ P. Consider the above reaction sequence and identify the major product P.

1. A.Methoxymethane
2. B.Methanoic acid
3. C.Methanal
4. D.Methane✓

Correct answer: D

Coagulation of egg, on heating is because of :

1. A.The secondary structure of protein remains unchanged
2. B.Denaturation of protein occurs✓
3. C.Biological property of protein remains unchanged
4. D.Breaking of the peptide linkage in the primary structure of protein occurs

Correct answer: B

In an atom, total number of electrons having quantum numbers n = 4, $|m_l|=1$ and $m_s=-\frac12$ is ______

Correct answer: 6

Number of compounds from the following with zero dipole moment is ______. HF, $\text{H}_2$, $\text{H}_2\text{S}$, $\text{CO}_2$, $\text{NH}_3$, $\text{BF}_3$, $\text{CH}_4$, $\text{CHCl}_3$, $\text{SiF}_4$, $\text{H}_2\text{O}$, $\text{BeF}_2$

Correct answer: 6

Combustion of 1 mole of benzene is expressed at $\text{C}_6\text{H}_6(l)+\frac{15}{2}\text{O}_2(g)\to6\text{CO}_2(g)+3\text{H}_2\text{O}(l)$. The standard enthalpy of combustion of 2 mol of benzene is −'x' kJ. x = ______. Given: 1. standard Enthalpy of formation of 1 mol of $\text{C}_6\text{H}_6(l)$ is 48.5 kJ mol$^{-1}$. 2. Standard Enthalpy of formation of 1 mol of $\text{CO}_2(g)$ is −393.5 kJ mol$^{-1}$. 3. Standard Enthalpy of formation of 1 mol of $\text{H}_2\text{O}(l)$ is −286 kJ mol$^{-1}$.

Correct answer: 6535

Using the given figure, the ratio of $R_f$ values of sample A and sample C is $x\times10^{-2}$. Value of x is ______

Correct answer: 50

Considering acetic acid dissociates in water, its dissociation constant is $6.25\times10^{-5}$. If 5 mL of acetic acid is dissolved in 1 litre water, the solution will freeze at $-x\times10^{-2}$°C, provided pure water freezes at 0°C. x = ______. (Nearest integer) Given : $(K_f)_{\text{water}}=1.86$ K kg mol$^{-1}$, density of acetic acid is 1.2 g mL$^{-1}$, molar mass of water = 18 g mol$^{-1}$, molar mass of acetic acid = 60 g mol$^{-1}$, density of water = 1 g cm$^{-3}$. Acetic acid dissociates as $\text{CH}_3\text{COOH}\rightleftharpoons\text{CH}_3\text{COO}^-+\text{H}^+$

Correct answer: 19

Consider the following single step reaction in gas phase at constant temperature. $2\text{A}_{(g)}+\text{B}_{(g)}\to\text{C}_{(g)}$. The initial rate of the reaction is recorded as $r_1$ when the reaction starts with 1.5 atm pressure of A and 0.7 atm pressure of B. After some time, the rate $r_2$ is recorded when the pressure of C becomes 0.5 atm. The ratio $r_1:r_2$ is ______ $\times10^{-1}$. (Nearest integer)

Correct answer: 315

The fusion of chromite ore with sodium carbonate in the presence of air leads to the formation of products A and B along with the evolution of $\text{CO}_2$. The sum of spin-only magnetic moment values of A and B is ______ B.M. (Nearest integer) [Given atomic number : C : 6, Na : 11, O : 8, Fe : 26, Cr : 24]

Correct answer: 6

In the Claisen-Schmidt reaction to prepare 351 g of dibenzalacetone using 87 g of acetone, the amount of benzaldehyde required is ______ g. (Nearest integer)

Correct answer: 318

The product (C) in the following sequence of reactions has ______ π bonds. Propylbenzene $\xrightarrow{\text{KMnO}_4-\text{KOH},\ \Delta}$ (A) $\xrightarrow{\text{H}_3\text{O}^+}$ (B) $\xrightarrow{\text{Br}_2/\text{FeBr}_3}$ (C)

Correct answer: 4

X g of ethanamine was subjected to reaction with $\text{NaNO}_2$/HCl followed by hydrolysis to liberate $\text{N}_2$ and HCl. The HCl generated was completely neutralised by 0.2 moles of NaOH. X is ______ g.

Correct answer: 9

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $50^{th}$ word is :

1. A.JBBOH
2. B.OBBJH✓
3. C.OBBHJ
4. D.HBBJO

Correct answer: B

For x ⩾ 0, the least value of K, for which $4^{1+x}+4^{1-x}$, $\frac K2$, $16^x+16^{-x}$ are three consecutive terms of an A.P., is equal to :

1. A.8
2. B.4
3. C.10✓
4. D.16

Correct answer: C

If the constant term in the expansion of $\left(\frac{\sqrt[5]3}{x}+\frac{2x}{\sqrt[3]5}\right)^{12}$, $x\ne0$, is $\alpha\times2^8\times\sqrt[5]3$, then 25α is equal to :

1. A.724
2. B.742
3. C.639
4. D.693✓

Correct answer: D

Let A(−1, 1) and B(2, 3) be two points and P be a variable point above the line AB such that the area of △PAB is 10. If the locus of P is ax + by = 15, then 5a + 2b is :

1. A.6
2. B.$-\frac65$
3. C.4
4. D.$-\frac{12}{5}$✓

Correct answer: D

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies:

1. A.r = 0
2. B.$2r^2-4r+1=0$
3. C.$2r^2-8r+7=0$
4. D.$r^2-8r+8=0$✓

Correct answer: D

Let the circle $C_1:x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at (−1, 0) and radius 2. If the line of the common chord of $C_1$ and $C_2$ intersects the y-axis at the point P, then the square of the distance of P from the centre of $C_1$ is :

1. A.2✓
2. B.1
3. C.4
4. D.6

Correct answer: A

Let the set S = {2, 4, 8, 16, ..., 512} be partitioned into 3 sets A, B, C with equal number of elements such that A ∪ B ∪ C = S and A ∩ B = B ∩ C = A ∩ C = φ. The maximum number of such possible partitions of S is equal to:

1. A.1680✓
2. B.1640
3. C.1520
4. D.1710

Correct answer: A

The values of m, n, for which the system of equations x + y + z = 4, 2x + 5y + 5z = 17, x + 2y + mz = n has infinitely many solutions, satisfy the equation :

1. A.$m^2+n^2-mn=39$✓
2. B.$m^2+n^2-m-n=46$
3. C.$m^2+n^2+m+n=64$
4. D.$m^2+n^2+mn=68$

Correct answer: A

Let $f,g:\mathbb{R}\to\mathbb{R}$ be defined as : $f(x)=|x-1|$ and $g(x)=\begin{cases}e^x, & x\ge0\\ x+1, & x\le0\end{cases}$. Then the function f(g(x)) is

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

Correct answer: A

Let $f:[-1,2]\to\mathbb{R}$ be given by $f(x)=2x^2+x+[x^2]-[x]$, where [t] denotes the greatest integer less than or equal to t. The number of points, where f is not continuous, is :

1. A.5
2. B.6
3. C.3
4. D.4✓

Correct answer: D

Let $\beta(m,n)=\int_0^1x^{m-1}(1-x)^{n-1}dx$, m, n > 0. If $\int_0^1(1-x^{10})^{20}dx=a\times\beta(b,c)$, then 100(a + b + c) equals

1. A.1021
2. B.2120✓
3. C.2012
4. D.1120

Correct answer: B

The area enclosed between the curves y = x|x| and y = x − |x| is :

1. A.$\frac43$✓
2. B.1
3. C.$\frac23$
4. D.$\frac83$

Correct answer: A

Consider three vectors $\vec a,\vec b,\vec c$. Let $|\vec a|=2$, $|\vec b|=3$ and $\vec a=\vec b\times\vec c$. If $\alpha\in\left[0,\frac\pi3\right]$ is the angle between the vectors $\vec b$ and $\vec c$, then the minimum value of $27|\vec c-\vec a|^2$ is equal to:

1. A.110
2. B.124✓
3. C.121
4. D.105

Correct answer: B

Let (α, β, γ) be the image of the point (8, 5, 7) in the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{5}$. Then α + β + γ is equal to :

1. A.16
2. B.20
3. C.14✓
4. D.18

Correct answer: C

The coefficients a, b, c in the quadratic equation $ax^2+bx+c=0$ are from the set {1, 2, 3, 4, 5, 6}. If the probability of this equation having one real root bigger than the other is p, then 216 p equals :

1. A.57
2. B.76
3. C.38✓
4. D.19

Correct answer: C

The number of real solutions of the equation x|x + 5| + 2|x + 7| − 2 = 0 is ______

Correct answer: 3

If $1+\frac{\sqrt3-\sqrt2}{2\sqrt3}+\frac{5-2\sqrt6}{18}+\frac{9\sqrt3-11\sqrt2}{36\sqrt3}+\frac{49-20\sqrt6}{180}+\ldots$ upto ∞ $=2+\left(\sqrt{\frac ba}+1\right)\log_e\left(\frac ab\right)$, where a and b are integers with gcd(a, b) = 1, then 11a + 18b is equal to ______

Correct answer: 76

The number of solutions of $\sin^2x+(2+2x-x^2)\sin x-3(x-1)^2=0$, where −π ≤ x ≤ π, is ______

Correct answer: 2

Let a line perpendicular to the line 2x − y = 10 touch the parabola $y^2=4(x-9)$ at the point P. The distance of the point P from the centre of the circle $x^2+y^2-14x-8y+56=0$ is ______

Correct answer: 10

Let a > 0 be a root of the equation $2x^2+x-2=0$. If $\lim_{x\to\frac1a}\frac{16(1-\cos(2+x-2x^2))}{(1-ax)^2}=\alpha+\beta\sqrt{17}$, where α, β ∈ Z, then α + β is equal to ______

Correct answer: 170

Let the mean and the standard deviation of the probability distribution | X | α | 1 | 0 | −3 | | P(X) | $\frac13$ | K | $\frac16$ | $\frac14$ | be μ and σ, respectively. If σ − μ = 2, then σ + μ is equal to ______

Correct answer: 5

Let the maximum and minimum values of $\left(\sqrt{8x-x^2-12}-4\right)^2+(x-7)^2$, $x\in\mathbb{R}$ be M and m, respectively. Then $M^2-m^2$ is equal to ______

Correct answer: 1600

If $f(t)=\int_0^\pi\frac{2x\,dx}{1-\cos^2t\sin^2x}$, 0 < t < π, then the value of $\int_0^{\pi/2}\frac{\pi^2dt}{f(t)}$ equals ______

Correct answer: 1

Let y = y(x) be the solution of the differential equation $\frac{dy}{dx}+\frac{2x}{(1+x^2)^2}y=xe^{\frac{1}{(1+x^2)}}$; y(0) = 0. Then the area enclosed by the curve $f(x)=y(x)e^{-\frac{1}{(1+x^2)}}$ and the line y − x = 4 is ______

Correct answer: 18

Let the point (−1, α, β) lie on the line of the shortest distance between the lines $\frac{x+2}{-3}=\frac{y-2}{4}=\frac{z-5}{2}$ and $\frac{x+2}{-1}=\frac{y+6}{2}=\frac{z-1}{0}$. Then $(\alpha-\beta)^2$ is equal to ______

Correct answer: 25