COMEDK 2026 Afternoon Shift

In the equation $X=G^{-1 / 2} h^{1 / 2} c^{5 / 2}$, where G- universal gravitation constant, $h$ - Planck's constant and c - velocity of light, the dimensions of X are that of

A boy, standing at a certain height, kicks a football horizontally with a velocity of $19.6 \mathrm{~ms}^{-1}$. What will be the ratio of horizontal and vertical components of velocities after $2 s$ ?

An object is dropped from a certain point A at a height ' h ' from the ground. During it's journey straight downwards, the object passes points $B$ and $C$ such that the ratio of time taken $t_1$ to cover $A B$ and $t_2$ to cover $B C$ is $1:(\sqrt{2}-1)$. What is the ratio of distances $A B: B C$ ?

A particle moves along a parabolic path $y=9 x^2$ in such a way that the x component of velocity remains constant. If, the acceleration of the particle is $2 j m s^{-2}$, find the x component of velocity.

A block of metal, of 25 g mass moves down without acceleration when the plane is inclined at an angle of $30^{\circ}$. When the inclination is increased by $30^{\circ}$, find the downward acceleration of the block.

A block of mass 1.5 kg moves along the floor of a hall with a speed of $5 \mathrm{~ms}^{-1}$. It strikes an uncompressed spring and compresses it till the block becomes motionless. If the force constant of the spring is $10000 \mathrm{Nm}^{-1}$ and the spring is compressed by 5 cm , calculate the effective force of kinetic friction.

A bullet, fired into a door gets embedded exactly at it's centre, causing the door to rotate about it's vertical axis, practically without friction, with an angular velocity of $0.625 \mathrm{rads}^{-1}$. The door is 1.0 m wide and weighs 12 kg . If the mass of the bullet is 10 g , find the speed with which it was fired. (Hint: The moment of inertia of the door about the vertical axis at one end is $\frac{M L^2}{3}$.

A singer, during his performance, stands on the edge of a circular turntable, and begins to walk along its edge with a speed of $1.5 \mathrm{~ms}^{-1}$ relative to the ground. The turn table is mounted on a frictionless vertical axle. Its radius R =3m and its moment of inertia about the axle is $150 \mathrm{~kg} \mathrm{~m}^2$. It is initially at rest. If the mass of the singer is 75 kg , the time taken by the man to complete one revolution is:

A wire, made of a certain material of length-l and area of cross section-a can withstand a maximum load $=\mathrm{W}$ without breaking. If, another wire of the same material and crosssectional area is used with double the original length, what will be the maximum load that the wire can withstand, without breaking?

A vessel of volume $27 \times 10^4 \mathrm{cc}$, contains a mixture of Hydrogen (molar mass $=2 \mathrm{~g} \mathrm{~mol}^{-1}$ ) and oxygen (molar mass $=32 \mathrm{~g} \mathrm{~mol}^{-1}$ ) gas at standard temperature and pressure. If the mass of hydrogen is 16 g , find the mass of oxygen gas contained in the vessel.

Calculate the vapour pressure that can help the formation of a spherical droplet of water of radius $6.25 \times 10^{-5} \mathrm{~m}$ at $22^{\circ} \mathrm{C}$. Given: The surface tension of water at the given temperature is $7.28 \times 10^{-2} \mathrm{Nm}^{-1}$.

A block of a certain material is heated to a temperature of $500^{\circ} \mathrm{C}$ and then placed on a large ice block. If 1.455 kg of ice melts, find the mass of the block.

Specific heat of the material is $0.39 \mathrm{Jg}^{-1} \mathrm{C}^{-1}$ and heat of fusion of water is $335 \mathrm{Jg}^{-1}$.

Following graph shows four different processes, adiabatic, isothermal, isobaric and isochoric for an ideal gas, from the same initial state. Study the graph carefully and state which of the following statements is correct?

Find the mass of oxygen gas with which $1.882 \times 10^{23}$ degrees of freedom are associated at N.T.P. Given: Molar mass of diatomic gas, oxygen is $32 \mathrm{~g} \mathrm{~mol}^{-1}$ and oxygen molecule possess three translational and two rotational degrees of freedom.

Two neutral bodies of masses $m_1$ and $m_2$ are kept at a distance of $r \mathrm{~cm}$ from one another in a vacuum medium. A gravitational force Fracts between the two bodies. The entire set-up is then transferred, as it is, to a water medium. The force between the bodies is then $\mathrm{F}_{\mathrm{w}}$. The ratio of $F_v$ to $F_w$ is:

Two particles, one heavy and the other light, placed at 50 cm from each other, are under the influence of gravitational force of one another. If mass of the heavier particle is 4 kg and its acceleration under the influence of gravitational force is $5 \times 10^{-10} \mathrm{~ms}^{-2}$, find the mass of the lighter particle.

Force constant of interatomic bond, in a certain element, is $7.1 \mathrm{Nm}^{-1}$. If the atom oscillates in SHM in a certain direction, what is its frequency?

Given: Mole weight of the given element is 108 g and Avagadro's number $=6.023 \times 10^{23} \mathrm{~g} \mathrm{~mol}^{-1}$

Resonance is produced between a turning fork and a resonance column tube with upper end open and lower end closed by water surface. If the frequency of the tuning fork is 800 Hz , and first two resonances are observed at lengths $9.75 \mathrm{~cm}, 31.25 \mathrm{~cm}$,

Find the length at which the third resonance occurs? Also find the speed of sound.

If the intensity of the central maximum in the Young's double slit experiment is $\mathrm{I}_0$, what will be the intensity at the same region when one of the slits is blocked by an opaque object?

In a single slit diffraction experiment, the diffraction pattern is observed on a screen placed at a distance of 2 m from the slit of 1 mm width. If the distance between the first dark fringe on either side of the central to right fringe is 2.2 mm . what is the wave length of the monochromatic light from a distance source, used in this experiment?

The width of the fringes obtained with a light of wave length $6.2 \times 10^{-8} \mathrm{~m}$ is 1.82 mm . If the whole apparatus is immersed in a liquid of refractive index 1.3 , what will be the width of the resulting fringe?

The ratio of the angle of deviation produced by a thin prism, when it is placed in air to the angle of deviation produced when it is immersed in water of refractive index $\frac{4}{3}$ is:

The distance between the objective and eye piece of astronomical telescope in normal adjustment is 27 cm and its magnifying power is 8 . What is the focal length of the eye piece?

The critical angle for a typical glass air interface is $42^{\circ}$. If a ray of light falls normally on one of the faces of the prism of angle $45^{\circ}$. The emergent ray will:

An object placed 40 cm in front of a thin convex lens is moved to 60 cm from the lens. If the focal length of the lens is 30 cm the ratio of magnification of the image at the initial position to the final position is:

A capacitor of capacitance $8 \mu \mathrm{~F}$ is fully charged by connecting it to a source of 200 V . It is then disconnected from the supply and connected to an uncharged capacitor of capacitance $4 \mu \mathrm{~F}$. The electrostatic energy lost in this sharing is:

A metallic circular loop is placed with its plane perpendicular to a uniform magnetic field of 0.3 T . If the radius of the loop decreases at a constant rate of $2 \mathrm{~mm} \mathrm{~s}^{-1}$, what will be the induced emf in the loop, when the radius of the loop becomes 5 cm .

When a current of 2.5 A passes through the primary coil of a transformer of 200 number turns, the magnetic flux linked with the secondary coil having 400 turns is $600 \times 10^{-6} \mathrm{~T} \mathrm{~m}^2$. Find the induced emf in the secondary coil, when the current in the primary coil increases at a rate of $0.2 \mathrm{As}^{-1}$

A pith ball of mass ' $m$ ' gram and charge ' $Q$ ' is suspended using a mass less silk thread near a large charged conducting metal sheet of area ' $A$ ' and surface charged density ' $\sigma$ '. If the silk thread makes an angle $\Theta$ with the metal sheet, then:

Two point charges $P=+25 \mu C$ and $Q=-16 \mu C$ are placed 5 cm apart. Find the position of the point at which the resultant electric field is zero:

Bodies $P, Q, R, S$ are labelled as having charges $Q_P=0.5 \times 10^{-19} C, Q_Q=0.7 \times 10^{-19} C$, $Q_R=2.1 \times 10^{-19} C, Q_S=4.8 \times 10^{-19} C$ respectively.

Select the body having the correct charge.

[Given electronic charge $=e=1.6 \times 10^{-19} \mathrm{C}$ ]

In a uniform electric field $10^6 N C^{-1}$ an electric dipole of length 4 cm is placed with its axis making an angle $60^{\circ}$ with the electric field. If the dipole experiences a torque of $8 \sqrt{3} \mathrm{~N} \mathrm{~m}$.

Find the potential energy of the dipole.

Uniform electric field of $5 \times 10^3 \mathrm{NC}^{-1}$ is maintained in the positive Y direction. Now a point charge of $2 \times 10^{-4} \mathrm{C}$ at rest is released from the origin. The kinetic energy attained by the charge when it is 5 m from the origin is:

A charge of $5 \mu \mathrm{C}$ is placed at the centre of a spherical shell $S_1$ of radius 10 cm . Now this system is enclosed inside another spherical shell $S_2$ of radius 20 cm . The ratio of the electrical flux through the surface $S_2$ to $S_1$ is :

The current through the circular coil is halved and the radius of the coil is doubled. If $B_1$ and $B_2$ are respectively the initial and final magnetic field strength, then:

An electric field and magnetic field $1.8 \times 10^4 \mathrm{Vm}^{-1}$ and $6 \times 10^{-3} \mathrm{~T}$ respectively are applied simultaneously on an electron beam such that path of the beam remains undeviated, then the speed of the electron will be:

A circular coil of radius $r=10 c m$ having 300 turns carries a current of 2 A . The coil is suspended vertically in a uniform magnetic field of strength 0.7 T . If the plane of the coil makes an angle $30^{\circ}$ with the magnetic field, the torque needed to prevent it from turning is:

A galvanometer of resistance $50 \Omega$ is having 30 divisions and a current sensitivity $10 \mathrm{~mA} /$ div. What should be the shunt resistance so that it can be converted into an ammeter of range 10 A ?

In a given circuit, the instantaneous values of the alternating voltage and current are $V=0.5 \sin \left(80 \pi t+\frac{\pi}{3}\right)$ volt and $I=0.5 \sin (80 \pi t)$ ampere respectively. Find the average power consumed in that circuit.

A source of alternating emf $\varepsilon=\varepsilon_0 \sin (\omega t)$ is connected to a capacitor. Then the instantaneous current in the circuit is: ˋ

An electronic device operates at 2 MHz . The oscillating circuit has an inductance $20 \times 10^{-5} \mathrm{H}$. What is the capacitive reactance of the resonant circuit?

An electric coil is rated $400 \mathrm{~W}, 200 \mathrm{~V}$. It is cut into two equal parts and connected in parallel to the same source of 200 V . Calculate the percentage increase in energy produced per second.

On both sides of a magnetic needle, two short magnets A and B are placed on the same horizontal line which is perpendicular to the magnetic meridian. The south poles of $A$ and $B$ are facing each other, which are 10 cm and 20 cm respectively from the magnetic needle. If the needle remains undeflected, the ratio of the magnetic moment of $A$ to that $B$ is:

The material selected for making a permanent magnet should have:

A parallel combination of ' $n$ ' cells of emf ' $E$ ' and internal resistance ' $r$ ' each, are connected across the external resistance ' $R$ '. If the external resistance ' $R$ ' is negligibly small, then the current ' $I$ ' through the external resistance is:

The dimensional formula for specific resistance is:

The voltage - current graph for a metal wire of uniform area of cross section at two different temp $T$ and $T^{\prime}$ is shown.

Then choose the correct statement:

Pick out the correct statement from the following;

Find the current through the $40 \Omega$ resistor in the given circuit having a diode, three resistors and two cells.

What is the minimum wavelength of radiation required to detect a p-n junction diode made of a semiconductor having band gap 3.3 eV .

[Planck's constant $h=6.6 \times 10^{34} \mathrm{~J} . \mathrm{s}$ ]

If the ratio of the nuclear radii of two atoms is $2: 3$ then the ratio of their mass numbers is:

What is the frequency of the electron in the first orbit of hydrogen atom of orbital radius $0.5 \times 10^{-10} \mathrm{~m}$, if its orbital velocity in that orbit is $2.2 \times 10^6 \mathrm{~ms}^{-1}$.

The main function of cadmium used in the nuclear reactor is:

A hydrogen atom absorbs energy and rises to $n=3$ state from its ground state $n=1$. If the potential energy of the atom at its ground state is -13.6 eV , find the wave length emitted by it when it returns to its ground state:

{Planck's constant $=6.6 \times 10^{34} \mathrm{~J} \mathrm{~s}$ }

The atomic mass of an element $10 X^{20}$ is 19.98170 amu. The binding energy per nucleon of that element is: [given mass of neutron = 1.00867amu and mass of proton = 1.00783 amu and 1amu = 931 MeV ]

The basic principle used behind the working of electron microscope is:

When metal of work function 1.4 eV is exposed to a radiation, the maximum kinetic energy of the electron emitted is 0.4 eV . The stopping potential required is:

The variation of the stopping potential ( $\mathrm{V}_{\mathrm{o}}$ ) with the frequency of incident radiation( $n$ ) is as given below.

[no - threshold frequency. h - Planck's constant e-electronic charge] then the slope of the graph AB with the frequency axis is:

The frequency of photon which is emitted during a transition of electron of $\mathrm{He}^{+}$ion from fifth energy level to third energy level will be:

How many molecules of $\mathrm{CO}_2(\mathrm{~g})$ are obtained on reaction of 24 grams of methane with 4 moles of oxygen?

500 mL of an aqueous solution of glucose $\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6$ (Molar mass $180 \mathrm{gmol}^{-1}$ ) contains $6.02 \times 10^{22}$ molecules.

The concentration of the solution will be:

Which of the following statement is correct?

The standard enthalpies of formation of $\mathrm{CH}_4(\mathrm{~g}), \mathrm{CO}_2(\mathrm{~g})$ and $\mathrm{H}_2 \mathrm{O}(\mathrm{l})$ are $-74.8 \mathrm{~kJ} \mathrm{~mol}^{-1},-393.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $-285.8 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively. Then the enthalpy change for the given reaction in $\mathrm{kJ} \mathrm{mol}^{-1}$ will be:

$$ 2 \mathrm{CH}_4(\mathrm{~g})+4 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_2(\mathrm{~g})+4 \mathrm{H}_2 \mathrm{O}(\mathrm{l}) $$

For an ideal gas undergoing an isothermal change, there is $\_\_\_\_$

Identify the INCORRECT statement

An equilibrium mixture taken in 2 litre vessel of the reaction: $2 \mathrm{SO}_2(\mathrm{~g})+\mathrm{O}_2(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_3(\mathrm{~g})$ has 4 moles of $\mathrm{SO}_2, 3$ moles of $\mathrm{O}_2$ and 6 moles of $\mathrm{SO}_3$ then the value of equilibrium constant $\left(\mathrm{K}_c\right)$ will be:

At $30^{\circ} \mathrm{C}$ the solubility of $\mathrm{PbI}_2$ salt in 0.2 M KI solution will be $X$, if the solubility product of $\mathrm{PbI}_2$ at $30^{\circ} \mathrm{C}$ is $2.4 \times 10^{-8}$. Identify the value of $X$.

When 1 mole of benzene is mixed with 1 mole of toluene, the vapours will contain

Identify the law which is stated as "For any solutions, the partial vapour pressure of each volatile component in the solution is directly proportional to its mole fraction".

The system that forms maximum boiling azeotrope is:

A $5 \%$ solution (by mass) of cane sugar in water has a freezing point of 271 K . The freezing point of a $5 \%$ solution (by mass) of glucose in water is: [freezing point of pure water: 273.15 K ]

In the redox reaction, taking place in acidic medium: $\mathrm{X} \mathrm{MnO}_4^{-}(\mathrm{aq})+\mathrm{YSO}_2(\mathrm{~g}) \rightarrow \mathrm{Mn}^{+2}(\mathrm{aq})+\mathrm{HSO}_4^{-}(\mathrm{aq})$, the ratio of $X: Y$ in a stoichiometrically balanced equation will be

$x$ moles of $\mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7$ oxidises 1 mole of ferrous oxalate, in acidic medium. Hence ' $x$ ' is:

$$ \begin{aligned} &\text { Using the data given below, the strongest reducing agent is: }\\ &\begin{array}{ll} \mathrm{E}_{\mathrm{Cr}_2 \mathrm{O}_7}^{\mathrm{o}}{ }^{2-} / \mathrm{Cr}^{3+}=1.33 \mathrm{~V} & \mathrm{E}_{\mathrm{MnO}_4^{-} / \mathrm{Mn}^{2+}}^{\mathrm{o}}=1.51 \mathrm{~V} \\ \mathrm{E}_{\mathrm{Cl}_2 / \mathrm{Cl}^{-}}^{\mathrm{O}}=1.36 \mathrm{~V} & \mathrm{E}_{\mathrm{Cr}^{3+} / \mathrm{Cr}}^{\mathrm{O}}=-0.74 \mathrm{~V} \end{array} \end{aligned} $$

The quantity of Ca that can be produced from molten $\mathrm{CaCl}_2$, with the same quantity of electricity (in coulombs) required to produce 4.8 g of Mg from molten $\mathrm{MgCl}_2$ is: [Atomic mass of $\mathrm{Mg}=24 \mathrm{u}$; Atomic mass of $\mathrm{Ca}=40 \mathrm{u}$ ]

The $\Delta \mathrm{G}^{\circ}$ for the reaction, $C d^{2+}(a q)+Z n(s) \rightarrow Z n^{2+}(a q)+C d(s)$ is:

Which of the following is always true about a spontaneous cell reaction in a galvanic cell?

Resistance of 0.2 M solution of an electrolyte is $50 \Omega$. The conductivity of the solution is $1.3 \mathrm{Sm}^{-1}$. If the resistance of 0.4 M solution of the same electrolyte is $260 \Omega$, its molar conductivity is:

The initial pressure of the system before decomposition for a first order gas phase reaction $A_{(g)} \rightarrow B_{(g)}+C_{(g)}$ was $\mathrm{P}_{\mathrm{i}}$. After lapse of time ' t ', total pressure of the system increased by ' x ' units and became $\mathrm{P}_{\mathrm{t}}$. The rate constant ' k ' for the reaction is given as:

A first order reaction is $50 \%$ complete in 30 minutes at 300 K and in 10 minutes at 320 K . The activation energy of the reaction ( $E_a$ ) is: $\left[R=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} ; \log 2=0.3010 ; \log 3=0.4771\right]$

Which of the following factors is altered by the addition of a catalyst during a chemical reaction?

In the synthesis of $\mathrm{NH}_3$ from $\mathrm{H}_2$ and $\mathrm{N}_2$, if $6 \times 10^{-2}$ mole of hydrogen disappears in 10 minutes, the number of moles of $\mathrm{NH}_3$ formed in 0.3 minutes is:

$$ \text { When the concentration of the reactant in a given reaction is halved and if the rate of reaction is halved, the order of the reaction is: } $$

Which of the following is an INCORRECT match?

The number of bond pairs and lone pairs of electrons in the molecule $I F_5$ is:

According to Molecular orbital theory, which of the following is correct with respect to bond order?

The element with the highest third ionisation enthalpy is:

Pick out the correct option

Assertion(A): Mercury is not considered as a transition element

Reason (R): Mercury is a liquid

The product and its colour when $\mathrm{MnO}_2$ is fused with KOH in presence of $\mathrm{O}_2$ :

Which of the following does not correctly represent the order of the property indicated against it?

Which one of the following complex-isomerism pair matches correctly?

Identify the complex which exhibits all 3 characteristics; paramagnetic; high spin configuration; octahedral geometry

Which of the following is an INCORRECT statement?

$$ \text { What is the major product }[Z] \text { formed when compound }[C] \text { undergoes the following reactions: } $$

$$ \text { Among the given resonating structures of molecules negative mesomeric effect is represented by: } $$

$$ \text { Identify the aromatic compounds among the given set based on Huckel's rule: } $$

The statements given below contains assertion and reason. Choose the correct option

Assertion (A): Propene reacts with HBr in presence of organic peroxide gives 1-bromopropane.

Reason(R): The reaction occurs through carbocation intermediate

$$ \begin{aligned} &\text { What is the major product }\left[\mathrm{P}_3\right] \text { formed when } n \text { - Hexane undergoes the given series of reactions: }\\ &\begin{aligned} \text { n-Hexane } & \frac{\mathrm{V}_2 \mathrm{O}_5}{20 \mathrm{~atm}, 773 \mathrm{k}}\left[\mathrm{P}_1\right] \\ {\left[\mathrm{P}_1\right] } & \xrightarrow[\triangle]{\mathrm{C}_2 \mathrm{H}_5 \mathrm{Cl}, \text { Anhyd.AlCl }}\left[\mathrm{P}_2\right] \\ {\left[\mathrm{P}_2\right] } & \xrightarrow[333 \mathrm{~K}]{\text { Conc. } \mathrm{HNO}_3+\mathrm{H}_2 \mathrm{SO}_4}\left[\mathrm{P}_3\right] \text { Major product } \end{aligned} \end{aligned} $$

An unsaturated organic compound $\left(\mathrm{C}_3 \mathrm{H}_6\right)$, undergoes the following series of reactions:

Identify compound [D]

Arrange the given alkanes in increasing order of their boiling points.

(A) 2, 2-dimethylpropane

(B) 2-methylbutane

(C) n-pentane

(D) n-butane

With reference to the two statements Assertion and Reason, choose the correct option.

Assertion: The order of reactivity towards $\mathrm{S}_{\mathrm{N}} 1$ reaction is: $\mathrm{C}_6 \mathrm{H}_5-\mathrm{CH}_2 \mathrm{Br}>\left(\mathrm{CH}_3\right)_3-\mathrm{C}-\mathrm{Br}>\left(\mathrm{CH}_3\right)_2-\mathrm{CH}-\mathrm{Br}$.

Reason: Among the given 3 compounds, the Benzyl carbocation formed is the most stable while Isopropyl carbocation is the least stable one.

$$ \text { Identify the correct structure of o-ethyl anisole. } $$

$$ \text { A compound }[X] \text { undergoes reactions as given. Identify compounds }[C] \text { and }[D] \text { formed in these reactions. } $$

$$ [\mathrm{A}] \xrightarrow{\mathrm{Cr}_2 \mathrm{O}_7^{2-} / \mathrm{H}^{+}}[\mathrm{C}] $$

$$ [\text { B }] \xrightarrow[\text { (ii) } \mathrm{Na}_2 \mathrm{CO}_{3(\text { aq })}+\mathrm{I}_2]{\text { (i)aq. } \mathrm{KOH}} [\mathrm{D}] \quad+\mathrm{CH}_3 \mathrm{COONa}$$

Choose the incorrect statement.

Two statements $[A]$ and $[B]$ are given below. Choose the correct option.

A) Protonated $\mathrm{R}-\mathrm{C} \mathrm{H}_2-\mathrm{OH}$ can serve as Electrophiles while neutral $\mathrm{R}-\mathrm{OH}$ acts as a Nucleophile.

B) The bond between $\mathrm{O}-\mathrm{H}$ cleaves when $\mathrm{R}-\mathrm{CH}_2-\mathrm{OH}$ acts as Electrophiles and the bond between $\mathrm{C}-\mathrm{O}$ cleaves when they act as Nucleophiles.

Choose the correct reason:

o-hydroxybenzaldehyde is a liquid at room temperature while $p$-hydroxy-benzaldehyde is a high melting solid, because

$$ \text { Match the reactions in List I with the final products formed as given in List II. } $$

Which one of the following is the correct statement?

$$ \text { The product formed in the following reaction is: } $$

$$ \text { Consider the series of reactions given and identify the final product [Z]. } $$

$$ [X] \xrightarrow{\mathrm{CrO}_3}[Y] \quad \xrightarrow[\text { (ii) } \mathrm{H}_2 \mathrm{O}]{\text { (i) } \mathrm{CH}_3 \mathrm{MgBr}}[Z] $$

$$ \begin{aligned} &\text { Identify products }\left[\mathbf{P}_{\mathbf{1}}\right] \text { and }\left[\mathbf{P}_{\mathbf{2}}\right] \text { formed when Benzonitrile undergoes the following reactions. }\\ &\text { Benzo nitrile } \xrightarrow[\text { (ii) } \mathrm{H}_3 \mathrm{O}^{+}]{\text {(i) } \mathrm{SnCl}_2 / \mathrm{HCl} / \text { ether }}[\mathrm{X}] \end{aligned} $$

A Nitrogen containing organic compound with molecular formula $\mathrm{C}_3 \mathrm{H}_5 \mathrm{~N}$ undergoes the following reactions.

$$ \begin{aligned} & \mathrm{C}_3 \mathrm{H}_5 \mathrm{~N}+\mathrm{Na} / \mathrm{Hg} \text { in } \mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} \rightarrow[\mathrm{~A}] \\ & {[\mathrm{A}]+\mathrm{CH}_3 \mathrm{COCl} / \mathrm{NaOH}(\mathrm{aq}) \rightarrow[\mathrm{B}]} \\ & {[\mathrm{A}]+\mathrm{CHCl}_3 / \mathrm{KOH} \text { on heating } \rightarrow[\mathrm{C}] \text { (foul smelling compound) }} \end{aligned} $$

Identify compounds $[\mathrm{B}]$ and $[\mathrm{C}]$

Choose the incorrect statement.

$$ \text { Identify the reagents, }[A] \text { and }[B] \text { used up when Salicylic acid undergoes the following reactions: } $$

Identify the final product formed when benzenamine reacts with the given reagents in the sequential order as:

i. $\quad\left(\mathrm{CH}_3-\mathrm{CO}\right)_2 \mathrm{O} /$ Pyridine.

ii.      Conc. $\mathrm{HNO}_3+\mathrm{H}_2 \mathrm{SO}_4$ \& then reacted with $\mathrm{H}_3 \mathrm{O}^{+}$

iii. $\quad \mathrm{NaNO}_2 \mathrm{HCl}(273 \mathrm{~K})$ \& then reacted with $\mathrm{H}_3 \mathrm{PO}_2(\mathrm{aq})$.

A small segment of a polypeptide gave on complete hydrolysis 3 molecules of alanine, 2 molecules of glycine and 3 molecules of cysteine. What is the number of peptide linkages in the segment of the polypeptide?

$$ \text { Match the characteristic from Col. I with the Vitamins given in Col. II. } $$

Identify the correct statement.

$$ \text { The range of the relation } R=\left\{(x, y): y=x+\frac{6}{x} \text {; where } x, y \in \mathbb{N} \text { and } x<6\right\} \text { is: } $$

A student needs to buy notebooks $(n)$ for a semester. Double the number of notebooks plus 5 must strictly exceed 15 , but the number of notebooks plus 10 must be no more than 22 . What is the range of notebooks they can buy?

Let $A$ and $B$ be two subsets of $\xi=\{\mathbf{1}, \mathbf{2}, \mathbf{3},-------, \mathbf{4 4}, \mathbf{4 5}\}$ such that

$A=\{x: x$ is divisible by 3 and 4$\}$

$B=\{x: x$ is a perfect square number $\}$

Then $n(B-A)$ equals

Given the sets $A=\{1,2,3\} ; B=\{2,3,5\}$ and $C=\{4,5,6\}$ identify which of the following statement is incorrect.

The conjugate of the multiplicative inverse of the complex number $\boldsymbol{z}=\frac{\mathbf{1}+\mathbf{7} \boldsymbol{i}}{\mathbf{3}+\boldsymbol{i}}$ is:

Every term of a geometric progression is positive, and every term is the sum of the two preceding terms. Then the common ratio of the geometric progression is:

If $\boldsymbol{k}$ is the arithmetic mean of two given quantities and $\boldsymbol{p}, \boldsymbol{q}$ are the geometric means between the same two quantities, then $\boldsymbol{p}^{\mathbf{3}}+\boldsymbol{q}^{\mathbf{3}}$ is:

A coach needs to select a $\mathbf{4}$-player starting lineup from a pool of $\mathbf{1 0}$ players:

• 5 guards

• 3 forwards

• 2 centres

Find the number of different selections if the 4-player starting lineup must include:

• At least 1 guard

• At most 1 forward

• Exactly 1 centre

$$ \text { If }{ }^{\mathrm{n}} C_{13},{ }^{\mathrm{n}} C_{14} \text { and }{ }^{\mathrm{n}} C_{15} \text { are in arithmetic progression, then the positive integer value of ' } \mathbf{n} \text { ' can be } $$

Vishnu has two jars of marbles, Jar A and Jar B.

• Jar A contains 3 yellow marbles and 2 green marbles.

• Jar B contains 4 yellow marbles and 3 green marbles.

Vishnu flips a fair coin.

• If it lands heads, he picks two marbles at random without replacement from Jar A.

• If it lands tails, he picks two marbles at random with replacement from Jar B.

Given that Vishnu picked one yellow and one green marble, what is the probability that they came from Jar B?

A company is migrating its database, and two software engineers, Ishaan and Kavya, take turns running a data-sync script that has a constant success rate of $\frac{3}{8}$ per attempt.

If Ishaan initiates the first attempt and they persist until the migration is successful, what is the probability that Kavya is the one who initiates the successful sync?

A coffee roaster has $\mathbf{1 2}$ rare coffee beans with intensity scores ranked from $\mathbf{1}$ (mildest) to $\mathbf{1 2}$ (strongest).

You choose 7 beans at random and line them up from mildest to strongest:

$$ C_1< C_2< C_3< C_4< C_5< C_6< C_7 $$

What is the probability that the third bean $\left(C_3\right)$ has an intensity score of exactly 4 ?

If $P(A \cup B)=0.85, P(B)=0.50$ and $P(A \cap B)=0.30$. Then $P\left(A \cap B^{\prime}\right)=$

Cards are numbered from 12 to 51 . Two cards are drawn one after the other without replacement. Find the probability that one card is a multiple of $\mathbf{6}$ and the other card is a multiple of $\mathbf{8}$.

If the coefficients of $x^2$ and $x^3$ in the expansion of $(3+k x)^9$ are equal, then the value of ' $\boldsymbol{k}$ ' is

$$ \text { If the projection of } \vec{a}=5 \hat{\imath}+\hat{\jmath}+\lambda \hat{k} \text { on } \vec{b}=2 \hat{\imath}+6 \hat{\jmath}+3 \hat{k} \text { is } 4 \text { units, then } \lambda= $$

$$ \text { If }(\vec{a}+\vec{b}) \perp \vec{b} \text { and }(\vec{a}+2 \vec{b}) \perp \vec{a} \text {, then } $$

The equation of the perpendicular drawn from the point $A(6,1,3)$ to the line $\frac{x-1}{2}=\frac{2-y}{-1}=\frac{z-3}{2}$ is $\frac{x-6}{\boldsymbol{a}}=\frac{y-1}{\boldsymbol{b}}=\frac{z-3}{\boldsymbol{c}}$. If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are the possible integers such that $\boldsymbol{a}<0$, then the value of $\boldsymbol{a}-\boldsymbol{b}+\mathbf{5} \boldsymbol{c}$ is:

$$ \begin{aligned} &\text { Consider two skew lines in 3D space. }\\ &M_1: \frac{x-1}{1}=\frac{2-y}{1}=\frac{z-5}{1} \text { and } M_2: \frac{x+3}{1}=\frac{y-7}{2}=\frac{z+4}{1} \end{aligned} $$

Let $L_1$ be the line of shortest distance (common perpendicular) between $M_1$ and $M_2$

If $L_2$ is a line parallel to the vector $\vec{b}=\hat{\jmath}+\hat{k}$,

Then the acute angle $\boldsymbol{\theta}$ between the lines $L_1$ and $L_2$ is:

Consider the lines $L_1$ and $L_2$ given by the following vector equations:

$$ L_1: \vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(3 \hat{i}+\boldsymbol{t} \hat{j}) \quad L_2: \vec{r}=(4 \hat{i}+\boldsymbol{a} \hat{j}-\hat{k})+\mu(2 \hat{i}+3 \hat{k}) $$

If $\boldsymbol{a}=-2$ and the lines intersect, then the value of ' $\mathbf{t}$ ' is:

Matrix $A=\left[\begin{array}{ccc}1 & 1 & 2 \\ 1 & -2 & 2 \\ 1 & 0 & -1\end{array}\right]$,

Given $\boldsymbol{M}_{\mathbf{2 2}}$ and $\boldsymbol{A}_{\mathbf{3 2}}$ are the minor and cofactor of the adjoint matrix of $\boldsymbol{A}$ respectively then the value of the expression $\boldsymbol{M}_{\mathbf{2 2}}+\boldsymbol{A}_{\mathbf{3 2}}-|\boldsymbol{a} \boldsymbol{d} \boldsymbol{j}|$ is:

Given $A=\left[\begin{array}{lll}x & 1 & -2\end{array}\right]$ and $B=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$ If $\boldsymbol{A} \boldsymbol{B} \boldsymbol{A}^{\boldsymbol{t}}=[-\mathbf{2 0}]$ then the value of $\boldsymbol{x}$ is:

Let A be a square matrix of order $3 \times 3$. If $|A|=-4$, then the value of $\left|\frac{A^{-1}}{-2}\right|$ is:

Let $A=\left[a_{i j}\right]$ be a square matrix of order $3 \times 3$, where the elements are defined as $a_{i j}=\left\{\begin{array}{ll}i-2 j & \text { if } i=j \\ 0 & \text { if } i> j \\ 1 & \text { if } i < j\end{array} \quad\right.$ then the value of $\left|A^t\right|$ is

If the matrix $M=\left[\begin{array}{ccc}x+5 & a & -4 \\ -2 & 0 & b \\ c & 6 & y+1\end{array}\right]$ is a skew symmetric matrix, the value of the expression $\boldsymbol{a} \boldsymbol{b}+\boldsymbol{c}^{\mathbf{2}}-\boldsymbol{x} \boldsymbol{y}$ is:

The variance of a set of 20 observations is 16 . If 7 is added to each observation, and then $\mathbf{5}$ is subtracted from each resulting observation, what will be the new standard deviation?

The feasible region represented by the constraints:

$$ \begin{aligned} & x+2 y \leq 120 \\ & x+y \geq 60 \\ & x-2 y \geq 0 \\ & x \geq 0 \text { and } y \geq 0 \end{aligned} $$

Suppose 'a' and 'b' are non-zero constants satisfying the following system of equations $\boldsymbol{a} \sin ^3 x+\boldsymbol{b} \cos ^3 x=\sin x \cos x$ and $\mathbf{a} \sin x-\boldsymbol{b} \cos x=0$, then $\mathbf{2}\left(\boldsymbol{a}^6+\boldsymbol{b}^6\right)-\mathbf{3}\left(\boldsymbol{a}^4+\boldsymbol{b}^4\right)+\mathbf{1}=$

If $2 \sin \theta=\left(x+\frac{1}{x}\right)$, then $\sin 3 \theta+\frac{1}{2}\left(x^3+\frac{1}{x^3}\right)=$

$$ \text { The expression } \frac{\tan \left(x-\frac{\pi}{2}\right) \cos \left(\frac{3 \pi}{2}+x\right)-\sin ^3\left(\frac{7 \pi}{2}-x\right)}{\cos \left(x-\frac{\pi}{2}\right) \tan \left(\frac{3 \pi}{2}+x\right)} \text { simplifies to: } $$

$$ \text { The domain of the function } f(x)=\sin ^{-1}(\sqrt{x-1}) $$

If $X=\tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]+\cos ^{-1}\left[\cos \left(\frac{7 \pi}{6}\right)\right]$ and $Y=\sin ^{-1}\left[\sin \left(\frac{11 \pi}{6}\right)\right]+\tan ^{-1}\left[\tan \left(\frac{4 \pi}{3}\right)\right]$ then the value of $\mathbf{2} \boldsymbol{X}-\boldsymbol{Y}$ is:

If the two ends of the major axis of an ellipse are $(5,0)$ and $(-5,0)$ and one focus lies on the line $3 x-5 y-9=0$, then its equation is

Let point Q be the image of point $P(2,-1)$ in the line $3 x+5=4 y$.

Find the area of the circle that has the segment PQ as the diameter.

The foci of a hyperbola are the same as those of the ellipse with equation $9 x^2+16 y^2=144$.

If the length of the transverse axis of this hyperbola is $2 \cos \alpha$, then its equation is:

Let the line $L_1$ be a line passing through the point $(\mathbf{0},-\mathbf{6})$ and making an angle of $\mathbf{1 5 0}^{\circ}$ with the positive $x$-axis. Then the equation of a line $L_2$ parallel to $L_1$ and crossing the $y$-axis 2 units below the origin is:

A line L passes through the point of intersection of the lines $3 x+y-10=0$ and $x-y-2=0$.

If the perpendicular distance of the line $L$ from the point $(5,1)$ is exactly $\frac{2}{\sqrt{5}}$ units, which of the following represents the correct equation for line L ?

If $f(x)=x^3+\frac{3}{2} x^2+3 x+3$, then $f(x)$ is

A movie screen on a wall is $\mathbf{2 0}$ feet high and $\mathbf{1 0}$ feet above the floor. What is the maximum viewing angle $\boldsymbol{\theta}$ (in radians) that can be achieved by positioning yourself at the optimal distance from the wall?

The absolute maximum and minimum values of the function $f(x)=\sin x+\sqrt{3} \cos x$ in $[0, \pi]$ are

The function $f(x)=e^{a x}+e^{-a x}, x \in \mathbb{R}$ and $a<0$, is strictly decreasing for all values of ' $x$ ', where

An open hemispherical storage tank has radius 13 m . Oil flows into the tank such that the depth ' $\boldsymbol{h}$ ' of oil in the tank changes at the rate of $3 \mathrm{~m} / \mathrm{hr}$. When the depth $\boldsymbol{h}=1 \mathrm{~m}$, the rate of change of the area of the top surface of the oil is

$$ \text { The degree of the differential equation } \sqrt{1+\left(\frac{d y}{d x}\right)^{1 / 3}}=\frac{d^2 y}{d x^2} \text { is: } $$

$$ \text { The particular solution of the differential equation }(x-y)(d x+d y)=(d x-d y) \text { when } y=-1 \text { and } x=0 \text { is } $$

Let the population of a species of birds surviving at a time ' $\boldsymbol{t}$ ' be governed by the differential equation $\frac{d p}{d t}-p=-100$. If $p(0)=50$, then $p\left(-\log _e 2\right)$ is equal to

$$ \int \tan ^{-1}\left(\sqrt{\frac{1-\sin x}{1+\sin x}}\right) d x= $$

The function $\boldsymbol{x}+\boldsymbol{y}=\boldsymbol{\operatorname { t a n }}^{-\mathbf{1}} \boldsymbol{y}$ is the solution of which of the following differential equations?

$$ \int\left(\sin ^6 x+\cos ^6 x+3 \sin ^2 x \cos ^2 x\right) d x= $$

$$ \int \frac{d x}{x \sqrt{x^2+4}}= $$

$$ \int \frac{\log x}{(1+x)^2} d x $$

The function $\boldsymbol{f}(\boldsymbol{x})=|\boldsymbol{x}|+|\boldsymbol{x}-\mathbf{1}|$ is:

$$ \mathop {\lim }\limits_{x \to {\pi \over 2}}\left(\frac{1-\sin x}{\cos x}\right) \text { is equal to } $$

If $\mathop {\lim }\limits_{x \to 0}\left(\frac{p \sin 2 x+1-\cos 2 x}{x+\tan x}\right)=1$ then the value of ' $p$ ' is

If $\log y=\log (\sin x)-x^2$, then $\frac{d^2 y}{d x^2}+\mathbf{4} x \frac{d y}{d x}+\mathbf{4} x^2 y=$

$$ \text { The second derivative of } \sin 3 \boldsymbol{x} \boldsymbol{\operatorname { c o s }} \mathbf{5 x} \text { is: } $$

$$ \text { If } y=\tan ^{-1}\left(\frac{\sqrt{1+x^3}+\sqrt{1-x^3}}{\sqrt{1+x^3}-\sqrt{1-x^3}}\right) \text { then } \frac{\boldsymbol{d} \boldsymbol{y}}{\boldsymbol{d x}}= $$

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{x}{x^2+1} \quad \forall x \in \mathbb{R}$ is

Find the area bounded by the curve $y=|2-x|$, the $x$-axis, and the lines $x=0$ and $x=5$

$$ \int_0^{\frac{\pi}{2}} \frac{3 \sin x+4 \cos x}{\sin x+\cos x} d x= $$

The area enclosed by the curve $y=-x^2$ and the line $x+y+2=0$ is