24 jan shift 2

Question Paper

Physics Single Correct (Maximum Marks: 80)

Question No. 1

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

A metallic rod of length $L$ is rotated with an angular speed of $\omega$ normal to a uniform magnetic field $B$ about an axis passing through one end of rod as shown in figure. The induced emf will be :

.

$\frac{1}{2}{B}^2{L}^2\omega$
$\frac{1}{4}B{L}^2\omega$
$\frac{1}{2}B{L}^2\omega$
$\frac{1}{4}{B}^{2}L\omega$

Question No. 2

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The logic gate equivalent to the given circuit diagram is :

OR
NAND
NOR
AND

Question No. 3

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The frequency $\left(\nu \right)$ of an oscillating liquid drop may depend upon radius $\left(r\right)$ of the drop, density $\left(\rho \right)$ of liquid and the surface tension $\left(s\right)$ of the liquid as: $\nu =r^a{\rho }^b{s}^c$. The values of $a, b$ and $c$ respectively are 

$\left(-\frac{3}{2}, -\frac{{\displaystyle 1}}{{\displaystyle 2}}, \frac{{\displaystyle 1}}{{\displaystyle 2}}\right)$
$\left(-\frac{3}{2}, \frac{{\displaystyle 1}}{{\displaystyle 2}}, \frac{{\displaystyle 1}}{{\displaystyle 2}}\right)$
$\left(\frac{3}{2}, \frac{{\displaystyle 1}}{{\displaystyle 2}},-\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)$
$\left(\frac{3}{2}, -\frac{{\displaystyle 1}}{{\displaystyle 2}}, \frac{{\displaystyle 1}}{{\displaystyle 2}}\right)$

Question No. 4

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Given below are two statements:

Statement I: Acceleration due to earth's gravity decreases as you go 'up' or 'down' from earth's surface.

Statement II: Acceleration due to earth's gravity is same at a height '$h$' and depth '$d$' from earth's surface, if $h = d$.

In the light of above statements, choose the most appropriate answer form the options given below

Statement I is incorrect but statement II is correct
Both Statement I and Statement II are incorrect
Statement I is correct but statement II is incorrect
Both Statement I and II are correct

Question No. 5

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The electric field and magnetic field components of an electromagnetic wave going through vacuum is described by

 $E_x=E_0\sin \left(kz-\omega t\right)$

$B_y=B_0\sin \left(kz-\omega t\right)$

Then the correct relation between $E_0$ and $B_0$ is given by 

$E_0{B}_0=\omega k$
$E_0=k{B}_0$
$k{E}_0=\omega B_0$
$\omega E_0=k{B}_0$

Question No. 6

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The velocity-time graph of a body moving in a straight line is shown in figure.

 

The ratio of displacement and distance travelled by the body in time $0$ to $10 \mathrm{s}$ is

$1:1$
$1:2$
$1:3$
$1:4$

Question No. 7

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

A tuning fork vibrates with frequency $256 \mathrm{~Hz}$ and gives one best per second with the third normal mode of vibration of an open pipe. What is the length of the pipe? (Speed of sound of air is $340 \mathrm{~ms}^{-1}$ )

$190 \mathrm{~cm}$
$180 \mathrm{~cm}$
$220 \mathrm{~cm}$
$200 \mathrm{~cm}$

Question No. 8

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Assertion $A$: A pendulum clock when taken to Mount Everest becomes fast.

Reason $R$: The value of g (acceleration due to gravity) is less at Mount Everest than its value on the surface of earth.

In the light of the above statements, choose the most appropriate answer from the options given below 

Both $A$ and $R$ are correct and $R$ is the correct explanation of $A$
Both $A$ and $R$ are correct but $R$ is not the correct explanation of $A$
$A$ is correct but $R$ is not correct
$A$ is not correct but $R$ is correct

Question No. 9

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

If the distance of the earth from Sun is $1.5\times {10}^6 \mathrm{km}$, then the distance of an imaginary planet from Sun, if its period of revolution is $2.83$ years is:

$6 \times {10}^7 \mathrm{km}$
$6 \times {10}^6 \mathrm{km}$
$3 \times {10}^6 \mathrm{km}$
$3 \times {10}^7 \mathrm{km}$

Question No. 10

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

An $\alpha$ particle, a proton and an electron have the same kinetic energy. Which one of the following is correct in case of their de-Broglie wavelength:

${\lambda }_{\alpha }>{\lambda }_p>{\lambda }_e$
${\lambda }_{\alpha }<{\lambda }_p<{\lambda }_e$
${\lambda }_{\alpha }={\lambda }_p={\lambda }_e$
${\lambda }_{\alpha }>{\lambda }_p<{\lambda }_e$

Question No. 11

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The electric potential at the centre of two concentric half rings of radii $R_1$ and $R_2$, having same linear charge density $\lambda$ is

.

$\frac{2\lambda }{{\in }_0}$
$\frac{\lambda }{2{\in }_0}$
$\frac{\lambda }{4{\in }_0}$
$\frac{\lambda }{{\in }_0}$

Question No. 12

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

If two vectors $\overrightarrow{P}=\hat{\mathrm{i}}+2m\hat{\mathrm{j}}+m\hat{\mathrm{k}}$  and $\overrightarrow{Q}=4\hat{\mathrm{i}}-2\hat{\mathrm{j}}+m\hat{\mathrm{k}}$  are perpendicular to each other. Then, the value of $m$ will be 

$-1$
$2$
$3$
$1$

Question No. 13

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

A long solenoid is formed by winding $70$ turns ${\mathrm{cm}}^{-1}$ . If $2.0 \mathrm{A}$ current flows, then the magnetic field produced inside the solenoid is ______________.

$({\mu }_0= 4\mathrm{\pi }\times {10}^{–7} \mathrm{T} \mathrm{m} {\mathrm{A}}^{–1})$

$1232\times {10}^{–4} \mathrm{T}$
$176\times {10}^{–4} \mathrm{T}$
$352\times {10}^{–4} \mathrm{T}$
$88\times {10}^{–4} \mathrm{T}$

Question No. 14

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

When a beam of white light is allowed to pass through convex lens parallel to principal axis, the different colours of light converge at different point on the principle axis after refraction. This is called :

Scattering
Chromatic aberration 
Spherical aberration
Polarisation

Question No. 15

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

In an Isothermal change, the change in pressure and volume of a gas can be represented for three different temperature; $T_3>T_2>T_1$ as:

Question No. 16

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Assertion (A) : Steel is used in the construction of buildings and bridges.
Reason (R) : Steel is more elastic and its elastic limit is high. 

In the light of above statements, choose the most appropriate answer from the options given below

Both $A$ and $R$ are correct and $R$ is the correct explanation of $A$
Both $A$ and $R$ are correct but $R$ is NOT the correct explanation of $A$
$A$ is correct but $R$ is not correct
$A$ is not correct but $R$ is correct

Question No. 17

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

A photon is emitted in transition from $n=4 \mathrm{to} n=1$ level in hydrogen atom. The corresponding wavelength for this transition is (given, $h=4\times {10}^{-15} \mathrm{eV} \mathrm{s}$)

$941 \mathrm{nm}$
$974 \mathrm{nm}$
$99.3 \mathrm{nm}$
$94.1 \mathrm{nm}$

Question No. 18

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

A cell of emf $90 \mathrm{V}$ is connected across series combination of two resistors each of $100 \mathrm{\Omega }$ resistance. A voltmeter of resistance $400 \mathrm{\Omega }$ is used to measure the potential difference across each resistor. The reading of the voltmeter will be:

$40 \mathrm{V}$
$45 \mathrm{V}$
$80 \mathrm{V}$
$90 \mathrm{V}$

Question No. 19

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Let ${\gamma }_1$ be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and ${\gamma }_2$ be the similar ratio of diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio $\frac{{\gamma }_1}{{\gamma }_2}$ is:

$\frac{27}{35}$
$\frac{35}{27}$
$\frac{25}{21}$
$\frac{21}{25}$

Question No. 20

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

A body of mass $200 \mathrm{g}$ is tied to a spring of spring constant $12.5 \mathrm{N} {\mathrm{m}}^{-1}$, while the other end of spring is fixed at point $O$. If the body moves about $O$ in a circular path on a smooth horizontal surface with constant angular speed $5 \mathrm{rad} {\mathrm{s}}^{-1}$, then the ratio of extension in the spring to its natural length will be :

$1:2$
$1:1$
$2:3$
$2:5$

Physics Numerical (Maximum Marks: 20)

Question No. 1

Numerical Type
The answer has to be filled into the input box provided below.

The energy released per fission of nucleus of $\mathrm{X}_{240}$ is $200 \mathrm{MeV}$. The energy released if all the atoms in $120 \mathrm{g}$ of pure $\mathrm{X}_{240}$ undergo fission is _____$\times {10}^{25} \mathrm{MeV}$.  

(Given $N_A=6\times {10}^{23}$)   

Question No. 2

Numerical Type
The answer has to be filled into the input box provided below.

A body of mass $1 \mathrm{kg}$ begins to move under the action of a time dependent force $\overrightarrow{F}=\left(t\hat{\mathrm{i}}+3t^2\hat{\mathrm{j}}\right) \mathrm{N}$, where $\hat{\mathrm{i}}$ and $\hat{\mathrm{j}}$ are the unit vectors along $x$ and $y$ axis. The power developed by above force, at the time $t=2 \mathrm{s}$, will be _______$\mathrm{W}$. 

Question No. 3

Numerical Type
The answer has to be filled into the input box provided below.

A parallel plate capacitor with air between the plate has a capacitance of $15 \mathrm{pF}$. The separation between the plate becomes twice and the space between them is filled with a medium of dielectric constant $3.5$. Then the capacitance becomes $\frac{\mathrm{x}}{4} \mathrm{pF}$. The value of $x$ is____.

Question No. 4

Numerical Type
The answer has to be filled into the input box provided below.

Three identical resistors with resistance $R=12 \mathrm{\Omega }$ and two identical inductors with sell inductance $L=5 \mathrm{mH}$ are connected to an ideal battery with emf of $12 \mathrm{V}$ as shown in figure. The current through the battery long after the switch has been closed will be________$\mathrm{A}$.

Question No. 5

Numerical Type
The answer has to be filled into the input box provided below.

A single turn current loop in the shape of a right angle triangle with sides $5 \mathrm{cm}, 12 \mathrm{cm}, 13 \mathrm{cm}$ is carrying a current of $2 \mathrm{A}$. The loop is in a uniform magnetic field of magnitude $0.75 \mathrm{T}$ whose direction is parallel to the current in the $13 \mathrm{cm}$ side of the loop. The magnitude of the magnetic force on the $5 \mathrm{cm}$ side will be $\frac{\mathrm{x}}{130} \mathrm{N}$. The value of $x$ is _______.

Question No. 6

Numerical Type
The answer has to be filled into the input box provided below.

A spherical ball of radius $1 \mathrm{mm}$ and density $10.5 \mathrm{g} {\mathrm{cc}}^{-1}$ is dropped in glycerine of coefficient of viscosity $9.8$ poise and density $1.5 \mathrm{g} {\mathrm{cc}}^{-1}$. Viscous force on the ball when it attains constant velocity is $3696\times {10}^{-x} \mathrm{N}$. The value of $x$ is

(Given, $g=9.8 \mathrm{m} {\mathrm{s}}^{-2} \text{and} \mathrm{\pi }=\frac{22}{7}$)

Question No. 7

Numerical Type
The answer has to be filled into the input box provided below.

A convex lens of refractive index $1.5$ and focal length $18 \mathrm{cm}$ in air is immersed in water. The change in focal length of the lens will be_____$\mathrm{cm}$. 

(Given refractive index of water$=\frac{4}{3}$)

Question No. 8

Numerical Type
The answer has to be filled into the input box provided below.

If a copper wire is stretched to increase its length by $20\%$. The percentage increase in resistance of the wire is _____%.

Question No. 9

Numerical Type
The answer has to be filled into the input box provided below.

A mass $m$ attached to free end of a spring executes SHM with a period of $1 \mathrm{s}$. If the mass is increased by $3 \mathrm{kg}$ the period of oscillation increases by one second, the value of mass $m$ is ________ $\mathrm{kg}$. 

Question No. 10

Numerical Type
The answer has to be filled into the input box provided below.

A uniform solid cylinder with radius $R$ and length $L$ has moment of inertia $I_1$, about the axis of cylinder. A concentric solid cylinder of radius $R^{\text{'}}=\frac{R}{2}$  and length $L^{\text{'}}=\frac{L}{2}$ is carved out of the original cylinder. If $I_2$ is the moment of inertia of the carved out portion of the cylinder then $\frac{I_1}{I_2}=$  __________.

(Both $I_1$ and $I_2$ are about the axis of the cylinder)   

Chemistry Single Correct (Maximum Marks: 80)

Question No. 1

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Given below are two statements :

Statement I :

 under Clemmensen reduction conditions will give 

Statement II :

 under Wolff-Kishner reduction condition will give 

In the light of the above statements, choose the correct answer from the options given below :

Statement I is false but Statement II is true
Both Statement I and Statement II are false
Statement I is true but Statement II is false
Both Statement I and Statement II are true

Question No. 2

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Which one amongst the following are good oxidising agents?

(a) ${\mathrm{Sm}}^{2+}$

(b) ${\mathrm{Ce}}^{2+}$

(c) ${\mathrm{Ce}}^{4+}$

(d) ${\mathrm{Tb}}^{4+}$

Choose the most appropriate answer from the options given below :

C only
D only
A and B only
C and D only

Question No. 3

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Given below are two statements, one is labelled as

Assertion A and the other is labelled as Reason R.

Assertion A : Benzene is more stable than hypothetical cyclohexatriene.

Reason R : The delocalized $\mathrm{\pi }$ electron cloud is attracted more strongly by nuclei of carbon atoms.

In the light of the above statements, choose the correct answer from the options given below:

A is true but R is false.
A is false but R is true.
Both A and R are correct and R is the correct explanation of A.
Both A and R are correct but R is NOT the correct explanation of A.

Question No. 4

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Which of the following cannot be explained by crystal field theory?

The order of spectrochemical series
Magnetic properties of transition metal complexes
Colour of metal complexes
Stability of metal complexes

Question No. 5

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Find out the major products from the following reactions.

Question No. 6

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

lodoform can be prepared from all except :

Ethyl methyl ketone
Isopropyl alcohol
$3-$Methyl$-2-$ butanone
Isobutyl alcohol

Question No. 7

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The catenation tendency of $\mathrm{C}, \mathrm{Si}$ and $\mathrm{Ge}$ is in the order $\mathrm{Ge} < \mathrm{Si} < \mathrm{C}$. The bond energies (in $\mathrm{kJ} \mathrm{~mol}^{-1}$ ) of $\mathrm{C}-\mathrm{C}, \mathrm{Si}-\mathrm{Si}$ and $\mathrm{Ge}-\mathrm{Ge}$ bonds are respectively ;

$348,297,260$
$297,348,260$
$348,260,297$
$260,297,348$

Question No. 8

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

What is the number of unpaired electron(s) in the highest occupied molecular orbital of the following species : ${\mathrm{N}}_2, \mathrm{N}{ }_2^{+}, {\mathrm{O}}_2, {\mathrm{O}}_2^{+}$ ?

$0, 1, 2, 1$
$2, 1, 2, 1$
$0, 1, 0, 1$
$2, 1, 0, 1$

Question No. 9

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Given below are two statements :

Statement I : Pure Aniline and other arylamines are usually colourless.

Statement II : Arylamines get coloured on storage due to atmospheric reduction.

In the light of the above statements, choose the most appropriate answer from the options given below :

Both Statement I and Statement II are incorrect
Both Statement I and Statement II are correct
Statement I is correct but Statement II is incorrect
Statement I is incorrect but Statement II is correct

Question No. 10

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Choose the correct colour of the product for the following reaction.

 

Yellow
White
Red
Blue

Question No. 11

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Which will undergo deprotonation most readily in basic medium?

a only
c only
Both a and c
b only

Question No. 12

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

${\mathrm{K}}_2{\mathrm{Cr}}_2{\mathrm{O}}_7$ paper acidified with dilute ${\mathrm{H}}_2{\mathrm{SO}}_4$ turns green when exposed to

Carbon dioxide
Sulphur trioxide
Hydrogen sulphide
Sulphur dioxide

Question No. 13

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The hybridization and magnetic behaviour of cobalt ion in $\left[\mathrm{Co}\right({\mathrm{NH}}_3{)}_6{]}^{3+}$ complex, respectively is

${\mathrm{sp}}^3{\mathrm{d}}^2$ and diamagnetic
${\mathrm{d}}^2{\mathrm{sp}}^3$ and paramagnetic
${\mathrm{d}}^2{\mathrm{sp}}^3$ and diamagnetic
${\mathrm{sp}}^3{\mathrm{d}}^2$ and paramagnetic

Question No. 14

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Which of the following forms stable $+4$ oxidation state?

$\operatorname{La}(Z=57)$
$\operatorname{Eu}(Z=63)$
$\mathrm{Ce}(Z=58)$
$\operatorname{Gd}(Z=64)$

Question No. 15

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Glycosidic linkage is actually an :

Carbonyl bond
Ether bond
Ester bond
Amide bond

Question No. 16

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The ratio of masses of oxygen and nitrogen in a particular gaseous mixture is 1 : 4. The ratio of number of their molecules is:

1 : 4
7 : 32
1 : 8
3 : 16

Question No. 17

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

A student has studied the decomposition of a gas $A{B}_3$ at $25°C$. He obtained the following data

$p (mm Hg)$	$50$	$100$	$200$	$400$
Relative $t_{\frac{1}{2}}\left(s\right)$	$4$	$2$	$1$	$0.5$

The order of the reaction is

$0.5$
$2$
$1$
$0$ (zero)

Question No. 18

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Which one of the following molecules is expected to exhibit paramagnetic behaviour?

O₂
O₂^-2
C₂
N₂

Question No. 19

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Choose the correct representation of conductometric titration of benzoic acid vs sodium hydroxide.

Question No. 20

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Given :
$\mathrm{XNa}_2 \mathrm{HAsO}_3+\mathrm{YNaBrO}_3+\mathrm{ZHCl} \rightarrow \mathrm{NaBr}+\mathrm{H}_3 \mathrm{AsO}_4+\mathrm{NaCl}$
The values of $\mathrm{X}, \mathrm{Y}$ and $\mathrm{Z}$ in the above redox reaction are respectively :

$2,1,2$
$2,1,3$
$3,1,6$
$3,1,4$

Chemistry Numerical (Maximum Marks: 20)

Question No. 1

Numerical Type
The answer has to be filled into the input box provided below.

One mole of an ideal monoatomic gas is subjected to changes as shown in the graph. The magnitude of the work done (by the system or on the system) is _______ J (nearest integer)

Given : log 2 = 0.3, ln 10 = 2.3

Question No. 2

Numerical Type
The answer has to be filled into the input box provided below.

Maximum number of isomeric monochloro derivatives which can be obtained from 2,2,5,5- tetramethylhexane by chlorination is _________

Question No. 3

Numerical Type
The answer has to be filled into the input box provided below.

The rate constant of a zero order reaction is $2.0 \times$ $10^{-2} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$. If the concentration of the reactant after 25 seconds is $0.5 \mathrm{M}$. What is the initial concentration?

Question No. 4

Numerical Type
The answer has to be filled into the input box provided below.

Following figure shows spectrum of an ideal black body at four different temperatures. The number of correct statement/s from the following is _______.

A. T₄ > T₃ > T₂ > T₁
 B. The black body consists of particles performing simple harmonic motion.
 C. The peak of the spectrum shifts to shorter wavelength as temperature increases.
 D. not equal to constant
 E. The given spectrum could be explained using quantisation of energy

Question No. 5

Numerical Type
The answer has to be filled into the input box provided below.

Sum of $\pi$-bonds present in peroxodisulphuric acid and pyrosulphuric acid is

Question No. 6

Numerical Type
The answer has to be filled into the input box provided below.

The Total pressure observed by mixing two liquid $\mathrm{A} \text{and} \mathrm{B}$ is $350 \mathrm{mm} \mathrm{Hg}$ when their mole fractions are $0.7 \text{and} 0.3$ respectively. The Total pressure becomes$410 \mathrm{mm} \mathrm{Hg}$ if the mole fractions are changed to$0.2 \text{and} 0.8$ respectively for $\mathrm{A} \text{and} \mathrm{B}$. The vapour pressure of pure $\mathrm{A}$ is ______ $\mathrm{mm} \mathrm{Hg}$. (Nearest integer) Consider the liquids and solutions behave ideally

Question No. 7

Numerical Type
The answer has to be filled into the input box provided below.

The number of units, which are used to express concentration of solutions from the following is _________.

(Mass percent, Mole, Mole fraction, Molarity, ppm, Molality.)

Question No. 8

Numerical Type
The answer has to be filled into the input box provided below.

Total number of tripeptides possible by mixing of valine and proline is ________ .

Question No. 9

Numerical Type
The answer has to be filled into the input box provided below.

A current of $10.0 \mathrm{A}$ flows for $2.00 \mathrm{h}$ through an electrolytic cell containing a molten salt of metal $\mathrm{X}$. This results in the decomposition of $0.250 \mathrm{mol}$ of metal $\mathrm{X}$ at the cathode. The oxidation state of $\mathrm{X}$ in the molten salt is:

$\left(\mathrm{F}=96,500 \mathrm{C}\right)$

Question No. 10

Numerical Type
The answer has to be filled into the input box provided below.

If the $\mathrm{pKa}$ of lactic acid is $5$, then the $\mathrm{pH}$ of $0.005 \mathrm{M}$ calcium lactate solution at $25° \mathrm{C}$ is __________ $\times {10}^{–1}$ (Nearest integer)
 

Mathematics Single Correct (Maximum Marks: 80)

Question No. 1

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Let $\mathrm{y}=\mathrm{y}\left(\mathrm{x}\right)$ be the solution of the differential equation $({\mathrm{x}}^2– 3{\mathrm{y}}^2)\mathrm{dx}+3\mathrm{xy} \mathrm{dy}=0, \mathrm{y}\left(1\right)=1$. Then $6{\mathrm{y}}^2\left(\mathrm{e}\right)$ is equal to

$3{\mathrm{e}}^2$
${\mathrm{e}}^2$
$2{\mathrm{e}}^2$
$\frac{3{\mathrm{e}}^2}{2}$

Question No. 2

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The value of ${\left(\frac{1+\sin \frac{2\mathrm{\pi }}{9}+i\cos \frac{2\mathrm{\pi }}{9}}{1+\sin \frac{2\mathrm{\pi }}{9}-i\cos \frac{2\mathrm{\pi }}{9}}\right)}^3$ is

$\frac{-1}{2}\left(1-i\sqrt{3}\right)$
$\frac{1}{2}\left(1-i\sqrt{3}\right)$
$\frac{-1}{2}\left(\sqrt{3}-i\right)$
$\frac{1}{2}\left(\sqrt{3}+i\right)$

Question No. 3

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The set of values of $a$ for which $\underset{x\to a}{\lim }\left(\left[x-5\right]-\left[2x+2\right]\right)=0$, where, $\left[\zeta \right]$ denotes the greatest integer less than or equal to $\zeta$ is equal to

$\left(-7.5,-6.5\right)$
$(-7.5,-6.5]$
$\left[-7.5,-6.5\right]$
$[-7.5,-6.5)$

Question No. 4

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

If $f\left(x\right)=\frac{2^{2x}}{2^{2x}+2}$, $x\in \mathrm{R}$,  then $f\left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+f\left(\frac{3}{2023}\right).........f\left(\frac{2022}{2023}\right)$ is equal to 

$2011$
$1010$
$2010$
$1011$

Question No. 5

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Let $\overrightarrow{\alpha }=4\hat{\mathrm{i}}+3\hat{\mathrm{j}}+5\hat{\mathrm{k}}$  and $\overrightarrow{\beta }=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-4\hat{\mathrm{k}}$. Let ${\overrightarrow{\beta }}_1$ be parallel to $\overrightarrow{\alpha }$ and ${\overrightarrow{\beta }}_2$ be perpendicular to $\overrightarrow{\alpha }$. If $\overrightarrow{\beta }={\overrightarrow{\beta }}_1+{\overrightarrow{\beta }}_2$, then the value of $5{\overrightarrow{\beta }}_2·\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)$ is

$6$
$11$
$7$
$9$

Question No. 6

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The number of real solutions of the equation $3\left({\mathrm{x}}^2+\frac{1}{{\mathrm{x}}^2}\right)-2\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)+5=0$, is

$4$
$0$
$3$
$2$

Question No. 7

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

$ABC$ is a triangle in a plane with vertices $A\left(2, 3, 5\right), B\left(-1, 3, 2\right)$ and $C\left(\lambda , 5, \mu \right)$ . If the median through $A$ is equally inclined to the coordinate axes, then the value of $\left({\lambda }^3+{\mu }^3+5\right)$ is

$1130$
$1348$
$1077$
$676$

Question No. 8

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Let $f\left(x\right)$ be a function such that $f\left(x+y\right)=f\left(x\right)·f\left(y\right)$ for all $x,y\in \mathrm{N}$, If  $f\left(1\right)=3$ and $\sum _{k=1}^{n}f\left(k\right)=3279$, then the value of $n$ is

$6$
$8$
$7$
$9$

Question No. 9

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The number of square matrices of order $5$ with entries from the set $\{0,1\}$, such that the sum of all the elements in each row is $1$ and the sum of all the elements in each column is also $1$, is 

$225$
$120$
$150$
$125$

Question No. 10

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

If ${\left(27\right)}^{999}$ is divided by $7$, then the remainder is

$3$
$1$
$6$
$2$

Question No. 11

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

If $f\left(x\right)=x^3-x^2{f}^{\text{'}}\left(1\right)+x{f}^{"}\left(2\right)-f^{\text{'}\text{'}\text{'}}\left(3\right)$, $x\in \mathrm{R}$, then

$3f\left(1\right)+f\left(2\right)=f\left(3\right)$
$f\left(3\right)-f\left(2\right)=f\left(1\right)$
$2f\left(0\right)-f\left(1\right)+f\left(3\right)=f\left(2\right)$
$f\left(1\right)+f\left(2\right)+f\left(3\right)=f\left(0\right)$

Question No. 12

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The number of integers, greater than $7000$ that can be formed, using the digits $3,5,6,7,8$ without repetition is

$120$
$168$
$220$
$48$

Question No. 13

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The locus of the middle points of the chords of the circle $C_1: {\left(x-4\right)}^2+{\left(y-5\right)}^2=4$ which subtend an angle ${\theta }_i$ at the centre of the circle $C_i$, is a circle of radius $r_i$. If ${\theta }_1=\frac{\mathrm{\pi }}{3}$, ${\theta }_3=\frac{2\mathrm{\pi }}{3}$ and ${r_1}^2={r_2}^2+{r_3}^2$, then ${\theta }_2$ is equal to

$\frac{\mathrm{\pi }}{4}$
$\frac{3\mathrm{\pi }}{4}$
$\frac{\mathrm{\pi }}{6}$
$\frac{\mathrm{\pi }}{2}$

Question No. 14

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Let $A$ be a $3\times 3$ matrix such that $\left|adj\left(adj\left(adj.A\right)\right)\right|={12}^4$. Then $\left|A^{-1}adj A\right|$ is equal to

$2\sqrt{3}$
$\sqrt{6}$
$12$
$1$

Question No. 15

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

If $a, b, c$ are sides of a scalene triangle, then the value of $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ is :

non - negative
negative
positive
non-positive

Question No. 16

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

The coefficient of $x^{10}$ in the expansion of $(1+x)^2$ $\left(1+x^2\right)^3\left(1+x^3\right)^4$ is equal to

52
44
50
56

Question No. 17

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

A hyperbola whose transverse axis is along the major axis of the conic $\frac{x^2}{3}+\frac{y^2}{4}=4$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $\frac{3}{2}$, then which of the following points does not lie on the hyperbola $?$

$\left(\sqrt{5},2\sqrt{2}\right)$
$\left(0,2\right)$
$\left(5,2\sqrt{3}\right)$
$\left(\sqrt{10},2\sqrt{3}\right)$

Question No. 18

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

A variable circle passes through the fixed point $A(p, q)$ and touches $x$-axis. The locus of the other end of the diameter through $\mathrm{A}$ is

$(x-p)^2=4 q y$
$(x-q)^2=4 p y$
$(y-p)^2=4 q x$
$(y-q)^2=4 p x$

Question No. 19

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Let the six numbers $a_1, a_2, ..., a_6$ be in $\mathrm{A}.\mathrm{P}.$ and $a_1+a_3=10$.If the mean of these six numbers is $\frac{19}{2}$ and their variance is ${\sigma }^2$, then $8{\sigma }^2$ is equal to

$220$
$210$
$200$
$105$

Question No. 20

Only One Option Correct Type
Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

${\int }_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}}\frac{48}{\sqrt{9-4x^2}}dx$ is equal to

$\frac{\mathrm{\pi }}{3}$
$\frac{\mathrm{\pi }}{2}$
$\frac{\mathrm{\pi }}{6}$
$2\mathrm{\pi }$

Mathematics Numerical (Maximum Marks: 20)

Question No. 1

Numerical Type
The answer has to be filled into the input box provided below.

Let $x, y, z$ be positive real numbers such that $x+y+z=12$ and $x^3{y}^4{z}^5=\left(0.1\right){\left(600\right)}^3.$ Then $x^3+y^3+z^3$ is equal to

Question No. 2

Numerical Type
The answer has to be filled into the input box provided below.

Let he sum of the coefficient of first three terms in the expansion of ${\left(x-\frac{3}{x^2}\right)}^n; x=0, n\in N$ be $376$. Then, the coefficient of $x^4$ is equal to:

Question No. 3

Numerical Type
The answer has to be filled into the input box provided below.

If the shortest distance between the lines  $\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}$ and $\frac{x-\lambda }{3}=\frac{y-2\sqrt{6}}{4}=\frac{z+2\sqrt{6}}{5}$ is $6$, then square of sum of all possible values(s) of $\lambda$ is

Question No. 4

Numerical Type
The answer has to be filled into the input box provided below.

If the area of the region bounded by the curves $y^2-2y=-x$ and $x+y=0$ is $A$, then $8A=$

Question No. 5

Numerical Type
The answer has to be filled into the input box provided below.

Let $f_k\left(x\right)=\frac{1}{k}\left({\sin }^{k }x+{\cos }^{k }x\right)$ where $x\in R$ and $k\ge 1$. Then $f_4\left(x\right)-f_6\left(x\right)$ equals M, find $\frac{1}{M}$.

Question No. 6

Numerical Type
The answer has to be filled into the input box provided below.

The urns $A, B$ and $C$ contains $4$ red, $6$ black; $5$ red, $5$ black and $\lambda$ red, $4$ black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn $C$ is $0.4$, then the square of length of the side of largest equilateral triangle, inscribed in the parabola $y^2=\lambda x$ with one vertex at vertex of parabola is

Question No. 7

Numerical Type
The answer has to be filled into the input box provided below.

Let $\overrightarrow{a}=\hat{i}+2\hat{j}+\lambda \hat{k}$, $\overrightarrow{b}=3\hat{i}-5\hat{j}-\lambda \hat{k}$, $\overrightarrow{a}·\overrightarrow{c}=7$, $2\left(\overrightarrow{b}·\overrightarrow{c}\right)+43=0$, $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{c}$, then $\overrightarrow{a}·\overrightarrow{b}$ is equal to

Question No. 8

Numerical Type
The answer has to be filled into the input box provided below.

The equations of the sides $AB, BC \& CA$ of a triangle $ABC$ are $2x+y=0$, $x+py=21a$ $\left(a\ne 0\right)$ and $x-y=3$ respectively. Let $P\left(2,a\right)$ be the centroid of the triangle $ABC$, then ${\left(BC\right)}^2$ is equal to

Question No. 9

Numerical Type
The answer has to be filled into the input box provided below.

The minimum number of elements that must be added to relation $R=\left\{\left(a,b\right),\left(b,c\right),\left(b,d\right)\right\}$ on the set $\left\{a,b,c,d\right\}$, so that it is an equivalence relation is

Question No. 10

Numerical Type
The answer has to be filled into the input box provided below.

Let $f$ be a differentiable function defined on $\left[0,\frac{\mathrm{\pi }}{2}\right]$ such that $f\left(x\right)>0$ and $f\left(x\right)+{\int }_0^{x}f\left(t\right)\sqrt{1-{\left({\log }_e\left(f\left(t\right)\right)\right)}^2}dt=e$ $\forall x\in \left[0,\frac{\mathrm{\pi }}{2}\right]$, then ${\left\{6{\log }_e\left(f\left(\frac{\mathrm{\pi }}{6}\right)\right)\right\}}^2$ is equal to