2023 25 Jan Shift 1 Previous Year paper

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.

Match List I with List II

List - I		List - II
A	Surface tension	I.	$\mathrm{kg} {\mathrm{m}}^{-1} {\mathrm{s}}^{-1}$
B	Pressure	II.	$\mathrm{kg} \mathrm{m} {\mathrm{s}}^{-1}$
C	Viscosity	III.	$\mathrm{kg} {\mathrm{m}}^{-1} {\mathrm{s}}^{-2}$
D	Impulse	IV.	$\mathrm{kg} {\mathrm{s}}^{-2}$

Choose the correct answer from the options given below:

A-IV, B-III, C-II, D-I
A-IV, B-III, C-I, D-II
A-III, B-IV, C-I, D-II
A-II, B-I, C-III, D-IV

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.

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is $100 \mathrm{g}$. The time period of the motion of the particle will be (approximately)

(take $g=10 {\mathrm{ms}}^{-2}$, radius of earth $=6400 \mathrm{km}$)

$24$ hours
$1$ hour $24$ minutes
$1$ hour $40$ minutes
$12$ hours

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.

Match List I with List II

List – I (Current configuration)		List – II (Magnetic field at point $O$)
A		I.	$B_0=\frac{{\mu }_{0}I}{4\pi r}[\pi +2]$
B		II.	$B_0=\frac{{\mu }_0}{4}\frac{I}{r}$
C		III.	$B_0=\frac{{\mu }_{0}I}{2\pi r}[\pi -1]$
D		IV.	$B_0=\frac{{\mu }_{0}I}{4\pi r}[\pi +1]$

Choose the correct answer from the option given below:

A-III, B-IV, C-I, D-II
A-I, B-III, C-IV, D-II
A-III, B-I, C-IV, D-II
A-II, B-I, C-IV, D-III

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.

An ideal gas goes through a reversible cycle $a\to b\to c\to d\to a$ has the V - T diagram shown below. Process $d\to a and b\to c$ are adiabatic.

The corresponding P - V diagram for the process is (all figures are schematic and not drawn to scale) :

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.

Electron beam used in an electron microscope, when accelerated by a voltage of $20 \mathrm{kV}$ has a de-Broglie wavelength of ${\lambda }_0$. If the voltage is increased to $40 \mathrm{kV}$ then the de-Broglie wavelength associated with the electron beam would be:

$3{\lambda }_0$
$9{\lambda }_0$
$\frac{{\lambda }_0}{2}$
$\frac{{\lambda }_0}{\sqrt{2}}$

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.

In an LC oscillator, if values of inductance and capacitance become twice and eight times, respectively, then the resonant frequency of oscillator becomes $x$ times its initial resonant frequency ${\omega }_0$. The value of $x$ is:

$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$
$16$
$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$16$}\right.$
$4$

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.

Three rods of Copper, Brass and Steel are welded together to form a Y-shaped structure. Area of cross-section of each rod is $4 {\mathrm{cm}}^2$. End of copper rod is maintained at $100°\mathrm{C}$. Where as ends of brass and steel are kept at $0°\mathrm{C}$. Lengths of the copper, brass and steel rods are $46, 13$and $12 \mathrm{cms}$ respectively. The rods are thermally insulated from surroundings except at ends. Thermal conductivities of copper, brass and steel are $0.92, 0.26$ and $0.12$ CGS units respectively. Rate of heat flow through copper rod is :

$1.2 \mathrm{Cal}/\mathrm{s}$
$2.4 \mathrm{Cal}/\mathrm{s}$
$4.8 \mathrm{Cal}/\mathrm{s}$
$6.0 \mathrm{Cal}/\mathrm{s}$

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.

$T$ is the time period of simple pendulum on the earth's surface. Its time period becomes $xT$ when taken to a height $R$ (equal to earth's radius) above the earth's surface. Then, the value of $x$ will be:

$4$
$2$
$\frac{1}{2}$
$\frac{1}{4}$

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.

A car travels a distance of $x$ with speed $v_1$ and then same distance $x$ with speed $v_2$ in the same direction. The average speed of the car is:

$\frac{v_1{v}_2}{2\left(v_1+v_2\right)}$
$\frac{v_1+v_2}{2}$
$\frac{2x}{v_1+v_2}$
$\frac{2v_1{v}_2}{v_1+v_2}$

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.

The ratio of the density of oxygen nucleus $\left(\mathrm{O}_8^{16}\right)$ and helium nucleus $\left(\mathrm{He}_2^4\right)$ is

$4:1$
$8:1$
$1:1$
$2:1$

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.

A car is moving with a constant speed of $20 \mathrm{m} {\mathrm{s}}^{-1}$ in a circular horizontal track of radius $40 \mathrm{m}$. A bob is suspended from the roof of the car by a massless string. The angle made by the string with the vertical will be : (Take $\mathrm{g}=10 \mathrm{m} {\mathrm{s}}^{-2}$)

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

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.

All electromagnetic wave is transporting energy in the negative $z$ direction. At a certain point and certain time the direction of electric field of the wave is along positive $y$ direction. What will be the direction of the magnetic field of the wave at that point and instant?

Positive direction of $x$
Positive direction of $z$
Negative direction of $x$
Negative direction of $y$

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 uniform metallic wire carries a current $2 \mathrm{A}$, when $3.4 \mathrm{V}$ battery is connected across it. The mass of uniform metallic wire is $8.92\times {10}^{-3} \mathrm{kg}$, density is $8.92\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$ and resistivity is $1.7\times {10}^{-8} \mathrm{\Omega }-\mathrm{m}$  The length of wire is : 

 $l=6.8 \mathrm{m}$

 $l=10 \mathrm{m}$  

$l=5 \mathrm{m}$  

$l=100 \mathrm{m}$

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.

The root mean square velocity of molecules of gas is

Proportional to square of temperature ($T^2$).
Inversely proportional to square root of temperature ($\sqrt{\frac{1}{T}}$).
Proportional to square root of temperature $\sqrt{T}$.
Proportional to temperature $\left(T\right)$.

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 Young's double slits experiment, the position of 5^th bright fringe from the central maximum is $5 \mathrm{cm}$. The distance between slits and screen is $1 \mathrm{m}$ and wavelength of used monochromatic light is $600 \mathrm{nm}$. The separation between the slits is:

$60 \mathrm{\mu m}$
$48 \mathrm{\mu m}$
$12 \mathrm{\mu m}$
$36 \mathrm{\mu m}$

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.

An object of mass $8 \mathrm{kg}$ is hanging from one end of a uniform rod $CD$ of mass $2 \mathrm{kg}$ and length $1 \mathrm{m}$ pivoted at its end $C$ on a vertical wall as shown in figure. It is supported by a cable $AB$ such that the system is in equilibrium. The tension in the cable is:

(Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

 

$240 \mathrm{N}$
$90 \mathrm{N}$
$300 \mathrm{N}$
$30 \mathrm{N}$

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.

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

Assertion A: Photodiodes are used in forward bias usually for measuring the light intensity.

Reason R: For a $p-n$ junction diode, at applied voltage $V$ the current in the forward bias is more than the current in the reverse bias for $\left|V_z\right|>\pm V\ge \left|V_0\right|$ where $V_0$ is the threshold voltage and $V_z$ is the breakdown voltage.

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

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

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 solenoid of $1200$ turns is wound uniformly in a single layer on a glass tube $2 \mathrm{m}$ long and $0.2 \mathrm{m}$ in diameter. The magnetic intensity at the center of the solenoid when a current of $2 \mathrm{A}$ flows through it is:

$2.4\times {10}^3 \mathrm{A} {\mathrm{m}}^{-1}$
$1.2\times {10}^3 \mathrm{A} {\mathrm{m}}^{-1}$
$1 \mathrm{A} {\mathrm{m}}^{-1}$
$2.4\times {10}^{-3} \mathrm{A} {\mathrm{m}}^{-1}$

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.

A parallel plate capacitor has plate area $40 {\mathrm{cm}}^2$ and plates separation $2 \mathrm{mm}$. The space between the plates is filled with a dielectric medium of a thickness $1 \mathrm{mm}$ and dielectric constant $5$. The capacitance of the system is:

$24{\epsilon }_0 \mathrm{F}$
$\frac{3}{10}{\epsilon }_0 \mathrm{F}$
$\frac{10}{3}{\epsilon }_0 \mathrm{F}$
$10{\epsilon }_0 \mathrm{F}$

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.

Which of the following expressions corresponds to simple harmonic motion along a straight line, where $\mathrm{x}$ is the displacement and $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are positive constants?

$a+b x-c x^2$
$\mathrm{bx}^2$
$a-b x+c x^2$
$-b x$

Physics Numerical (Maximum Marks: 20)

Question No. 1

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

An object of mass $m$ initially at rest on a smooth horizontal plane starts moving under the action of force $F=2 \mathrm{N}$. In the process of its linear motion, the angle $\theta$ (as shown in figure) between the direction of force and horizontal varies as $\theta =kx$, where $k$ is a constant and $x$ is the distance covered by the object from its initial position. The expression of kinetic energy of the object will be $\mathrm{E}=\frac{\mathit{n}}{\mathit{k}}\sin \theta$. The value of $n$ is ______.

Question No. 2

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

In the given circuit, the equivalent resistance between the terminal A and B is ____ $\mathrm{\Omega }$

Question No. 3

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

As shown in the figure, in an experiment to determine Young's modulus of a wire, the extension-load curve is plotted. The curve is a straight line passing through the origin and makes an angle of $45°$ with the load axis. The length of wire is $62.8 \mathrm{cm}$ and its diameter is $4 \mathrm{mm}$. The Young's modulus is found to be  $\mathrm{x}\times {10}^4 \mathrm{N} {\mathrm{m}}^{-2}$

The value of $x$ is _______.

Question No. 4

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

A ray of light is incident from air on a glass plate having thickness $\sqrt{3} \mathrm{cm}$ and refractive index $\sqrt{2}$. The angle of incidence of a ray is equal to the critical angle for glass-air interface. The lateral displacement of the ray when it passes through the plate is $\_\_\_\_\times {{10}^{-}}^2\mathrm{cm}$ . (given $\sin 15°=0.26$)

Question No. 5

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

${\mathrm{I}}_{\mathrm{CM}}$ is moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc. ${\mathrm{I}}_{\mathrm{AB}}$ is its moment of inertia about an axis AB perpendicular to plane and parallel to axis CM at a distance $\frac{2}{3}\mathrm{R}$ from center, where R is the radius of the disc. The ratio of ${\mathrm{I}}_{\mathrm{AB}}$ and ${\mathrm{I}}_{\mathrm{CM}}$ is $x:9$. The value of $x$ is ______.

Question No. 6

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

The distance between two consecutive points with phase difference of $60°$ in a wave of frequency $500 \mathrm{Hz}$ is $6.0 \mathrm{m}$. The velocity with which wave is travelling is ______ $\mathrm{km} {\mathrm{s}}^{-1}$.

Question No. 7

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

If $\overrightarrow{P}=3\hat{\mathrm{i}}+\sqrt{3}\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and $\overrightarrow{Q}=4\hat{\mathrm{i}}+\sqrt{3}\hat{\mathrm{j}}+2.5\hat{\mathrm{k}}$  then, the unit vector in the direction of $\overrightarrow{P}\times \overrightarrow{Q}$ is $\frac{1}{x}(\sqrt{3}\hat{\mathrm{i}}+\hat{\mathrm{j}}-2\sqrt{3}\hat{\mathrm{k}})$. The value of $x$ is

Question No. 8

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

An LCR series circuit of capacitance $62.5 \mathrm{nF}$ and resistance of $50 \mathrm{\Omega }$, is connected to an A.C. source of frequency $2.0 \mathrm{kHz}$. For maximum value of amplitude of current in circuit, the value of inductance is ____$\mathrm{mH}$.$\left.(\text{Take} {\mathrm{\pi }}^2=10\right)$

Question No. 9

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

The wavelength of the radiation emitted is ${\lambda }_0$ when an electron jumps from the second excited state to the first excited state of hydrogen atom. If the electron jumps from the third excited state to the second orbit of the hydrogen atom, the wavelength of the radiation emitted will be $\frac{20}{x}{\lambda }_0$. The value of $x$ is________.

Question No. 10

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

A uniform electric field of $10 \mathrm{N} {\mathrm{C}}^{-1}$ is created between two parallel charged plates (as shown in figure). An electron enters the field symmetrically between the plates with a kinetic energy $0.5 \mathrm{eV}.$ The length of each plate is $10 \mathrm{cm}$. The angle $\left(\theta \right)$ of deviation of the path of electron as it comes out of the field is ______(in degree).

 

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
${\text{E}}_{{\text{Cr}}^{3+}/\text{Cr}}^{\mathrm{o}}=-0.74{\text{ V; E}}_{{\text{MnO}}_4^{-}/{\text{Mn}}^{2+}}^{\mathrm{o}}=1.51V$
${\text{E}}_{{\text{Cr}}_2{\text{O}}_7^{2-}/{\text{Cr}}^{3+}}^{\mathrm{o}}=1\mathrm{.}33{\text{ V; E}}_{{\mathrm{Cl}}_2|{\mathrm{Cl}}^{-}}^{\mathrm{o}}=1.36V$
Based on the data given above, strongest oxidising agent will be:

${\text{Mn}}^{2+}$
${\text{MnO}}_4^{-}$
${\text{Cl}}^{-}$
${\text{Cr}}^{3+}$

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.

Assertion: Among the carbon allotropes, diamond is an insulator, whereas, graphite is a good conductor of electricity.
Reason: Hybridization of carbon in diamond and graphite are $\mathrm{s}{\mathrm{p}}^3$ and $\mathrm{s}{\mathrm{p}}^2$ respectively.

Both assertion and reason are correct, but the reason is not the correct explanation for the assertion.
Both assertion and reason are correct, but the reason is the correct explanation for the assertion.
Both assertion and reason are incorrect.
Assertion is incorrect statement, but the reason is correct.

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 radius of the 2^nd orbit of ${\mathrm{Li}}^{2+}$ is $\mathrm{x}$. The expected radius of the 3^rd orbit of ${\mathrm{Be}}^{3+}$ is

$\frac{9}{4}\mathrm{x}$
$\frac{4}{9}\mathrm{x}$
$\frac{27}{16}\mathrm{x}$
$\frac{16}{27}\mathrm{x}$

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.

The correct order in aqueous medium of basic strength in case of methyl substituted amines is:

${\mathrm{Me}}_2\mathrm{NH}>{\mathrm{MeNH}}_2>{\mathrm{Me}}_3\mathrm{N}>{\mathrm{NH}}_3$
${\mathrm{Me}}_2\mathrm{NH}>{\mathrm{Me}}_3\mathrm{N}>{\mathrm{MeNH}}_2>{\mathrm{NH}}_3$
${\mathrm{NH}}_3>{\mathrm{Me}}_3\mathrm{N}>{\mathrm{MeNH}}_2>{\mathrm{Me}}_2\mathrm{NH}$
${\mathrm{Me}}_3\mathrm{N}>{\mathrm{Me}}_2\mathrm{NH}>{\mathrm{MeNH}}_2>{\mathrm{NH}}_3$

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.

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

Assertion A: Acetal/Ketal is stable in basic medium.

Reason R: The high leaving tendency of alkoxide ion gives the stability to acetal/ketal in basic medium.

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

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.

In the cumene to phenol preparation in presence of air, the intermediate is

 

 

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 rate coefficient $(k)$ for a particular reactions is $1.3 \times 10^{-4} \mathrm{M}^{-1} \mathrm{~s}^{-1}$ at $100^{\circ} \mathrm{C}$, and $1.3 \times 10^{-3}$ $\mathrm{M}^{-1} \mathrm{~s}^{-1}$ at $150^{\circ} \mathrm{C}$. What is the energy of activation $\left(\mathrm{E}_{\mathrm{A}}\right)$ (in $\left.\mathrm{kJ}\right)$ for this reaction? $(\mathrm{R}=$ molar gas constant $=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ )

16
60
99
132

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.

Match items of Row I with those of Row II.

Row I:

(P)

(Q)

(R)

(S)

Row II:

(i) $\mathrm{\alpha }‐\mathrm{D}‐(—)$ Fructofuranose.

(ii) $\mathrm{\beta }‐\mathrm{D}‐(—)$ Fructofuranose.

(iii) $\mathrm{\alpha }‐\mathrm{D}‐(‐)$ Glucopyranose.

(iv) $\mathrm{\beta }‐\mathrm{D}‐(‐)$ Glucopyranose.

Correct match is

P→iv, Q→iii, R→i, S→ii
P→i, Q→ii, R→iii, S→iv
P→iii, Q→iv, R→ii, S→i
P→iii, Q→iv, R→i, S→ii

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 compound which will have the lowest rate towards nucleophilic aromatic substitution on treatment with ${\mathrm{OH}}^{–}$ is

 

 

 

 

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.

Identify the product formed ($A$ and $E$)

 

 

 

 

 

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.

Stability of the species  ${\text{Li}}_2\text{,}{\text{Li}}_2^{-}\text{and}{\text{Li}}_2^{+}$ increases in the order of

${\text{Li}}_2<{\text{Li}}_2^{-}<{\text{Li}}_2^{+}$
${\text{Li}}_2^{-}<{\text{Li}}_2<{\text{Li}}_2^{+}$
${\text{Li}}_2<{\text{Li}}_2^{+}<{\text{Li}}_2^{-}$
${\text{Li}}_2^{-}<{\text{Li}}_2^{+}<{\text{Li}}_2$

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.

Consider the reaction :
${\text{H}}_2{\text{SO}}_{3\left(\text{aq}\right)}+{\text{Sn}}_{\left(\text{aq}\right)}^{4+}+{\text{H}}_2{\text{O}}_{\left(l\right)}\to {\text{Sn}}_{\left(\text{aq}\right)}^{2+}+{\text{HSO}}_{4\left(\text{aq}\right)}^{-}+3{\text{H}}_{\left(\text{aq}\right)}^{+}$
Which of the following statements is correct ?

Sn⁴⁺ is the oxidizing agent because it undergoes oxidation
Sn⁴⁺ is the reducing agent because it undergoes oxidation
H₂SO₃ is the reducing agent because it undergoes reduction
H₂SO₃ is the reducing agent because it undergoes oxidation

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 following statements concern elements in the periodic table. Which of the following is true?

For group $15$ elements, the stability of $+5$ oxidation state increases down the group.
Elements of group $16$ have lower ionization enthalpy values compared to those of group $15$ in the corresponding periods.
The group $13$ elements are all metals.
All the elements in group $17$ are gases.

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.

A gaseous compound of nitrogen and hydrogen contains $12.5 \%$ (by mass) of hydrogen. The density of the compound relative to hydrogen is 16. The molecular formula of the compound is:

$\mathrm{NH}_2$
$\mathrm{N}_3 \mathrm{H}$
$\mathrm{NH}_3$
$\mathrm{N}_2 \mathrm{H}_4$

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.

Which of the following statements/relationships is not correct in thermodynamic changes ?

$\Delta \mathrm{U}=0$ (isothermal reversible expansion of a gas)
$\mathrm{w}=-\mathrm{nRT} \ln \frac{\mathrm{V}_2}{\mathrm{~V}_1}$ (isothermal reversible expansion of an ideal gas)
$\mathrm{w}=\mathrm{nRT} \ln \frac{\mathrm{V}_2}{\mathrm{~V}_1}$ (isothermal reversible expansion of an ideal gas)
For a system of constant volume heat involved directly changes to internal energy.

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.

Match the List-I with List-II :

Cations	Group reaction
$\mathrm{P}\to {\mathrm{Pb}}^{2+},{\mathrm{Cu}}^{2+}$	${\mathrm{H}}_2 \mathrm{S}$ gas in presence of dilute $\mathrm{HCl}$
$\mathrm{Q}\to {\mathrm{Al}}^{3+},{\mathrm{Fe}}^{3+}$	${\left({\mathrm{NH}}_4\right)}_2{\mathrm{CO}}_3$ in presence of ${\mathrm{NH}}_4\mathrm{OH}$
$\mathrm{R}\to {\mathrm{Co}}^{2+},{\mathrm{Ni}}^{2+}$	${\mathrm{NH}}_4\mathrm{OH}$ in presence of ${\mathrm{NH}}_4\mathrm{CI}$
$\mathrm{S}\to {\mathrm{Ba}}^{2+},{\mathrm{Ca}}^{2+}$	${\mathrm{H}}_2 \mathrm{S}$ in presence of ${\mathrm{NH}}_4\mathrm{OH}$

 

 

$\mathrm{P}\to \mathrm{i},\mathrm{Q}\to \mathrm{iii},\mathrm{R}\to \mathrm{ii},\mathrm{S}\to \mathrm{iv}$
$\mathrm{P}\to \mathrm{iv},\mathrm{Q}\to \mathrm{ii},\mathrm{R}\to \mathrm{iii},\mathrm{S}\to \mathrm{i}$
$\mathrm{P}\to \mathrm{iii},\mathrm{Q}\to \mathrm{i},\mathrm{R}\to \mathrm{iv},\mathrm{S}\to \mathrm{ii}$
$\mathrm{P}\to \mathrm{i},\mathrm{Q}\to \mathrm{iii},\mathrm{R}\to \mathrm{iv},\mathrm{S}\to \mathrm{ii}$

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.

Inert gases have positive electron gain enthalpy. Its correct order is

$\mathrm{Xe}<\mathrm{Kr}<\mathrm{Ne}<\mathrm{He}$
$\mathrm{He}<\mathrm{Ne}<\mathrm{Kr}<\mathrm{Xe}$
$\mathrm{He}<\mathrm{Xe}<\mathrm{Kr}<\mathrm{Ne}$
$\mathrm{He}<\mathrm{Kr}<\mathrm{Xe}<\mathrm{Ne}$

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 of the following conformations will be the most stable?

 

 

 

 

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.

In which of the following exothermic reactions, the heat liberated per mole is the highest ?

$\mathrm{CaO}+\mathrm{H}_2 \mathrm{O} \rightarrow \mathrm{Ca}(\mathrm{OH})_2$
$\mathrm{SrO}+\mathrm{H}_2 \mathrm{O} \rightarrow \mathrm{Sr}(\mathrm{OH})_2$
$\mathrm{BaO}+\mathrm{H}_2 \mathrm{O} \rightarrow \mathrm{Ba}(\mathrm{OH})_2$
$\mathrm{MgO}+\mathrm{H}_2 \mathrm{O} \rightarrow \mathrm{Mg}(\mathrm{OH})_2$

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.

The correct sequence of reagents for the preparation of $Q$ and $R$ is:

(i) ${\mathrm{Cr}}_2{\mathrm{O}}_3, 770 \mathrm{K}, 20 \mathrm{atm}$; (ii) ${\mathrm{CrO}}_2{\mathrm{Cl}}_2, {\mathrm{H}}_3{\mathrm{O}}^{+}$; (iii) $\mathrm{NaOH}$; (iv) ${\mathrm{H}}_3{\mathrm{O}}^{+}$
(i) ${\mathrm{CrO}}_2{\mathrm{Cl}}_2, {\mathrm{H}}_3{\mathrm{O}}^{+}$; (ii) ${\mathrm{Cr}}_2{\mathrm{O}}_3, 770 \mathrm{K}, 20 \mathrm{atm}$; (iii) $\mathrm{NaOH}$; (iv) ${\mathrm{H}}_3{\mathrm{O}}^{+}$
(i) ${\mathrm{KMnO}}_4, {\mathrm{OH}}^{-}$; (ii) ${\mathrm{Mo}}_2{\mathrm{O}}_3, \mathrm{\Delta }$; (iii) $\mathrm{NaOH}$; (iv) ${\mathrm{H}}_3{\mathrm{O}}^{+}$
(i) ${\mathrm{Mo}}_2{\mathrm{O}}_3, \mathrm{\Delta }$; (ii) ${\mathrm{CrO}}_2{\mathrm{Cl}}_2, {\mathrm{H}}_3{\mathrm{O}}^{+}$; (iii) $\mathrm{NaOH}$; (iv) ${\mathrm{H}}_3{\mathrm{O}}^{+}$

Chemistry Numerical (Maximum Marks: 20)

Question No. 1

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

Consider the cell $\mathrm{Pt}\left(\mathrm{s}\right)\left|{\mathrm{H}}_2( \mathrm{s})\left(\mathrm{latm}\right)\right|{\mathrm{H}}^{+}\left(\mathrm{aq},\left[{\mathrm{H}}^{+}\right]=1\right)\left|\right|{\mathrm{Fe}}^{3+}\left(\mathrm{aq}\right),{\mathrm{Fe}}^{2+}\left(\mathrm{aq}\right)\mid Pt\left(\mathrm{s}\right)$

Given: ${\mathrm{E}}_{{\mathrm{Fe}}^{3+}/{\mathrm{Fe}}^{2+}}°=0.771 \mathrm{V}$ and ${\mathrm{E}}_{{\mathrm{H}}^{+}/\frac{1}{2}{\mathrm{H}}_2}°=0 \mathrm{V}, \mathrm{T}=298 \mathrm{K}$

If the potential of the cell is $0.712 \mathrm{V}$ the ratio of concentration of ${\mathrm{Fe}}^{2+}$ to ${\mathrm{Fe}}^{3+}$ is
(Nearest integer)

Question No. 2

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

In sulphur estimation. $0.471 \mathrm{g}$ of an organic compound gave $1.4439 \mathrm{g}$ of barium sulphate. The percentage of sulphur in the compound is _______(Nearest Integer)

(Given: Atomic mass Ba: $137\mathrm{u}:\mathrm{S}:32\mathrm{u},\mathrm{O}:16\mathrm{u}$ )

Question No. 3

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

The density of a monobasic strong acid (Molar mass $24.2 \mathrm{g} \mathrm{mol})$ is $1.21 \mathrm{kg} \mathrm{L}$. The volume of its solution required for the complete neutralization of $25 \mathrm{mL}$ of $0.24 \mathrm{M} \mathrm{NaOH}$ is ${10}^{-2} \mathrm{mL}$ (Nearest integer)

Question No. 4

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

The osmotic pressure of solutions of PVC in cyclohexanone at $300 \mathrm{K}$ are plotted on the graph. The molar mass of PVC is  $\mathrm{g} {\mathrm{mol}}^{-1}$ (Nearest integer)

(Given : $\mathrm{R}=0.083 \mathrm{L} \mathrm{atm} {\mathrm{K}}^{-1} {\mathrm{mol}}^{-1}$ )

 

Question No. 5

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

The number of paramagnetic species from the following is

${\left[\mathrm{Ni}(\mathrm{CN}{)}_4\right]}^{2-},\left[\mathrm{Ni}(\mathrm{CO}{)}_4\right],{\left[{\mathrm{NiCl}}_4\right]}^{2-}$

${\left[\mathrm{Fe}(\mathrm{CN}{)}_6\right]}^{4-},{\left[\mathrm{Cu}{\left({\mathrm{NH}}_3\right)}_4\right]}^{2+}$

${\left[\mathrm{Fe}(\mathrm{CN}{)}_6\right]}^{3-}\text{ and }{\left[\mathrm{Fe}{\left({\mathrm{H}}_2\mathrm{O}\right)}_6\right]}^{2+}$

Question No. 6

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

For the first order reaction $\mathrm{A}\to \mathrm{B}$  the half life is $30 \min$. The time taken for $75\%$ completion of the reaction is mm. (Nearest integer)

Given : $\log 2=0.3010$

$\log 3=0.4771$

$\log 5=0.6989$

Question No. 7

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

How many of the following metal ions have similar value of spin only magnetic moment in gaseous state? ________

(Given: Atomic number$:\mathrm{V},23; \mathrm{Cr},24; \mathrm{Fe},26; \mathrm{Ni},28)$

${\mathrm{V}}^{3+}, {\mathrm{Cr}}^{3+}, {\mathrm{Fe}}^{2+}, {\mathrm{Ni}}^{3+}$

Question No. 8

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

The total number of lone pairs of electrons on oxygen atoms of ozone is

Question No. 9

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

A litre of buffer solution contains $0.1$ mole of each of ${\mathrm{NH}}_3$ and ${\mathrm{NH}}_4\mathrm{Cl}$. On the addition of $0.02$ mole of $\mathrm{HCl}$ by dissolving gaseous $\mathrm{HCl}$, the $\mathrm{pH}$ of the solution is found to be $\_\_\_\_\_\times {10}^{-3}$ (Nearest integer)

Given: ${\mathrm{pK}}_{\mathrm{b}}\left({\mathrm{NH}}_3\right)=4.745$

$\log 2=0.301$

$\log 3=0.477$

$\mathrm{T}=298 \mathrm{K}]$

Question No. 10

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

An athlete is given $100 \mathrm{g}$ of glucose $\left({\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6\right)$ for energy. This is equivalent to $1800 \mathrm{kJ}$ of energy. The $50\%$ of this energy gained is utilized by the athlete for sports activities at the event. In order to avoid storage of energy, the weight of extra water he would need to perspire is $\mathrm{\_\_\_\_\_g}$ (Nearest integer) Assume that there is no other way of consuming stored energy.

Given : The enthalpy of evaporation of water is $45 \mathrm{kJ} {\mathrm{mol}}^{-1}$

Molar mass of $\mathrm{C},\mathrm{H}\&\mathrm{O}$ are $12.1$ and $16 \mathrm{g} {\mathrm{mol}}^{-1}$.

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 $z_1=2+3i$ and $z_2=3+4i$. The set $\mathrm{S}=\left\{\mathrm{z}\in \mathrm{C}:{\left|\mathrm{z}-{\mathrm{z}}_1\right|}^2-{\left|\mathrm{z}-{\mathrm{z}}_2\right|}^2={\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}^2\right\}$ represents a

straight line with sum of its intercepts on the coordinate axes equals $14$
hyperbola with the length of the transverse axis $7$
straight line with the sum of its intercepts on the coordinate axes equals $-18$
hyperbola with eccentricity $2$

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 number of distinct real roots of the equation, $\left|\begin{array}{ccc}\cos x& \sin x& \sin x\\ \sin x& \cos x& \sin x\\ \sin x& \sin x& \cos x\end{array}\right|=0$ in the interval $\left[-\frac{\pi }{4},\frac{\pi }{4}\right]$ is :

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

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 value of

$\underset{n\to \infty }{\lim }\frac{1+2-3+4+5-6+\dots +(3n-2)+(3n-1)-3n}{\sqrt{2n^4+4n+3-}\sqrt{n^4+5n+4}}$ is

$\frac{\sqrt{2}+1}{2}$
$3(\sqrt{2}+1)$
$\frac{3}{2}(\sqrt{2}+1)$
$\frac{3}{2\sqrt{2}}$

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.

Let $f:\left(0,1\right)\to \mathrm{ℝ}$ be a function defined by $f\left(x\right)=\frac{1}{1-e^{-x}}$, and $g\left(x\right)=\left(f\left(-x\right)-f\left(x\right)\right)$. Consider two statements

(I) $g$ is an increasing function in $\left(0,1\right)$

(II) $g$ is one-one in $\left(0,1\right)$
Then,

Only (I) is true
Only (II) is true
Neither (I) nor (II) is true
Both (I) and (II) are true

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 minimum value of the function $f\left(x\right)={\int }_0^2{e}^{\left|x-t\right|}dt$ is

$2\left(e-1\right)$
$2e-1$
$2$
$e\left(e-1\right)$

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.

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is:

${47}^{th}$
${45}^{th}$
${46}^{th}$
${44}^{th}$

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.

If the vectors $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mu \hat{\mathrm{k}}$ are mutually orthogonal, then $(\lambda, \mu)=$

$(2,-3)$
$(-2,3)$
$(3,-2)$
$(-3,2)$

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.

The vector $\overrightarrow{a}=-\hat{i}+2\hat{j}+\hat{k}$ is rotated through a right angle, passing through the $y$-axis in its way and the resulting vector is $\overrightarrow{b}$. Then the projection of $3\overrightarrow{a}+\sqrt{2}\overrightarrow{b}$ on $\overrightarrow{c}=5\hat{i}+4\hat{j}+3\hat{k}$ is

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

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 integral $\int_{7 \pi / 4}^{7 \pi / 3} \sqrt{\tan ^2 x} d x$ is equal to :

$\log 2 \sqrt{2}$
$\log 2$
$2 \log 2$
$\log \sqrt{2}$

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.

Let $x=2$ be a local minima of the function $f\left(x\right)=2x^4-18x^2+8x+12,x\in \left(-4,4\right)$. If $M$ is local maximum value of the function $f$ in $\left(-4,4\right)$, then $M=$

$12\sqrt{6}-\frac{33}{2}$
$12\sqrt{6}-\frac{31}{2}$
$18\sqrt{6}-\frac{33}{2}$
$18\sqrt{6}-\frac{31}{2}$

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.

The points of intersection of the line $ax + by=0$, $(\mathrm{a}\ne \mathrm{b})$ and the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}=0$ are $A(\alpha ,0)$ and $B(1,\beta )$. The image of the circle with $AB$ as a diameter in the line $\mathrm{x}+\mathrm{y}+2=0$ is :

$x^2+y^2+5x+5y+12=0$
$x^2+y^2+3x+5y+8=0$
$x^2+y^2+3x+3y+4=0$
$x^2+y^2-5x-5y+12=0$

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.

Let $f\left(x\right)=\int \frac{2x}{\left(x^2+1\right)\left(x^2+3\right)}dx$. If $f\left(3\right)=\frac{1}{2}\left({\log }_{e}5-{\log }_{e}6\right)$, then $f\left(4\right)$ is equal to

$\frac{1}{2}\left({\log }_{\mathrm{e}}17-{\log }_{\mathrm{c}}19\right)$
${\log }_{\mathrm{e}}17-{\log }_{\mathrm{e}}18$
$\frac{1}{2}\left({\log }_{\mathrm{c}}19-{\log }_{\mathrm{c}}17\right)$
${\log }_{\mathrm{c}}19-{\log }_{\mathrm{c}}20$

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 area of triangle formed by the lines joining the vertex of the parabola, $x^2=8 y$, to the extremities of its latus rectum is

2
8
1
4

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.

The mean and variance of the marks obtained by the students in a test are $10$ and $4$ respectively. Later, the marks of one of the students is increased from $8$ to $12$ . If the new mean of the marks is $10.2$. then their new variance is equal to:

$4.04$
$4.08$
$3.96$
$3.92$

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.

The distance of the point $P\left(4,6,-2\right)$ from the line passing through the point $\left(-3,2,3\right)$ and parallel to a line with direction ratios $3,3,-1$ is equal to:

$3$
$\sqrt{6}$
$2\sqrt{3}$
$\sqrt{14}$

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.

Consider the lines $L_1$ and $L_2$ given by

$L_1:\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2}$

$L_2:\frac{x-2}{1}=\frac{y-2}{2}=\frac{z-3}{3}$

A line $L_3$ having direction ratios $1,-1,-2$, intersects $L_1$ and $L_2$ at the points $P$ and $Q$ respectively. Then the length of line segment $PQ$ is

$2\sqrt{6}$
$3\sqrt{2}$
$4\sqrt{3}$
$4$

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.

Let $\mathrm{y}=\mathrm{y}\left(\mathrm{x}\right)$  be the solution curve of the differential equation $\frac{dy}{dx}=\frac{y}{x}\left(1+{\mathrm{xy}}^2\left(1+{\log }_{e}x\right)\right)$, $\mathrm{x}>0,\mathrm{y}\left(1\right)=3$. Then $\frac{{\mathrm{y}}^2\left(\mathrm{x}\right)}{9}$ is equal to :

$\frac{x^2}{5-2x^3\left(2+{\log }_e{x}^3\right)}$
$\frac{x^2}{2x^3\left(2+{\log }_e{x}^3\right)-3}$
$\frac{x^2}{3x^3\left(1+{\log }_e{x}^2\right)-2}$
$\frac{x^2}{7-3x^3\left(2+{\log }_e{x}^2\right)}$

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.

Let $x,y,z>1$ and $A=\left[\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 2& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 3\end{array}\right]$. Then $\left|adj\left(adj{\mathrm{A}}^2\right)\right|$ is equal to

$6^4$
$2^8$
$4^8$
$2^4$

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 M be the maximum value of the product of two positive integers when their sum is $66$ . Let the sample space $S=\left\{x\in Z:x(66-x)\ge \frac{5}{9}M\right\}$ and the event $\mathrm{A}=\{\mathrm{x}\in \mathrm{S}:\mathrm{x}$ is a multiple of $3$}. Then $\mathrm{P}\left(\mathrm{A}\right)$ is equal to

$\frac{15}{44}$
$\frac{1}{3}$
$\frac{1}{5}$
$\frac{7}{22}$

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.

Let $y\left(x\right)=\left(1+x\right)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)$. Then $y^{\text{'}}-y^{"}$ at $x=-1$ is equal to

$976$
$464$
$496$
$944$

Mathematics Numerical (Maximum Marks: 20)

Question No. 1

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

If the sum of all the solutions of
${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)+{\cot }^{-1}\left(\frac{1-x^2}{2x}\right)=\frac{\pi }{3}$, $-1<x<1,x\ne 0$, is $\alpha -\frac{4}{\sqrt{3}}$, then $\alpha$ is equal to _____ .

Question No. 2

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

Let $S=\left\{1,2,3,5,7,10,11\right\}$. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of $3$, is _____ .

Question No. 3

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

Let $x$ and $y$ be distinct integers where $1\le x\le 25$ and $1\le y\le 25$. Then, the number of ways of choosing $x$ and $y$, such that $x+y$ is divisible by $5$ , is _____ .

Question No. 4

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

If $\underset{1}{\overset{2}{\int }}\frac{dx}{{\left(x^2-2x+4\right)}^{\frac{3}{2}}}=\frac{k}{k+5}$, then $k$ is equal to

Question No. 5

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

The constant term in the expansion of
${\left(2x+\frac{1}{x^7}+3x^2\right)}^5$ is _____ .

Question No. 6

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

If the area enclosed by the parabolas $P_1:2y=5x^2$ and $P_2:x^2-y+6=0$ is equal to the area enclosed by $P_1$ and $y=\alpha x,\alpha >0$, then ${\alpha }^3$ is equal to _____ .

Question No. 7

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

If the length of the latus rectum of an ellipse is $4$ units and the distance between a focus and its nearest vertex on the major axis is $\frac{3}{2}$ units, then its eccentricity is $e$, find $\frac{1}{e}$.

Question No. 8

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

For some $a,b,c\in \mathrm{\mathbb{N}}$, let $f\left(x\right)=ax-3$ and $g\left(x\right)=x^b+c,x\in \mathrm{ℝ}$. If ${\left(fog\right)}^{-1} \left(x\right)={\left(\frac{x-7}{2}\right)}^{\frac{1}{3}}$, then $\left(f\circ g\right)\left(ac\right)+\left(g\circ f\right)\left(b\right)$ is equal to _____ .

Question No. 9

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

Let
$S=\left\{\alpha :{\log }_2\left(9^{2\alpha -4}+13\right)-{\log }_2\left(\frac{5}{2}·3^{2\alpha -4}+1\right)=2\right\}.$ Then the maximum value of $\beta$ for which the equation ${\mathrm{x}}^2-2{\left(\sum _{\alpha \in s}\alpha \right)}^2\mathrm{x}+\sum _{a\in s}{\left(\alpha +1\right)}^2\beta =0$ has real roots, is _____ .

Question No. 10

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

Let $A_1,A_2,A_3$ be the three A.P. with the same common difference $d$ and having their first terms as $A,A+1, A+2$, respectively. Let $a,b,c$ be the $7^{\text{th }},9^{\text{th }},{17}^{\text{th }}$ terms of $A_1, A_2, A_3$, respectively such that $\left|\begin{array}{ccc}a& 7& 1\\ 2b& 17& 1\\ c& 17& 1\end{array}\right|+70=0$. If $a=29$, then the sum of first $20$ terms of an AP whose first term is $c-a-b$ and common difference is $\frac{d}{12}$, is equal to _____ .