JEE Main 2023 Question Paper Feb 1 Shift 1- Download PDF

The value of \(\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots +\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}\) is :

• (1) \(\frac{2^{51}}{50 !}\)
• (2) \(\frac{2^{50}}{50 !}\)
• (3) \(\frac{2^{51}}{51 !}\)
• (4) \(\frac{2^{50}}{51 !}\)

Let \(S\) be the set of all solutions of the equation \(\cos ^{-1}(2 x)-2 \cos ^{-1}\left(\sqrt{1-x^2}\right)=\pi\), \(x \in\left[-\frac{1}{2}, \frac{1}{2}\right]\) Then \(\displaystyle\sum_{x \in S} 2 \sin ^{-1}\left(x^2-1\right)\) is equal to

• (1) 0
• (2) \(\frac{-2 \pi}{3}\)
• (3) \(\pi-2 \sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)\)
• (4) \(\pi-\sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)\)

Let \(R\) be a relation on \(R\), given by \(R=\{(a, b): 3 a-3 b+\sqrt{7}\) is an irrational number \(\}\)Then \(R\) is

• (1) reflexive and symmetric but not transitive
• (2) reflexive and transitive but not symmetric
• (3) reflexive but neither symmetric nor transitive
• (4) an equivalence relation

The shortest distance between the lines \(\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-4}{-3}\) and \(\frac{x+3}{1}=\frac{y+5}{4}=\frac{z-1}{-5}\) is

• (1) \(7 \sqrt{3}\)
• (2) \(6 \sqrt{3}\)
• (3) \(4 \sqrt{3}\)
• (4) \(5 \sqrt{3}\)

Let \(S\) denote the set of all real values of \(\lambda\) such that the system of equations
\( \lambda x+y+z=1 \)
\(x+\lambda y+z=1 \)
\(x+y+\lambda z=1\)
is inconsistent, then \(\displaystyle \sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)\) is equal to

• (1) 12
• (2) 4
• (3) 2
• (4) 6

Let \(S=x: x∈R\) and \(\left(\sqrt{3}+\sqrt{2})^{x^2-4}+(\sqrt{3}-\sqrt{2})^{x^2-4}=10\right\}\)Then \(n(S)\) is equal to

• (1) 2
• (2) 4
• (3) 0
• (4) 6

If the center and radius of the circle \(\left|\frac{z-2}{z-3}\right|=2\) are respectively \((\alpha, \beta)\) and \(\gamma\), then \(3(\alpha+\beta+\gamma)\) is equal to

• (1) 12
• (2) 11
• (3) 10
• (4) 9

The mean and variance of 7 observations are 8 and 16, respectively. If one observation 14 is omitted and a and b are respectively the mean and variance of the remaining 6 observations, then \(a+3b−5\) is equal to

Let \(f(x)=2 x+\tan ^{-1} x\) and \(g(x)=\log _e\left(\sqrt{1+x^2}+x\right), x \in[0,3]\) Then

• (1) \(\min f^{\prime}(x)=1+\max g^{\prime}(x)\)
• (2) there exist \(0 \lt x_1 \lt x_2 \lt 3\) such that \(f(x) \lt g(x), \forall x \in\left(x_1, x_2\right)\)
• (3) \(\max f(x)\gt \max g(x)\)
• (4) there exists \(\hat{x} \in[0,3]\) such that \(f^{\prime}(\hat{x}) \lt g^{\prime}(\hat{x})\)

Let \(f(x)= \begin{vmatrix} 1+\sin ^2 x & \cos ^2 x & \sin 2 x \\ \sin ^2 x & 1+\cos ^2 x & \sin 2 x \\ \sin ^2 x & \cos ^2 x & 1+\sin 2 x\end{vmatrix}, x \in\left[\frac{\pi}{6}, \frac{\pi}{3}\right] \) If \(\alpha\) and \(\beta\) respectively are the maximum and the minimum values of \(f\), then

• (1) \(\beta^2+2 \sqrt{\alpha}=\frac{19}{4}\)
• (2) \(\alpha^2+\beta^2=\frac{9}{2}\)
• (3) \(\alpha^2-\beta^2=4 \sqrt{3}\)
• (4) \(\beta^2-2 \sqrt{\alpha}=\frac{19}{4}\)

The area enclosed by the closed curve \(C\) given by the differential equation \(\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0\) is \(4 \pi\).

Let \(P\) and \(Q\) be the points of intersection of the curve \(C\) and the \(y\)-axis If normals at \(P\) and \(Q\) on the curve \(C\) intersect \(x\)-axis at points \(R\) and \(S\) respectively, then the length of the line segment \(R S\) is 

• (1) \(\frac{4 \sqrt{3}}{3}\)
• (2) \(\frac{2 \sqrt{3}}{3}\)
• (3) 2
• (4) \(\quad 2 \sqrt{3}\)

If the orthocentre of the triangle, whose vertices are \((1,2),(2,3)\) and \((3,1)\) is \((\alpha, \beta)\), then the quadratic equation whose roots are \(\alpha+4 \beta\) and \(4 \alpha+\beta\), is

• (1) \(x^2-19 x+90=0\)
• (2) \(x^2-20 x+99=0\)
• (3) \(x^2-18 x+80=0\)
• (4) \(x^2-22 x+120=0\)

If \(y=y(x)\) is the solution curve of the differential equation \(\frac{d y}{d x}+y \tan x=x \sec x, 0 \leq x \leq \frac{\pi}{3}, y(0)=1\) then \(y\left(\frac{\pi}{6}\right)\) is equal to

• (1) \(\frac{\pi}{12}-\frac{\sqrt{3}}{2} \log _e\left(\frac{2}{e \sqrt{3}}\right)\)
• (2) \(\frac{\pi}{12}+\frac{\sqrt{3}}{2} \log _e\left(\frac{2}{e \sqrt{3}}\right)\)
• (3) \(\frac{\pi}{12}+\frac{\sqrt{3}}{2} \log _e\left(\frac{2 \sqrt{3}}{e}\right)\)
• (4) \(\frac{\pi}{12}-\frac{\sqrt{3}}{2} \log _e\left(\frac{2 \sqrt{3}}{e}\right)\)

\(\displaystyle \lim _{n \rightarrow \infty}\left[\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+\ldots+\frac{1}{2 n}\right]\) is equal to

• (1) \(\log _e2\)
• (2) \(\log _e\left(\frac{2}{3}\right)\)
• (3) 0
• (4) \(\log _e\left(\frac{3}{2}\right)\)

The sum to \(10\) terms of the series \(\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\ldots \) is

• (1) \(\frac{56}{111}\)
• (2) \(\frac{58}{111}\)
• (3) \(\frac{55}{111}\)
• (4) \(\frac{59}{111}\)

The combined equation of the two lines \(a x+b y+c=0\) and \(a^{\prime} x+b^{\prime} y+c^{\prime}=0\) can be written as \((a x+b y+c)\left(a^{\prime} x+b^{\prime} y+c^{\prime}\right)=0\)
The equation of the angle bisectors of the lines represented by the equation \(2 x^2+x y-3 y^2=0\) is

• (1) \(3 x^2+5 x y+2 y^2=0\)
• (2) \(x^2-y^2-10 x y=0\)
• (3) \(3 x^2+x y-2 y^2=0\)
• (4) \(x^2-y^2+10 x y=0\)

In a binomial distribution \(B(n, p)\), the sum and the product of the mean and the variance are 5 and 6 respectively, then \(6(n+p-q)\) is equal to

• (1) 50
• (2) 53
• (3) 52
• (4) 51

The negation of the expression \(q \vee((\sim q) \wedge p)\) is equivalent to

• (1) \(p \wedge(\sim q)\)
• (2) \((\sim p) \wedge(\sim q)\)
• (3) \((\sim p) \vee(\sim q)\)
• (4) \((\sim p) \vee q\)

For a triangle \(A B C\), the value of \(\cos 2 A+\cos 2 B+\cos 2 C\) is least If its inradius is \(3\) and incentre is \(M\), then which of the following is NOT correct?

• (1) area of \(\triangle A B C\) is \(\frac{27 \sqrt{3}}{2}\)
• (2) \(\sin 2 A+\sin 2 B+\sin 2 C=\sin A+\sin B+\sin C\)
• (3) perimeter of \(\triangle A B C\) is \(18 \sqrt{3}\)
• (4) \(\overrightarrow{M A} \cdot \overrightarrow{M B}=-18\)

Let the image of the point \(P(2,-1,3)\) in the plane \(x+2 y-z=0\) be \(Q\) Then the distance of the plane \(3 x+2 y+z+29=0\) from the point \(Q\) is

• (1) \(2 \sqrt{14}\)
• (2) \(\frac{22 \sqrt{2}}{7}\)
• (3) \(3 \sqrt{14}\)
• (4) \(\frac{24 \sqrt{2}}{7}\)

If \(\int\limits_0^1\left(x^{21}+x^{14}+x^7\right)\left(2 x^{14}+3 x^7+6\right)^{1 / 7} d x=\frac{1}{l}(11)^{m / n}\) where \(l, m, n \in N , m\) and \(n\) are coprime then \(l+m+n\) is equal to ______

Let \(a_1=8, a_2, a_3, \ldots, a_n\) be an AP If the sum of its first four terms is \(50\) and the sum of its last four terms is \(170\) , then the product of its middle two terms is______

Let \(f: R \rightarrow R\) be a differentiable function such that \(f^{\prime}(x)+f(x)=\int\limits_0^2 f(t) d t\)If \(f(0)=e^{-2}\), then \(2 f(0)-f(2)\) is equal to_____

The remainder, when \(19^{200}+23^{200}\) is divided by \(49\) , is_______

The number of \(3\)-digit numbers, that are divisible by either \(2\) or \(3\) but not divisible by \(7\) , is_________

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is ___________.

Let \(A\) be the area bounded by the curve \(y=x|x-3|\), the \(x\)-axis and the ordinates \(x=-1\) and \(x=2\) Then \(12 A\) is equal to _____

\(A(2,6,2), B(-4,0, \lambda), C(2,3,-1)\) and \(D(4,5,0),|\lambda| \leq 5\) are the vertices of a quadrilateral \(A B C D\) If its area is 18 square units, then \(5-6 \lambda\) is equal to________

If \(f(x)=x^2+g^{\prime}(1) x+g^{\prime \prime}(2)\) and \(g(x)=f(1) x^2+x f^{\prime}(x)+f^{\prime \prime}(x)\), then the value of \(f(4)-g(4)\) is equal to ______

Let \(\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}\) and \(\vec{u}\) be a vector such that \(|\vec{u}|=\alpha>\) If the minimum value of the scalar triple product \([\vec{u}\,\, \vec{v}\,\, \vec{w}]\) is \(-\alpha \sqrt{3401}\), and \(|\vec{u} \cdot \hat{i}|^2=\frac{m}{n}\) where \(m\) and \(n\) are coprime natural numbers, then \(m+n\) is equal to _____

Let \(\sigma\) be the uniform surface charge density of two infinite thin plane sheets shown in figure Then the electric fields in three different region \(E_I, E_{I I}\) and \(E_{I I I}\) are:

• (1) \(\vec{E}_I=0, \vec{E}_{I I}=\frac{\sigma}{\epsilon_0} \hat{n}, E_{I I I}=0\)
• (2) \(\vec{E}_I=\frac{2 \sigma}{\epsilon_0} \hat{n}, \vec{E}_{I I}=0, \vec{E}_{I I I}=\frac{2 \sigma}{\epsilon_0} \hat{n}\)
• (3) \(\vec{E}_I=-\frac{\sigma}{\epsilon_0} \hat{n}, \vec{E}_{I I}=0, \vec{E}_{I I I}=\frac{\sigma}{\epsilon_0} \hat{n}\)
• (4) \(\vec{E}_I=\frac{\sigma}{2 \epsilon_0} \hat{n}, \vec{E}_{I I}=0, \vec{E}_{I I I}=\frac{\sigma}{2 \epsilon_0} \hat{n}\)

' \(n\) ' polarizing sheets are arranged such that each makes an angle \(45^{\circ}\) with the preceeding sheet An unpolarized light of intensity I is incident into this arrangement The output intensity is found to be \(I / 64\) The value of \(n\) will be:

• (1) 6
• (2) 5
• (3) 3
• (4) 4

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

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

Find the magnetic field at the point \(P\) in figure. The curved portion is a semicircle connected to two long straight wires

• (1) \(\frac{\mu_0 i}{2 r}\left(1+\frac{1}{\pi}\right)\)
• (2) \(\frac{\mu_0 i}{2 r}\left(1+\frac{2}{\pi}\right)\)
• (3) \(\frac{\mu_0 i}{2 r}\left(\frac{1}{2}+\frac{1}{\pi}\right)\)
• (4) \(\frac{\mu_0 i}{2 r}\left(\frac{1}{2}+\frac{1}{2 \pi}\right)\)

The equivalent resistance between \(A\) and \(B\) of the network shown in figure:

• (1) \(11 \frac{2}{3} R\)
• (2) \(14 R\)
• (3) \(\frac{8}{3} R\)
• (4) \(21 R\)

If a planet has mass equal to 16 times the mass of earth, and radius equal to 4 times that of earth. The ratio of escape speed of a planet to that of earth is?

• (1) 2:1
• (2) 1:2
• (3) √2:1
• (4) 4:1

A steel wire with mass per unit length \(70 \times 10^{-3} kg\, m ^{-1}\) is under tension of \(70 \,N\).
The speed of transverse waves in the wire will be:

• (1) \(10 \,m / s\)
• (2) \(100\, m / s\)
• (3) \(200 \,\pi m / s\)
• (4) \(50\, m / s\)

An object moves with speed \(v_1, v_2\) and \(v_3\) along a line segment \(AB , BC\) and \(C D\) respectively as shown in figure Where \(AB = BC\) and \(AD =3 AB\), then average speed of the object will be:

• (1) \(\frac{\left(v_1+v_2+v_3\right)}{3}\)
• (2) \(\frac{3 v_1 v_2 v_3}{\left(v_1 v_2+v_2 v_3+v_3 v_1\right)}\)
• (3) \(\frac{\left(v_1+v_2+v_3\right)}{3 v_1 v_2 v_3}\)
• (4) \(\frac{v_1 v_2 v_3}{3\left(v_1 v_2+v_2 v_3+v_3 v_1\right)}\)

A child stands on the edge of the cliff \(10 m\) above the ground and throws a stone horizontally with an initial speed of \(5 \,ms ^{-1}\) Neglecting the air resistance, the speed with which the stone hits the ground will be ______\(ms ^{-1}\) (given, \(g=10 \,ms ^{-2}\) )

• (1) 15
• (2) 20
• (3) 30
• (4) 25

A proton moving with one tenth of velocity of light has a certain de Broglie wavelength of \(\lambda\) An alpha particle having certain kinetic energy has the same de-Brogle wavelength \(\lambda\) The ratio of kinetic energy of proton and that of alpha particle is:

• (1) \(2: 1\)
• (2) \(4: 1\)
• (3) \(1: 4\)
• (4) \(1: 2\)

Given below are two statements:
Statement I : Acceleration due to gravity is different at different places on the surface of earth
Statement II : Acceleration due to gravity increases as we go down below the earth's surface
In the light of the above statements, choose the correct answer from the options given below

• (1) Both Statement I and Statement II are false
• (2) Statement \(I\) is false but Statement II is true
• (3) Statement I is true but Statement II is false
• (4) Both Statement I and Statement II are true

A sample of gas at temperature \(T\) is adiabatically expanded to double its volume The work done by the gas in the process is (given, \(\gamma=\frac{3}{2}\))

• (1) \(W=T R[\sqrt{2}-2]\)
• (2) \(W=\frac{R}{T}[2-\sqrt{2}]\)
• (3) \(W=R T[2-\sqrt{2}]\)
• (4) \(W=\frac{T}{R}[\sqrt{2}-2]\)

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

The mass of proton, neutron and helium nucleus are respectively \(10073\, u\), \(10087\, u\) and \(40015\, u\) The binding energy of helium nucleus is:

• (1) \(14.2 \, MeV\)
• (2) \(7.1 \, MeV\)
• (3) \(56.8\, MeV\)
• (4) \(28.4 \, MeV\)

\(\left(P+\frac{a}{V^2}\right)(V-b)=R T\) represents the equation of state of some gases Where \(P\) is the pressure, \(V\) is the volume, \(T\) is the temperature and \(a, b, R\) are the constants The physical quantity, which has dimensional formula as that of \(\frac{b^2}{a}\), will be:

• (1) Bulk modulus
• (2) Compressibility
• (3) Modulus of rigidity
• (4) Energy density

The average kinetic energy of a molecule of the gas is

• (1) proportional to absolute temperature
• (2) proportional to volume
• (3) dependent on the nature of the gas
• (4) proportional to pressure

A mercury drop of radius \(10^{-3} m\) is broken into \(125\) equal size droplets Surface tension of mercury is \(045\, Nm ^{-1}\) The gain in surface energy is:

• (1) \(5 \times 10^{-5} J\)
• (2) \(17.5 \times 10^{-5} J\)
• (3) \(2.26 \times 10^{-5} J\)
• (4) \(28 \times 10^{-5} J\)

Which of the following frequencies does not belong to FM broadcast?

• (1) \(64 \,MHz\)
• (2) \(106 \,MHz\)
• (3) \(99\, MHz\)
• (4) \(89 \,MHz\)

Two equal positive point charges are separated by a distance \(2 a\) The distance of a point from the centre of the line joining two charges on the equatorial line (perpendicular bisector) at which force experienced by a test charge \(q_0\) becomes maximum is \(\frac{a}{\sqrt{x}}\) The value of \(x\) is ______

A charge particle of \(2 \,\mu C\) accelerated by a potential difference of \(100\, V\) enters a region of uniform magnetic field of magnitude \(4 \,mT\) at right angle to the direction of field The charge particle completes semicircle of radius \(3 \,cm\) inside magnetic field The mass of the charge particle is _______ \(\times 10^{-18} kg\)

A light of energy 12.75 eV is incident on a hydrogen atom in its ground state. The atom absorbs the radiation and reaches to one of its excited states. The angular momentum of the atom in the excited state is \(\frac{x}{π} \times 10^{-17} eVs\). The value of x __________ is (use \(h=4.14 \times 10^{-15} eVs\), \(c=3 \times 108 ms^{-1}\)).

A thin cylindrical rod of length \(10\, cm\) is placed horizontally on the principle axis of a concave mirror of focal length \(20 \,cm\) The rod is placed in a such a way that mid point of the rod is at \(40 \,cm\) from the pole of mirror The length of the image formed by the mirror will be \(\frac{x}{3} cm\) The value of \(x\) is ______

A small particle moves to position \(5 \hat{i}-2 \hat{j}+\hat{k}\) from its initial position \(2 \hat{i}+3 \hat{j}-4 \hat{k}\) under the action of force \(5 \hat{i}+2 \hat{j}+7 \hat{k} N\). The value of work done will be ________\(J\)

A series LCR circuit is connected to an ac source of \(220\, V , 50\, Hz\) The circuit contain a resistance \(R =100\, \Omega\) and an inductor of inductive reactance \(X _{ L }=796\, \Omega\) The capacitance of the capacitor needed to maximize the average rate at which energy is supplied will be ________\(\mu F\)

A solid cylinder is released from rest from the top of an inclined plane of inclination 30° and length 60 cm If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is ________ \(ms ^{-1}\)(Given \(g =10 \,ms ^{-2}\) )

The amplitude of a particle executing SHM is \(3 \,cm\) The displacement at which its kinetic energy will be \(25 \%\) more than the potential energy is:_______ \(cm\)

A certain pressure ' \(P\) ' is applied to \(1\) litre of water and \(2\) litre of a liquid separately Water gets compressed to \(0.01 \%\) whereas the liquid gets compressed to \(0.03 \%\) The ratio of Bulk modulus of water to that of the liquid is \(\frac{3}{x}\) The value of \(x\) is ________

In an experiment to find emf of a cell using potentiometer, the length of null point for a cell of emf \(15 \,V\) is found to be \(60 \,cm\) If this cell is replaced by another cell of emf \(E\), the length-of null point increases by \(40 \,cm\) The value of \(E\) is \(\frac{x}{10} V\) The value of \(x\) is _______

Which of the following complex will show largest splitting of \(d\)-orbitals?

• (1) \(\left[ F _{ e }\left( C _2 O _4\right)_3\right]^{3-}\)
• (2) \(\left[ F _{ e }( CN )_6\right]^{3-}\)
• (3) \(\left[ F _{ e }\left( NH _3\right)_6\right]^{3+}\)
• (4) \(\left[ F _{ e } F _6\right]^{3-}\)

Which of the following are the example of double salt?
A \(FeSO _4 \left( NH _4\right)_2 SO _4 \cdot 6 H _2 O\)
B \(CuSO _4 4 NH _3 H _2 O\)
C \(K _2 SO _4 \cdot Al _2\left( SO _4\right)_3 \cdot 24 H _2 O\)
D \(Fe ( CN )_2 \cdot 4 KCN\)
Choose the correct answer

• (1) A and B only
• (2) A, B and D only
• (3) A and C only
• (4) B and D only

Highest oxidation state of \(Mn\) is exhibited in \(Mn _2 O _7\) The correct statements about \(Mn _2 O _7\) are
(A) Mn is tetrahedrally surrounded by oxygen atoms
(B) \(Mn\) is octahedrally surrounded by oxygen atoms
(C) Contains \(Mn-O-Mn\) bridge
(D) Contains \(Mn - Mn\) bond
Choose the correct answer from the options given below:

• (1) A and D only
• (2) A and C only
• (3) B and C only
• (4) B and D only

The correct representation in six-membered pyranose form for the following sugar \( [X]\) is 

• (1)
• (2)
• (3)
• (4)

But-2-yne is reacted separately with one mole of Hydrogen as shown below: 

A. A is more soluble than B
B. B The boiling point & melting point of \(A\) are higher and lower than \(B\) respectively 
C. A is more polar than \(B\) because dipole moment of \(A\) is zero 
D. \(Br _2\) adds easily to \(B\) than \(A\) 
Identify the incorrect statements from the options given below:

• (1) A, C & D only
• (2) A and B only
• (3) B, C & D only
• (4) B and C only

Resonance in carbonate ion \(\left( CO _3{ }^{2-}\right)\) is

• (1) All these structures are in dynamic equilibrium with each other.
• (2) Each structure exists for equal amount of time.
• (3) \(CO _3{ }^{2-}\) has a single structure i.e., resonance hybrid of the above three structures.
• (4) It is possible to identify each structure individually by some physical or chemical method.

Match List I with List II

Choose the correct answer from the options given below:

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

In the following reaction, ' \(A\) ' is
 

• (1)
• (2)
• (3)
• (4)

A solution of \(FeCl _3\) when treated with \(K _{ t }\left[ Fe ( CN )_6\right]\) gives a prussiun blue precipitate due to the formation of

• (1) \(Fe \left[ Fe ( CN )_6\right]\)
• (2) \(K \left[ Fe _2( CN )_6\right]\)
• (3) \(Fe _4\left[ Fe ( CN )_6\right]_3\)
• (4) \(Fe _3\left[ Fe ( CN )_6\right]_2\)

Identify the incorrect option from the following:

• (1)

• (2)

• (3)

• (4)

Decreasing order of dehydration of the following alcohols is

• (1) \(a\gt d\gt b\gt c\)
• (2) \(d \gt b \gt c \gt a\)
• (3) \(b \gt a \gt d \gt c\)
• (4) \(b\gt d\gt c\gt a\)

How can photochemical smog be controlled?

• (1) By using catalytic convertors in the automobiles/industry.
• (2) By complete combustion of fuel.
• (3) By using catalyst.
• (4) By using tall chimneys.

Match List I with List II

Choose the correct answer from the options given below:

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

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

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: In an Ellingham diagram, the oxidation of carbon to carbon monoxide shows a negative slope with respect to temperature. 
Reason R: CO tends to get decomposed at higher temperature. 
In the light of the above statements, choose the correct answer from the options given below 

• (1) Both A and R are correct and R is the correct explanation of A
• (2) Both A and R are correct but R is NOT the correct explanation of A
• (3) A is correct but R is not correct
• (4) A is not correct but R is correct

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\)
Assertion A : Hydrogen is an environment-friendly fuel
Reason R : Atomic number of hydrogen is 1 and it is a very light element
In the light of the above statements, choose the correct answer from the options given below

• (1) Both \(A\) and \(R\) are true and \(R\) is the correct explanation of \(A\)
• (2) A is true but \(R\) is false
• (3) \(A\) is false but \(R\) is true
• (4) Both \(A\) and \(R\) are true but \(R\) is NOT the correct explanation of \(A\)

Which of the following represents the lattice structure of \(A _{0.95} O\) containing \(A ^{2+}, A ^{3+}\) and \(O ^{2-}\) ions?

• (1) A and B only
• (2) B and C only
• (3) A only
• (4) B only

Choose the correct statement(s):
A Beryllium oxide is purely acidic in nature
B Beryllium carbonate is kept in the atmosphere of \(CO _2\)
C Beryllium sulphate is readily soluble in water
D Beryllium shows anomalous behavior
Choose the correct answer from the options given below:

• (1) A, B and C only
• (2) B, C and D only
• (3) A and B only
• (4) A only

Given below are two statements:
Statement I : Chlorine can easily combine with oxygen to form oxides; and the product has a tendency to explode
Statement II : Chemical reactivity of an element can be determined by its reaction with oxygen and halogens
In the light of the above statements, choose the correct answer from the options given below

• (1) Both the Statements I and II are false
• (2) Both the Statements I and II are true
• (3) Statement I is false but Statement II is true
• (4) Statement I is true but Statement II is false

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason \(R\)
Assertion A: Amongst \(He, Ne, Ar\) and \(Kr\); \(1 g\) of activated charcoal adsorbs more of \(Kr\)
Reason R: The critical volume \(V_c\left( cm ^3 mol ^{-1}\right)\) and critical pressure \(P_e\) (atm) is highest for Krypton but the compressibility factor at critical point \(Z_c\) is lowest for Krypton 
In the light of the above statements, choose the correct answer from the options given below

• (1) Both \(A\) and \(R\) are true but \(R\) is NOT the correct explanation of \(A\)
• (2) Both \(A\) and \(R\) are true and \(R\) is the correct explanation of \(A\)
• (3) A is false but \(R\) is true
• (4) \(A\) is true but \(R\) is false

\(25 \,mL\) of an aqueous solution of \(KCl\) was found to require \(20\, mL\) of \(1\, M\) \(AgNO _3\) solution when titrated using \(K _2 CrO _4\) as an indicator What is the depression in freezing point of \(KCl\) solution of the given concentration? _________ (Nearest integer) (Given: \(K _f=20 \,K \,kg \,mol ^{-1}\)) 
Assume 1) \(100 \%\) ionization and 
2) density of the aqueous solution as \(1 \,g \,mL ^{-1}\)

Electrons in a cathode ray tube have been emitted with a velocity of \(1000\, m s ^{-1}\). The number of following statements which is/are true about the emitted radiation is _______ Given : \(h =6 \times 10^{-34} J s , m _e=9 \times 10^{-31} kg\) 
(A) The deBroglie wavelength of the electron emitted is \(66667 nm\)
(B) The characteristic of electrons emitted depend upon the material of the electrodes of the cathode ray tube 
(C) The cathode rays start from cathode and move towards anode 
(D) The nature of the emitted electrons depends on the nature of the gas present in cathode ray tube

At what \(pH\), given half cell \(MnO _4^{-}(01 M ) \mid Mn ^{2+}(0001 M )\) will have electrode potential of \(1.282\, V\)? (Nearest Integer)
(Given: \(E _{ MnO _{4}^-| Mn ^{2+}}^{ o }=154 \,V , \frac{2303 RT }{ F }=0059\, V\))

\(A\) and \(B\) are two substances undergoing radioactive decay in a container The half life of \(A\) is \(15 min\) and that of \(B\) is \(5\, min\) If the initial concentration of \(B\) is \(4\) times that of \(A\) and they both start decaying at the same time, how much time will it take for the concentration of both of them to be same? ______ min

At \(25^{\circ} C\), the enthalpy of the following processes are given: 
\(H _2( g )+ O _2( g ) \rightarrow 2 OH ( g ) \Delta H ^{\circ} =78 \,kJ \,mol ^{-1}\)
\(H _2( g )+1 / 2 O _2( g ) \rightarrow H _2 O ( g ) \Delta H ^{\circ} =-242 \,kJ \,mol ^{-1}\)
\(H _2( g ) \rightarrow 2 H ( g ) \Delta H ^{\circ} =436\, kJ \,mol ^{-1}\)
\(1 / 2 O _2( g ) \rightarrow O ( g ) \Delta H ^{\circ} =249\, kJ\, mol ^{-1}\) 
What would be the value of \(X\) for the following reaction?______ (Nearest integer) 
\(H _2 O ( g ) \rightarrow H ( g )+ OH ( g ) \Delta H ^{\circ}= X\, kJ\, mol -1\)

Sum of oxidation states of bromine in bromic acid and perbromic acid is ——-.

Number of isomeric compounds with molecular formula \(C _9 H _{10} O\) which 
(i) do not dissolve in \(NaOH\)
(ii) do not dissolve in \(HCl\)
(iii) do not give orange precipitate with 2,4-DNP 
(iv) on hydrogenation give identical compound with molecular formula \(C _9 H _{12} O\) is _________

The total number of chiral compound/s from the following is ________

(i) \(X ( g ) \rightleftharpoons Y ( g )+ Z\) (g) \(K _{ p 1}=3\)
(ii)\(A ( g ) \rightleftharpoons 2 B ( g )\,\, K _{ p 2}=1\)
If the degree of dissociation and initial concentration of both the reactants \(X ( g )\) and \(A ( g )\) are equal, then the ratio of the total pressure at equilibrium \(\left(\frac{p_1}{p_2}\right)\) is equal to \(x : 1\). The value of \(x\) is _________ (Nearest integer)

The density of \(3 M\) solution of \(NaCl\) is \(10 \,g \,mL -1\). Molality of the solution is _______ \(\times 10^{-2} m\) (Nearest integer). Given: Molar mass of Na and \(Cl\) is \(23\) and \(355 \,g \,mol ^{-1}\) respectively.