JEE Main 2023 Jan 29 Shift 2 Question Paper with Solution PDF- Download Here

Let \(𝑦=𝑦(𝑥)\) be the solution of the differential equation \(𝑥\;𝑙𝑜𝑔_𝑒⁡𝑥\frac{𝑑𝑦}{ 𝑑𝑥} +𝑦=𝑥^2𝑙𝑜𝑔_𝑒⁡𝑥,(𝑥>1)\).  If \(𝑦(2)=2\), then \(𝑦(𝑒)\) is equal to 

• (1) \(\frac{2+e^2}{2}\)
• (2) \(\frac{1+e^2}{2}\)
• (3) \(\frac{1+e^2}{4}\)
• (4) \(\frac{4+e^2}{4}\)

Let R be a relation defined on N as a ⁡R⁡b if 2a+3b is a multiple of 5 ,a , b∈N. Then R is

• (1) transitive but not symmetric
• (2) an equivalence relation
• (3) not reflexive
• (4) symmetric but not transitive

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is

• (1) 79
• (2) 86
• (3) 84
• (4) 89

If \(\overrightarrow a =\hat i+2\hat k,\overrightarrow b =\hat i+\hat j+\hat k,\overrightarrow c =7\hat i−3\hat j+4\hat k,r ×\hat b +\hat b ×\overrightarrow c =\overrightarrow0\) and \(\overrightarrow r .\overrightarrow a =0\). Then \(\overrightarrow r. \overrightarrow c\)  is equal to 

• (1) 30
• (2) 32
• (3) 36
• (4) 34

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is

• (1) 507
• (2) 432
• (3) 472
• (4) 400

Let \(𝑆={𝑤_1,𝑤_2,…….}\) be the sample space associated to a random experiment. Let \(𝑃(𝑤_𝑛)= \frac{𝑃(𝑤_{𝑛−1})}{2} ,𝑛≥2.\) Let 𝐴=\({(2𝑘+3𝑙:k,𝑙∈ℕ)}\) and \(𝐵={𝑤_𝑛:𝑛∈𝐴}\). Then 𝑃( 𝐵) is equal to

• (1) \(\frac3{64}\)
• (2) \(\frac1{16}\)
• (3) \(\frac1{32}\)
• (4) \(\frac3{32}\)

Consider a function f:N→R, satisfying
\( f(1)+2f(2)+3f(3)+⋯+xf(x)=x(x+1)f(x);x≥2 with f(1)=1.\)
Then \(\frac{1}{f(2022)} + \frac{1}{f(2028)}\) is equal to

• (1) 8100
• (2) 8400
• (3) 8000
• (4) 8200

The set of all values of \(λ\) for which the equation \(cos^2⁡2x−2sin^4⁡x−2cos^2⁡x=λ\) has a real solution x, is 

• (1) \([-1,-\frac{1}{2}]\)
• (2) \([-\frac{3}{2},-1]\)
• (3) \([-\frac{3}{2},-1]\)
• (4) [-2,-1]

The area of the region A = \({(x,y):|cos⁡x−sin⁡x|≤y≤sin⁡x,0≤x≤\frac{π}{ 2} }\) is

• (1) \(\sqrt 5−2\sqrt 2+1\)
• (2) \(1-\frac{3}{\sqrt 2}+\frac{4}{\sqrt 5}\)
• (3) \(\frac{3}{\sqrt 5}-\frac{3}{\sqrt 2}+1\)
• (4) \(\sqrt 5+2\sqrt 2−4.5\)

Let \(\overrightarrow 𝑎 =4\hat 𝑖+3\hat 𝑗ˆ\) and \(\overrightarrow 𝑏 =3\hat 𝑖−4\hat 𝑗\)+\(5\hat 𝑘\). If \(\overrightarrow 𝑐\)  is a vector such that \(\overrightarrow 𝑐 ⋅(\overrightarrow 𝑎 ×\overrightarrow𝑏⃗ )+25=0\), \(\overrightarrow𝑐 ⋅(\hat𝑖+\hat𝑗+ \hat𝑘)=4\), and projection of \(\overrightarrow𝑐\)  on \(\overrightarrow𝑎\)  is \(1\) , then the projection of \(\overrightarrow 𝑐\)  on \(\overrightarrow 𝑏\) equals 

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

The value of the integral \(\int^2_1\bigg(\frac{t^4+1}{t^6+1}\bigg)\)dt is 

• (1) \(tan^{-1}2-\frac{1}{3}tan^{-1}8+\frac{\pi}{3}\)
• (2) \(tan^{-1}2-\frac{1}{3}tan^{-1}8-\frac{\pi}{3}\)
• (3) \(tan^{-1}\frac{1}{2}+\frac{1}{3}tan^{-1}8-\frac{\pi}{3}\)
• (4) \(tan^{-1}\frac{1}{2}-\frac{1}{3}tan^{-1}8+\frac{\pi}{3}\)

Let 𝐾 be the sum of the coefficients of the odd powers of 𝑥 in the expansion of \((1+𝑥)^{99}\). Let 𝑎 be the middle term in the expansion of \((2+\frac{1}{\sqrt 2} )^{200}\). If \(\frac{⁡^{200}𝐶_{99} 𝐾}{𝑎} =\frac{2^𝑙 𝑚}{𝑛} \), where 𝑚 and 𝑛 are odd numbers, then the ordered pair (𝑙, 𝑛) is equal to

• (1) (50,51)
• (2) (50,101)
• (3) (51,99)
• (4) (51,101)

The value of the integral \(∫_{\frac 12}^2 \frac{tan^{-1}x}{x}dx\) is equal to

• (1) \( \frac{π}2 log_e⁡2\)
• (2) \( 2 log_e⁡2\)
• (3) \( \frac{1}2 log_e⁡2\)
• (4) \( \frac{π}4 log_e⁡2\)

The shortest distance between the lines \(\frac{x−1}2 =\frac{y−2}1 =\frac{z−6}{-3} \) and \(\frac{x−1}2 =\frac{y+8}{-7} =\frac{z−4}{5} \) is

• (1) \(5\sqrt 3\)
• (2) \(2\sqrt 3\)
• (3) \(3\sqrt 3\)
• (4) \(4\sqrt 3\)

Let f and g be twice differentiable functions on R such that
f''(x)=g''(x)+6x
f''(1)=4g'(1)-3=9
f(2)=3g(2)=12.
Then which of the following is NOT true?

• (1) If \(−1<𝑥<2\), then \(|𝑓(𝑥)−𝑔(𝑥)|<8\)
• (2) \(|𝑓'(𝑥)−𝑔'(𝑥)|<6⇒−1<𝑥<1\)
• (3) \(𝑔(−2)−𝑓(−2)=20\)
• (4) There exists \(𝑥_0∈(1,\frac{3}{2})\) such that \(𝑓(𝑥0)=𝑔(𝑥0)\)

The statement B⇒((∼A) ∨ B) is equivalent to :

• (1) B⇒(A⇒B)
• (2) A⇒((∼A)⇒B)
• (3) A⇒(A⇒B)
• (4) B⇒((∼A)⇒B)

The plane \(2x−y+z=4\) intersects the line segment joining the points \(A(a,−2,4)\) and \(B(2,b,−3)\) at the point C in the ratio \(2:1\) and the distance of the point C from the origin is \(\sqrt5\). If \(ab<0 \) and \(P\) is the point \((a−b,b,2b−a)\). Then \(CP^2 \) is equal to

• (1) \(\frac{16}{3}\)
• (2) \(\frac{17}{3}\)
• (3) \(\frac{73}{3}\)
• (4) \(\frac{97}{3}\)

If the tangent at a point 𝑃 on the parabola \(𝑦^2=3𝑥\) is parallel to the line \(𝑥+2𝑦=1\) and the tangents at the points 𝑄 and 𝑅 on the ellipse \(\frac{𝑥^2}{ 4} +\frac{𝑦^2}{ 1} =1\) are perpendicular to the line 𝑥−𝑦=2, then the area of the triangle PQR is : 

• (1) \(\frac{3}{2}\sqrt 5\)
• (2) \(5\sqrt 3\)
• (3) \(3\sqrt 5\)
• (4) \(\frac{9}{\sqrt 5}\)

If the lines \(\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{1}\)  and \(\frac{x-a}{2}=\frac{y+2}{3}=\frac{z-3}{1}\) intersect at the point 𝑃, then the distance of the point 𝑃 from the plane 𝑧=𝑎 is : 

• (1) 10
• (2) 22
• (3) 28
• (4) 16

The set of all values of \(t∈R\), for which the matrix \(\begin{bmatrix} e^t & e^{−t}(sin⁡\ t−2cos\ ⁡t) & e^{−t}(−2sin⁡\ t−cos\ ⁡t) \\[0.3em] e^t & e^{−t}(2sin\ ⁡t+cos\ ⁡t) & e^{−t}(sin\ ⁡t−2cos\ ⁡t) \\[0.3em] e^t & e^{−tcos\ ⁡t}& e^{−tsin\ ⁡t} \end{bmatrix}\) is invertible, is

• (1) \( \mathbb{R} \)
• (2) \( \{(2k+1)\frac \pi2 ,k∈Z\}\)
• (3) \(\{k\pi+\frac \pi4 ,k∈Z\}\)
• (4) \(\{k\pi,k∈Z\}\)

The total number of 4-digit numbers whose greatest common divisor with 54 is 2 , is ____.

Let \(𝑋={11,12,13,…,40,41}\) and \(𝑌={61,62,63,…,90,91}\) be the two sets of observations. If \(\bar 𝑥\) and \(\bar 𝑦\) are their respective means and \(σ^2\) is the variance of all the observations in \(𝑋∪𝑌\), then \(|\bar 𝑥+\bar 𝑦−σ^2|\) is equal to ____ .

Let \(α=8−14𝑖\), \(𝐴=\{z∈C:\frac {αz−\bar α\bar z}{z^2−(\bar z)^2−112i }=1\}\) and \(𝐵={𝑧∈C:|𝑧+3𝑖|=4}\). Then \(∑_{𝑧∈𝐴∩𝐵 }(\text {𝑅𝑒}\ ⁡𝑧−\text {𝐼𝑚⁡}\ 𝑧)\) is equal to _____ .

Let \( \alpha_1, \alpha_2, \dots, \alpha_7 \) be the roots of the equation \( x^7 + 3x^5 - 13x^3 - 15x = 0 \) and \( |\alpha_1| \geq |\alpha_2| \geq \dots \geq |\alpha_7| \). Then \( \alpha_1 \alpha_2 - \alpha_3 \alpha_4 + \alpha_5 \alpha_6 \) is equal to:

Let \({a_k}\) and \({b_k}\), \(𝑘∈N\), be two G.P.s with common ratios \(r_1\) and \(r_2\) respectively such that \(a_1= b_1=4\) and \(r_1<r_2\). Let \(c_k=a_k+b_k\), \(k∈N\). If \(c_2=5\) and \(c_3=\frac {13}{4}\), then \( \displaystyle\sum_{k=1}^{∞ }c_k−(12a_6+8b_4)\) is equal to _____ .

Let \(A\) be a symmetric matrix such that \(|A|=2\) and \(\begin{bmatrix} 2 & 1 \\[0.3em] 3 & \frac 32 \\[0.3em] \end{bmatrix}𝐴−\begin{bmatrix} 1 & 2 \\[0.3em] α & β \\[0.3em] \end{bmatrix}\). If the sum of the diagonal elements of \(A\) is \(s\), then \(\frac {βs}{α^2}\) is equal to ____ .

A circle with centre \((2,3) \) and radius \(4\) intersects the line \(𝑥+𝑦=3\) at the points \(𝑃\) and \(𝑄\). If the tangents at \(𝑃\) and \(𝑄\) intersect at the point \(S(α,β)\), then \(4α−7β\) is equal to _____ .

A triangle is formed by the tangents at the point \((2,2)\) on the curves \(𝑦^2=2𝑥\) and \(𝑥^2+𝑦^2=4𝑥\), and the line \(𝑥+𝑦+2=0\). If \(𝑟\) is the radius of its circumcircle, then \(𝑟^2\) is equal to ____ .

Let \(𝑎_1=𝑏_1=1\)and \(𝑎_𝑛=𝑎_{𝑛−1}+(𝑛−1)\), \(𝑏_𝑛=𝑏_{𝑛−1}+𝑎_{𝑛−1}\), \(∀𝑛≥2\). If \(𝑆= \displaystyle\sum_{ 𝑛=1}^{10}\frac {𝑏_𝑛}{2^𝑛}\) and \(𝑇= \displaystyle\sum_{𝑛=1}^{8} \frac {𝑛}{2^{𝑛−1}}\), then \(2^7(2𝑆−𝑇)\)  is equal to ____ .

If the equation of the normal to the curve \(𝑦= 𝑥−𝑎 (𝑥+𝑏)(𝑥−2)\) at the point \((1,−3)\) is \(𝑥−4𝑦=13\), then the value of \(𝑎+𝑏\) is equal to ___ .

The ratio of de-Broglie wavelength of an \(\alpha\) particle and a proton accelerated from rest by the same potential is \(\frac {1}{\sqrt 𝑚}\) , the value of \(m\) is-

• (1) 16
• (2) 4
• (3) 2
• (4) 8

A force acts for 20 s on a body of mass 20 kg, starting from rest, after which the force ceases and then body describes 50 m in the next 10 s. The value of force will be:

• (1) 40 N
• (2) 5 N
• (3) 20 N
• (4) 10 N

The equation of a circle is given by \(𝑥^2+𝑦^2=𝑎^2\), where 𝑎 is the radius. If the equation is modified to change the origin other than (0,0) then. find out the correct dimensions of A and B in a new equation : \((𝑥−𝐴𝑡)^2+(𝑦−𝑡 𝐵 )^2 =𝑎^2\). The dimensions of 𝑡 is given as \([T^{−1}]\).

• (1) \(A=[LT],B=[L^{−1} T^{−1}] \)
• (2) \(A=[L^{−1} T^{−1}],B=[LT] \)
• (3) \(A=[L^{−1} T],B=[LT^{−1}] \)
• (4) \(A=[L^{−1} T^{−1}],B=[LT^{−1}]\)

A square loop of area \(25\ cm^2\) has a resistance of \(10\ Ω\). The loop is placed in uniform magnetic field of magnitude \(40.0 \ T\). The plane of loop is perpendicular to the magnetic field. The work done in pulling the loop out of the magnetic field slowly and uniformly in \(1.0\) sec, will be

• (1) \(1.0×10^{-3} J \)
• (2) \( 2.5×10^{−3} J \)
• (3) \( 5×10^{−3} J \)
• (4) \( 1.0×10^{−4} J \)

Heat energy of 184 kJ is given to ice of mass \(600 g\) at −12°C. Specific heat of ice is \( 2222.3 J kg^{−1}C^{−1}\) and latent heat of ice in \( 336 kJkg^{−1} \)
A. Final temperature of system will be 0°C.
B. Final temperature of the system will be greater than 0°C.  
C. The final system will have a mixture of ice and water in the ratio of 5:1.  
D. The final system will have a mixture of ice and water in the ratio of 1:5. 
E. The final system will have water only.
Choose the correct answer from the options given below:

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

Identify the correct statements from the following:
A. Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative.
B. Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative.  
C. Work done by friction on a body sliding down an inclined plane is positive.  
D. Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity in zero.  
E. Work done by the air resistance on an oscillating pendulum in negative.  
Choose the correct answer from the options given below:

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

A scientist is observing a bacteria through a compound microscope. For better analysis and to improve its resolving power he should. (Select the best option)

• (1) Increase the refractive index of the medium between the object and objective lens
• (2) Decrease the diameter of the objective lens
• (3) Increase the wave length of the light
• (4) Decrease the focal length of the eye piece.

Given below are two statements: 
Statement I: Electromagnetic waves are not deflected by electric and magnetic field.
Statement II: The amplitude of electric field and the magnetic field in electromagnetic waves are related to each other as \(E_0 = \sqrt{\frac{\mu_0}{E_0}}B_0\).  
In the light of the above statements, choose the correct answer from the options given below:

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

The time taken by an object to slide down 45° rough inclined plane is n times as it takes to slide down a perfectly smooth 45° incline plane. The coefficient of kinetic friction between the object and the incline plane is:

• (1) \(\sqrt{1−\frac1{𝑛_2} }\)
• (2) \(1+ \frac{1}{𝑛_2} \)
• (3) \(1- \frac{1}{𝑛_2} \)
• (4) \(\sqrt{ \frac{1}{1-𝑛^2}}\)

For the given figures, choose the correct options:

• (1) At resonance, current in (b) is less than that in (a)
• (2) The rms current in circuit (b) can never be larger than that in (a)
• (3) The rms current in figure(a) is always equal to that in figure (b)
• (4) The rms current in circuit (b) can be larger than that in (a)

An object moves at a constant speed along a circular path in a horizontal plane with center at the origin. When the object is at =+2 m, its velocity is \(−4\hat{j}m/s\). The object's velocity (v) and acceleration (𝑎) at 𝑥=−2 m will be

• (1) \(𝑣=−4\hat{i}\,m/s ,\,𝑎=−8\hat{j} m/s^2\)
• (2) \(𝑣=−4\hat{i}\,m/s ,\,𝑎=8\hat{j} m/s^2\)
• (3) \(𝑣=4\hat{i}\,m/s ,\,𝑎=8\hat{j} m/s^2\)
• (4) \(𝑣=−4\hat{i}\,m/s ,\,𝑎=8\hat{j} m/s^2\)

The time period of a satellite of earth is 24 hours. If the separation between the earth and the satellite is decreased to one fourth of the previous value, then its new time period will become.

• (1) 4 hours
• (2) 6 hours
• (3) 3 hours
• (4) 12 hours

The electric current in a circular coil of four turns produces a magnetic induction 32 T at its centre. The coil is unwound and is rewound into a circular coil of single turn, the magnetic induction at the centre of the coil by the same current will be :

• (1) 16 T
• (2) 2 T
• (3) 8 T
• (4) 4 T

A fully loaded boeing aircraft has a mass of \(5.4×10^5 \) kg. Its total wing area is \(500 m^2\). It is in level flight with a speed of 1080 km/h. If the density of air 𝜌 is \(1.2 kg m^{−3}\), the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface in percentage will be. (\(𝑔=10 m/s^2\))

• (1) 16
• (2) 10
• (3) 8
• (4) 6

For the given logic gates combination, the correct truth table will be

• (1)

  A	B	X
  0 0 1 1	0 1 0 1	1 0 0 0

• (2)

  A	B	X
  0 0 1 1	0 1 0 1	0 1 1 1

• (3)

  A	B	X
  0 0 1 1	0 1 0 1	0 1 1 0

• (4)

  A	B	X
  0 0 1 1	0 1 0 1	1 0 1 0

At \(300 K\) the rms speed of oxygen molecules is \(\sqrt {\frac {\alpha+5}{\alpha}}\) times to that of its average speed in the gas. Then, the value of \(\alpha\) will be
(used \(\pi=\frac {22}{7}\))

• (1) 28
• (2) 24
• (3) 32
• (4) 27

A point charge \(2×10^{−2}C\) is moved from P to S in a uniform electric field of \(30NC^{−1}\) directed along positive x-axis. If coordinates of P and S are \((1,2,0) m \) and \((0,0,0) m\) respectively, the work done by electric field will be

• (1) 1200 mJ
• (2) −1200 mJ
• (3) −600 mJ
• (4) 600 mJ

With the help of potentiometer, we can determine the value of emf of a given cell. The sensitivity of the potentiometer is
(A) directly proportional to the length of the potentiometer wire
(B) directly proportional to the potential gradient of the wire 
(C) inversely proportional to the potential gradient of the wire 
(D) inversely proportional to the length of the potentiometer wire
Choose the correct option for the above statements:

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

Substance A has atomic mass number 16 and half-life of 1 day. Another substance 𝐵 has atomic mass number 32 and half life of \(\frac1{ 2}\) day. If both 𝐴 and 𝐵 simultaneously start undergo radio activity at the same time with initial mass 320 g each, how many total atoms of A and B combined would be left after 2 days.

• (1) \(3.38×10^{24 }\)
• (2) \( 1.69×10^{24 } \)
• (3) \( 6.76×10^{24 } \)
• (4) \( 6.76×10^{23} \)

The modulation index for an A.M. wave having maximum and minimum peak-to-peak voltages of 14 mV and 6 mV respectively is:

• (1) 0.4
• (2) 0.6
• (3) 0.2
• (4) 1.4

A particle of mass 100 g is projected at time \(𝑡=0\) with a speed \(20\,ms^{−1}\) at an angle \(45°\) to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time \(𝑡=2 s\) is found to be \(\sqrt{K}\  kgm^2/s.\) The value of \(K\) is ________. (Take \(g=10\) \(ms^{−2}\))

In an experiment of measuring the refractive index of a glass slab using travelling microscope in physics lab, a student measures real thickness of the glass slab as 5.25 mm and apparent thickness of the glass slab as 5.00 mm. Travelling microscope has 20 divisions in one cm on main scale and 50 divisions on vernier scale is equal to 49 divisions on main scale. The estimated uncertainty in the measurement of refractive index of the slab is \(\frac{𝑥}{10} ×10^{−3}\), where 𝑥 is _______.

An inductor of inductance \(2\mu H\) is connected in series with a resistance, a variable capacitor and an AC source of frequency \(7\ kHz\). The value of capacitance for which maximum current is drawn into the circuit is \(\frac 1x\) F, where the value of \(𝑥 \) is ______. (Take \(\pi=\frac {22}{7}\) )

A particle of mass 250 g executes a simple harmonic motion under a periodic force F=(−25𝑥)N. The particle attains a maximum speed of 4 m/s during its oscillation. The amplitude of the motion is ______ cm.

A metal block of base area \(0.20 \,m^2\) is placed on a table, as shown in figure. A liquid film of thickness \(0.25\, mm\) is inserted between the block and the table. The block is pushed by a horizontal force of 0.1 N and moves with a constant speed. If the viscosity of the liquid is \(5.0×10^{−3}Pl\), the speed of block is  ________ \(×10^{−3}m/s\)

A car is moving on a circular path of radius \(600\ m\) such that the magnitudes of the tangential acceleration and centripetal acceleration are equal. The time taken by the car to complete first quarter of revolution, if it is moving with an initial speed of \(54 \ km/hr\) is \(𝑡(1−𝑒^{−\frac \pi2})\ 𝑠\). The value of \(t\) is _____ .

Unpolarised light is incident on the boundary between two dielectric media, whose dielectric constants are 2.8 (medium −1 ) and 6.8 (medium −2 ), respectively. To satisfy the condition, so that the reflected and refracted rays are perpendicular to each other, the angle of incidence should be \( tan^{−1}(1+\frac{10}\theta )^{\frac{1} 2}\), then the value of 𝜃 is _______.
(Given for dielectric media, \( \mu_r=1\) )

For a charged spherical ball, electrostatic potential inside the ball varies with 𝑟 as \(V=2𝑎𝑟^2+𝑏\). Here, 𝑎 and 𝑏 are constant and r is the distance from the center. The volume charge density inside the ball is −𝜆𝑎𝜀. The value of 𝜆 is ______. 
(𝜀= permittivity of the medium)

When two resistances 𝑅₁ and 𝑅₂ connected in series and introduced into the left gap of a meter bridge and a resistance of 10Ω is introduced into the right gap, a null point is found at 60 cm from left side. When 𝑅₁ and 𝑅₂ are connected in parallel and introduced into the left gap, a resistance of 3Ω is introduced into the right-gap to get null point at 40 cm from left end. The product of 𝑅₁𝑅₂ is _______ Ω².

A null point is found at 200 cm in potentiometer when cell in secondary circuit is shunted by 5Ω. When a resistance of 15Ω is used for shunting, null point moves to 300 cm. The internal resistance of the cell is _______ Ω.

Given below are two statements :
Statement I : The decrease in first ionization enthalpy from B to Al is much larger than that from Al to Ga.
Statement II : The d orbitals in Ga are completely filled. In the light of the above statements,
choose the most appropriate answer from the options given below

• (1) Statement I is incorrect but statement II is correct
• (2) Both the statements I and II are correct
• (3) Both the statements I and II are incorrect
• (4) Statement I is correct but statement II is incorrect

Given below are two statements: 
Statement I: Nickel is being used as the catalyst for producing syn gas and edible fats. 
Statement II : Silicon forms both electron rich and electron deficient hydrides. 
In the light of the above statements, choose the most appropriate answer from the options given below:

• (1) Statement I is correct but statement II is incorrect
• (2) Both the statements I and II are incorrect
• (3) Statement I is incorrect but statement II is correct
• (4) Both the statements I and II are correct

Which of the following relations are correct? 
(A) \(ΔU=q+pΔV\)
(B) \(ΔG=ΔH−TΔS\)
(C) \(ΔS=\frac{𝑞_{𝑟𝑒𝑣}}{𝑇}\)
(D) \(ΔH=ΔU−ΔnRT\)
Choose the most appropriate answer from the options given below:

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

A solution of \(C_rO_5 \) in amyl alcohol has a _____ colour.

• (1) Yellow
• (2) Green
• (3) Blue
• (4) Orange-Red

According to MO theory the bond orders for \(O_2^{2−},CO\,\, \text {and} \,\,NO^+\) respectively, are

• (1) 1, 2 and 3
• (2) 1,3 and 2
• (3) 2,3 and 3
• (4) 1, 3 and 3

When a hydrocarbon A undergoes combustion in the presence of air, it requirs 9.5 equivalents of oxygen and produces 3 equivalents of water. What is the molecular formula of A?

• (1) \( C_9H_9 \)
• (2) \( C_8H_6 \)
• (3) \( C_9H6 \)
• (4) \( C_6H_6 \)

The set of correct statements is :
(i) Manganese exhibits +7 oxidation state in its oxide. 
(ii) Ruthenium and Osmium exhibit +8 oxidation in their oxides. 
(iii) Sc shows +4 oxidation state which is oxidizing in nature. 
(iv) Cr shows oxidising nature in +6 oxidation state.

• (1) (ii) and (iii)
• (2) (i), (ii) and (iv)
• (3) (ii), (iii) and (iv)
• (4) (i) and (iii)

Correct order of spin only magnetic moment of the following complex ions is:
(Given At.no. Fe: 26, Co:27)

• (1) \( [FeF_6]^{3−} \gt [Co(C_2O_4)_3]^{3−}\gt [CoF_6]^{3−}\)
• (2) \( [FeF_6]^{3−}>[CoF_6]^{3−}>[Co(C_2O_4)_3]^{3−} \)
• (3) \( [Co(C_2O_4)_3]^{3−}\gt [CoF_6]^3−\gt [FeF_6]^{3−} \)
• (4) \( [CoF_6]^{3−}\gt [FeF_6]^{3−}\gt [Co(C_2O_4)_3]^{3−} \)

Reaction of propanamide with \(Br_2/KOH(aq)\) produces :

• (1) Propylamine
• (2) Ethylnitrile
• (3) Propanenitrile
• (4) Ethylamine

Following tetrapeptide can be represented as

(F, L, D, Y, I, Q, P are one letter codes for amino acids)

• (1) PLDY
• (2) FIQY
• (3) YQLF
• (4) FLDY

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

The one giving maximum number of isomeric alkenes on dehydrohalogenation reaction is
(excluding rearrangement)

• (1) 2-Bromopropane
• (2) 2-Bromo-3,3-dimethylpentane
• (3) 1-Bromo-2-methylbutane
• (4) 2-Bromopentane

An indicator ' X ' is used for studying the effect of variation in concentration of iodide : on the rate of reaction of iodide ion with \(H_2O_2 \) at room temp. The indicator ' X ' forms blue colored complex with compound ' A ' present in the solution. The indicator ' X ' and compound 'A' respectively are

• (1) Methyl orange and \(H_2O_2\)
• (2) Starch and iodine
• (3) Starch and \(H_2O_2\)
• (4) Methyl orange and iodine

Find out the major product for the following reaction.

• (1)
• (2)
• (3)
• (4)

Find out the major products from the following reaction sequence.

• (1)
• (2)
• (3)
• (4)

A doctor prescribed the drug Equanil to a patient. The patient was likely to have symptoms of which disease?

• (1) Hyperacidity
• (2) Anxiety and stress
• (3) Depression and hypertension
• (4) Stomach ulcers

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

The major component of which of the following ore is sulphide based mineral?

• (1) Siderite
• (2) Sphalerite
• (3) Malachite
• (4) Calamine

The concentration of dissolved Oxygen in water for growth of fish should be more than 𝑋 ppm and Biochemical Oxygen Demand in clean water should be less than 𝑌 ppm. X and Y in ppm are, respectively.

• (1) X Y
  4  8
• (2) X Y
  6  5
• (3) X Y
  4  15
• (4) X Y
  6  12

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

The volume of HCl, containing \(73 g L^{−1}\), required to completely neutralise NaOH obtained by reacting \(0.69 g\) of metallic sodium with water, is _______ mL.( Nearest Integer)
(Given : molar Masses of Na, Cl, O, H, are 23,35.5,16 and 1 \(g mol^{−1}\) respectively)

The equilibrium constant for the reaction
\(Zn(s)+Sn^{2+}(aq)⇌Zn^{2+}(aq)+Sn(s)\, is\, 1×10^{20}\) at \(298 K\). 
The magnitude of standard electrode potential of \(Sn/Sn^{2+}\) if \(E^{\degree}_{ Zn^{2+}/Zn}= –0.76V \) is _______ \(×10^{−2} V\). (Nearest integer).
(Given : \(\frac{2.303RT}{ F}\) \(=0.059 V \))

At \(298\ K\)
\(N_2( g)+3H_2( g)⇌2NH_3( g),K_1=4×10^5\)
\(N_2( g)+O_2( g)⇌2NO(g),K_2=1.6×10^{12}\)
\(H_2( g)+1 2O_2( g)⇌H_2O(g),K_3=1.0×10^{−13}\)
Based on above equilibria, the equilibrium constant of the reaction, 
\(2NH_3( g)\)+\(\frac5 {2}\) \(O_2( g)⇌2NO(g)+3H_2O(g)\) is _______ \(×10^{−33 }\). (Nearest integer)

The denticity of the ligand present in the Fehling's reagent is ____ .

When 0.01 mol of an organic compound containing 60% carbon was burnt completely, 4.4 g of \(CO_2\) was produced. The molar mass of compound is _______ \(gmol^{−1}\). (Nearest integer).

Total number of acidic oxides among \(N_2O_3,NO_2, N_2O,Cl_2O_7,SO_2,CO,CaO,Na_2O\) and NO is ____ .

Assume that the radius of the first Bohr orbit of hydrogen atom is \(0.6Å\). The radius of the third Bohr orbit of \(He^+\)is _______ picometer. (Nearest Integer)

For conversion of compound \(A→B\), the rate constant of the reaction was found to be \(4.6×10^{−5} L mol^{−1} s^{−1}\). The order of the reaction is _____ .

A metal 𝑀 forms hexagonal close-packed structure. The total number of voids in 0.02 mol of it is _______ \(×10^{21}\). (Nearest integer).
( Given \(N_A=6.02×10^{23}\))

On heating, \(LiNO_3\) gives how many compounds among the following?
\(Li_2O,N_2,O_2,LiNO_2,NO_2 \)