JEE Main 2024 29 Jan Shift 1 Question Paper (Available)- Download Solution PDF with Answer Key

Suppose\(f(x)=\frac{(2^x+2^{-x})tanx\sqrt{tan^{-1}(x^2-x+1)}}{(7x^2+3x+1)^{3}}\) then the value of \(f'(0)\) is equal to

• (1) \(\pi\)
• (2) 0
• (3) \(\sqrt\pi\)
• (4) \(\frac{\pi}{2}\)

Let \(\overrightarrow{a},\overrightarrow{b}\) and \(\overrightarrow{c}\) be three non-zero vectors such that \(\overrightarrow{b}\) and \(\overrightarrow{c}\) are non-collinear. If \(\overrightarrow{a}+5\overrightarrow{b}\) is collinear with \(\overrightarrow{c},\overrightarrow{b}+6\overrightarrow{c}\) is collinear with \(\overrightarrow{a}\) and \(\overrightarrow{a}+ α\overrightarrow{b} + β\overrightarrow{c} = 0\), then α + β is equal to

• (1) 35
• (2) 30
• (3) -30
• (4) -25

Let\((5,\frac{a}{4})\),be the circumcenter of a triangle with vertices A (a, -2),B (a, 6)and C \((\frac{a}{4},-2)\).Let \(\alpha\) denote the circumradius, \(\beta\) denote the area and \(\gamma\) denote the perimeter of the triangle. Then \(\alpha+\beta+\gamma\) is

• (1) 60
• (2) 53
• (3) 62
• (4) 30

If\(\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}\)is the solution of\( 4cos\theta+ 5sin\theta=1\)then the value of \(tan\alpha\) is

• (1) \(\frac{10-\sqrt10}{6}\)
• (2) \(\frac{10-\sqrt10}{12}\)
• (3) \(\frac{\sqrt10-10}{12}\)
• (4) \(\frac{\sqrt10-10}{6}\)

If in a G.P. of64terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to:

• (1) 7
• (2) 4
• (3) 5
• (4) 6

A fair die is thrown until the number 2 appears. What is the probability that 2 appears in an even number of throws?

• (1) \(\frac{5}{6}\)
• (2) \(\frac{1}{6}\)
• (3) \(\frac{5}{11}\)
• (4) \(\frac{6}{11}\)

In an A.P., the sixth term a_6​=2. If the product a₁​a₄​a₅ is the greatest, then the common difference of the A.P. is equal to:

• (1) \(\frac{3}{2}\)
• (2) \(\frac{8}{5}\)
• (3) \(\frac{2}{3}\)
• (4) \(\frac{5}{8}\)

Let

\( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix} \)

and \(|2A|^3 = 2^{21}\) where \(\alpha, \beta \in \mathbb{Z}\). Then a value of \(\alpha\) is:

• (1) 5
• (2) 3
• (3) 9
• (4) 17

 Let \( R \) be a relation on \( \mathbb{Z} \times \mathbb{Z} \) defined by \((a, b) R (c, d)\) if and only if \(ad - bc\) is divisible by 5.
Then \( R \) is:

• (1) Reflexive but neither symmetric nor transitive
• (2) Reflexive and symmetric but not transitive
• (3) Reflexive, symmetric and transitive
• (4) Reflexive and transitive but not symmetric

If \(f(x)=\begin{cases}2+2x, & -1\le x<0\\ 1-\dfrac{x}{3}, & 0\le x\le 3\end{cases}\) ; \(g(x)=\begin{cases}-x, & -3\le x\le 0\\ x, & 0<x\le 1\end{cases}\), then the range of \((f\circ g)(x)\) is

• (1) \([0,1)\)
• (2) \([0,3)\)
• (3) \((0,1]\)
• (4) \([0,1]\)

Let O be the origin and the position vector of A and B be \(2\hat{i}+2\hat{j}+\hat{k}\) and \(2\hat{i}+4\hat{j}+4\hat{k}\) respectively. If the internal bisector of\(\angle AOB\) meets the line AB at C, then the length of OC is

• (1) \(\frac{2}{3}\sqrt31\)
• (2) \(\frac{2}{3}\sqrt34\)
• (3) \(\frac{3}{4}\sqrt34\)
• (4) \(\frac{3}{2}\sqrt31\)

In a △ABC, suppose y = x is the equation of the bisector of the angle B and the equation of the side AC is 2x−y = 2. If 2AB = BC and the points A and B are respectively (4, 6) and (α, β), then α + 2β is equal to:

• (1) 42
• (2) 39
• (3) 48
• (4) 45

For \( x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), if\[y(x) = \int \frac{\csc x + \sin x}{\csc x \sec x + \tan x \sin^2 x} \, dx\]and\[\lim_{x \to -\frac{\pi}{2}} y(x) = 0\]then \( y\left(\frac{\pi}{4}\right) \) is equal to

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

Let A be a square matrix such that \(AA^T=I\).Then \(\frac{1}{2}A[(A+A^T)^2+(A-A^T)^2]\) is equal to

• (1) A²+I
• (2) A³+I
• (3) A²+A^T
• (4) A³+A^T

If \( z = \frac{1}{2} - 2i \), is such that \( |z + 1| = \alpha z + \beta (1 + i) \), \( i = \sqrt{-1} \) and \( \alpha, \beta \in \mathbb{R} \), then \( \alpha + \beta \) is equal to:

• (1) -4
• (2) 3
• (3) 2
• (4) -1

Consider the function \(f :[\frac{1}{2},1]\)⇢R defined by  \(f(x)=4\sqrt2x^3-3\sqrt2x-1\).Consider the statements
(1)The curve y=f(x) intersect the x-axis exactly at one point
(2)The curve y=f(x) intersect the x-axis at \(x=cos\frac{\pi}{12}\)
Then 

• (1) Only (II) is correct
• (2) Both (I) and (II) are incorrect
• (3) Only (I) is correct
• (4) Both (I) and (II) are correct

Let PQR be a triangle with R (-1,4, 2). Suppose M(2, 1, 2) is the mid point of PQ. The distance of the centroid of \(\triangle PQR\) from the point of intersection of the line\(\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1}\) and \(\frac{x-1}{1}=\frac{y+3}{-3}=\frac{z+1}{1}\) is

• (1) 69
• (2) 9
• (3) \(\sqrt{69}\)
• (4) \(\sqrt{99}\)

\(\displaystyle\lim_{x\to\frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^2}\int_{x^3}^{\left(\frac{\pi}{2}\right)^3}\cos\left(t^{1/3}\right)dt\right)\) is equal to

• (1) \(\dfrac{3\pi^2}{4}\)
• (2) \(\dfrac{3\pi^2}{8}\)
• (3) \(\dfrac{3\pi}{4}\)
• (4) \(\dfrac{3\pi}{8}\)

A function \(y=f(x)\) satisfies\(f (x)sin2x+sinx-(1+cos^2x) f'(x)=0\) with condition\(f(0)=0\).Then\(f(\frac{\pi}{2})\) equals to

• (1) 1
• (2) 0
• (3) -1
• (4) 2

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

The area (in square units) of the part of the circle \(x^2 + y^2 = 169\) which is below the line \( 5x - y = 13 \) is\[\frac{\pi \alpha}{2 \beta} - \frac{65}{2} + \frac{\alpha}{\beta} \sin^{-1} \left( \frac{12}{13} \right)\]where \( \alpha \) and \( \beta \) are coprime numbers. Then \( \alpha + \beta \) is equal to

If the mean and variance of the data \( 65, 68, 58, 44, 48, 45, 60, \alpha, \beta, 60 \) where \( \alpha > \beta \) are \( 56 \) and \( 66.2 \) respectively, then \( \alpha^2 + \beta^2 \) is equal to

Equation of two diameters of a circle are \(2x-3y=5\) and \(3x-4y=7\).The line joining the points \((-\frac{22}{7},-4)\) and \((-\frac{1}{7},3)\) intersects the circle at only one point \(P(\alpha,\beta)\).Then \(17\beta-\alpha\) is equal to.

A line with direction ratios \( 2, 1, 2 \) meets the lines \(x = y + 2 = z\) and \(x + 2 = 2y = 2z\) respectively at the points \( P \) and \( Q \). If the length of the perpendicular from the point \( (1, 2, 12) \) to the line \( PQ \) is \( l \), then \( l^2 \) is

If \(\frac{{^{11}C_1}}{2} + \frac{{^{11}C_2}}{3} + \ldots + \frac{{^{11}C_9}}{10} = \frac{n}{m}\) with \(\gcd(n, m) = 1\), then \(n + m\) is equal to:

If the points of intersection of two distinct conics \(x^2 + y^2 = 4b\) and \(\frac{x^2}{16} + \frac{y^2}{b^2} = 1\) lie on the curve \(y^2 = 3x^2\) then \( 3\sqrt{3} \) times the area of the rectangle formed by the intersection points is __.

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - x + 2 = 0 \) with \( \text{Im}(\alpha) > \text{Im}(\beta) \). Then \( \alpha^6 + \alpha^4 + \beta^4 - 5 \alpha^2 \) is equal to

Let \( f(x) = 2^x - x^2, \, x \in \mathbb{R} \). If \( m \) and \( n \) are respectively the number of points at which the curves \( y = f(x) \) and \( y = f'(x) \) intersect the x-axis, then the value of \( m + n \) is

If the solution curve \( y = y \, x \) of the differential equation \((1 + y^2) \left(1 + \log_e x\right) dx + x \, dy = 0, \quad x > 0\) passes through the point \( (1, 1) \) and\[y(e) = \frac{\alpha - \tan\left(\frac{3}{2}\right)}{\beta + \tan\left(\frac{3}{2}\right)},\]then \( \alpha + 2\beta \) is

Voltage and resistance for a resistor are measured as \(V = 200±5V \;\&\; R = 20 + 0.2 \;Ω\). Find the percentage error in the current \(I = \frac{V }{R}\)

• (1) \( \frac{1}{1} \)
• (2) \( \frac{1}{8} \)
• (3) 8 : 1
• (4) \( \frac{1}{4} \)

• (1) −15 cm
• (2) 45 cm
• (3) −45 cm
• (4) 15 cm

The deflection in a moving coil galvanometer falls from 25 divisions to 5 divisions when a shunt of \( 24 \, \Omega \) is applied. The resistance of the galvanometer coil will be:

• (1) \( 12 \, \Omega \)
• (2) \( 96 \, \Omega \)
• (3) \( 48 \, \Omega \)
• (4) \( 100 \, \Omega \)

• (1) 2 A
• (2) 1.2 A
• (3) 2.4 A
• (4) 1.5 A

At what distance above and below the surface of the earth a body will have the same weight (take radius of earth as R)?

• (1) \(\frac{R}{2}\)
• (2) \((\sqrt5-1)\frac{R}{2}\)
• (3) \((\sqrt3-1)\frac{R}{2}\)
• (4) \((\sqrt5-1)R\)

• (1) \(10^{- 3}\; Ω\)
• (2) \(7.5 \times 10^{- 3}\; Ω\)
• (3) \(6.75 \times 10^{- 3}\; Ω\)
• (4) \(3.75 \times 10^{- 3}\; Ω\)

A thermodynamic system is taken from an original state A to an intermediate state B by a linear process as shown in the figure. It's volume is then reduced to the original value from B to C by an isobaric process. The total work done by the gas from A to B and B to C would be :

• (1) 33800 J
• (2) 2200 J
• (3) 0 J
• (4) 1200 J

• (1) \(\frac{2Q}{\varepsilon_0}\)
• (2) \(\frac{5Q}{\varepsilon_0}\)
• (3) \(\frac{7Q}{\varepsilon_0}\)
• (4) \(\frac{3Q}{\varepsilon_0}\)

A biconvex lens of refractive index 1.5 has a focal length of 20 cm in air. Its focal length when immersed in a liquid of refractive index 1.6 will be:

• (1) -160 cm
• (2) 160 cm
• (3) 16 cm
• (4) -16 cm

• (1) 4500 J
• (2) 50000 J
• (3) 4200 J
• (4) 4000 J

Given below are two statements:
Statement 1: If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in hot water.
Statement 2: If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in cold water.
In the light of the above statements, choose the most appropriate option from the options given below:

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

If the radius of curvature of the path of two particles of the same mass are in the ratio \( 3:4 \), then in order to have constant centripetal force, their velocities will be in the ratio of:

• (1) \( \sqrt{3} : 2 \)
• (2) \( 1 : \sqrt{3} \)
• (3) \( \sqrt{3} : 1 \)
• (4) \( 2 : \sqrt{3} \)

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

• (1) 16
• (2) 8
• (3) 4
• (4) 32

In the given circuit, the breakdown voltage of the Zener diode is 3.0 V. What is the value of I_z ?

• (1) 7 mA
• (2) 1.5 mA
• (3) 5.5 mA
• (4) 10 mA

• (1) \(\sqrt{2p^2 - 2p + 1} \, s\)
• (2) \((2p + 1) \, s\)
• (3) \((2p - 1) \, s\)
• (4) \((2p^2 - 2p + 1) \, s\)

The electric current through a wire varies with time as \(I = I_0 + \beta t\) where \( I_0 = 20 \, \text{A} \) and \( \beta = 3 \, \text{A/s} \). The amount of electric charge that crosses through a section of the wire in \( 20 \, \text{s} \) is:

• (1) 80C
• (2) 1000C
• (3) 800C
• (4) 1600C

The potential energy function (in joules) of a particle in a region of space is given as\[U = (2x^2 + 3y^3 + 2z).\]Here \( x \), \( y \), and \( z \) are in meters. The magnitude of the \( x \)-component of force (in newtons) acting on the particle at point \( P (1, 2, 3) \) m is:

• (1) 2
• (2) 6
• (3) 4
• (4) 8

The explosive in a hydrogen bomb is a mixture of \( ^2\text{H} \), \( ^3\text{H} \), and \( ^6\text{Li} \) in some condensed form. The chain reaction is given by\[_3^6\text{Li} + _0^1\text{n} \rightarrow _2^4\text{He} + _1^3\text{H}\]\[_1^2\text{H} + _1^3\text{H} \rightarrow _2^4\text{He} + _0^1\text{n}\]During the explosion, the energy released is approximately ____.[Given: \( M(\text{Li}) = 6.01690 \, \text{amu} \), \( M(^3\text{H}) = 2.01471 \, \text{amu} \), \( M(_2^4\text{He}) = 4.00388 \, \text{amu} \), and \( 1 \, \text{amu} = 931.5 \, \text{MeV} \)]

• (1) 28.12 MeV
• (2) 12.64 MeV
• (3) 16.48 MeV
• (4) 22.22 MeV

In a double slit experiment shown in figure, when light of wavelength 400 nm is used, dark fringe is observed at P. If D = 0.2 m. the minimum distance between the slits S1 and S2 is ______ mm.

A cylinder is rolling down an inclined plane of inclination \( 60^\circ \). Its acceleration during rolling down will be \( \frac{x}{\sqrt{3}} \, \text{m/s}^2 \), where \( x = \) _____. (Use \( g = 10 \, \text{m/s}^2 \)).

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet \( S \) having surface charge density \( +\sigma \). The electron at \( t = 0 \) is at a distance of 1 m from \( S \) and has a speed of 1 m/s. The maximum value of \( \sigma \) if the electron strikes \( S \) at \( t = 1 \, \text{s} \) is \( \alpha \left[ \frac{m \epsilon_0}{e} \right] \frac{\text{C}}{\text{m}^2} \). The value of \( \alpha \) is ______.

When a hydrogen atom going from \( n = 2 \) to \( n = 1 \) emits a photon, its recoil speed is \( \frac{x}{5} \) m/s. Where \( x = \, \) _____. (Use: mass of hydrogen atom \( = 1.6 \times 10^{-27} \, \text{kg} \))

A square loop of side \( 10 \, \text{cm} \) and resistance \( 0.7 \, \Omega \) is placed vertically in the east-west plane. A uniform magnetic field of \( 0.20 \, \text{T} \) is set up across the plane in the northeast direction. The magnetic field is decreased to zero in \( 1 \, \text{s} \) at a steady rate. Then, the magnitude of the induced emf is \( \sqrt{x} \times 10^{-3} \, \text{V} \). The value of \( x \) is _____.

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of 20 cm from its center is \( 1.5 \times 10^{-5} \, \text{Tm} \). The magnetic moment of the dipole is _______ \( \text{Am}^2 \).
(Given: \( \frac{\mu_0}{4\pi} = 10^{-7} \, \text{TmA}^{-1} \))

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is \(\frac{x}{ 8}\) , where x = _________.

A ball rolls off the top of a stairway with horizontal velocity \( u \). The steps are \( 0.1 \, \text{m} \) high and \( 0.1 \, \text{m} \) wide. The minimum velocity \( u \) with which the ball just hits the step 5 of the stairway will be \( \sqrt{x} \, \text{m/s} \) where \( x = \) _____. (Use \( g = 10 \, \text{m/s}^2 \)).

In a test experiment on a model aeroplane in wind tunnel, the flow speeds on the upper and lower surfaces of the wings are 70 ms^–1 and 65 ms^–1 respectively. If the wing area is 2 m2 the lift of the wing is _______ N.
(Given density of air = 1.2 kg m^-3)

Identify the incorrect pair from the following:

• (1) Fluorspar - \( \text{BF}_3 \)
• (2) Cryolite - \( \text{Na}_3\text{AlF}_6 \)
• (3) Fluoroapatite - \( 3\text{Ca}_3(\text{PO}_4)_2 \cdot \text{CaF}_2 \)
• (4) Carnallite - \( \text{KCl} \cdot \text{MgCl}_2 \cdot 6\text{H}_2\text{O} \)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R:
Assertion A: The first ionisation enthalpy decreases across a period.
Reason R: The increasing nuclear charge outweighs the shielding across the period.
In the light of the above statements, choose the most appropriate 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

The major product(P) in the following reaction is

• (1)

• (2)

• (3)

• (4)

Identify product A and product B :

• (1)

• (2)

• (3)

• (4)

The arenium ion which is not involved in the bromination of Aniline is .

• (1)

• (2)

• (3)

• (4)

Given below are two statements:
Statement I: The electronegativity of group 14 elements from Si to Pb gradually decreases.
Statement II: Group 14 contains non-metallic, metallic, as well as metalloid elements.
In the light of the above statements, choose the most appropriate from the options given below:

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

Chlorine undergoes disproportionation in alkaline medium as shown below:
\(a \, \text{Cl}_2(g) + b \, \text{OH}^-(\text{aq}) \rightarrow c \, \text{ClO}^-(\text{aq}) + d \, \text{Cl}^-(\text{aq}) + e \, \text{H}_2\text{O}(\text{l})\)
The values of \( a \), \( b \), \( c \), and \( d \) in a balanced redox reaction are respectively:

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

In chromyl chloride test for confirmation of Cl^-, a yellow solution is obtained. Acidification of the solution and addition of amyl alcohol and 10% H₂O₂​ turns organic layer blue indicating formation of chromium pentoxide. The oxidation state of chromium in that is

• (1) \(+6\)
• (2) \(+5\)
• (3) \(+10\)
• (4) \(+3\)

Type of amino acids obtained by hydrolysis of proteins is:

• (1) \(α\)- Amino acids
• (2) \(β\)- Amino acids
• (3) \(γ\)- Amino acids
• (4) \(δ\)- Amino acids

The correct set of four quantum numbers for the valence electron of rubidium atom (Z = 37) is:

• (1) \( 5, 0, 0, +\frac{1}{2} \)
• (2) \( 5, 0, 1, +\frac{1}{2} \)
• (3) \( 5, 1, 0, +\frac{1}{2} \)
• (4) \( 5, 1, 1, +\frac{1}{2} \)

The difference in energy between the actual structure and the lowest energy resonance structure for the given compound is

• (1) Electromeric energy
• (2) Resonance energy
• (3) Ionization energy
• (4) Hyperconjugation energy

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

The final product A formed in the following multistep reaction sequence is

• (1)

• (2)

• (3)

• (4)

Which of the following is not correct?

• (1) ΔG is negative for a spontaneous reaction
• (2) ΔG is zero for a reversible reaction
• (3) ΔG is positive for a spontaneous reaction
• (4) ΔG is positive for a non-spontaneous reaction

The interaction between π bond and lone pair of electrons present on an adjacent atom is responsible for

• (1) Resonance effect
• (2) Hyperconjugation
• (3) Inductive effect
• (4) Electromeric effect

In which one of the following metal carbonyls, CO forms a bridge between metal atoms?

• (1) \(\text{[Co}_2(\text{CO})_8]\)
• (2) \(\text{[Mn}_2(\text{CO})_{10}]\)
• (3) \(\text{[Os}_3(\text{CO})_{12}]\)
• (4) \(\text{[Ru}_3(\text{CO})_{12}]\)

KMnO₄ decomposes on heating at 513 K to form O₂ along with

• (1) \(MnO_4^- , MnO_2\)
• (2) \(K_2MnO_4 , MnO_2\)
• (3) \(KMnO_4 , MnO_2\)
• (4) \(MnO_2, H_2O\)

• (1) N & S
• (2) Br
• (3) S
• (4) N

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R :
Assertion A : Aryl halides cannot be prepared by replacement of hydroxyl group of phenol by halogen atom.
Reason R : Phenols react with halogen acids violently. In the light of the above statements.
Choose the most appropriate from the options given below:

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

In alkaline medium, \( \text{MnO}_4^- \) oxidizes \( \text{I}^- \) to:

• (1) \( \text{IO}_4^- \)
• (2) \( \text{IO}^- \)
• (3) \( \text{I}_2 \)
• (4) \( \text{IO}_3^- \)

The mass of zinc produced by the electrolysis of zinc sulfate solution with a steady current of 0.015 A for 15 minutes is _____ \( \times \, 10^{-4} \) g.
(Atomic mass of zinc = 65.4 amu)

A solution of \( \text{H}_2\text{SO}_4 \) is 31.4% \( \text{H}_2\text{SO}_4 \) by mass and has a density of \( 1.25 \, \text{g/mL} \). The molarity of the \( \text{H}_2\text{SO}_4 \) solution is _____ M (nearest integer).
[Given molar mass of \( \text{H}_2\text{SO}_4 = 98 \, \text{g/mol} \)]

Consider the given reaction. The total number of oxygen atoms present per molecule of the product (P) is____.

For the reaction \( \text{N}_2\text{O}_4(g) \rightleftharpoons 2\text{NO}_2(g) \), \( K_p = 0.492 \, \text{atm} \) at 300 K. \( K_c \) for the reaction at the same temperature is _____ \( \times \, 10^{-2} \).
(Given: \( R = 0.082 \, \text{L atm mol}^{-1} \text{K}^{-1} \))

The number of species from the following which are paramagnetic and with bond order equal to one is ___.
Species: H₂, He₂⁺, O₂^-, N₂, O₂^2-, F₂, Ne₂⁺, B₂

For a reaction taking place in three steps at the same temperature, the overall rate constant \( K = \frac{K_1 K_2}{K_3} \). If \( E_{a1} \), \( E_{a2} \), and \( E_{a3} \) are 40, 50, and 60 kJ/mol respectively, the overall \( E_a \) is ____ kJ/mol.

From the compounds given below, the number of compounds which give a positive Fehling’s test is____.
Compounds: Benzaldehyde, Acetaldehyde, Acetone, Acetophenone, Methanal, 4-nitrobenzaldehyde, cyclohexane carbaldehyde.

Number of compounds with one lone pair of electrons on central atom amongst following is
\(CIF_3, XeO_3, BrF_5, XeF_4, O_3, NH_3\),\( H_2O\)

The osmotic pressure of a dilute solution is \(7 × 10^5\) Pa at \(273K\). Osmotic pressure of the same solution at\(283K\) is _____\(× 10^4 Nm^2\)