JEE Main 2025 Jan 23 Shift 2 — Question Paper

Let $A = \{(x, y) \in \mathbf{R} \times \mathbf{R} : |x+y| \geq 3\}$ and $B = \{(x, y) \in \mathbf{R} \times \mathbf{R} : |x| + |y| \leq 3\}$. If $C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\}$, then $\sum_{(x, y) \in C} |x+y|$ is :

Let $X = \mathbf{R} \times \mathbf{R}$. Define a relation R on X as :

$(a_1, b_1) \text{ R } (a_2, b_2) \Leftrightarrow b_1 = b_2$.

Statement I : R is an equivalence relation.

Statement II : For some $(a, b) \in X$, the set $S = \{(x, y) \in X : (x, y) \text{ R } (a, b)\}$ represents a line parallel to $y = x$.

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

The number of complex numbers $z$, satisfying $|z| = 1$ and $\left|\dfrac{z}{\bar{z}} + \dfrac{\bar{z}}{z}\right| = 1$, is :

The system of equations $$x + y + z = 6,$$ $$x + 2y + 5z = 9,$$ $$x + 5y + \lambda z = \mu,$$ has no solution if

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix such that $A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$, $A \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$ and $A \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, then $a_{23}$ equals :

If in the expansion of $(1+x)^{\mathrm{p}} (1-x)^{\mathrm{q}}$, the coefficients of $x$ and $x^2$ are 1 and $-2$, respectively, then $\mathrm{p}^2 + \mathrm{q}^2$ is equal to :

A board has 16 squares as shown in the figure :

[Figure: a 4 × 4 grid of 16 squares]

Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :

A rod of length eight units moves such that its ends A and B always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$, respectively. If the locus of the point P, that divides the rod AB internally in the ratio $2 : 1$ is $9(x^2 + \alpha y^2 + \beta xy + \gamma x + 28y) - 76 = 0$, then $\alpha - \beta - \gamma$ is equal to :

The length of the chord of the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{2} = 1$, whose mid-point is $\left(1, \dfrac{1}{2}\right)$, is :

Let the shortest distance from $(a, 0)$, $a > 0$, to the parabola $y^2 = 4x$ be 4. Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is :

Let the range of the function $f(x) = 6 + 16\cos x \cdot \cos\left(\dfrac{\pi}{3} - x\right) \cdot \cos\left(\dfrac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x$, $x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3x + 4y + 12 = 0$ is :

The distance of the line $\dfrac{x-2}{2} = \dfrac{y-6}{3} = \dfrac{z-3}{4}$ from the point $(1, 4, 0)$ along the line $\dfrac{x}{1} = \dfrac{y-2}{2} = \dfrac{z+3}{3}$ is :

If the square of the shortest distance between the lines $\dfrac{x-2}{1} = \dfrac{y-1}{2} = \dfrac{z+3}{-3}$ and $\dfrac{x+1}{2} = \dfrac{y+3}{4} = \dfrac{z+5}{-5}$ is $\dfrac{\mathrm{m}}{\mathrm{n}}$, where m, n are coprime numbers, then $\mathrm{m} + \mathrm{n}$ is equal to :

$\lim\limits_{x \to \infty} \dfrac{(2x^2 - 3x + 5)(3x - 1)^{\frac{x}{2}}}{(3x^2 + 5x + 4)\sqrt{(3x + 2)^x}}$ is equal to :

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of $81\ \mathrm{cm}^3/\text{min}$ and the thickness of the ice-cream layer decreases at the rate of $\dfrac{1}{4\pi}$ cm/min. The surface area (in $\mathrm{cm}^2$) of the chocolate ball (without the ice-cream layer) is :

If the area of the region $\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq \mathrm{a} + \mathrm{e}^{|x|} - \mathrm{e}^{-x}, \mathrm{a} > 0\}$ is $\dfrac{\mathrm{e}^2 + 8\mathrm{e} + 1}{\mathrm{e}}$, then the value of a is :

Let $\int x^3 \sin x\, \mathrm{d}x = g(x) + C$, where C is the constant of integration. If $8\left(g\left(\dfrac{\pi}{2}\right) + g'\left(\dfrac{\pi}{2}\right)\right) = \alpha\pi^3 + \beta\pi^2 + \gamma$, $\alpha, \beta, \gamma \in \mathbf{Z}$, then $\alpha + \beta - \gamma$ equals :

If $\mathrm{I} = \displaystyle\int_0^{\frac{\pi}{2}} \dfrac{\sin^{\frac{3}{2}} x}{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x}\, \mathrm{d}x$, then $\displaystyle\int_0^{2\mathrm{I}} \dfrac{x \sin x \cos x}{\sin^4 x + \cos^4 x}\, \mathrm{d}x$ equals :

Let $x = x(y)$ be the solution of the differential equation $y = \left(x - y \dfrac{\mathrm{d}x}{\mathrm{d}y}\right) \sin\left(\dfrac{x}{y}\right)$, $y > 0$ and $x(1) = \dfrac{\pi}{2}$. Then $\cos(x(2))$ is equal to :

Let the point A divide the line segment joining the points $P(-1, -1, 2)$ and $Q(5, 5, 10)$ internally in the ratio $r : 1$ $(r > 0)$. If O is the origin and $\left(\overrightarrow{\mathrm{OQ}} \cdot \overrightarrow{\mathrm{OA}}\right) - \dfrac{1}{5}\left|\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OA}}\right|^2 = 10$, then the value of $r$ is :

The roots of the quadratic equation $3x^2 - \mathrm{p}x + \mathrm{q} = 0$ are $10^{\text{th}}$ and $11^{\text{th}}$ terms of an arithmetic progression with common difference $\dfrac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88, then $\mathrm{q} - 2\mathrm{p}$ is equal to _________.

Let $\alpha, \beta$ be the roots of the equation $x^2 - \mathrm{a}x - \mathrm{b} = 0$ with $\text{Im}(\alpha) < \text{Im}(\beta)$. Let $\mathrm{P_n} = \alpha^{\mathrm{n}} - \beta^{\mathrm{n}}$. If $\mathrm{P_3} = -5\sqrt{7}\, i$, $\mathrm{P_4} = -3\sqrt{7}\, i$, $\mathrm{P_5} = 11\sqrt{7}\, i$ and $\mathrm{P_6} = 45\sqrt{7}\, i$, then $|\alpha^4 + \beta^4|$ is equal to _________.

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is _________.

The variance of the numbers $8, 21, 34, 47, \ldots, 320$ is _________.

The focus of the parabola $y^2 = 4x + 16$ is the centre of the circle C of radius 5. If the values of $\lambda$, for which C passes through the point of intersection of the lines $3x - y = 0$ and $x + \lambda y = 4$, are $\lambda_1$ and $\lambda_2$, $\lambda_1 < \lambda_2$, then $12\lambda_1 + 29\lambda_2$ is equal to _________.

Match List - I with List - II.

	List - I		List - II
(A)	Permeability of free space	(I)	$[\mathrm{M\,L^2\,T^{-2}}]$
(B)	Magnetic field	(II)	$[\mathrm{M\,T^{-2}\,A^{-1}}]$
(C)	Magnetic moment	(III)	$[\mathrm{M\,L\,T^{-2}\,A^{-2}}]$
(D)	Torsion constant	(IV)	$[\mathrm{L^2\,A}]$

Choose the correct answer from the options given below :

The energy of a system is given as $E(t) = \alpha^3 e^{-\beta t}$, where $t$ is the time and $\beta = 0.3~\mathrm{s^{-1}}$. The errors in the measurement of $\alpha$ and $t$ are $1.2\%$ and $1.6\%$, respectively. At $t = 5$ s, maximum percentage error in the energy is :

A ball having kinetic energy KE, is projected at an angle of $60°$ from the horizontal. What will be the kinetic energy of ball at the highest point of its flight ?

A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that $\theta(t) = 5t^2 - 8t$, where $\theta(t)$ is the angular position of the rotating disc as a function of time t.

How much power is delivered by the applied torque, when $t = 2$ s ?

If a satellite orbiting the Earth is 9 times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon $= 27$ days and gravitational attraction between the satellite and the moon is neglected.

A massless spring gets elongated by amount $x_1$ under a tension of 5 N. Its elongation is $x_2$ under the tension of 7 N. For the elongation of $(5x_1 - 2x_2)$, the tension in the spring will be :

Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure gauge attached to the pipe is $P_1$. The reading of the pressure gauge falls to $P_2$ when the valve is opened. The speed of water flowing in the pipe is proportional to

[Figure: P-V diagram with points A $(2V_0, P_0)$, B $(3V_0, P_0)$, C $(3V_0, 2P_0)$ and D $(V_0, 2P_0)$; the path goes A → B (horizontal at $P_0$), B → C (vertical at $3V_0$), C → D (horizontal at $2P_0$)]

Using the given P-V diagram, the work done by an ideal gas along the path ABCD is :

Water of mass m gram is slowly heated to increase the temperature from $T_1$ to $T_2$. The change in entropy of the water, given specific heat of water is $1~\mathrm{J\,kg^{-1}\,K^{-1}}$, is :

The equation of a transverse wave travelling along a string is $y(x, t) = 4.0 \sin\left[20 \times 10^{-3} x + 600 t\right]$ mm, where $x$ is in mm and $t$ is in second. The velocity of the wave is :

Two point charges $-4~\mu c$ and $4~\mu c$, constituting an electric dipole, are placed at $(-9, 0, 0)$ cm and $(9, 0, 0)$ cm in a uniform electric field of strength $10^4~\mathrm{NC^{-1}}$. The work done on the dipole in rotating it from the equilibrium through $180°$ is :

A galvanometer having a coil of resistance $30~\Omega$ need 20 mA of current for full-scale deflection. If a maximum current of 3 A is to be measured using this galvanometer, the resistance of the shunt to be added to the galvanometer should be $\frac{30}{X}~\Omega$, where X is

Two charges $7~\mu c$ and $-4~\mu c$ are placed at $(-7~\mathrm{cm}, 0, 0)$ and $(7~\mathrm{cm}, 0, 0)$ respectively. Given, $\epsilon_0 = 8.85 \times 10^{-12}~\mathrm{C^2\,N^{-1}\,m^{-2}}$, the electrostatic potential energy of the charge configuration is :

A plane electromagnetic wave of frequency 20 MHz travels in free space along the $+x$ direction. At a particular point in space and time, the electric field vector of the wave is $E_y = 9.3~\mathrm{Vm^{-1}}$. Then, the magnetic field vector of the wave at that point is

The refractive index of the material of a glass prism is $\sqrt{3}$. The angle of minimum deviation is equal to the angle of the prism. What is the angle of the prism ?

The width of one of the two slits in Young's double slit experiment is $d$ while that of the other slit is $xd$. If the ratio of the maximum to the minimum intensity in the interference pattern on the screen is $9 : 4$ then what is the value of $x$ ?

(Assume that the field strength varies according to the slit width.)

A concave mirror of focal length $f$ in air is dipped in a liquid of refractive index $\mu$. Its focal length in the liquid will be :

In photoelectric effect an em-wave is incident on a metal surface and electrons are ejected from the surface. If the work function of the metal is 2.14 eV and stopping potential is 2V, what is the wavelength of the em-wave ?

(Given hc = 1242 eVnm where h is the Planck's constant and c is the speed of light in vaccum.)

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : The binding energy per nucleon is found to be practically independent of the atomic number A, for nuclei with mass numbers between 30 and 170.

Reason (R) : Nuclear force is long range.

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

What is the current through the battery in the circuit shown below ?

[Figure: circuit with two parallel branches, each containing a diode in series with a $20~\Omega$ resistor; both diodes are oriented in the same direction; the parallel combination is connected across a 5 V battery]

A satellite of mass $\frac{M}{2}$ is revolving around the earth in a circular orbit at a height of $\frac{R}{3}$ from the surface of the earth. The angular momentum of the satellite is $M\sqrt{\frac{GMR}{x}}$. The value of $x$ is __________, where M and R are the mass and radius of the earth, respectively.

(G is the gravitational constant)

An air bubble of radius 1.0 mm is observed at a depth of 20 cm below the free surface of a liquid having surface tension $0.095~\mathrm{J/m^2}$ and density $10^3~\mathrm{kg/m^3}$. The difference between pressure inside the bubble and atmospheric pressure is __________ $\mathrm{N/m^2}$.

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

A time varying potential difference is applied between the plates of a parallel plate capacitor of capacitance $2.5~\mu\mathrm{F}$. The dielectric constant of the medium between the capacitor plates is 1. It produces an instantaneous displacement current of 0.25 mA in the intervening space between the capacitor plates, the magnitude of the rate of change of the potential difference will be __________ $\mathrm{Vs^{-1}}$.

In a series LCR circuit, a resistor of $300~\Omega$, a capacitor of 25 nF and an inductor of 100 mH are used. For maximum current in the circuit, the angular frequency of the ac source is __________ $\times 10^4$ radians $\mathrm{s^{-1}}$.

At steady state the charge on the capacitor, as shown in the circuit below, is __________ $\mu\mathrm{C}$.

[Figure: circuit with an $8~\mu\mathrm{F}$ capacitor connected in parallel with a $10~\Omega$ resistor; this combination is in series with a $15~\Omega$ resistor and a 5 V battery]

Given below are two statements :

Statement (I) : For a given shell, the total number of allowed orbitals is given by $n^2$.

Statement (II) : For any subshell, the spatial orientation of the orbitals is given by $-l$ to $+l$ values including zero.

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

The effect of temperature on spontaneity of reactions are represented as :

	$\Delta H$	$\Delta S$	Temperature	Spontaneity
(A)	$+$	$-$	any T	Non spontaneous
(B)	$+$	$+$	low T	spontaneous
(C)	$-$	$-$	low T	Non spontaneous
(D)	$-$	$+$	any T	spontaneous

The incorrect combinations are :

Standard electrode potentials for a few half cells are mentioned below :

$$E^{\circ}_{Cu^{2+}/Cu} = 0.34\ \mathrm{V},\quad E^{\circ}_{Zn^{2+}/Zn} = -0.76\ \mathrm{V}$$

$$E^{\circ}_{Ag^{+}/Ag} = 0.80\ \mathrm{V},\quad E^{\circ}_{Mg^{2+}/Mg} = -2.37\ \mathrm{V}$$

Which one of the following cells gives the most negative value of $\Delta G^{\circ}$ ?

When a non-volatile solute is added to the solvent, the vapour pressure of the solvent decreases by 10 mm of Hg. The mole fraction of the solute in the solution is 0.2. What would be the mole fraction of the solvent if decrease in vapour pressure is 20 mm of Hg ?

Consider a binary solution of two volatile liquid components 1 and 2. $x_1$ and $y_1$ are the mole fractions of component 1 in liquid and vapour phase, respectively. The slope and intercept of the linear plot of $\dfrac{1}{x_1}$ vs $\dfrac{1}{y_1}$ are given respectively as :

Consider the reaction

$$X_2Y(g) \rightleftharpoons X_2(g) + \tfrac{1}{2}Y_2(g)$$

The equation representing correct relationship between the degree of dissociation $(x)$ of $X_2Y(g)$ with its equilibrium constant $K_p$ is ________.

Assume $x$ to be very very small.

Which of the following graphs most appropriately represents a zero order reaction ?

pH of water is 7 at $25^{\circ}\mathrm{C}$. If water is heated to $80^{\circ}\mathrm{C}$, it's pH will :

Given below are two statements about X-ray spectra of elements :

Statement (I) : A plot of $\sqrt{\nu}$ ($\nu =$ frequency of X-rays emitted) vs atomic mass is a straight line.

Statement (II) : $\nu$ ($\nu =$ frequency of X-rays emitted) vs atomic number is a straight line.

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

Below are given the atomic numbers of a few elements of group 14. The atomic number of the element with the lowest melting point is :

Consider the following reactions

$$\mathrm{K_2Cr_2O_7} \xrightarrow[-H_2O]{\text{KOH}} [A] \xrightarrow[-H_2O]{\text{H}_2\text{SO}_4} [B] + \mathrm{K_2SO_4}$$

The products [A] and [B], respectively are :

Match List-I with List-II.

	List-I		List-II
(A)	Bronze	(I)	Cu, Ni
(B)	Brass	(II)	Fe, Cr, Ni, C
(C)	UK silver coin	(III)	Cu, Zn
(D)	Stainless steel	(IV)	Cu, Sn

Choose the correct answer from the options given below :

Identify the coordination complexes in which the central metal ion has $d^4$ configuration.

(A) $\mathrm{[FeO_4]^{2-}}$

(B) $\mathrm{[Mn(CN)_6]^{3-}}$

(C) $\mathrm{[Fe(CN)_6]^{3-}}$

(D) $\mathrm{Cr_2(O{-}\overset{\overset{\textstyle O}{\|}}{C}{-}Me)_4(H_2O)_2}$ (chromium(II) acetate dimer)

(E) $\mathrm{[NiF_6]^{2-}}$

Choose the correct answer from the options given below :

Identify A, B and C in the given below reaction sequence

$$[A] \xrightarrow{\text{HNO}_3} \mathrm{Pb(NO_3)_2} \xrightarrow{\text{H}_2\text{SO}_4} [B] \xrightarrow[\substack{(2)\ \text{Acetic acid}\\ (3)\ \mathrm{K_2CrO_4}}]{(1)\ \text{Ammonium acetate}} [C]\ (\text{Yellow ppt})$$

Consider the following reaction

$$\mathrm{R{-}\overset{\overset{\textstyle O}{\|}}{C}{-}R} + H_2O \underset{}{\overset{K}{\rightleftharpoons}} \mathrm{R{-}\underset{\underset{\textstyle OH}{|}}{\overset{\overset{\textstyle OH}{|}}{C}}{-}R}$$

Statement (I) : In the case of formaldehyde ($\mathrm{H{-}\overset{\overset{\textstyle O}{\|}}{C}{-}H}$), K is about 2280, due to small substituents, hydration is faster.

Statement (II) : In the case of trichloro acetaldehyde ($\mathrm{H{-}\overset{\overset{\textstyle O}{\|}}{C}{-}CCl_3}$), K is about 2000 due to $-I$ effect of $-Cl$.

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

Match List-I (Isomers of $\mathrm{C_{10}H_{14}}$) with List-II (Ozonolysis product).

[Figure: List-I shows four bicyclic hydrocarbon structures (methyl-substituted hydrindane/indane-type fused 6- and 5-membered rings, each with one ring double bond) labelled (A), (B), (C), (D); List-II shows four open-chain poly-carbonyl ozonolysis products labelled (I)-(IV).]

• (A) 2-methyl bicyclic isomer with double bond in the six-membered ring
• (B) 2-methyl isomer with double bond at the ring fusion / five-membered ring
• (C) 6-methyl isomer with double bond in the six-membered ring
• (D) 1-methyl isomer with the double bond conjugated to the ring junction

List-II products:

• (I) $\mathrm{H{-}CO{-}CH(CH_3){-}CH_2{-}CO{-}(CH_2)_3{-}CO{-}CHO}$ type dialdehyde/diketone with a branch methyl
• (II) $\mathrm{CH_3{-}CO{-}CO{-}(CH_2)_4{-}CO{-}CHO}$ type
• (III) $\mathrm{H{-}CO{-}CH_2CH_2{-}CO{-}CH(CH_3){-}CO{-}CHO}$ type (two terminal CHO, a mid ketone and a branch methyl)
• (IV) $\mathrm{H{-}CO{-}CO{-}(CH_2)_4{-}CO{-}CO{-}CH_3}$ type

Choose the correct answer from the options given below :

The ascending order of relative rate of solvolysis of following compounds is :

[Figure: four bromo compounds]

• (A) 1-bromo-3,3,5,5-tetramethylcyclohexane (tertiary bromide on a gem-dimethyl-substituted cyclohexane ring)
• (B) 1-bromo-1,2,3,4-tetrahydronaphthalene (benzylic secondary bromide, with a methyl at the 4-position)
• (C) bromodiphenylmethane, $\mathrm{(C_6H_5)_2CHBr}$ (dibenzylic secondary bromide)
• (D) 3-bromo-2-methylpentane (a secondary alkyl bromide)

Identify the products [A] and [B], respectively in the following reaction :

$$\text{Chlorobenzene} \xrightarrow[\text{(ii) } H^{+}]{\text{(i) NaOH, 623 K, 300 atm}} [A] \xrightarrow[\text{H}_2\text{SO}_4]{\text{Na}_2\text{Cr}_2\text{O}_7} [B]$$

Given below are two statements :

Statement (I) : The boiling points of alcohols and phenols increase with increase in the number of C-atoms.

Statement (II) : The boiling points of alcohols and phenols are higher in comparison to other class of compounds such as ethers, haloalkanes.

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

The $\alpha$-Helix and $\beta$-Pleated sheet structures of protein are associated with its :

When 81.0 g of aluminium is allowed to react with 128.0 g of oxygen gas, the mass of aluminium oxide produced in grams is ________. (Nearest integer)

Given :

Molar mass of Al is $27.0\ \mathrm{g\ mol^{-1}}$

Molar mass of O is $16.0\ \mathrm{g\ mol^{-1}}$

The bond dissociation enthalpy of $X_2$, $\Delta H^{\circ}_{\text{bond}}$ calculated from the given data is ________ $\mathrm{kJ\ mol^{-1}}$. (Nearest integer)

$$\mathrm{M^{+}X^{-}(s) \rightarrow M^{+}(g) + X^{-}(g)} \quad \Delta H^{\circ}_{\text{lattice}} = 800\ \mathrm{kJ\ mol^{-1}}$$

$$\mathrm{M(s) \rightarrow M(g)} \quad \Delta H^{\circ}_{\text{sub}} = 100\ \mathrm{kJ\ mol^{-1}}$$

$$\mathrm{M(g) \rightarrow M^{+}(g) + e^{-}(g)} \quad \Delta H^{\circ}_{i} = 500\ \mathrm{kJ\ mol^{-1}}$$

$$\mathrm{X(g) + e^{-}(g) \rightarrow X^{-}(g)} \quad \Delta H^{\circ}_{\text{eg}} = -300\ \mathrm{kJ\ mol^{-1}}$$

$$\mathrm{M(s) + \tfrac{1}{2}X_2(g) \rightarrow M^{+}X^{-}(s)} \quad \Delta H^{\circ}_{f} = -400\ \mathrm{kJ\ mol^{-1}}$$

[Given : $\mathrm{M^{+}X^{-}}$ is a pure ionic compound and X forms a diatomic molecule $X_2$ in gaseous state]

0.01 mole of an organic compound (X) containing 10% hydrogen, on complete combustion produced 0.9 g $\mathrm{H_2O}$. Molar mass of (X) is ________ $\mathrm{g\ mol^{-1}}$.

A compound 'X' absorbs 2 moles of hydrogen and 'X' upon oxidation with $\mathrm{KMnO_4\,|\,H^{+}}$ gives $\mathrm{CH_3{-}\overset{\overset{\textstyle O}{\|}}{C}{-}CH_3}$, $\mathrm{CH_3{-}\overset{\overset{\textstyle O}{\|}}{C}{-}OH}$ and $\mathrm{CH_3{-}\overset{\overset{\textstyle O}{\|}}{C}{-}CH_2CH_2{-}\overset{\overset{\textstyle O}{\|}}{C}{-}OH}$.

The total number of $\sigma$ bonds present in the compound 'X' is ________.