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All the letters of the word "GTWENTY" are written in all possible ways with or without meaning, and these words are arranged as in a dictionary. The serial number of the word "GTWENTY" is:

Mathematics
permutations and combinations

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A rectangular loop of sides 12 cm and 5 cm, with its sides parallel to the x-axis and y-axis respectively, moves with a velocity of 5 cm/s in the positive x-axis direction, in a space containing a variable magnetic field in the positive z direction.
The field has a gradient of \(10^{-3}\) T/cm along the negative x direction, and it is decreasing with time at the rate of \(10^{-3}\) T/s. If the resistance of the loop is 6 mΩ, the power dissipated by the loop as heat is \(\_\_\_\_\_\_ \times 10^{-9}\) W.

Physics
Electromagnetic induction

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Two resistances of 100 \(\Omega\) and 200 \(\Omega\) are connected in series across a 20 V battery as shown in the figure below. The reading in a 200 \(\Omega\) voltmeter connected across the 200 \(\Omega\) esistance is _______.

Fill in the blank with the correct answer from the options given below

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An artificial cell is made by encapsulating \(0.2 \, \text{M}\) glucose solution within a semipermeable membrane. The osmotic pressure developed when the artificial cell is placed within a \(0.05 \, \text{M}\) solution of NaCl at 300 K is ______ \( \times 10^{-1} \, \text{bar} \). (Nearest Integer)
Given: \(R = 0.083 \, \text{L bar mol}^{-1} \text{K}^{-1}\)
Assume complete dissociation of NaCl.

Chemistry
Colligative Properties

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An infinite plane sheet of charge having uniform surface charge density \( +\sigma_s \, \text{C/m}^2 \) is placed on the \( x \)-\( y \) plane. Another infinitely long line charge having uniform linear charge density \( +\lambda_e \, \text{C/m} \) is placed at \( z = 4 \, \text{m} \) plane and parallel to the \( y \)-axis. If the magnitude values \( |\sigma_s| = 2 |\lambda_e| \), then at point \( (0, 0, 2) \), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is \( \pi \sqrt{n} : 1 \).The value of \( n \) is ______.

Physics
Electrostatics

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Assertion: The angular velocity of the moon revolving around the earth is more than that of the earth revolving around the Sun.
Reason: The time taken by the moon to revolve around the earth is less than the time taken by the earth to revolve around the sun.

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Given below are two statements:
Statement I: In C₃ plants, some O₂ binds to RuBisCO, hence CO, fixation is decreased.
Statement II: In C₄plants, mesophyll cells show very little photorespiration while bundle sheath cells do not show photorespiration.
In the light of the above statements, choose the correct answer from the options given below:

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Let \(\vec{a} = \hat{i} - 3\hat{j} + 7\hat{k}, \quad \vec{b} = 2\hat{i} - \hat{j} + \hat{k}, \quad \text{and} \quad \vec{c} \text{ be a vector such that}\) \((\vec{a} + 2\vec{b}) \times \vec{c} = 3(\vec{c} \times \vec{a}).\)
If \(\vec{a} \cdot \vec{c} = 130\), then \(\vec{b} \cdot \vec{c}\) is equal to \(\_\_\_\_\_\_\_\_ .\)

Mathematics
Vector Algebra

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If the plates of a parallel plate capacitor connected to a battery are moved close to each other, then
A. the charge stored in it, increases.
B. the energy stored in it, decreases.
C. its capacitance increases.
D. the ratio of charge to its potential remains the same.
E. the product of charge and voltage increases.
Choose the most appropriate answer from the options given below :

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Consider the following reaction:\[3\text{PbCl}_2 + 2(\text{NH}_4)_3\text{PO}_4 \rightarrow \text{Pb}_3(\text{PO}_4)_2 + 6\text{NH}_4\text{Cl}\]If 72 mmol of PbCl\(_2\) is mixed with 50 mmol of (\(\text{NH}_4\))\(_3\)PO\(_4\), then the amount of \(\text{Pb}_3(\text{PO}_4)_2\) formed is ______ mmol (nearest integer).

Chemistry
Stoichiometry and Stoichiometric Calculations

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\(L_1: γ^-=(i+2j+3k)+λ(i-j+k)\);
\(L_2: γ^-=(4i+5j+6k)+μ(i+j-k)\),
intersect \(L_1\) and \(L_2\) at P and Q respectively. If \((α, β, γ)\) is the mid point of the line segment PQ, then \(2(α, β, γ\)) is equal to

Mathematics
distance between two points

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When a tall pea plant with round seeds was selfed, it produced the progeny of
(a) Tall plants with round seeds and
(b) Tall plants with wrinkled seeds.
Identify the genotype of the parent plant.

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