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$dm=\left(\frac{m}{L}\right)dx$

$\therefore T=\frac{m{\omega }^2}{2L}\left(L^2-x^2\right)$

$\therefore ?L={\int }_0^L\frac{m{\omega }^2}{2L\pi r^{2}Y}\left(L^2-x^2\right)dx$

$=?L=\frac{m{\omega }^2{L}^2}{3\pi r^{2}Y}$

$i=\frac{v_0}{R}\left(1-e^{\frac{tR}{L}}\right)$

$U=\frac{1}{2}L{i}^2$

$\frac{dU}{dt}=Li\frac{di}{dt}$

$2L\times \frac{R}{L}\times \frac{v_0^2}{R^2}\left(1-e^{\frac{tR}{L}}\right)e^{\frac{tR}{L}}=\frac{v_0^2}{R}\left(1-e^{\frac{tR}{L}}\right)$

$t=\frac{L}{R}In2$

The GATE 2026 exam pattern for CS, EC, EE, and ME can be checked below. The general aptitude section (GA) carries 15 marks, while the subject-based questions carries 85 marks, out of which engineering mathematics section consists of 13 marks. The GATE question paper will be in English language and the time duration is three hours. 

Number of moles of C = Number of moles of CO₂ = $\frac{330}{44}$ moles

Number of moles of H = 2 × no. of moles of H₂O = $\left(\frac{270}{18}\times 2\right)$ moles

 Mass of C = $\frac{330}{44}\times 12$ gm = 90 gm

 Mass of H = $\frac{270}{18}\times 2\times 1gm=30gm$

$\begin{array}{l}\mathrm{\%}ofC=\frac{90}{120}\times 100\mathrm{\%}=75\mathrm{\%}\\ \mathrm{\%}ofH=\frac{30}{120}\times 100\mathrm{\%}=25\mathrm{\%}\end{array}$  

$y=A\sin ?\frac{2\pi t}{T}$

$t_2-t_1=\frac{T}{2\pi }\left. [{\sin }^{-1}?\frac{x_1}{A}-{\sin }^{-1}?\frac{x_2}{A}\right]$

Deformation energy per unit volume,

$U=\int dU=\int \frac{P^2}{2B}Ady$

$=\int \frac{{\rho }^2{g}^{2}y}{2B}Ady$

$U=\frac{{\rho }^2{g}^2}{2B}\frac{h^3}{3}A$

$=\frac{{\rho }^2{g}^2{h}^2}{6B}\left(volume\right)$

$=1.5\mu J$

GATE exam duration is three hours. It will be conducted in two shift timings. GATE exam slot timings are-

• Forenoon - 9:30 am to 12:30 pm
• Afternoon - 2:30 pm to 5:30 pm

WBJEE 2026 application form has to be submitted before final date. Authorities will announce the registration last date soon. After last date, the authorities will not accept any online applications.

WBJEE OMR sheets will be available after the provisional answer key is released. The responses recorded in the entrance examination will be available through the OMR sheets. Students can check the OMR sheets to know which options they had marked in the examination.

WBJEE 2026 official exam date has not been announced yet. However, it is expected to be held in the fourth week of April 2026.

WBJEE 2026 seat allotment dates have not been announced yet. Students will have to log in to check their allotment status and to download the allocation letter for the admission process. Separate allotment results will be announced for the different rounds of counselling.

$U=a{P}^4$

$\frac{PV}{\gamma -1}=a{P}^4$

$P^{-3}V=a\left(\gamma -1\right)$

$P{V}^{-1/3}=constant$

$ComparingwithP{V}^x=constant$

$x=-\frac{1}{3}$

$C=\frac{R}{\gamma -1}+\frac{R}{1-x}=\frac{R}{\frac{5}{3}-1}+\frac{R}{1+\frac{1}{3}}=\frac{3R}{2}+\frac{3R}{4}$

$C=\frac{9R}{4}$

WBJEE 2026 application form date has not been announced yet. It may be released in February. To be registered, candidates will have to fill form with required details and submit it online. No other mode will be used to accept applications.

The GATE 2026 exam pattern of Life Sciences (XL) is as follows -

• General aptitude  - 15 marks
• B1 section (reasoning and comprehension) carries 16 questions of 25 marks
• Optional section from C1 to C6 carries 39 questions of 60 marks. 

Candidates must go through the GATE pattern and marking scheme before attempting the questions. 

$b=a\sqrt[]{1-e^2}$

$\frac{dA}{dt}=\frac{L}{2m}$

$\frac{A}{T}=\frac{L}{2m}$

$T=\frac{2mA}{L}=\frac{2m\pi ab}{L}=\frac{2m\pi a^2\sqrt[]{1-e^2}}{L}$

The GATE paper code of Engineering Sciences is XE. This year, IIT Guwahati has introduced a new sectional paper under XE, naming Energy Science (XE-I). The optional section (where the applicant has to choose any two) carries 70 marks. The GATE optional sections are as follows -

• Fluid Mechanics
• Materials Science
• Solid Mechanics
• Thermodynamics
• Polymer Science and Engineering
• Food Technology
• Atmospheric and Oceanic Sciences
• Energy Science

There is no limit to attempt questions. Candidates can answer as many questions as they can out of 65 questions. However, they must note that GATE offers negative marking to incorrect responses. 

$f\left(A,B\right)=\frac{1-sinB+sinA-sinAsinB+cosAcosB}{cosA\left(1-sinB\right)}$

$\begin{array}{l}\Rightarrow f\left(A,B\right)\left(cosA-cosAsinB-cosB+sinAcosB\right)\\ =2sinA-2sinB\end{array}$

$\Rightarrow f\left(A,B\right)=\frac{2\left(sinA-sinB\right)}{cosA-cosB+sin\left(A-B\right)}=2g\left(A,B\right)$