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This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat\underset{x\to 0}{lim}\frac{sin\left(\alpha +\beta \right)x-sin\left(\alpha -\beta \right)x+sin2\alpha x}{co{s}^2\beta x-co{s}^2\alpha x}.x\\ =\underset{x\to 0}{lim}\frac{\left[2sin\alpha x.cos\beta x+sin2\alpha x\right].x}{2sin\left(\alpha +\beta \right)x.sin\left(\alpha -\beta \right)x}\\ =\underset{x\to 0}{lim}\frac{\left[2sin\alpha x.cos\beta x+2sin\alpha x.cos\alpha x\right].x}{2sin\left(\alpha +\beta \right)x.sin\left(\alpha -\beta \right)x}\\ =\underset{x\to 0}{lim}\frac{2sin\alpha x\left(cos\beta x+cos\alpha x\right).x}{2sin\left(\alpha +\beta \right)x.sin\left(\alpha -\beta \right)x}\\ =\underset{x\to 0}{lim}\frac{sin\alpha x\left[2cos\left(\frac{\alpha +\beta }{2}\right)x.cos\left(\frac{\alpha -\beta }{2}\right)x\right].x}{sin\left(\alpha +\beta \right)x.sin\left(\alpha -\beta \right)x}\\ =\underset{x\to 0}{lim}\frac{sin\alpha x\left[2cos\left(\frac{\alpha +\beta }{2}\right)x.cos\left(\frac{\alpha -\beta }{2}\right)x\right].x}{2sin\left(\frac{\alpha +\beta }{2}\right)x.cos\left(\frac{\alpha +\beta }{2}\right)x.2sin\left(\frac{\alpha -\beta }{2}\right)x.cos\left(\frac{\alpha -\beta }{2}\right)x}\\ =\underset{x\to 0}{lim}\frac{sin\alpha x.x}{2sin\left(\frac{\alpha +\beta }{2}\right)xsin\left(\frac{\alpha -\beta }{2}\right)x}\\ =\underset{x\to 0}{lim}\frac{1}{2}\frac{\frac{sin\alpha x}{\alpha x}.\left(\alpha x\right).x}{\left[\frac{sin\left(\frac{\alpha +\beta }{2}\right)x}{\left(\frac{\alpha +\beta }{2}\right)x}\times \left(\frac{\alpha +\beta }{2}\right).x\right]\left[\frac{sin\left(\frac{\alpha -\beta }{2}\right)x}{\left(\frac{\alpha -\beta }{2}\right)x}\times \left(\frac{\alpha -\beta }{2}\right).x\right]}\\ =\frac{1}{2}.\frac{\alpha x^2}{\left(\frac{\alpha +\beta }{2}\right)x\left(\frac{\alpha -\beta }{2}\right)x}=\frac{1}{2}.\left[\frac{\alpha }{\left(\frac{\alpha +\beta }{2}\right)\left(\frac{\alpha -\beta }{2}\right)}\right]\\ =\frac{1}{2}.\frac{4\alpha }{{\alpha }^2-{\beta }^2}=\frac{2\alpha }{{\alpha }^2-{\beta }^2}\end{array}$

GIET Engineering College (Godavari Global University) assigns a certain number of seats to the MBA programme according to the college regulations. The total untake capacity if of 60 seats. It should be noted that the information about seat availability is taken from official sources and could be subject to revision. 

BAU Ranchi accepted entrance exams are ICAR AIEEA and others. The exam pattern of the ICAR AIEEA PG exam is mentioned in the table below.  

Yes, the curriculum designed by Patanjali University for BA courses is credit-based. Each subject included in the course syllabus is allocated a certain credit. Students who pass the subject paper in end-semester examinations are awarded the respective credits. At the end of the program, students must have attained a certain number of credits to get the degree. 

Birsa Agricultural University rankings are not visible on the official website. Students must keep visiting the official website for updated information. Birsa Agriculture University is dedicated to advancing Animal Husbandry, Agriculture, Forestry, Dairy, and the Fishery sectors, especially for the resource-poor farmers present in Jharkhand. 

Yes, students of IIM Rohtak can access digital resources remotely via the RemoteXs service. The Knowledge Resource Centre (KRC) of IIM Rohtak is open 24/7 to its students. The Knowledge Resource Centre (KRC) of IIM Rohtak caters to the academic needs of the students and the faculty members. It has learning resources in the areas of management and related disciplines. KRC has a huge collection of books, journals, reports, e-databases, e-journals, and other digital resources that are internationally recognised as important to the business and management community.

International students are required to meet certain admission requirements to apply to Northeastern University undergraduate courses. Below are these mentioned: 

• Submit a secondary school report
• Minimum 2.0 GPA required
• Submission of standardized test scores
• English language requirements
• Academic Transcripts for each year
• Evidence of English language proficiency
• ACT/SAT score

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat\underset{y\to 0}{lim}\frac{\left(x+y\right)sec\left(x+y\right)-xsecx}{y}\\ =\underset{y\to 0}{lim}\frac{xsec\left(x+y\right)+ysec\left(x+y\right)-xsecx}{y}\\ =\underset{y\to 0}{lim}\frac{\left[xsec\left(x+y\right)-xsecx\right]}{y}+\underset{y\to 0}{lim}\frac{ysec\left(x+y\right)}{y}\\ =\underset{y\to 0}{lim}\frac{x\left[sec\left(x+y\right)-secx\right]}{y}+\underset{y\to 0}{lim}sec\left(x+y\right)\\ =\underset{y\to 0}{lim}\frac{x\left[\frac{1}{cos\left(x+y\right)}-\frac{1}{cosx}\right]}{y}+\underset{y\to 0}{lim}sec\left(x+y\right)\\ =\underset{y\to 0}{lim}x\left[\frac{cosx-cos\left(x+y\right)}{y.cosx.cos\left(x+y\right)}\right]+\underset{y\to 0}{lim}sec\left(x+y\right)\\ =\underset{y\to 0}{lim}x\left[\frac{-2sin\left(\frac{x+x+y}{2}\right).sin\left(\frac{x-x-y}{2}\right)}{y.cosx.cos\left(x+y\right)}\right]+\underset{y\to 0}{lim}sec\left(x+y\right)\\ =\underset{y\to 0}{lim}x\left[\frac{-2sin\left(x+\frac{y}{2}\right).sin\left(\frac{-y}{2}\right)}{y.cosx.cos\left(x+y\right)}\right]+\underset{y\to 0}{lim}sec\left(x+y\right)\\ =\underset{\begin{array}{l}y\to 0\\ \therefore \frac{y}{2}\to 0\end{array}}{lim}x\left[\frac{2sin\left(x+\frac{y}{2}\right).sin\left(\frac{y}{2}\right)}{cosx.cos\left(x+y\right).\left(\frac{y}{2}\right).2}\right]+\underset{y\to 0}{lim}sec\left(x+y\right)\\ Takingwehave\\ =x\left[sinx.\frac{1}{cosx.cosx}\right]+secx\\ =xsecxtanx+secx=secx\left(xtanx+1\right)\end{array}$

The hostel fees of IIM Rohtak depend on the type of accommodation. As per an unofficial source, IIM Rohtak hostel fees are INR 25,000 per academic year. However, this charge does not include mess and other charges. As per the 2025 brochure available on the official website of the institute, some of the additional charges are provided below, which will be paid by students who opt for hostel accommodation. Electricity, maintenance, and other charges will be paid by the students as per the usage.

• Hostel usage charge: INR 36,000 (Academic year)
• Mess charge: INR 53,000 (Academic year)

Top autonomous engineering colleges in Pune include:

1. College of Engineering Pune (COEP Tech)

2. Vishwakarma Institute of Technology (VIT)

3. MIT World Peace University

4. Pimpri Chinchwad College of Engineering (PCCOE)

5. Cummins College of Engineering for Women

6. AISSMS College of Engineering

They offer good placements, faculty, and infrastructure.

The following scholarships are provided by Birsa Agricultural University:

• MHRD Scholarships
• National Scholarships Portal
• DAAD Scholarships
• Prime Minister's Research Fellowship
• UGC Scholarships-and-Fellowships
• Fulbright Fellowships
• DST Fellowship
• JSPS Postdoctoral Fellowships
• Hubert H. Humphrey Fellowship
• National Overseas Scholarship Scheme, and others.

For 100% scholarship infomration, students must visit the official website of BAU. 

Yes, IIM Rohtak provides hostel accomodation to all the students who are persuing residential flagship courses, such as MBA/ PGDM. In total, IIM Rohtak has 12 hostel buildings. The hostels are seperate for girls and boys. IIM Rohtak hostels are very comfortable and have all the basic ameneties, including study table, chair, bed, fans, wardrobes, etc. 

Northeastern University is very popular among international students, so naturally, this university can be considered a little expensive. Northeastern University, Boston, fees for international students vary between INR 48.77L and 55.72 L for the UG level, from INR 11.06 L to 58.46 L for the PG level. This should be noted that this is only the first- year fee without the inclusion of living and accommodation. The table below includes the Northeastern University fees for the first year:

IIM Rohtak has a sprawling camous with all the modern infrastructure. The institute has state-of-the-art classrooms with the latest technology to cater to academic and research needs of the fcaulty nd the students. It has dedicated hostels and guest houses for students and faculty members. Some of the facilities available at the institute campus include an auditorium, labs, a guest house, an amphitheatre, etc. 

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Lety=x cos x\dots \left(i\right)\\ \Rightarrow y+\Delta y=\left(x+\Delta x\right)cos\left(x+\Delta x\right)\dots \left(ii\right)\\ Subtractingeqn.\left(i\right)fromeqn.\left(ii\right)\\ y+\Delta y-y=\left(x+\Delta x\right)cos\left(x+\Delta x\right)-x cos x\\ \Rightarrow \Delta y=xcos\left(x+\Delta x\right)+\Delta xcos\left(x+\Delta x\right)-x cos x\\ Dividingbothsidesby\Delta xandtakethe\text{\hspace{0.17em}limit}weget\\ \underset{\Delta x\to 0}{lim}\frac{\Delta y}{\Delta x}=\underset{\Delta x\to 0}{lim}\frac{xcos\left(x+\Delta x\right)-x cos x+\Delta xcos\left(x+\Delta x\right)}{\Delta x}\\ \frac{dy}{dx}=\underset{\Delta x\to 0}{lim}\frac{x\left[cos\left(x+\Delta x\right)- cos x\right]}{\Delta x}+\underset{\Delta x\to 0}{lim}\frac{\Delta xcos\left(x+\Delta x\right)}{\Delta x}\\ =\underset{\Delta x\to 0}{lim}\frac{x\left[-2sin\frac{\left(x+\Delta x+x\right)}{2}.sin\frac{\left(x+\Delta x-x\right)}{2}\right]}{\Delta x}+\underset{\Delta x\to 0}{lim}cos\left(x+\Delta x\right)\\ =\underset{\begin{array}{l}\Delta x\to 0\\ \therefore \frac{\Delta x}{2}\to 0\end{array}}{lim}\frac{x\left[-2sin\left(x+\frac{\Delta x}{2}\right).sin\frac{\Delta x}{2}\right]}{2\times \frac{\Delta x}{2}}+\underset{\Delta x\to 0}{lim}cos\left(x+\Delta x\right)\\ \therefore \frac{\Delta x}{2}\to 0Taking\text{\hspace{0.17em}limits}wehave\\ =x\left[-sinx\right]+cosx\left[?\underset{\frac{\Delta x}{2}\to 0}{lim}\frac{sin\frac{\Delta x}{2}}{\frac{\Delta x}{2}}=1\right]\\ =-xsinx+cosx\end{array}$

The highest and average salary trends witnessed during NIT Delhi MTech placements are mentioned below:

No. Only 15% of management quota seats are open to candidates from all over India. While the remaining 85% of seats are reserved for Telangana local candidates.

Applicants are advised to first check the seat matrix released during Telangana NEET counselling process to get a seat of their choice.