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This is a Objective answer type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}LetI={\displaystyle \underset{-\frac{\pi }{4}}{\overset{\frac{\pi }{4}}{\int }}\frac{dx}{1+cos2x}}\\ ={\displaystyle \underset{-\frac{\pi }{4}}{\overset{\frac{\pi }{4}}{\int }}\frac{dx}{2co{s}^{2}x}}=\frac{1}{2}{\displaystyle \underset{-\frac{\pi }{4}}{\overset{\frac{\pi }{4}}{\int }}se{c}^{2}xdx}=\frac{1}{2}{\left[tanx\right]}_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\\ =\frac{1}{2}\left[tan\frac{\pi }{4}-tan\left(-\frac{\pi }{4}\right)\right]=\frac{1}{2}\left[1+1\right]=\frac{1}{2}\times 2=1\\ Hence,thecorrectoptionis\left(a\right).\end{array}$

4.20 Kindly go through the solution

Manipur Board class 12 syllabus is sometimes updated to match new education policies or reduce student stress. This year and also last year, the board din not make any changes in the Manipur 12th syllabus.Students should regularly visit the official website of COHSEM for the latest syllabus updates. Schools also provide the revised Manipur HSC syllabus at the beginning of the session. Students should obtain the updated Manipur 12th syllabus and follow it for the preparation of the exams.

The total tuition fee for the BBA programme at NIU ranges from INR 3.3 Lacs to INR 4.4 Lakh. This amount includes tuition fees, exam charges, administrative costs, and a security deposit. The final fee may vary depending on the chosen specialisation and academic year. Fees are subject to change.

This is a Objective answer type Questions as classified in NCERT Exemplar

Appreciation of meritorious excellence is important for the students to motivate them to do better. GIET Engineering College (GGU) BTech Scholarships might be given to students as a token of recognition. Sometimes these are also merit cum means scholarships which helps the students to pursue their higher education without any financial constraints.

This is a Objective answer type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}LetI=\int \frac{x+sinx}{1+cosx}dx \\ ={\displaystyle \int \frac{x}{1+cosx}dx+}{\displaystyle \int \frac{sinx}{1+cosx}dx}\\ ={\displaystyle \int \frac{x}{2co{s}^2\frac{x}{2}}dx+}{\displaystyle \int \frac{2sin\frac{x}{2}cos\frac{x}{2}}{2co{s}^2\frac{x}{2}}dx}\\ =\frac{1}{2}{\displaystyle \int x.}se{c}^2\frac{x}{2}dx+{\displaystyle \int tan\frac{x}{2}}dx\\ =\frac{1}{2}\left[x.{\displaystyle \int se{c}^2\frac{x}{2}dx-{\displaystyle \int \left(D\left(x\right).{\displaystyle \int se{c}^2\frac{x}{2}dx}\right)dx}}\right]+{\displaystyle \int tan\frac{x}{2}}dx\\ =\frac{1}{2}\left[x.2tan\frac{x}{2}-{\displaystyle \int 2tan\frac{x}{2}dx}\right]+{\displaystyle \int tan\frac{x}{2}}dx\\ =xtan\frac{x}{2}-{\displaystyle \int tan\frac{x}{2}dx+}{\displaystyle \int tan\frac{x}{2}dx+}C\\ \therefore I=xtan\frac{x}{2}+C\\ Hence,thecorrectoptionis\left(d\right).\end{array}$

Admission to M.Tech course is based on entrance exam at Chouksey Engineering College. Master of Technology, a PG-level course in engineering and technology that generally spans two years. Students who have completed their bachelor's degree in a relevant discipline can pursue an M.Tech degree to gain advanced technical knowledge and research experience in a specific domain, leading to better career prospects. MTech courses strongly focus on research, authenticity, and an analytical approach.

Check out the table to know the total number of students placed across different countries in placement season 2025 at ITM Institute of Hotel Management, Navi Mumbai:

This is a Objective answer type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}LetI={\displaystyle \int \frac{x^3}{x+1}}dx \\ \therefore I={\displaystyle \int \left(x^2-x+1-\frac{1}{x+1}\right)dx=\frac{x^3}{3}}-\frac{x^2}{2}+x-log\left|x+1\right|+C\\ =x-\frac{x^2}{2}+\frac{x^3}{3}-log\left|x+1\right|+C\\ Hence,thecorrectoptionis\left(d\right).\end{array}$

To enroll in a M.Tech course at Chouksey Engineering College, candidates must meet the eligibility criteria set by the institute. The institute offers full-time M.Tech courses lasting two years. Students are selected based on entrance exam. In addition to the selection criteria, students must pass B.E/BTech (common eligibility).

The cost of studying at Kochi Business School may depend on individual choice and career goals. The total tuition fee for the MBA course at Kochi Business School is around INR 3.7 Lakhs. Students must note that the fee may include multiple components, such as tuition fees, security deposit, hostel fees, mess fees, and more. 

4.14. Kindly go through the solution

Amal Jyothi College of Engineering admission 2025 last date has been not declared yet. Applicants should be frequently checking the official site of the company. Based on government or management quota, the deadlines vary. The early application is recommended to have a seat of choice.

The ITM Institute of Hotel Management, Navi Mumbai brochure is out on the official website for the year 2025. Check out the tabulated data given below to know more about ITM Institute of Hotel Management, Navi Mumbai placement for 2025:

This is a Objective answer type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}LetI={\displaystyle \int \frac{dx}{\left(x+2\right)\left(x^2+1\right)}}=alog\left|1+x^2\right|+bta{n}^{-1}x+\frac{1}{5}log\left|x+2\right|+C\\ Letusresolvethegiven\text{\hspace{0.17em}integratedinto}partialfractions\\ Put\frac{1}{\left(x+2\right)\left(x^2+1\right)}=\frac{A}{\left(x+2\right)}+\frac{Bx+C}{\left(x^2+1\right)}\\ 1=A\left(x^2+1\right)+\left(x+2\right)\left(Bx+C\right)\\ 1=A{x}^2+A+B{x}^2+Cx+2Bx+2C\\ 1=\left(A+B\right)x^2+\left(C+2B\right)x+\left(A+2C\right)\\ Comparingtheliketerms,wehave\\ A+B=0\dots \left(i\right)\\ 2B+C=0\dots \left(ii\right)\\ A+2C=1\dots \left(iii\right)\\ Subtracting\left(i\right)from\left(ii\right)weget\\ 2C-B=1\therefore B=2C-1\\ PuttingthevalueofBineq.\left(ii\right)wehave\\ 2\left(2C-1\right)+C=0\Rightarrow 4C-2+C=0\\ 5C=2\therefore C=\frac{2}{5}\\ \therefore B=2\left(\frac{2}{5}\right)-1=-\frac{1}{5}andA=\frac{1}{5}\\ \therefore {\displaystyle \int \frac{dx}{\left(x+2\right)\left(x^2+1\right)}}={\displaystyle \int \frac{\frac{1}{5}}{\left(x+2\right)}dx+{\displaystyle \int \frac{-\frac{1}{5}x+\frac{2}{5}}{\left(x^2+1\right)}}}dx\\ =\frac{1}{5}{\displaystyle \int \frac{1}{\left(x+2\right)}dx}-\frac{1}{5}{\displaystyle \int \frac{x-2}{\left(x^2+1\right)}dx}\\ =\frac{1}{5}{\displaystyle \int \frac{1}{\left(x+2\right)}dx}-\frac{1}{5}{\displaystyle \int \frac{x}{\left(x^2+1\right)}dx}+\frac{2}{5}{\displaystyle \int \frac{1}{\left(x^2+1\right)}dx}\\ \text{\&thi}\end{array}$

The highest domestic package offered at ITM Institute of Hotel Management, Navi Mumbai placements stood at over INR 32.64 LPA. Refer to the table to know the international the highest package same:

Amal Jyothi College of engineering admissions are not direct admissions without an entrance test. In case they have a valid entrance score, however, candidates can also apply under the management quota. Admissions are carried out through merit and eligibility only. One shall submit an application online and then verify documents.

Admission to NEHU Shillong is conducted based on JEE Main scores. Candidates should have valid JEE Maim scores to get into NEHU Shillong. Moreover, 70k rank in JEE Main is sufficient to get admission at NEHU Shillong. As per the NEHU Shillong cutoff 2025 for the last-round seat allotment result, the cutoff ranged between 70898 and 79008 for the General AI category candidates. The courses available under 70k rank can be visible below in the table: