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 $\frac{dy}{dx}=\frac{1}{1+sin2x}$

$dy=\frac{se{c}^{2}xdx}{{\left(1+tanx\right)}^2}$

$\Rightarrow y=-\frac{1}{1+tanx}+c$

When

$x=\frac{\pi }{4},y=\frac{1}{2}$ gives c = 1

So

$x+\frac{\pi }{4}=\frac{5\pi }{6}or\frac{13\pi }{6}\Rightarrow x=\frac{7\pi }{12}or\frac{23\pi }{12}$

sum of all solutions =

$\pi +\frac{7\pi }{12}+\frac{23\pi }{12}=\frac{42\pi }{12}$

Hence k = 42

The  Vivekananda Institute of Management Studies the highest package has not been disclosed yet. Once released it will be updated here. Meanwhile, students can refer to the median package given in the table below:

NOTE: The above data is obtained from the NIRF report 2025

To join the courses offered at Aptech Aviation and Hospitality Academy, one must go through a multi-step admission process. This includes filling out the registration form, connecting with the admission counsellors and completing the admission process. Candidates are selected based on the admission criteria which include securing specified marks in Class 12 with relevant subjects. Physical fitness along with having the specified credentials/ qualifications. Selected students are then required to pay the Aptech Aviation and Hospitality Academy admission fees to secure their spots at the academy.

Each element of ordered pair (i, j) is either present in A or in B.

So, A + B = Sum of all elements of all ordered pairs {i, j} for $1\le i\le 10$ and $1\le j\le 10$

= 20 (1 + 2 + 3 + … + 10) = 1100

UNIPUNE offers admission to UG and PG level courses on merit basis and entrance exams. Students should secure minimum marks in Class 12 for UG level admissions. For PG level courses, Students have graduation degree with the set minimum marks. In addition to this, valid scores of accepted entrance exams is a must. Students should meet the cutoff rank for final admission. 

$l=\frac{48}{{\pi }^4}{\displaystyle {\int }_0^{\pi }\left[{\left(\frac{\pi }{2}-x\right)}^3-\frac{3{\pi }^2}{4}\left(\frac{\pi }{2}-x\right)+\frac{{\pi }^3}{4}\right]\frac{sinxdx}{1+co{s}^{2}x}}$

Using

${\displaystyle {\int }_a^{b}f\left(x\right)dx={\displaystyle {\int }_a^{b}f\left(a+b-x\right)dx}}$

we get

$l=\frac{48}{{\pi }^4}{\displaystyle {\int }_0^{\pi }\left[-{\left(\frac{\pi }{2}-x\right)}^3+\frac{3{\pi }^2}{4}\left(\frac{\pi }{2}-x\right)+\frac{{\pi }^3}{4}\right]\frac{sinxdx}{1+co{s}^{2}x}}$

Adding these two equations, we get

$\Rightarrow l=\frac{12}{\pi }{\left[-ta{n}^{-1}\left(cosx\right)\right]}_0^{\pi }=\frac{12}{\pi }.\frac{\pi }{2}=6$

LEAD College of Management offers a wide range of programmes to students in the field of Management in affiliation with the University of Calicut. The MBA programme is the most popular course offered by LEAD College of Management. It is available in full-time mode and includes specialisations in Human Resources, Finance, Marketing, and Tourism. Further, the fee for an MBA at LEAD College of Management ranges around INR 8 lakh. 

To enrol for a course at RIMT, students must meet the eligibility criteria set by the college. The institute offers various courses such as BBA, MBA, PG Diploma, etc. The RIMT admits students based on merit. To get admission at The RIMT, candidates must meet the eligibility criteria set by the college. Students can check the table below to know course-wise eligibility:

Sum of all elements of $A\cap B=2$  [Sum of natural number upto 100 which are neither divisible by 3 nor by 5]

$=2\left[\frac{100\times 101}{2}-3\left(\frac{33\times 34}{2}\right)-5\left(\frac{20\times 21}{2}\right)+15\left(\frac{6\times 7}{2}\right)\right]$

= 10100 – 3366 – 2100 + 630

= 5264

Check out table below for BTech branch-wise placement rate of NIT Agartala placements 2023:

 Vivekananda Institute of Management Studies has not released its placement report yet. However, the NIRF report for 2025 is out. Students can refer to the placement highlights given in the table below:

NOTE: The above data is obtained from the NIRF report 2025

Uni of Bath receives more 30,000 applications each year from prospective students, of which usually more than 4,000 students get selected. As a result, the Bath University acceptance rate usually stays between 12-14%, making the admission process of the university extremely competitive. Given below are the enrollment statistics over the years:

$\left(sin10°.sin50°.sin70°\right).\left(sin10°.sin20°.sin40°\right)$

$=\left(\frac{1}{4}sin30°\right).\left[\frac{1}{2}sin10°\left(cos20°-cos60°\right)\right]$

$=\frac{1}{32}\left[sin30°-sin10°-sin10°\right]$

$\frac{1}{64}-\frac{1}{16}sin10°$

Clearly $\alpha =\frac{1}{64}$

Hence 16 + a^-1 = 80

 Vivekananda Institute of Management Studies has been providing successful placements over the years. NIRF Report for 2025 has been released Check out the table below to know the key highlights of the  Vivekananda Institute of Management Studies placements 2024:

NOTE: The above data is obtained from the NIRF report 2025

To enrol for a course, students must meet the eligibility criteria set by the college. Candidates should pass Class 12/Class 10 or Class 8 as per the course requirement. Candidates are selected based on merit. The Renaissance Institute of Management and Technology admissions are currently open for all courses. 

 Vivekananda Institute of Management Studies provides good placements to its graduating students. The students are given excellent placement support every year placing them in top companies. Check out the table below to know the key highlights of the  Vivekananda Institute of Management Studies placements 2024:

NOTE: The above data is obtained from the NIRF report 2025

 $\left(4+x^2\right)dy-2x\left(x^2+3y+4\right)dx=0$

$\Rightarrow \frac{dy}{dx}=\left(\frac{6x}{x^2+4}\right)y+2x$

$e^{-3\mathcal{l}n\left(x^2+4\right)}=\frac{1}{{\left(x^2+4\right)}^3}$

So

$\frac{y}{{\left(x^2+4\right)}^3}={\displaystyle \int \frac{2x}{{\left(x^2+4\right)}^3}dx+c}$

$\Rightarrow y=-\frac{1}{2}\left(x^2+4\right)+c{\left(x^2+4\right)}^3$

When x = 0, y = 0 gives

$c=\frac{1}{32}$

So, for x = 2, y = 12