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NUALS Kochi accepts the CLAT cutoff ranks for admission to the BA LLB (Hons) and LLM courses for various categories under the All India quota.

As per the NUALS Kochi CLAT Cutoff 2025 for the General AI category, in 2025, the cutoff was 1154. In 2024 and 2023, the cutoff was 1341 and 1136, respectively. This difference in the ranks suggests that the competition for admission to the BA LLB (Hons) course has been fluctuating over the years. 

Students can check the table below to know the year-wise round 2 cutoff ranks for the General AI category for the BA LLB (Hons) course.

 The college offers courses such as BSc, BA, MSc, Diploma, etc. The selection criteria are merit-based. To get admitted to PTVASC, candidates must meet the eligibility criteria set by the institute. The courses are offered in various streams such as Arts, Mass Communication, Science, etc.

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6. Let (r + 1)th be the general term of ( $x^2- yx){}^{12}$

So, T_r-1 = ¹²C_r (x²)^12–r (–yx)^r

= (–1)^r12C_rx^24–2ry^rx^r

= (–1)^r12C_r $x^{24-2r+r}{y}^r$

= (-1)^r12C­_r $x^{24-r}{y}^r$

121. Kindly go through the solution

Candidates willing to take admission to the BSc programme of Chettinad Academy of Research and Education (CARE) must intiate the admission process by registering their candidature. For this, candidates must visit the university's official website. After completing the registration process, they must submit a duly filled application form.

The university shortlists prospective candidates for BSc admission based on the eligibility and merit in the qualifying exam/ entrance exam. Selected candidates must confirm their admission by submitting the applicable CARE BSc fees.

With acceptance rate of 50%, Niigata University is least selective MBA university in Japan. On average, 50 out of every 100 applicants secure admission to university. Some other best MBA universities in Japan, along with their acceptance rates are as follows:

In (D), each of the terms has a degree 2.

Hence, (D) is homogenous

$\therefore$ Option (D) is correct.

For admission to the MA course at Parle Tilak Vidyalaya Association's Sathaye College, students need to fulfil the minimum eligibility criteria in the first place. The selection criteria for this course is merit-based and candidates must pass graduation. The duration of the course is two years.

5. Let (r + 1)^th term be the general term of (x²–y)⁶.

So, T_r-1 = ⁶C_r (x²)^6-r (-y)^r

= (–1)^r .⁶C_r . $x^{12-r}$ . $y^r$

120. Let $y={\left(5x\right)}^{3cos2x}$

Taking log,

$logy=3cos2x\left[log5x\right]$

Differentiating w r t. x,

$\frac{1}{y}\frac{dy}{dx}=3cos2x\frac{d}{dx}log5x+3log5x\frac{d}{dx}cos2x$

$=3\left[cos2x\cdot \frac{1}{5x}\frac{d}{dx}\left(5x\right)-log5x\times sin2x\frac{d}{dx}2x\right]$

$\frac{dy}{dx}=3y\left[cos2x\times \frac{5}{5x}-log5x\cdot sin2x\cdot 2\right]$

$=3{\left(5x\right)}^{3cos2x}\left[\frac{cos2x}{x}-2sin2xlog5x\right]$

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For a homogenous D.E. of the formula $f\left(\frac{y}{x}\right)$

We put,  $\frac{x}{y}=0=x=vy$

$\therefore$ Option (c) is correct.

The given D.E. is

$\begin{array}{l}2xy+y^2-2x^2\frac{dy}{dx}=0\\ =2x^2\frac{dy}{dx}=2xy+y^2\\ =\frac{dy}{dx}=\frac{2xy+y^2}{2x^2}=\frac{y}{x}+\frac{1}{2}{\left(\frac{y}{x}\right)}^2=f\left(\frac{y}{x}\right)\end{array}$

i.e, the given is homogenous.

Let, $y=vx$ $=\frac{y}{x}=v$ so that $\frac{dy}{dx}=v+x\frac{dv}{dx}$ is the D.E.

Then, $v+x\frac{dv}{dx}=v+\frac{1}{2}{v}^2$

$\begin{array}{l}=x\frac{dv}{dx}=\frac{1}{2}{v}^2\\ =\frac{dv}{v^2}=\frac{dx}{2x}\end{array}$

Now, $={\displaystyle \int \frac{dv}{v^2}={\displaystyle \int \frac{dx}{2x}}}$

$\begin{array}{l}=\frac{v-2+1}{-2+1}=\frac{1}{2}log\left|x\right|+c\\ =\frac{-1}{v}=\frac{1}{2}log\left|x\right|+c\end{array}$

Putting back $\frac{y}{x}=v$ we get,

$=-\frac{x}{y}=\frac{1}{2}log\left|x\right|+c$

$Given\frac{y}{x}=vwhenx=1$ and y= 2

$\begin{array}{l}=-\frac{1}{2}=\frac{1}{2}log\left|1\right|+c\\ =c=-\frac{1}{2}\end{array}$

$\therefore$ The particular solution is,

$\begin{array}{l}=-\frac{x}{y}=\frac{1}{2}log\left|x\right|-\frac{1}{2}\\ =-\frac{2x}{y}=log\left|x\right|-1\\ =y=\frac{-2x}{log\left|x\right|-1}=\frac{2x}{1-log\left|x\right|}\end{array}$

For MA course admission, candidates must meet the eligibility criteria set by the institute. Aspirants should pass graduation in any stream. The admission criteria for the course is merit-based at Parle Tilak Vidyalaya Association's Sathaye College. MA (Master of Arts) is one of the most popular postgraduate-level courses undertaken by students after their bachelor's degree. Aspiring Candidates can choose any subject specialisation of their choice which can give them an edge over others, and eventually helps them decide their career paths accordingly.

119. Let $y=si{n}^{3}x+co{s}^{6}x$

So,  $\frac{dy}{dx}=\frac{d}{dx}\left(si{n}^{3}x+co{s}^{6}x\right)$

$=3si{n}^{2}x\frac{d}{dx}sinx+6co{s}^{5}x\frac{d}{dx}cosx$

$=3si{n}^{2}xcosx-6co{s}^{5}xsinx$

$=3sinxcosx\left(sinx-2co{s}^4\right)$

The given D.E.is

$\begin{array}{l}\frac{dy}{dx}-\frac{y}{x}+cosec\left(\frac{y}{x}\right)=0\\ =\frac{dy}{dx}=\frac{y}{x}-cosec\left(\frac{y}{x}\right)=f\left(\frac{y}{x}\right)\end{array}$

i.e, the given D.E. is homogenous.

Let, $y=vx=\frac{y}{x}=v$ So that, $\frac{dy}{dx}=v+x\frac{dv}{dx}$ in the D.E

Then, $v+x\frac{dv}{dx}=v-cosecv$

$\begin{array}{l}=x\frac{dv}{dx}=-cosecv\\ =\frac{dv}{cosecv}=-\frac{dx}{x}\\ =sinvdv=-\frac{dx}{x}\end{array}$

Integrating both sides we get,

$\begin{array}{l}{\displaystyle \int sinvdv=-{\displaystyle \int \frac{dx}{x}}}\\ =-cosv=-log\left|x\right|+c\\ =cosv=log\left|x\right|-c\end{array}$

Putting back $v=\frac{y}{x}$ we get,

$=cos\frac{y}{x}=log\left|x\right|-c$

Given, $y=0,when,x=1$

$\begin{array}{l}=cos0=log1-c\\ =c=-1\end{array}$

$\therefore$ The required particular solution is

$\begin{array}{l}cos\left(\frac{y}{x}\right)=log\left|x\right|+1=log\left|x\right|+log\left|c\right|\\ =cos\left(\frac{y}{x}\right)=log\left|cx\right|\end{array}$

Post graduation from top MBA university in Japan, graduates secure good job opportunities in global companies. Mentioned below are average placement salaries offered to students at top universities for MBA in Japan-