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f : N ->N

f (n + 1) = f (n) + f (1)

Let f (1) = a, a  $\in$ N

f (2) = f (1) + f (1) = 2a

f (3) = f (2) + f (1) = 3a

and so on

->f (m) = ma, m, a  $\in$ N

->f is one – one, Þ option (2) is true.

Suppose f (g (x) is one-one

then f (g (x₁) $\ne$ f (g (x₂) for x₁ $\ne$  x₂

->g (x₁) $\ne$ g (x₂) (as f is one-one)

->g is one – one

Virohan: GH Raisoni Skilltech University is one of the renowned colleges in Nagpur to offer quality education to students in the field of Medicine and Healthcare. The college offers a BSc programme to students to bridge the gap between education and employment. Further, the fee structure of Virohan: GH Raisoni Skilltech University Nagpur ranges between INR 5.4 Lacs and INR 6.4 lakh. 

Students should compare aspects like available specialisations, faculty experience, practical training opportunities, placement support, infrastructure, and industry exposure. Additionally, factors such as seat intake, college reputation, location, and student reviews can help determine whether ITM Institute of Hotel Management offers a better fit than other institutions for the BSc programme.

$z^2+\alpha z+\beta =0,\alpha ,\beta \in R$

roots : 1 – 2i, 1 + 2i

Sum of roots = 2 = -α and product of roots = 5 = β

α - β = -2 - 5 = -7

As per questions

$\frac{dy}{dx}=\frac{x^2-4x+y+8}{x-2}$           

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

$\frac{dy}{dx}=\left(x-2\right)+\frac{y+4}{x-2}$                       .(i)

Let $\frac{y+4}{x-2}=t$  

(y + 4) = t(x – 2)

Putting in equation (i)

$\left(x-2\right)\frac{dt}{dx}+t=\left(x-2\right)+t$        

    $\frac{dt}{dx}=1$        

dt = dx

Integrating on both the sides t = x + c

$\frac{y+4}{x-2}=x+c$  

Passing through origin C = -2

  $\therefore$         equation of curve $\frac{y+4}{x-2}=x-2$

Combined equation of pair of lines OP and OQ is

$x^2+2y^2=2{\left(x+y\right)}^2$

$\Rightarrow x^2+4xy=0\Rightarrow x\left(x+4y\right)=0\Rightarrow \{\begin{array}{l}x=0\left(lineOP\right)\\ y=-\frac{x}{4}\left(lineOQ\right)\end{array}$

$tan\left(90°+\theta \right)=-\frac{1}{4}$

$-cot\theta =-\frac{1}{4}\Rightarrow tan\theta =4$

$\theta =ta{n}^{-1}4=co{t}^{-1}\frac{1}{4}=\frac{\pi }{2}-ta{n}^{-1}\frac{1}{4}$

$\angle POQ=\pi -\theta =\frac{\pi }{2}+ta{n}^{-1}\frac{1}{4}$

Life at NIPS Hotel Management Ranchi is quite good. Practical teaching methods are followed, in which on-the-job training is given to students. NIPS Hotel Management Ranchi has a state-of-the-art infrastructure with various facilities such as a library, conference room, and others. The classrooms are equipped with all the needed facilities. Hence, the college tries to make student learning environment quite easy.

Symbiosis Centre for Information Technology Pune offers MBA at PG level in full-time mode for 2 years. The Institute offers MBA based on minimum aggregate of 50% in graduation and valid scores in SNAP or MAH CET. It is ranked by various ranking bodies such as Business Today 2024 and Outlook ICare. 

Yes, the India Today 2024 rankings for Hotel/Hospitality Management included the BSc programme offered by ITM Institute of Hotel Management, placing the institute at the 53rd spot nationally. This recognition from a reputable ranking body reinforces the programme's credibility and the institute's dedication to delivering high-quality education in the hospitality and culinary arts fields.

$l_n={\displaystyle \underset{\pi /4}{\overset{\pi /2}{\int }}co{t}^{n}xdx=}{\displaystyle \underset{\pi /4}{\overset{\pi /2}{\int }}co{t}^{n-2}x\left(cose{c}^{2}x-1\right)dx}$

$={\left[\frac{-co{t}^{n-1}x}{n-1}\right]}_{\pi /4}^{\pi /2}-l_{n-2}=\frac{1}{n-1}-l_{n-2}$

$l_n+l_{n-2}=\frac{1}{n-1}$

$\begin{array}{l}n=4\Rightarrow l_4+l_2=\frac{1}{3}\\ n=5\Rightarrow l_5+l_3=\frac{1}{4}\\ n=6\Rightarrow l_6+l_4=\frac{1}{5}\end{array}\}\Rightarrow \frac{1}{l_2+l_4},\frac{1}{l_3+l_5},\frac{1}{l_4+l_6}$ are in A.P.

The Product Design duration in India differs on the basis of the course level at which you're planning to pursue the course. The UG Product Design courses are usually 3 to 4 years long, while for PG courses, you need to invest at least 2 years at college. A PhD in Product Design will take you around 3 to 5 years to complete.

Further for diploma and certification Product Design courses, the duration can range from 1 month to 1 year, based on the course type, course focus and the college.

The tuition fee structure followed at Amrita School of Dentistry is a per annum structure. The total tuition fee at Amrita School of Dentistry is INR 14 Lacs which is divided into 4 instalments of INR 3.5 Lacs each. The duration of Amrita School of Dentistry BDS courseis 5 years where, 4 years are spent on covering the BDS syllabus and the leftover year is spent on internships.

The Product Design course fees at top design institutes differ on the basis of the college offering the course, its ranking, location, student popularity, placements, and the course level. The average fees for Product Design courses is INR 5 Lacs to INR 25 lakh.

You must go to the college websites to have better clarity on their respective Product Design fee structures to ensure a smooth admission process and academic journey.

Pursuing MBA at Vinayaka Mission's Kirupananda Variyar Engineering College, Vinayaka Mission's Research Foundation is generally considered a good choice. Candidates must consider factors such as infrastructure, faculty experience, hostel and mess facilities, curriculum, etc. while taking admission. 

There are many platforms where you can find Product Design courses online even for free. Some of the top student choices are Coursera, Udemy, and edX. Most of the students find these platforms good and affordable that is why they are the go-to options for learning each and everything about product designing.

Hope this helps!!!

A Product Design portfolio is important because it is the documentation of your skill set, certifications, projects, internship experience, and basically the overall value-add that you can bring to the table if hired by any recruiter. A strong Product Design portfolio gives every recruiter an indication of your creative abilities and can land you a promising job offer with a decent salary package.

2x + 3y + 2z = 9 . (i)

3x + 2y + 2z = 9     . (ii)

x – y + 4z = 8          . (iii)

(ii) – (i) ⇒ x = y

Then (iii) ⇒ z = 2

(i) ⇒ 5x + 4 = 9 ⇒ (x = 1, y = 1, z = 2) unique solution

In 2024, the BSc programme at ITM Institute of Hotel Management contributed to the institute securing the 53rd position in the India Today rankings for Hotel/ Hospitality Management. This ranking highlights the programme's quality, strong curriculum, and industry relevance, reflecting ITM's commitment to preparing students for successful careers in the competitive hospitality sector.

When you ask about some Product Design examples in the real world setting, you can find them in both physical and digital form, with focus on functionality, aestheticism and user satisfaction. Here are a few Product Design examples for your reference:

Physical Products: Bedside Lamp, Ballpoint Pen, Coca-Cola Bottle, Ketchup Bottle, Kitchen Stand Mixer, Apple IPhone, etc.

Digital Form Products: Google Search Engine, Spotify, Airbnb, Netflix, Mobile Apps, etc.