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Let P (at², 2 at) where

$a=\frac{3}{2}$                

T : yt = x + at² so point Q is

$\left(-a,at-\frac{a}{t}\right)$                

N : y = -tx + 2at + at³ passes through (5, -8)

$-8=-5t+3t+\frac{3}{2}{t}^3$

⇒ $3t^3-4t+16=0$                

⇒ t = -2

So ordinate of point Q is $-\frac{9}{4}$  

$\frac{24}{\pi }{\displaystyle \underset{0}{\overset{\sqrt[]{2}}{\int }}\frac{\left(2-x^2\right)}{\left(x^2+2\right)\sqrt[]{4+x^4}}dx}$

$\frac{24}{\pi }{\displaystyle \underset{0}{\overset{\sqrt[]{2}}{\int }}\frac{x^2\left(\frac{2}{x^2}-1\right)dx}{x\left(x+\frac{2}{x}\right)\times x\sqrt[]{\frac{4}{x^2}+x^2}}}$

$x+\frac{2}{x}=t$

$dt=\left(1-\frac{2}{x^2}\right)dx$

$=-\frac{12}{\pi }\left[\frac{\pi }{4}-\frac{2\pi }{2\times 2}\right]=-\frac{12}{\pi }\left[-\frac{\pi }{4}\right]=3$

India is seeing a rapid rise in product focused MBA programs. Top institutes include:

• ISB Hyderabad / Mohali – Product Management specialization
• IIM Bangalore / IIM Ahmedabad – Electives in Product Strategy & Innovation
• SPJIMR, Mumbai – Digital Product Management focus
• IMT Ghaziabad – Marketing + Product electives
• IIT Delhi (DoMS), IIT Kharagpur – Tech and product strategy
• Great Lakes, Chennai – One-year PGPM with product focus
• upGrad x Duke / Deakin / NMIMS – Online Product Management MBA/PGPM
• Institute of Product Leadership (IPL), Bengaluru – India's 1st product-focused B-school

  $LetG.P.be{a}_1=a,a_2=ar,a_3=a{r}^2$ ${}_{}{}_{}{}_{}{}^{}$ , ….

$?3a_2+a_3=2a_4$

^{$\Rightarrow 3ar+a{r}^2=2a{r}^3$}

$\Rightarrow 2a{r}^2-r-3=0$

$\therefore a_2+a_4+2a_5=a\left(r+r^3+2r^4\right)$

$\frac{}{}$ $=\frac{8}{3}\left(\frac{3}{2}+\frac{27}{8}+\frac{81}{8}\right)$  = 40

Yes, there are great online or hybrid options for working professionals:

Some of the top Online Programs are mentioned below:

• Duke / upGrad Global MBA (Product stream)
• Great Learning's PGPM in Product Management
• Udacity – Product Manager Nanodegree
• Coursera (UVA, Boston Univ, IESE) Product Strategy specializations
• Institute of Product Leadership (IPL) – Executive format

These online MBAs are flexible, but don't offer campus placements, so great for people with some industry experience.

Kindly consider the following figure

Check table below to know GMDU GMET exam timing:

Yes, you may get direct admission at K.S Group of Institutions provided you qualify for the admission requirements set by the institute. The institiution mainly offers admissions through entrance basis, however, certain amount of total seats are reserved to be offered under the management quota. Interested students can connect with the admission team of the institute to know more about the admission process and chances.

36 = 2 * 2 * 3 * 3

Number should be odd multiple of 2 and does not having factor 3 and 9

Odd multiple of 2 are

102, 106, 110, 114….998 (225 no.)

No. of multiplies of 3 are

102, 114, 126 ….990 (75 no.)

Which are also included multiple of 9

Hence,

Required = 225 – 75 = 150

CII SOL, Amity University, Noida offers courses such as MBA for 2 years in full-time mode on the basis of valid scores of accepted entrance exams. The Institute offers MBA at a total tuition fee worth INR 10.5 Lakh. CII SOL, Amity University, Noida also offers scholarship and loan facilities to help the candidates pursue the choice of course without any hesitance. 

Equation of perpendicular bisector of AB is

$y-\frac{3}{2}=-\frac{1}{5}\left(x-\frac{5}{2}\right)\Rightarrow x+5y=10$

Solving it with equation of given circle,

${\left(x-5\right)}^2+{\left(\frac{10-x}{5}-1\right)}^2=\frac{13}{2}$

$\Rightarrow x-5=\pm \frac{5}{2}\Rightarrow x=\frac{5}{2}or\frac{15}{2}$

But

$x\ne \frac{5}{2}$

because AB is not the diameter.

So, centre will be

$\left(\frac{15}{2},\frac{1}{2}\right)$

Now,

$r^2={\left(\frac{15}{2}-2\right)}^2+{\left(\frac{1}{2}+1\right)}^2=\frac{65}{2}$

GNDU GMET 2026 marking scheme is as follows:

• For every correct answer, student will get one mark.
• Zero mark will be given for any wrong answer.
• Zero mark will be given for any not attempted question.

$?X=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$

$\therefore X^2=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$

$\therefore Y=\alpha l+\beta X+\gamma X^2=\left[\begin{array}{ccc}\hfill \alpha \hfill & \hfill \beta \hfill & \hfill \gamma \hfill \\ \hfill 0\hfill & \hfill \alpha \hfill & \hfill \beta \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \alpha \hfill \end{array}\right]$

$\begin{array}{l}\therefore \alpha =5,\beta =10,y=15\\ \therefore {\left(\alpha -\beta +y\right)}^2=100\end{array}$

K.S. Group of Institutions admits students based on preset admission criteria which is both merit and entrance-based. Considering the ever-increasing competition and limited number of seats,  K.S. Group of Institutions admission chances largely depend on student to student. However, with right prep + strategy, average students can also get a seat at the institute provided they satisfy the admission requirements. Students may connect with the admission desk of the institute to learn about the KSGI admission chances.

GNDU GMET question paper will consist of 200 questions divided aross four sections. Following table brings the breakdown of GNDU GMET questions:

$z^2+z+1=0\Rightarrow z=w,w^2$

$\left|{\displaystyle \sum _{n=1}^{15}{\left(z^n+{\left(-1\right)}^n\frac{1}{z^n}\right)}^2}\right|=\left|{\displaystyle \sum _{n=1}^{15}\left(z^{2n}+\frac{1}{z^{2n}}+2{\left(-1\right)}^n\right)}\right|=\left|{\displaystyle \sum _{n=1}^{15}{w}^{2n}+\frac{1}{w^{2n}}+2{\left(-1\right)}^n}\right|$

$=\left|\frac{w^2\left(1-1\right)}{1-w^2}+\frac{\frac{1}{w^2}\left(1-1\right)}{1-\frac{1}{w^2}}-2\right|=2$

Even if the candidate clears the sectional and overall cutoff in IPMAT exam but does not get shortlisted for the PI round, it means that the candidate is eligible but not among the top-ranked candidates based on the composite score. Such a candidate will not be considered for admission to the IPM programme.

Such candidates can opt for reappearing for the exam next year, as there is no provision of re-evaluation. Candidates can also consider applying to other institutes which accept IPMAT Indore scores, such as NIRMA University, TAPMI, etc. 

No, BBA is not compulsory to get admission to an MBA course at Athena School of Management. Candidates are required to complete any bachelor's degree from any stream with a decent score to get admission into the MBA course. Moreover, after completing the minimum eligibility requirements, applicants are required to appear and secure a valid score in the accepted entrance exam, followed by GD&PI/ counselling.

  $?p+q=3$ $$ ….(i)

and $p^4+q^4$ ${}^{}{}^{}$ = 369 ….(ii)

${\left\{{\left(p+q\right)}^2-2pq\right\}}^2-2p^2{q}^2=369$

or ${\left(9-2pq\right)}^2-2{\left(pq\right)}^2=369$ ${}^{}{}^{}$

or ${\left(pq\right)}^2-18pq-144=0$ ${}^{}$

$\therefore pq=-6or24$

But pq 24 is not possible

$\therefore pq=-6$

$Hence,{\left(\frac{1}{p}+\frac{1}{q}\right)}^{-2}={\left(\frac{pq}{p+q}\right)}^2={\left(-2\right)}^2=4$

With a JEE Main percentile of 78.9 and CRL rank around 3.1 lakh, getting into top NITs or IIITs might not be possible, but you can still aim for good private or state-level colleges. Some options to consider:

1. SRM Institute of Science and Technology

2. Chandigarh University

3. Graphic Era University

4. Sharda University

5. Amity University, Noida