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Candidates who have passed Class 12/ HSc examination in Arts, Commerce, Science stream or equivalent with at least 60% aggregate (55% aggregate for SC/ ST/ PwD/ Transgender) are eligible to apply for IPM at IIFT Kakinada. Aspirants should also have passed Class 10 examination with 60% aggregate (55% aggregate for SC/ ST/ PwD/ Transgender) for filling out the form. 

Aspirants who are likely to appear in Class 12/ HSc examination by the end of July 31, 2025 can also apply, subject to meeting the eligibility criteria mentioned above.

With an low score in MHT CET you still have chance to get into government colleges if you come under SC/ST category, also on branch preferences you would still have an chance , also on further rounds the vacant seats will be filled with low score students.

In 2024, the PGDM cutoff has been released for various MBA exams. The Jaipuria Institute of Management cutoff 2024 stood at 80, 80, and 70 for CMAT, MAT, and XAT.  So, analysing the Jaipuria Institute of Management CMAT year-wise cutoffs, there are a high chances that the cutoffs will stood the same in 2025 too.

58. Let G₁ and G₂ be the two numbers between 3 and 81 so that 3, G₁, G₂, 81 is in G.P.

So, a = 3

a₄ = ar³ = 81 (when r = common ratio)

$\Rightarrow$ $r^3=\frac{81}{a}=\frac{81}{3}$

$\Rightarrow$ r³ = 27

$\Rightarrow$ r³ = 3³

$\Rightarrow$ r = 3

So, G1 = ar = 33=9 and G₂ = ar² 3´ (3)² =27

IIFT Kakinada students can apply for Merit-cum-Means scholarship offered at the institute. Under this scheme, students get a 20% waiver in the tuition fee of the course. The scholarship is offered to 10% of the admitted students in all full-time/ regular programmes offered by IIFT. Besides, students can also apply for scholarships offered by the Government of India.

If you belong to the General AI category, the MAMC NEET SS cutoff 2025 ranged between 36 and 2725 in the first round. The cutoff range varies by rounds, courses, specialisations and categories. Candidates can refer to the table below to view the NEET SS 2025 cutoff ranks for the Round. The cutoff data below is for the General category. 

57. Let r be the common ratio of the G.P. then,

First term = a₁ = a = a

$a_2=ar=b$

$a_3=a{r}^2=c$

$a_4=a{r}^3=d$

So, L.H.S.= $\left(a^2+b^2+c^2\right)\left(b^2+c^2+d^2\right)$

= $\left[a^2+{\left(ar\right)}^2+{\left(a{r}^2\right)}^2\right]\left[{\left(ar\right)}^2+{\left(a{r}^2\right)}^2+{\left(a{r}^3\right)}^2\right]$

= $\left[a^2+a^2{r}^2+a^2{r}^4\right]\left[a^2{r}^2+a^2{r}^4+a^2{r}^6\right]$

= $a^2\left[1+r^2+r^4\right]a^2{r}^2\left[1+r^2+r^4\right]$

= $a^4{r}^2{\left[1+r^2+r^4\right]}^2$

R.H.S.= ${\left(ab+bc+cd\right)}^2$

= ${\left[a\times ar+ar.a{r}^2+a{r}^{2}a{r}^3\right]}^2$

= ${\left[a^{2}r+a^2{r}^3+a^2{r}^5\right]}^2$ $$

= { $a^{2}r{\left[1+r^2+r^4\right]}^2$ }

= $a^4{r}^2{\left[1+r^2+r^4\right]}^2$

L.H.S. = R.H.S.

IIFT Kakinada offers Merit-cum-Means scholarships to students pursuing a course. The institute specified the merit as well as income requirement that aspirants need to fulfill to apply for the scholarship. The family income of the candidate should not be more than INR 8 Lacs per year for the scholarship. The upper ceiling of INR 8 Lacs is subject to revision.

56. Let a and r be the first term and common difference of the G.P

Thus, Sum of the first on term, $S_n=\frac{a\left(1-r^n\right)}{1-r}$

Let Sn = sum of term from (n+1)^th to (2n)^th  term

= $a{r}^n+a{r}^{n-1}+......+a{r}^{2n-1}$

= $\frac{a{r}^n\left[1-r^n\right]}{1-r}$

[ $?$ the above is a G.P. with first term ar^x and common ratio = $\frac{a{r}^{n+1}}{a{r}^n}=r^{n+1-n}=r$

and number of term from (2n)^th to (n+1)^th = n

So, $\frac{S_n}{S_n}=\frac{\frac{a\left(1-r^n\right)}{1-r}}{\frac{a{r}^n\left(1-r^n\right)}{1-r}}$

= $\frac{a}{a{r}^n}$

= $\frac{1}{r^n}$

The eligibility criteria for admissions to MMS courses at Thakur Institute of Management Studies and Research are that candidates must complete their bachelor's degree with a minimum aggregate of 50%. Moreover, the accepted entrance examination scores for admissions to the MMS course are different for the candidates from Maharashtra and non-Maharashtra. Check out the process below:

For Maharashtra Candidates: candidates must have secured non-zero scores in the MAH MBA/MMS CET.

For Other Indian Candidates: Candidates must have secured valid scores in CAT/CMAT/XAT, ATMA/MAT/GMAT.

55. Given, first term and xth term and a and b

Let r be the common ratio of the G.P.

Then,

Product of x terms, p= $\left(a\right)$ $\left(ar\right)$ $\left(a{r}^2\right)$ …… $\left(a{r}^{n-1}\right)$

p= $\left(a\right)$ $\left(ar\right)\left(a{r}^2\right)$ …. $\left(a{r}^{n-2}\right)\left(a{r}^{n-1}\right)$

= (a ×a× ….n term)(r´r²´r³ …. $r^{n-2}\times r^{n-1}$ )

= $a^x$ $r^{\left[1+2+3+....\left(n-2\right)+\left(n-1\right)\right]}$

p = $a^n$ $r^{n\left(\frac{n-1}{2}\right)}$

So, $p^2={\left[a^n{r}^n\left(\frac{n-1}{2}\right)\right]}^2$ [We know that,

= $n\frac{\left(n-1\right)}{2}$ [ $1+2+3+...+n=\frac{\left(n-1\right)n}{2}$

= $a^{2n}{r}^{2n}\left(\frac{n-1}{2}\right)$ [So, $1+2+3+....+\left(n-1\right)=\frac{\left(n-1+1\right)\left(n-1\right)}{2}=\frac{n(n-1)}{2}$

= $a^{2n}\times r^{n\left(n-1\right)}$

= ${\left(a\times a{r}^{n-1}\right)}^n$ $[?$ $a{n}^{n-1}=$ last term = b (given)]

= ${\left(a\times b\right)}^n$

According to the statistics of 2023, the total number of international students enrolled at this university were 12,000+. India, China, South Korea, Taiwan, and Canada were some of the top origin countries of students. For the Class 2028, the total number of applicants at this university was 78,769 for UG class. International students for this class accounted to 20% hailing from 160+ countries. The top countries represented by students from China, India, South Korea, Canada, and Turkey.

Candidates applying at IIFT Kakinada need to pay an application fee. There is relaxation in the application fee for SC/ ST/ PwD category candidates. They need to pay INR 1,000 as the application fee. For aspirants belonging to General, OBC-NCL and EWS categories, need to pay INR 2,000 as the application fee. Candidates can pay the non-refundable fee via any of the online payment modes.

54. Let a and r be the first term and common ratio of the G.P.

Then, ap = a

$A{R}^{p-1}=a$ …… I

and aq = b

$A{R}^{q-1}=b$ …….II

Also, ar = c

$A{R}^{r-1}=c$ ….III

Given, L.H.S. = $a^{a-r}$ $b^{r-p}$ $c^{p-q}$

${\left(A{R}^{p-1}\right)}^{q-r}$ ${\left(A{R}^{q-1}\right)}^{r-p}$ $\left(A{R}^{r-1}\right)$ $?$ using I, II and III

$A^{q-r}$ $R^{\left(p-1\right)}$ $A^{r-p}$ $R^{\left(q-1\right)\left(r-p\right)}$ $A^{p-q}$ $R^{\left(r-1\right)\left(p-q\right)}$

$A^{\left[q-r+r-p+p-q\right]}$ $R^{\left[\left(p-1\right)\left(q-r\right)+\left(q-1\right)\left(n-p\right)+\left(x-1\right)\left(p-q\right)\right]}$

A0 R[pqprq+r+qrpqr+p+prqrp+q] $$

$1\times$ $R^{\left[pq-pq+pr-pr+q-q+r-r+qr-qr+p-p\right]}$

= R

= 1

= R.H.S

For an LLM in American Law at Boston Uni, students should have a strong academic background. The class size of this university is 110-120. A few of the application materials that have to be submitted are:

• Transcripts (A first degree in Law or equivalent)
• Letters of recommendation
• CV
• TOEFL (minimum score of 100) / IELTS score report (7.0)
• High level of English language proficiency is required
• Work experience is preferred, but not required