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 ${\displaystyle \underset{0}{\overset{1}{\int }}1.{\left(1-x^n\right)}^{2n+1}dx}$ using by parts we get,

$\left(2n^2+n+1\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left(1-x^n\right)}^{2n+1}dx=1177{\displaystyle \underset{0}{\overset{1}{\int }}{\left(1-x^n\right)}^{2n+1}dx}}$

$\Rightarrow 2n^2+n+1=1177\Rightarrow n=24or-\frac{49}{2}\therefore n=24$

The top recruiters for the Justice K S Hegde Institute of Management placements are given below:

Coefficient of x in ${\left(1+x\right)}^p{\left(1-x\right)}^q={-}^p{C_0}^q{C}_1{+}^p{C_1}^q{C}_0=-3\Rightarrow p-q=3$

Coefficient of x² in ${\left(1+x\right)}^p{\left(1-x\right)}^q{=}^p{C_0}^q{C}_2{-}^p{C_1}^q{C}_1{-}^p{C_2}^q{C}_0=-5$

$\Rightarrow \frac{q\left(q-1\right)}{2}-pq+\frac{p\left(p-1\right)}{2}=-5\Rightarrow \frac{q\left(q-1\right)}{2}-\left(q-3\right)q+\frac{\left(q-3\right)\left(q-4\right)}{2}=-5\Rightarrow q=11,p=8$

Coefficient of x³ in ${\left(1+x\right)}^8{\left(1-x\right)}^{11}={-}^{11}{C}_3{+}^8{C_1}^{11}{C}_2{-}^8{C_2}^{11}{C}_1{+}^8{C}_3=23$

According to the placement statistics of 2025, the highest package earned by a BTech student of IIIT Ranchi was INR 54 LPA. This figure isalmost double that of the previous year where the highest package stood at around INR 28 lakh.

The average package of INR 7.04 LPA was offered at Justice K S Hegde Institute of Management placements 2023. Check out the table for clarity:

 $x^8-x^7-x^6+x^5+3x^4-4x^3-2x^2+4x-1=0$

$\Rightarrow x^7\left(x-1\right)-x^5\left(x-1\right)+3x^3\left(x-1\right)-x\left(x^2-1\right)+2x\left(1-x\right)+\left(x-1\right)=0$

$\Rightarrow \left(x-1\right)\left(x^2-1\right)\left(x^5+3x-1\right)=0\therefore x=\pm 1$ are roots of above equation and x5 + 3x – 1 is a monotonic term hence vanishes at exactly one value of x other then 1 or 1.

$\therefore$ 3 real roots.

The UPTAC counselling eligibility criteria for BTech & M.Tech (Integrated) course is:

Passed 10+2 examination and obtained at least 45% marks (40% marks in case of candidates belonging to SC/ST category) in the subjects taken together.

OR

Passed D.Voc stream in the same or allied sector.

The Justice K S Hegde Institute of Management has not released the placement rate as of yet. For more information on placement packages and recruiters for the JKSHIM placements 2021-23 for MBA, refer to the table below:

Given series $\left\{3\times 1\right\},\left\{3\times 2,3\times 3,3\times 4\right\},\left\{3\times 5,3\times 6,3\times 7,3\times 8,3\times 9\right\}.........$

$\therefore$ 11th set will have 1 + (10)2 = 21 terms

Also up to 10th set total 3 × k type terms will be 1 + 3 + 5 + ……… +19 = 100 terms

$\therefore Set11=\left\{3\times 101,3\times 102,......3\times 121\right\}$ $\therefore$ Sum of elements = 3 × (101 + 102 + ….+121)

$=\frac{3\times 222\times 21}{2}=6993.$

v = $\omega \sqrt[]{A^2-x^2}$ $\text{exception 25:}$

at x = 5, A = 10

$v\text{'}=3v=3\omega \sqrt[]{A^2-5^2}=\omega \sqrt[]{A{\text{'}}^2-5^2}$

$=3\sqrt[]{A^2-5^2}=\sqrt[]{A{\text{'}}^2-5^2}$

$\sqrt[]{10^2-5^2}=\sqrt[]{A{\text{'}}^2-25}$

$A{\text{'}}^2=25+9\times 75$ $A{\text{'}}^2=700$

$A\text{'}=\sqrt[]{700}cm$

The JKSHIM has a dedicated Corporate Relations Team that looks after the placements and internships. The training programs conducted through the cell are as follows:

• Aptitude Test

• Group Discussion & PI

• Resume Preparation

• Current Affairs & Issues

A = $\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)$

$A^2=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill a^2+bc\hfill & \hfill ab+bd\hfill \\ \hfill ac+dc\hfill & \hfill ac+d^2\hfill \end{array}\right)$

a2 + bc = bc + d2 = 1 ………. (i)

and b (a + d) = c (a + d) = 0 ……… (ii)

Case 1

b = c = 0

then possible ordered pair of

(a, d) $\equiv$  (1, 1) (-1, -1) (-1, 1) (1, -1) total 4 possible case

Case 2

a = -d

then (a, d) $\equiv$  (-1, 1) (1, -1)

then bc = 0

now if b = 0

then possible choice for {-1, 0, 1, 2, …….10} = 12

Similarly if c = 0 then possible choice for $b\in \left\{-1,0,1,2,......10\right\}$ is = 12

but (0, 0) counted twice

$\therefore$ bc = 0 in (12 + 12 – 1) = 23 ways

$\therefore$ total number of ways = 2 × 23 = 46

$\therefore$ total number of required matrices = 46 + 4 = 50

Factors of 36 = 2².3².1

Five-digit combinations can be

(1, 2, 3, 3), (1, 4, 3, 1), (1, 9, 2, 1), (1, 4, 9, 11), (1, 2, 3, 6, 1), (1, 6, 1, 1)

i.e., total numbers $\frac{5!5!}{2!2!}+\frac{5!}{2!2!}+\frac{5!}{2!2!}+\frac{5!}{3!}+\frac{5!}{2!}+\frac{5!}{3!2!}=\left(30\times 3\right)+20+60+10=180.$

School of Law, Pandit Deendayal Energy University application window opens online on the official website of the institute. Below are the steps to apply for the SOL, PDEU admission:

STEP 1: Visit the official website, i.e., pdpu.nopaperforms.com/law-form

STEP 2: Click on 'Admission Form'.

STEP 3: Fill out the details on the application form and submit it.

The Justice K S Hegde Institute of Management provides decent placements to its MBA graduates.  For more information on the institute's placements during 2021-23 for MBA, refer to the table below:

The Kerala NEET counselling registration process is conducted before the NEET exam is conducted. Hence, after the result declaration, candidates are given a chance to upload their NEET scores. In case a candidate does not upload his/her scores, or does not qualify for NEET by securing the cutoff marks, then he/she will not be considered eligible for the Kerala NEET counselling process. 

$d_1=\frac{199-100}{2}\cancel{\in }l$

$d_2=\frac{199-100}{3}=33$

$d_3=\frac{199-100}{4}\cancel{\in }l$

$d_n=\frac{199-100}{i+1}\in l$

$\Rightarrow di=33+11or9$

$\therefore$ sum of common differences = 33 + 11 + 9 = 53

RRB NTPC post preferences cannot be changed once it is exercised by candidates. The posts announced under RRB NTPC exam are Junior Clerk cum Typist, Accounts Clerk cum Typist, Trains Clerk, Commercial cum Ticket Clerk, Goods Train Manager, Chief Commercial cum Ticket Supervisor, Senior Clerk cum Typist, Junior Account Assistant cum Typist, and Station Master in various Zonal Railways and Production Units of Indian Railways.

Yes, Netaji Subhas University of Technology offers scholarships. The scholarships are awarded to the students on the basis of their family income and the fee reimbursement is done for the meritorious students. NSUT offers Chhatra Vittiya Sahayta Evam Protsahan Kosh incentive scheme to its students. The incentive scheme is divided into three parts, mentioned in the table below along with the eligibility criteria:

Let and are the roots of $\left(p^2+q^2\right)x^2-2q\left(p+r\right)x+q^2+r^2=0$

$\therefore \alpha +\beta >0and\alpha \beta >0$ Also, it has a common root with x² + 2x – 8 = 0

$\therefore$ The common root between above two equations is 4.

$\Rightarrow 16\left(p^2+q^2\right)-8q\left(p+r\right)+q^2+r^2=0\Rightarrow \left(16p^2-8pq+q^2\right)+\left(16q^2-8qr+r^2\right)=0$

$\Rightarrow {\left(4p-q\right)}^2+{\left(4q-r\right)}^2=0\Rightarrow q=4pandr=16p\therefore \frac{q^2+r^2}{p^2}=\frac{16p^2+256p^2}{p^2}=272$