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33. In the word EQUATION, there are vowels (E, U, A, I, O) and 3 consonants (Q, T, N).

Treating vowels as a whole as 1st object and consonants as a whole as 2nd object, we can have an arrangement of 2! = 2.

Similarly, arrangement within the vowels = 5! = 5 * 4 * 3 * 2 * 1 = 120

And arrangement within the consonants = 3! = 3 * 2 * 1 = 6

Therefore, total number of possible arrangement = 2 * 120 * 6 = 1440

7.

Barack Obama, Summer Redstone, Cass R. Sunstein and Michelle Obama are some very famous Harvard Law School Graduates. This B-School is very well-renowned for its highly skilled graduates who are contributing significantly to the entire world. The Harvard Law School alumni network is of 40,000 students worldwide.

Completing a BA degree from Bharathiar University Distance Education can lead to various career paths, each with different salaries. Career prospects may differ from student to student based on their individual learnings and skills. Tabulated below are a few career prospects one can pursue after a BA and their respective average payouts:

Note: The above-mentioned are industry-based average payouts. The actual salary may differ.

32. In the word DAUGHTER, there are 3 vowels, namely A, U, E and 5 consonants, namely D, G, H, T, R.

The number of ways of selecting 2 vowels out of 3

= ³C₂

= $\frac{3!}{2!\left(3-2\right)!}$

= 3

The number of ways of selecting 3 consonants out of 5

= ⁵C₃

= $\frac{5!}{3!\left(5-3\right)!}$

= 5 × 2

= 10

Therefore, the number of combination of 3 consonants and 2 vowels is 3 × 10 = 30.

Each of these 30 combinations has 5 letters which can be arranged among themselves in 5! Ways.

Therefore, the required numbers of different words is

= 30 × 5! = 30 × 5 × 4 × 3 × 2 × 1 = 3600

Jharkhand Raksha Shakti University has a total of 50 seats for BBA programme. The university follows the rules of the state government for seat reservation. Hence, the seats are reserved for SC, ST, OBC, and other categories. It must be noted that the given seat count is as per the official website/ sanctioning body. It is still subject to changes and hence, is indicative.

6. $\underset{x\to 0}{lim}\frac{{(x+1)}^5-1}{x}=\underset{x\to 0}{lim}\frac{x^5+5x^4+10x^3+10x^2+5x+1-1}{}$

Q6. $\underset{x\to 0}{\lim }\frac{{\left(x+1\right)}^5-1}{x}$

6. $\underset{x\to 0}{lim}\frac{{(x+1)}^5-1}{x}=\underset{x\to 0}{lim}\frac{x^5+5x^4+10x^3+10x^2+5x+1-1}{}$ $=\underset{x\to 0}{lim}{x}^4+5x^3+10x^2+10x+5.$ (0)4 + 5(0)3 + 10(0)2 + 10(0) + 5 = 5.

31. A student can choose 5 out of 9 available courses. However 2 specific courses are made compulsory.

So, now a student has 3 choices out of the remaining 7 courses.

Therefore, the required number of ways

=⁷C₃

= $\frac{7!}{3!\left(7?3\right)!}$

= 35

5. $\underset{x\to -1}{lim}\frac{x^{10}+x^5+1}{x-1}=\frac{{(-1)}^{10}+{(-1)}^5+1}{(-1)-1}=\frac{1-1+1}{-2}=-\frac{1}{2}$

4. $\underset{x\to 4}{lim}\frac{4x+3}{x-2}=\frac{4\times 4+3}{4-2}=\frac{16+3}{2}=\frac{19}{2}$

3. $\underset{r\to 1}{lim}\pi r^2=\pi ·{(1)}^2=\pi$

There are a total of 50 seats for BBA in Aviation Management course at Jharkhand Raksha Shakti University. Candidates are admitted as per the sanctioned intake and selection procedure. During seat allotment, the rules of the state government are considered. It must be noted that the given seat count is as per the official website/ sanctioning body. It is still subject to changes and hence, is indicative. 

SRK Institute of Technology generally admits students based on entrance exam scores or via counselling. Hence, securing 75% in Class 12 may not be enough to secure a spot at the undergradaute courses offered at the institute. Students also need to provide valid exam scores or fulfil the selection criteria of the institute. Howevre, students may connect with the college's help desk to learn more about the admsision chances of the preferred course.

2. $\underset{x\to \pi }{lim}\left(x-\frac{22}{7}\right)=\pi -\frac{22}{7}$

30. We are given 5 black and 6 red balls of which 2 black and 3 red balls can be selected.

Thus the required number of ways

= ^5C₂×⁶C₃

= $\frac{5!}{2!\left(5-2\right)!}$ × $\frac{6!}{3!\left(6-3\right)!}$

= 10 × 20

= 200

Yes, there is negative marking in the UPSC IES/ESE 2025 Prelims exam. For each wrong answer, one-third marks will be deducted in the Prelims exam. In the Mains exam, there is no negative marking. 

The University of Illinois at Chicago secures a global rank #298 as part of the latest Global University Rankings, rolled out by Shiksha. Another prominent university ranking platform like QS, ranks it #334 as part of its own variation of Global Rankings for the year 2026.