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Candidates who have completed Class 12 or equivalent with Physics, Chemistry and Biology or PCM with at least 50% aggregate are eligible to apply for DPharm course at Motherhood University. There is a relaxation of 5% in the aggregate for SC/ ST candidates. Hence, aspirants with 60% in Class 12 can apply for the DPharm programme.

 70. Given, f (x) = $\frac{x^{2}cos\left(\frac{\pi }{4}\right)}{sinx}$

$=\frac{x^2}{sinx}\times cos\frac{\pi }{4}$

So, f? (x) =

$=cos\frac{\pi }{4}\left[\frac{2·xsinx-x^{2}cosx}{si{n}^{2}x}\right]$

$=xcos\frac{\pi }{4}\left[\frac{2sinx-xcosx}{si{n}^{2}x}\right].$

8. 3 (2 – x) ≥ 2 (1 – x)

6 – 3x ≥ 2 – 2x

6– 2 ≥ 3x– 2x

4 ≥ x

So, x ∈ (–∞, 4]

Jharkhand Raksha Shakti University BSc course admission process starts with the release of application forms. Candidates who meet the eligibility requirements can register and apply. The applications are further shortlisted based on merit in the last qualifying exam. Additionally, the CUET scores may also be considered for the selection purposes. Once selection is done, the seat allotment process is done. The candidates can then pay the admission fee amount to lock their seats. 

According to the MCNUJC, Bhopal eligibility criteria, the selection of candidates for the BTech programme is based on their performance in the last qualifying examination. Those who have completed their Class 12 in the relevant subjects from a recognised board are eligible to apply. However, it is essential for candidates to review the eligibility criteria before applying for the programme.

7. 3 (x –1) ≤ 2 (x –3)

3x– 3 ≤ 2x –6

=> 3x– 2x ≤ –6 + 3

x ≤ –3

So, x ∈ (–∞, – 3]

The total tuition fee for MBA at Motherhood University ranges between INR 90,000 and INR 1.5 lakh. This information is sourced from official website/ sanctioning body and is subject to change. The total fee of MBA includes various components such as registration fee, tuition fee, exam fee, etc. Some of the components are one-time payable while others have to paid annually. 

B.Sc admissions in 2025 will generally begin between April and June, varying by state and university. For example, Madhya Pradesh starts on May 15, Goa on April 11, and Patliputra University (Bihar) from June 13. Applications usually open after Class 12 results, with counselling and merit lists released in June–July before classes begin in July–August.

69. Given, f (x) = $\frac{4x+5sinx}{3x+7cosx}$

So,

$f^{\prime }(x)=\frac{(3x+7cosx)\frac{d}{dx}(4x+5sinx)-(4x+5sinx)\frac{d}{dx}(3x+7cosx)}{{(3x+7cosx)}^2}$

$=\frac{(3x+7cosx)(4x+5cosx)-(4x+5sinx)(3-7sinx)}{{(3x+7cosx)}^2}$

$\begin{array}{l}\underset{\_}{=x+xx+x+{}^{}x-x+xx-x+35si{n}^{2}x}\\ {\left(3x+7cosx\right)}^2\end{array}$

$=\frac{15xcosx+28cosx+28xsinx-15sinx+35\left(co{s}^{2}x+si{n}^{2}x\right)}{{(3x+7cosx)}^2}$

$=\frac{35+15xcosx+28cosx+28xsinx-15sinx}{{(3x+7cosx)}^2}$

6. Given, 3x – 7 > 5x – 1.

7+1 > 5x – 3x

6 > 2x

$\Rightarrow \frac{-6}{2}>\frac{2x}{2}$

–3 > x

So, x ∈ (–∞, – 3)

Aspirants who have passed CAT or MAT exams with a good score are eligible for direct admission in the MBA programme. The other ways to get admission in MBA is based on qualifying exam scores or appear for entrance test conducted by the university (if any). Besides, aspirants must also fulfill the eligibility requirements for admission to MBA programme.

For the students belonging to the General AI quota, the NEET SS Round 1 cutoff for DM in Paediatric Hepatology admission at ILBS Delhi was 126. This is the cutoff rank for the first round. The cutoff rank varies by rounds, specialisations and categories. 

68. Given, f (x) = (x + cos x) (x tan x)

So, f?(x) = (x + cos x) $\frac{d}{dx}-tanx)+\frac{d}{dx}(x+cosx)·(x-tanx)$

$=(x+cosx)\left(\frac{dx}{dx}-\frac{d}{dx}tanx\right)+\left(\frac{d}{dx}+\frac{dcosx}{dx}\right)(x-tanx)$

Let g (x) = tan x.

So, g?(x) =

$=\underset{h\to 0}{lim}\frac{1}{h}\left[\frac{sin(x+h)}{cos(x+h)}-\frac{sinx}{cosx}\right]$

$=\underset{h\to 0}{lim}\frac{1}{h}\left[\frac{sin(x+h)cosx-sinxcos(x+h)}{cos(x+h)·cosx}\right]$

$=\underset{h\to 0}{lim}\frac{1}{h}\left[\frac{sin(x+h-x)}{cosx·cos(x+h)}\right]$

$=\underset{h\to 0}{lim}\frac{1}{cosx·co}$$\frac{}{}$$\frac{s(x+h)}{}\times \underset{h\to 0}{lim}\frac{sinh}{h}$

$=\frac{1}{co{s}^{2}x}\times 1$

= sec²x ______ (2)

     Put (2) in (1) we get,

     f?(x) = (x + cos x) (1 - sec²x) +(1 - sin x) (x- tan x)

     We know that,

     1 + tan²x = sec²x

     Þ 1 - sec²x = - tan²x

     So, f?(x) = - tan²x(x + cos x) +(x- tan x) (1 - sin x).

Jharkhand Raksha Shakti University BSc admission dates may get published on the official website. Candidates seeking admission must check official website and reach out to the admission office for updates. Moreover, CUET scores may get accepted for admission. The CUET entrance exam was held from May 13 to Jun 3.

Candidates who have completed a bachelor's degree with a minimum of 50% aggregate (45% aggregate for SC/ ST) are eligible to apply for MBA at Motherhood University. Aspirants must hold the documents proving their eligibility for verification at the time of admission. Candidates can get admission in MBA based on their merit or entrance exam scores.

5. Given,

4 x + 3 < 5 x+ 7.

3 – 7 < 5 x – 4 x.

–4 < x.

So, x ( –4, ∞)

Yes, there are about 6 distance BA colleges in Kolkata. Some of them are mentioned below along with their tuition fees:

Disclaimer: This information is sourced from official website.

For Andhra Pradesh ICET 2025, 37,572 candidates applied for the exam out of which 34,131 students appeared for the exam on May 07, 2025. As per the Andhra Pradesh ICET results 2025, a total of 32,719 candidates qualified the exam. Andhra Pradesh ICET result 2025 was announced on May 20, 2025.

4. Given, 3x + 8 > 2

3x + 8 – 8 > 2 – 8

3x > – 6

$\Rightarrow \frac{3x}{3}>-\frac{6}{3}$

x > – 2

(i) when x is an integer, x ∈ z all integer greater than – 2 are the soln. so, x = { – 1, 0, 1, 2, 3, …}.

(ii) when x is a real number, all real number greater than – 2 are the soln. so, x ( –2, ∞  )

Students pursuing a course from Motherhood University can avail the scholarships at Motherhood University. The university offers merit-based scholarships to students. This scholarship is offered to students based on their qualifying exam scores. Below is the detailed criteria: