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14. Given, 37 – (3x + 5) ≥9x –8 (x –3)

37– 3x–5 ≥9x –8x+ 24.

37 – 24–5 ≥9x –8x+ 3x

8 ≥4x.

$\Rightarrow \frac{8}{4}\ge x$

2 ≥x

So, x∈ (–∞, 2]

13. Given 2 (2x + 3) –10 < 6 (x –2)

4x + 6 –10 < 6x– 12.

4x –4 < 6x –12

12 –4 < 6x –4x

8 < 2x

$\Rightarrow \frac{8}{2}<\frac{2x}{2}\Rightarrow 4<x$

So, x∈ (4, –∞)

St. Clair College placement rate stands at 84.4%, and the graduates are employed within 6 months of graduation. St. Clair College placement rate attracts students from all over the world. The College graduates are placed in top sectors such as Operations, Business Development, Sales, Marketing, and more.

There are more than 100 seats for the BSc/ BSc (H) courses offered by Jharkhand Raksha Shakti University. For both the BSc (H) programmes, there are 50 seats each. Students are admitted as per the sanctioned seat intake. This seat intake is as per the official website/ sanctioning body. It is still subject to changes and hence, is indicative. 

Yes some UK based institutions are known to offer free courses specially designed for international students, graduates and alumni.

The UK Government, Study UK (British Council) & the Open University are among some popualar institutions known for offering free courses that are mostly available to be pursued in online formats.

These courses help students attain basic skills needed to land a job & can be pursued from other countries as well.

For more information, students are advised to refer the pages of the institutes mentioned above.

12. Given, $\frac{1}{2}$ $\left(\frac{3x}{5}+4\right)\ge \frac{1}{3}(x-6)$

$\frac{6}{2}\left(\frac{3x}{5}+4\right)\ge \frac{6}{3}(x-6)$

$\Rightarrow 3\left(\frac{3x}{5}+4\right)\ge 2(x-6)$

$\Rightarrow \frac{9}{5}x+12\ge 2x-12$

$\Rightarrow \frac{9}{5}x-2x\ge -12-12.$

$\Rightarrow \frac{9x-10x}{5}\ge -24.$

$\Rightarrow \frac{-x\times 5\ge }{5}-24\times 5.$

x ≥–120.

(–1) x (–x)≤ ( –120)× (–1)

x ≤ 120.

So, x ∈ (–∞, 120].

The total tuition fee for BA at Motherhood University ranges between INR 34,500 and INR 94,000. This information is sourced from official website/ sanctioning body and is subject to change. Candidates who are selected for admission need to pay the course fee to confirm their seat. The total fee of BA includes various components such as tuition fee, exam fee, etc.,

73. Given, f (x) = $\frac{x}{si{n}^{n}x}$

So, f?(x) = $\frac{\left(si{n}^{x}x\frac{dx}{dx}\right)-\left(x\frac{d}{dx}si{n}^{x}x\right)}{{\left(si{n}^{x}x\right)}^2}$

= $\frac{si{n}^{x}x-x·n{(sinx)}^{x-1}\frac{d}{dx}sinx}{{(sinx)}^{2x}}$

$=\frac{{(sinx)}^n-x·x{(sinx)}^{n-1}·cosx}{{(sinx)}^{2n}}$

$=\frac{{(sinx)}^{n-1}[sinx-nx·cosx]}{{(sinx)}^{2n}}$

$=\frac{sinx-n·x·cosx}{{(sinx)}^{2n-(n-1)}}$

$=\frac{sinx-nxcosx}{{(sinx)}^{n+1}}$

The applicants of BSc at JRSU have to submit/ upload the important documents. These are required to verify that the applicant is eligible for the course admission. Students shall keep them handy while filling out the application and during counselling. Some of the important documents are mentioned below:

• Class 10 and Class 12 mark sheet
• Aadhaar card or other identity proof
• Migration certificate
• Category certificate (if applicable), etc.

Yes, Motherhood University offers BA for three-years at the UG level. The course is spread across six semesters. Candidates who have completed Class 12/ intermediate in any discipline with a minimum of 50% aggregate are eligible to apply for the BA programme. The selection for this course is merit-based at Motherhood University.

The Madras Medical Mission NEET SS cutoff 2024 was released for admission to various DNB course branches. In the last round for the General AI category, the cutoff ranged from 738 to 4568, wherein DNB in Critical Care Medicine was the most sought-after course for admission. 

On the other hand, DNB in Nephrology came out as the least competitive branch, with the cutoff being 4568. General AI category students can refer to the table below to know the round 1 and last round cutoff for various courses.

Please note that other rounds, categories, courses, etc, will have different cutoffs.

11. Given, $\frac{3\left(x-2\right)5}{5}\le \frac{5\left(2-x\right)}{3}$

=> 3 x 3 (x – 2) ≤ 5 x 5 (2 – x)

=> 9x – 18 ≤ 50 – 25x

=> 9x + 25x ≤ 50 + 18

=> 34x ≤ 68

=> $x=\frac{68}{34}$

=> x ≤ 2

So,  $x\in (-\infty ,2]$

The total tuition fee for DPharm at Motherhood University is INR 2 lakh. This information is sourced from official website/ sanctioning body and is subject to change. Candidates have to pay some part of the fee at the time of admission to confirm their seat. Besides, aspirants also need to pay a registration fee of INR 1,500 to apply for DPharm course.

72. Given, f (x) = (x + sec x) (x tan x)

So, f?(x) = (x + see x) $\frac{d}{dx}(x-tanx)+(x-tanx)\frac{d}{dx}(x+secx).$

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

Let g(x) = tan x.

So, g?(x) =

$=\underset{h\to 0}{lim}\frac{1}{h}[tan(x+h)-tanx]$

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

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

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

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

$=\frac{1}{co{s}^{2}x}=se{c}^{2}x$ . _____ (2)

And P(x) = see x.

$=\underset{h\to 0}{lim}\frac{sec(x+h)-secx}{h}$

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

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

$=\underset{h\to 0}{lim}\frac{1}{h}\left[\frac{-2sin\left(\frac{x+x+h}{2}\right)sin\left(\frac{x-(x+h)}{2}\right)}{cos(x+h)cosx}\right]$

$=\frac{sinx}{co{s}^{2}x}$ = tan x sec x. ____ (3)

   Putting (2) and (3) in (1) we get,

   f?(x) = (x + sec x) (1 - sec²x) + (x- tan x) (1 + tan x sec x)

Candidates who meet the eligibility requirements can proceed with application procedure. For information regarding offline application, students shall contact the admission office. Aspirants shall ensure that the details provided in the application are correct and updated. The step-by-step JRSU BSc application procedure is explained below:

1. Go to the official website.
2. Click on the "Apply Now or Application" widget.
3. Fill out details and upload documents.
4. Pay application fees and submit the form.

NOTE: The above-mentioned is a generic process. Hence, is indicative.

Yes, the age of the candidates applying for DPharm programme should be 17 years on or before December 31 of the admission year. Besides, aspirants must also meet the qualification requirements. Candidates need to complete Class 12 or equivalent with PCB/ PCM with at least 50% aggregate (45% aggregate for SC/ ST) to apply for the DPharm programme.

71. Given, f (x) = $\frac{x}{1+tanx}$

$=\frac{x}{1+\frac{sinx}{cosx}}$

$=\frac{cosx·x}{cosx+sinx}$

$So,f^{\prime }(x)=\frac{(cosx+sinx)\frac{d}{dx}(x\cdot cosx)-\left(x\cdot cosx\right)\frac{d}{dx}\left(cosx+sinx\right)}{{(cosx+sinx)}^2}$

$=\frac{(cosx+sinx)(-xsinx+cosx)-\left(-x·sinxcosx+xco{s}^{2}x\right)}{{(cosx+sinx)}^2}$

$=\frac{-xcosxsinx+co{s}^{2}x-xsi{n}^{2}x+sinxcosx+xsinxcosx-xco{s}^{2}x}{{(cosx+sinx)}^2}$

$=\frac{co{s}^{2}x+sinxcosx-x\left(si{n}^{2}x+co{s}^{2}x\right)}{{(cosx+csinx)}^2}$

Dividing numerator and denominator by cos2x we get,

$=\frac{\frac{co{s}^{2}x}{co{s}^{2}x}+\frac{sinx}{cosx}·\frac{cosx}{cosx}-x\frac{1}{co{s}^{2}x}}{{\left(\frac{cosx}{cosx}+\frac{sinx}{cosx}\right)}^2}$

$=\frac{1+tanx-xse{c}^{2}x}{{(1+tanx)}^2}$

9. x + $\frac{x}{2}+\frac{x}{3}$ < 11

$\Rightarrow \frac{6x+3x+2x}{6}<11$

$\Rightarrow \frac{(6x+3x+2x)\times 6}{6}<11\times 6$

11x < 66

$\frac{11x}{11}<\frac{66}{11}$

x < 6.

So, x ∈ (–∞, 6)

St. Clair College is among the top colleges in Canada. St. Clair College shapes its students in such a way that they excel in every field. St. Clair College placement rate stands at 84.4%, and the graduates are employed within 6 months of graduation. St. Clair College graduate satisfaction rate is 80%. St. Clair College graduates work for top sectors that are listed below:

• Operations
• Business Development
• Sales
• Customer Success and Support
• Information Technology
• Engineering
• Healthcare Services
• Education
• Arts and Design
• Community and Social Services
• Human Resources
• Finance
• Media and Communication
• Marketing
• Program and Project Management