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This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l}Letf\left(x\right)=\sqrt[]{a^2-x^2}\\ f\left(x\right)isdefinedif{a}^2-x^2\ge 0\\ \Rightarrow x^2-a^2\le 0\Rightarrow x^2\le a^2\\ \Rightarrow x\le \pm a\Rightarrow -a\le x\le a\\ \therefore Domainoff\left(x\right)=\left[-a,a\right]\\ Hence,thecorrectoptionis\left(b\right).\end{array}$

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l}Giventhat:f\left(x\right)=\sqrt[]{1+x^2}\Rightarrow f\left(xy\right)=\sqrt[]{1+x^2{y}^2}\\ andf\left(x\right).f\left(y\right)=\sqrt[]{1+x^2}.\sqrt[]{1+y^2}=\sqrt[]{1+x^2+y^2+x^2{y}^2}\\ ?\sqrt[]{1+x^2{y}^2}\le \sqrt[]{1+x^2+y^2+x^2{y}^2}\\ \Rightarrow f\left(xy\right)\le f\left(x\right).f\left(y\right)\\ Hence,thecorrectoptionis\left(c\right).\end{array}$

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat:acos2\theta +bsin2\theta =c\\ \Rightarrow a\left[\frac{1-ta{n}^2\theta }{1+ta{n}^2\theta }\right]+b\left[\frac{2tan\theta }{1+ta{n}^2\theta }\right]=c\\ \Rightarrow a-ata{n}^2\theta +2btan\theta =c\left(1+ta{n}^2\theta \right)\left[?cos2\theta =\frac{1-ta{n}^2\theta }{1+ta{n}^2\theta },sin2\theta =\frac{2tan\theta }{1+ta{n}^2\theta }\right]\\ \Rightarrow a-ata{n}^2\theta +2btan\theta -cta{n}^2\theta -c=0\\ \Rightarrow -\left(a+c\right)ta{n}^2\theta +2btan\theta +\left(a-c\right)=0\\ \Rightarrow \left(a+c\right)ta{n}^2\theta -2btan\theta +\left(c-a\right)=0\\ Since\alpha and\beta aretherootsofthisequation\\ \Rightarrow tan\alpha +tan\beta =\frac{-\left(-2b\right)}{a+c}\\ \Rightarrow tan\alpha +tan\beta =\frac{2b}{a+c}.Henceproved.\end{array}$

The admission steps for SOU Devibai Narayandas Chhabada Rural Education Society are as follows: 

Step 1: Fill out the application form on the official website and also fill the form for the required entrance. 

Step 2: Candidates are shortlisted based on their merit or entrance cutoff. 

Step 3: Candidates who get the final admission offer are required to pay the course fee to secure their seat. 

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l}Giventhat:f\left(x\right)=\frac{1}{1-2cosx}\\ Weknowthat-1\le cosx\le 1\\ \Rightarrow 1\ge cosx\ge -1\Rightarrow -1\le -cosx\le 1\\ \Rightarrow -2\le -2cosx\le 2\Rightarrow -2+1\le 1-2cosx\le 2+1\\ \Rightarrow -1\le 1-2cosx\le 3\Rightarrow -1\le \frac{1}{1-2cosx}\le \frac{1}{3}\\ \Rightarrow -1\le f\left(x\right)\le \frac{1}{3}\\ So,therangeoff\left(x\right)=\left[-1,\frac{1}{3}\right]\\ Hence,thecorrectoptionis\left(b\right).\end{array}$

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

          $\begin{array}{l}Giventhat:\\ 3\left[si{n}^4\left(\frac{3\pi }{2}-\alpha \right)+si{n}^4\left(3\pi +\alpha \right)\right]-2\left[si{n}^6\left(\frac{\pi }{2}+\alpha \right)+si{n}^6\left(5\pi -\alpha \right)\right]\\ =3\left[co{s}^4\alpha +si{n}^4\left(\pi +\alpha \right)\right]-2\left[co{s}^6\alpha +si{n}^6\left(\pi -\alpha \right)\right]\\ =3\left[co{s}^4\alpha +si{n}^4\alpha \right]-2\left[co{s}^6\alpha +si{n}^6\alpha \right]\\ =3\left[co{s}^4\alpha +si{n}^4\alpha +2si{n}^2\alpha co{s}^2\alpha -2si{n}^2\alpha co{s}^2\alpha \right]-2\left[\begin{array}{l}{\left(co{s}^2\alpha +si{n}^2\alpha \right)}^3\\ -3co{s}^2\alpha si{n}^2\alpha \left(co{s}^2\alpha +si{n}^2\alpha \right)\end{array}\right]\\ =3\left[{\left(co{s}^2\alpha +si{n}^2\alpha \right)}^2-2si{n}^2\alpha co{s}^2\alpha \right]-2\left[1-3co{s}^2\alpha si{n}^2\alpha \right]\\ =3\left[1-2si{n}^2\alpha co{s}^2\alpha \right]-2\left[1-3co{s}^2\alpha si{n}^2\alpha \right]\\ =3-6si{n}^2\alpha co{s}^2\alpha -2+6co{s}^2\alpha si{n}^2\alpha \\ =3-2=1\\ Hence,thevalueofthegivenexpressionis1.\end{array}$

The branch-wise number of students placed during IIT Palakkad M.Tech placements 2024 is presented below:

The key highlights of IIT Palakkad M.Tech placements 2024 in terms of placement rate is presented below:

In addition, the branch-wise salary package offered during IIT Palakkad M.Tech placements 2024 is presented below:

The branch-wise students placed during IIT Palakkad BTech placements 2024 is presented below:

The branch-wise placement rate recorded during IIT Palakkad BTech placements 2024 is presented below:

In addition, the branch-wise salary package offered during IIT Palakkad BTech placements 2024 is presented below:

The participating recruiters presented 1,486 job offers to the students of the BTech batch 2025. The total number of offers made during DTU placements between 2022 and 2024 are presented below:

SPACE Common Entrance Test (SPACE-CET) is an online or home-based entrance exam used by IIAEIT for admissions and scholarship distribution. It is conducted twice yearly mainly in April and July. It supports entry into BTech (Aerospace, Aeronautical, Avionics) and M.Tech programms. 

Yes, a candidate can apply for admission in M.Tech at Indian Institute for Aeronautical Engineering and information Technology even if the graduation results are pending. The institute can grant them provisional admission in this case. However, it is mandatory to submit the graduation marksheet / degree to the institute once received, then only the admission can be confirmed. 

Indian Institute for Aeronautical Engineering and information Technology (IIAEIT) offers the dual degree course to candidates for a duration of five years. This course includes BTech + M.Tech degree. The selection criteria for this course requires the candidates to appear for AERO-CET.  Interested candidates can apply online on the official website or contact the admission office of IIAEIT. 

The difference between a Traditional University & a Metropolitan University is mentioned below:

The fee range for the certificate programme at Gamics Institute ranges between INR 80,000-INR 3 lakh. The fee structure is sourced from unofficial sources, hence indicative. The fee cover various academic expenses like tuition fees, laboratory charges, examination fees, library fees, and other relevant charges. It is important to note that the fee structure is subject to change, and candidates are advised to refer to the official website of the university for the latest and most accurate fee structure.