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Since a is a odd natural number then $\left|{\displaystyle \underset{1}{\overset{3}{\int }}{y}^{a}dy}\right|=\frac{364}{3}\Rightarrow \left|{\left(\frac{y^{a+1}}{a+1}\right)}_1^3\right|=\frac{364}{3}\Rightarrow \frac{3^{a+1}}{a+1}=\frac{364}{3}$  

 Þ a = 5

$d=4\sqrt[]{3}$ cm (Lateral shift)

By Snell’s law

${\mu }_{air}sin60°={\mu }_{g}sin\theta$

θ = 30

$\Rightarrow 1\times \frac{\sqrt[]{3}}{2}=\sqrt[]{3}sin\theta$  

$sin\theta =\frac{1}{2}$  

t = 12 cm

The level of difficulty depends on personal calibre of the students and is highly subjective. However, the Electrical Engineering field is known to have a very challenging mathematics section. This is due to its heavy reliance on concepts like:

• Calculus  
• Differential Equations  
• Linear Algebra  
• Complex Numbers and Analysis
• Transforms (Fourier, Laplace)  
• Probability and Statistics

| (A + I) (adj A + I)| = 4 Þ |A adj A + A + Adj A + I| = 4 Þ | (A)I + A + adj A + I|= 4|A| = -1

Þ |A + adj A| = 4

$A=\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right]\Rightarrow adjA=\left[\begin{array}{cc}\hfill a\hfill & \hfill -b\hfill \\ \hfill -c\hfill & \hfill d\hfill \end{array}\right]\Rightarrow \left|\begin{array}{cc}\hfill \left(a+d\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \left(a+d\right)\hfill \end{array}\right|=4\Rightarrow a+d=\pm 2$  

Based on a combination of factors like the depth of knowledge required, the intensity of study, abstractness of concepts, and demanding practical applications, Medicine (specifically an MBBS or MD degree) is frequently cited as the toughest course in the world.

The Hillwoods Academy of Higher Education eligibility is to complete the last qualifying examination from a recognised board or university. Students can look at the table below to know about the eligibility criteria for the various programmes offered at the academy:

The label of the "the easiest" engineering discipline often depends on personal strengths and weaknesses. What one person finds challenging, another might find relatively easy. However, some students find certain engineering branches to be less challenging than others. These engineering branches might include:

• Industrial Engineering
• Agricultural Engineering
• Environmental Engineering

Hillwoods Academy of Higher Education offers three courses in full-time mode at UG and PG levels. These programmes are as follows:

• BCA
• MCA
• PGDM

$\Delta =\left|\begin{array}{ccc}\hfill 8\hfill & \hfill 1\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill \lambda \hfill & \hfill -3\hfill & \hfill 0\hfill \end{array}\right|=12-3\lambda$  

So for  $\lambda$ = 4, it is having infinitely many solutions. ${\Delta }_x=\left|\begin{array}{ccc}\hfill -2\hfill & \hfill 1\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill \mu \hfill & \hfill -3\hfill & \hfill 0\hfill \end{array}\right|$  = -6 - $3\mu =0\Rightarrow -6-3\mu =0$  

For  $\mu =-2$ distance of $\left(4,-2,-\frac{1}{2}\right)$ from 8x + y + 4z + 2= 0 $\left|\frac{32-2-2+2}{\sqrt[]{64+1+16}}\right|=\frac{10}{3}$  units

Yes, Hillwoods Academy of Higher Education admissions are open and interested students can visit the official website of the institute to apply online. Before applying, students can go through the brochure to know about the Hillwoods Academy of Higher Education admission requirements and process.

Yes, meeting the English language requirements is one of the major Karlsruhe Institute of Technology in Germany eligibility criteria for international students. KIT Germany University accepts major English exam scores as proof of English language proficiency. Students can refer to the table below for minimum score requirements:

English Proficiency Exam UG Minimum Score Requirements PG Minimum Score Requirements
IELTS

Overall score of 6.5 or higher

(with no band below 5.5)

$\frac{z_2-0}{\left(1+2i\right)-0}=\left|\frac{OB}{OA}\right|e^{\frac{i\pi }{4}}\Rightarrow z_2=\left(1+2i\right)\left(1+i\right)=-1+3i\therefore arg{z}_2=\pi -ta{n}^{-1}3and\left|z_2\right|=\sqrt[]{10}$

$z_1-2z_2=3-4i\therefore arg\left(z_1-2z_2\right)=-ta{n}^{-1}\frac{4}{3}\Rightarrow \left|z_1-2z_2\right|=\left|2+4i+1-3i\right|=\sqrt[]{10}$

When a candidate chooses to accept and freeze or accept with an upgrade decision for the first time, the applicant will be directed to the portal of the booking fee of INR 15,000 - this is for the general/OBC-NCL/EWS category, and INR 7,500 for the SC/ST/PwD category.

 f (3x)- f (x) = x

Replace  $x\to \frac{x}{3}\Rightarrow f\left(x\right)-f\left(\frac{x}{3}\right)=\frac{x}{3}$  

Again replace  $x\to \frac{x}{3}\Rightarrow f\left(\frac{x}{3}\right)-f\left(\frac{x}{3^2}\right)-f\left(\frac{x}{3^2}\right)=\frac{x}{3^2}$

$\Rightarrow f\left(3x\right)-f\left(0\right)=\frac{3x}{2}puttingx=\frac{8}{3}\Rightarrow f\left(8\right)-f\left(0\right)=4\therefore f\left(0\right)=3$  

Also putting x =  $\frac{14}{3}$ in f (3x) – 3 = $\frac{3x}{2}\Rightarrow$ F (14) – 3 = 7 Þ f (14) = 10

The hardest engineering branches in India involve chemical engineering, electrical engineering, biomedical engineering, aerospace engineering and computer engineering.

As per the student reviews, the top 5 most difficult engineering courses in the world are:

1. Nuclear engineering,
2. Chemical engineering,
3. Aerospace engineering,
4. Biomedical engineering, and
5. Civil engineering.

Yes if your rank 540 you can secure seat in most engineering programs of IIT Palakkad.

Computer Science and Engineering: 6199

- Data Science and Engineering: The cutoff was around 8084.
- Electrical Engineering: The cutoff was about 9766.
- Civil Engineering: The cutoff was as high as 16487.

There are several factors that can affect the relationship between marks and rank in IIT JAM:

Individual Score

Number of candidates applied

Difficulty Level of the Exam

Seat Availability

Category Reservation