$$a=\frac{GM}{r^2}$$
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Friday, November 20, 2015
Thursday, June 14, 2012
Application of the Cayley-Hamilton Theorem
Cayley-Hamilton Theorem, a basic theorem from Linear Algebra, is not a part of JEE syllabus. But this simple result comes in handy in JEE often.
According to the Cayley-Hamilton Theorem, every square matrix A satisfies its own Characteristic Equation.
For a matrix A of size n X n, the Characteristic Equation in variable is defined as,
,
where I is the identity matrix of size n X n.
So according to the Cayley-Hamilton Theorem, we have,
This can be illustrated with an example.
Consider,
Then,According to the Cayley-Hamilton Theorem, every square matrix A satisfies its own Characteristic Equation.
For a matrix A of size n X n, the Characteristic Equation in variable is defined as,
,
where I is the identity matrix of size n X n.
So according to the Cayley-Hamilton Theorem, we have,
This can be illustrated with an example.
Consider,
From the Cayley-Hamilton Theorem,
Problem 1
Let
Suppose
(IIT-JEE 2005)
Solution 1
The given equation gives,
By applying the Cayley-Hamilton Theorem on A, we have,
Direct comparison of (1) and (2) gives c=-6 and d=11
Thursday, June 7, 2012
Differentiation Under The Integral Sign
Problems related to differentiation under the integral sign are very common in IIT-JEE. We will see how to solve such problems using Leibniz's Rule given below.
Problem 1
If f(x) is differentiable and ,
find f(4/25)
(IIT-JEE 2004)
Solution
Differentiating both sides,
Hence, f(4/25)=2/5 for t=2/5
PS: Note that -2/5 is also a valid answer. It was not among the options for the MCQ.
Problem 2
Solve
This is in 0/0 form. So we can apply L'Hospital's Rule directly. Differentiating the numerator and the denominator separately, we have,
Problem 1
If f(x) is differentiable and ,
find f(4/25)
(IIT-JEE 2004)
Solution
Differentiating both sides,
Hence, f(4/25)=2/5 for t=2/5
PS: Note that -2/5 is also a valid answer. It was not among the options for the MCQ.
Problem 2
Solve
(IIT-JEE 2007)
Solution This is in 0/0 form. So we can apply L'Hospital's Rule directly. Differentiating the numerator and the denominator separately, we have,
Wednesday, June 6, 2012
Hello World :D
Hello !!!
I completed my Dual Degree (B Tech & M Tech) in Electrical Engineering from Indian Institute of Technology Madras aka IIT Madras in 2012.
I will be using this blog to help IIT-JEE aspirants with some tips.
You can contact me at : iqbal1729(at)gmail(dot)com.
I completed my Dual Degree (B Tech & M Tech) in Electrical Engineering from Indian Institute of Technology Madras aka IIT Madras in 2012.
I will be using this blog to help IIT-JEE aspirants with some tips.
You can contact me at : iqbal1729(at)gmail(dot)com.
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