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Total No. of Questions : 5
Total No. of Printed Pages : 4
EX-69
B.Tech. 1st Semester (CSE, IT & Electronics)
Examination, 2022-23
Engeening Mathematics-I
Paper - BE - 101
Time : 3 Hours
[Maximum Marks : 60]
Note : - Attempt all questions. All question carry equal marks.
Attempt any two from each questions.
1. (a)
Explain the concept of Evolutes and involutes with suit-able examples.
(b)
Prove that
β(m,n) = ∫01 xm-1 (1-x)n-1 dx = Γ(m)Γ(n) / Γ(m+n)
(c)
Test the convergence of
∫0∞ e-x dx
2. (a)
Find the stationary points for finding maxima and minima of the function f(x, y) = sin x. siny. sin (x + y).
(b)
Find Maclaurin series expansion for
x / &sqrt;(1-x²)
(c)
Use L'hospital's rule to find the limit of -
limz→0 (z² + e1/z) / (2z - e1/z)
3. (a)
Find the fourier sine series for the function
f(x) = ex for 0 < x < π
(b)
Test for the convergence of the series
1 / (1.2.3) + 1 / (2.3.4) + ....
(c)
Find the taylor series for
f(x) = 1/x² about x = -1
4. (a)
Prove that the set of all vectors in a plane over the field of real numbers is a vector space with respect to vector addition and scalar multiplication.
(b)
Find whether the set of vectors (2, 3, 1), (-1, 4, -2), (1, 18, -4) is linearly independent or not in R³.
(c)
Show that the "projection mapping" f : R³ → R³ into the xy - plane given by f (x, y, z) = (x, y, 0) is linear.
5. (a)
Determine the values of K such that the rank of the matrix A is 3 where -
A = ||1|K|9 |1|2|9 |-1|2|K |0|2|3||
(b)
Find the eigen values and corresponding eigen vectors of the matrix.
(c)
Find the eigen values of matrix
A = ||1|2 |4|3||
and verify Cayley Hamilton theorem for matrix A.