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Total No. of Questions : 5
[Total No. of Printed Pages : 4]
EXS-69
B.Tech. Ist Semester (CSE.IT & Elect.)
Examination, 2023
Paper - BE-101
Engg Mathematics-I
Time : 3 Hours
[Maximum Marks : 60]
Note :- Attempt all questions. All question carry equal marks.
Attempt any two parts from each question.
1. (a)
Define Evolutes and Involutes with suitable examples.
2. (a)
Prove that
$\int_0^{\pi/2} \sqrt{\sin \theta} \, d\theta \int_0^{\pi/2} \frac{1}{\sqrt{\sin \theta}} \, d\theta = \pi$
(b)
Prove that
$\int_0^{\pi/2} \sqrt{\tan x} \, dx = \frac{\pi}{\sqrt{2}}$
(c)
Test the convergency of the series :
$1 + \left(\frac{1}{2}\right)^2 + \left(\frac{2}{3}\right)^3 + \left(\frac{3}{4}\right)^4 + \ldots$
3. (a)
Find the Fourier series expansion for the periodic function
$f(x) = x; 0 < x < 2\pi$
(b)
Expand $f(x) = \pi x - x^2$ in a half-range sine series in the interval $(0, \pi)$ upto the first three terms.
(c)
Reduce the matrix A to echelon form and, hence, find its rank, where
$A = \begin{bmatrix} 1 & 2 & 1 & 2 \\ 1 & 3 & 2 & 2 \\ 2 & 4 & 3 & 4 \\ 3 & 7 & 4 & 6 \end{bmatrix}$
4.
$x + y + z = 6,$
$x + 2y + 3z = 14,$
$x + 4y + 7z = 30$
are Consistent and solve them.
(c)
Find the eigenvalues and any one eigenvector of the given matrix A, where
$A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}$
4. (a)
State Lagrange's Mean Value theorem and verify it for the function $f(x) = x^2$ in $(1,5)$.
(b)
Verify Taylor's theorem for $f(x) = (1-x)^{1/2}$ with Lagrange's form of remainder upto 2 terms in the interval $[0,1]$.
(c)
find the Shortest distance from origin too the surface $xyz^2=2$.
5. (a)
Prove that the set of all diagonal matrices over R is a vector space with respect to matrix addition and scalar multiplication.
(b)
Write the vector u = (2, -5, 4) as Linear Combination of vectors V1 = (1, -3, 2), and V2 = (2, -1, 1) in vector space V3(R).
(c)
Show that the function. T: R3 → R3 difined by T(x,y) = (x+1, 2y, x+y) is not a Linear transformation.