Mathematics - I (Calculus, Multivariable Calculus & Linear Algebra) - B.Tech. 1st Semester Examination, 2024 (Old)

Show that \( f(x)=\sin^{2}x \) is continuous for every value of x.

Verify Rolle's Theorem for the function \( f(x)=2+(x-1)^{2/3} \) where \( x\in[0,2] \).

Test the convergence of the integral \( \int_{0}^{\infty}\frac{\cos x}{1+x^{2}}dx \).

Evaluate \( \Gamma(-\frac{5}{2}) \).

Define Power series and Taylor's series.

Find the smallest positive period of the function \( \sin(\frac{n\pi x}{L}) \).

Are the following set of vectors \( [1, -1, 1], [1, 1, -1], [0, 1, 0] \) linearly independent or dependent?

Define rank and nullity of a matrix.

Find the spectral radius of the matrix \( A=\begin{bmatrix}0&1\\ -1&0\end{bmatrix} \).

Show that the determinant of an orthogonal matrix has the value +1 or -1.

Verify Lagrange's mean value theorem for \( f(x)=(x-1)(x-2)(x-3) \) in \( [0,4] \).

Find the maxima and minima values of the function \( f(x)=5x^{6}+18x^{5}+15x^{4}-10 \).

Find the evolute of the parabola \( y^{2}=4ax \).

Evaluate the integral \( \int_{0}^{\pi/2}\sqrt{\tan \theta}d\theta \).

Find the radius of convergence of the series \( \sum_{m=0}^{\infty}(m+1)m~x^{m} \).

Expand the function \( f(x)=2x^{3}+7x^{2}+x-6 \) in powers of \( (x-2) \).

Find the half range sine Fourier series of the periodic function \( f(x)=\pi-x, \quad 0 < x < \pi \).

Solve the system of equations by Cramer's rule:
\( -x_{1}+x_{2}+2x_{3}=2 \)
\( 3x_{1}-x_{2}+x_{3}=6 \)
\( -x_{1}+3x_{2}+4x_{3} =4 \)

Find the inverse of the matrix \( \begin{bmatrix}3&-1&5\\ 2&6&4\\ 5&5&9\end{bmatrix} \) by Gauss-Jordan Elimination.

Can we say all vectors in \( R^{3} \) such that \( 4v_{2}+v_{3}=k \) form a vector space? If yes, determine the dimension and find a basis.

Find the nullity of the matrix \( \begin{bmatrix}4&0&2&8\\ 5&7&3&1\\ 0&6&9&0\end{bmatrix} \).

Find a basis of eigenvectors of the matrix \( A=\begin{bmatrix}2&1\\ 2&1\end{bmatrix} \).

Find the eigenvalues and eigenvectors of the matrix \( A=\begin{bmatrix}5&4\\ 1&2\end{bmatrix} \).

Diagonalize the matrix \( A=\begin{bmatrix}-2&2&-3\\ 2&1&-6\\ -1&-2&0\end{bmatrix} \).

Mean value theorem

Beta Function

Vector Space

Gauss Elimination