Mathematics—I (Calculus and Linear Algebra)

If $ Y = \int_0^a t^4 e^{-2t^2} dt $, then the value of $ Y $ is

The maximum area of a sector whose perimeter is $ k $ is given by

The function $ f(x) = \begin{cases} x^2, & 0 \leq x \leq \frac{1}{2} \\ x, & \frac{1}{2} < x \leq 1 \end{cases} $, then $ \lim_{x \to \frac{1}{2}} f(x) $

If $ \alpha, \beta $ are the roots of the equation $ ax^2 + bx + c = 0 $, then $ \lim_{x \to \alpha} \frac{1 - \cos(\alpha x^2 + bx + c)}{(x - \alpha)^2} $ is equal to

The series $ x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ... $ converges for

If $ u_n = \sqrt{n+1} - \sqrt{n}, \quad v_n = \sqrt{n^4 + 1} - n^2 $, then

$ \lim_{(x, y) \to (0, 0)} \frac{1 - x - y}{x^2 + y^2} $ is

The gradient of the function $ f(x, y, z) = \log (x^2 + y^2 + z^2) $ at $ (3, -4, 5) $ is

If $ A $ and $ B $ are any two square matrices of order $ 2 \times 2 $, then $ (A + B)^2 $ is equal to

The system of equations $ x - y + 3z = 4, x + z = 2, x + y - z = 0 $ has

Evaluate $ \int_{0}^{\infty} x^{m} e^{-ax^n} dx $.

Find the surface of the solid formed by the revolution about the axis of $ y $, of the part of the curve $ ay^2 = x^3 $ from $ x = 0 $ to $ x = 4a $, which is above the $ x $-axis.

Verify the Rolle's theorem for the function $ f(x) = 2x^3 + x^2 - 4x - 2 $.

Find $ \lim_{x \to \frac{\pi}{2}} (\sin x)^{\tan x} $.

Discuss the convergence of the sequence whose $ n $ th term is $ a_n = \frac{1}{\log n} $.

Test the convergence of the following series : $ 1 + \frac{x}{1} + \frac{1}{2} \cdot \frac{x^3}{3} + \frac{1 \cdot 3 \cdot x^5}{2 \cdot 4 \cdot 5} + \frac{1 \cdot 3 \cdot 5 \cdot x^7}{2 \cdot 4 \cdot 5 \cdot 6 \cdot 7} + \cdots $

Find the Fourier series expansion of the function $ f(x) = \{ \pi + x \}, -\pi \leq x \leq \pi $. Hence deduce that $ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots $

Find the Fourier cosine series and Fourier sine series of the following function in given interval : $ f(x) = \{ x + x^2 \}, 0 < x < 1 $

Discuss continuity of the following function at the point $ (-1, c) $ : $ f(x, y) = \begin{cases} \frac{x^2 y}{(1 + x)}, & x \neq -1 \\ y, & (x, y) = (-1, c) \end{cases} $

Find the extreme value of $ xyz $, when $ x + y + z = a, a > 0 $.

Find the maximum value of the function $ f(x) = \frac{1}{x} $.

Find the equation of the tangent plane to the surface $ xy + yz + zx = -1 $, at the point $ (1, -1, 2) $.

Find the eigenvalues and eigenvectors of the following matrix : $ \begin{bmatrix} -2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{bmatrix} $

Verify Cayley-Hamilton theorem for the matrix $ \begin{bmatrix} 0 & c & b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix} $

Determine the range of the following linear transformation. Also find the rank of $ T $, where it exists, $ T: V_3 \rightarrow V_3 $, defined by $ T(x_1, x_2, x_3) = \left( \frac{1}{2} x_1 + x_2 + x_3, \, x_1 - \frac{1}{3} x_2, \, x_3 \right) $

If $ Y = \int_0^a t^4 e^{-2t^2} dt $, then the value of $ Y $ is

The maximum area of a sector whose perimeter is $ k $ is given by

The function $ f(x) = \begin{cases} x^2, & 0 \leq x \leq \frac{1}{2} \\ x, & \frac{1}{2} < x \leq 1 \end{cases} $, then $ \lim_{x \to \frac{1}{2}} f(x) $

If $ \alpha, \beta $ are the roots of the equation $ ax^2 + bx + c = 0 $, then $ \lim_{x \to \alpha} \frac{1 - \cos(\alpha x^2 + bx + c)}{(x - \alpha)^2} $ is equal to

The series $ x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ... $ converges for

If $ u_n = \sqrt{n+1} - \sqrt{n}, \quad v_n = \sqrt{n^4 + 1} - n^2 $, then

$ \lim_{(x, y) \to (0, 0)} \frac{1 - x - y}{x^2 + y^2} $ is

The gradient of the function $ f(x, y, z) = \log (x^2 + y^2 + z^2) $ at $ (3, -4, 5) $ is

If $ A $ and $ B $ are any two square matrices of order $ 2 \times 2 $, then $ (A + B)^2 $ is equal to

The system of equations $ x - y + 3z = 4, x + z = 2, x + y - z = 0 $ has

Evaluate $ \int_{0}^{\infty} x^{m} e^{-ax^n} dx $.

Find the surface of the solid formed by the revolution about the axis of $ y $, of the part of the curve $ ay^2 = x^3 $ from $ x = 0 $ to $ x = 4a $, which is above the $ x $-axis.

Verify the Rolle's theorem for the function $ f(x) = 2x^3 + x^2 - 4x - 2 $.

Find $ \lim_{x \to \frac{\pi}{2}} (\sin x)^{\tan x} $.

Discuss the convergence of the sequence whose $ n $ th term is $ a_n = \frac{1}{\log n} $.

Test the convergence of the following series : $ 1 + \frac{x}{1} + \frac{1}{2} \cdot \frac{x^3}{3} + \frac{1 \cdot 3 \cdot x^5}{2 \cdot 4 \cdot 5} + \frac{1 \cdot 3 \cdot 5 \cdot x^7}{2 \cdot 4 \cdot 5 \cdot 6 \cdot 7} + \cdots $

Find the Fourier series expansion of the function $ f(x) = \{ \pi + x \}, -\pi \leq x \leq \pi $. Hence deduce that $ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots $

Find the Fourier cosine series and Fourier sine series of the following function in given interval : $ f(x) = \{ x + x^2 \}, 0 < x < 1 $

Discuss continuity of the following function at the point $ (-1, c) $ : $ f(x, y) = \begin{cases} \frac{x^2 y}{(1 + x)}, & x \neq -1 \\ y, & (x, y) = (-1, c) \end{cases} $

Find the extreme value of $ xyz $, when $ x + y + z = a, a > 0 $.

Find the maximum value of the function $ f(x) = \frac{1}{x} $.

Find the equation of the tangent plane to the surface $ xy + yz + zx = -1 $, at the point $ (1, -1, 2) $.

Find the eigenvalues and eigenvectors of the following matrix : $ \begin{bmatrix} -2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{bmatrix} $

Verify Cayley-Hamilton theorem for the matrix $ \begin{bmatrix} 0 & c & b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix} $

Determine the range of the following linear transformation. Also find the rank of $ T $, where it exists, $ T: V_3 \rightarrow V_3 $, defined by $ T(x_1, x_2, x_3) = \left( \frac{1}{2} x_1 + x_2 + x_3, \, x_1 - \frac{1}{3} x_2, \, x_3 \right) $

If $ Y = \int_0^{\infty} \frac{x^a}{a^x} \, dx, \, a > 1 $, then the value of $ Y $ is

The area bounded by the axis of $ x $, and the curve and ordinates $ y = \cosh \frac{x}{c} $ from $ x = 0 $ to $ x = a $ is

Consider the following functions :
1. $ y = x \sin \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 $
2. $ y = x^2 \sin \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 $
3. $ y = x^2 \cos \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 $
4. $ y = x \cos \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 $
The functions, differentiable at $ x = 0 $, are

For a positive term series $ \sum a_n $, the ratio test states that

If $ \lim_{x \to \infty} \frac{\sin 2x + a \sin x}{x^3} = b $ where $ b $ is finite, then the values of $ a $ and $ b $ respectively will be

The expansion of $ \tan x $ in powers of $ x $ by Maclaurin's theorem is valid in the interval

The value of $ \lim_{(x, y) \to (k, 0)} \left(1 + \frac{x}{y}\right)^y $ is

The gradient of the function $ f(x, y, z) = \sin(xyz) $, at $ (1, -1, \pi) $, is

If det(A) = 7, where $ A = \begin{bmatrix} a & b & c \\ 1 & 1 & g \\ g & \omega & 1 \end{bmatrix} $, then det(2A)^{-1} is equal to

If $ 3x + 2y + z = 0 $, $ x + 4y + z = 0 $ and $ 2x + y + 4z = 0 $ be a system of equations, then

Evaluate $ \int_{0}^{\infty} \log \left( x + \frac{1}{x} \right) \frac{dx}{1 + x^2} $

Find the volume of the solid generated by rotating completely about the x-axis where the area enclosed between $ y^2 = x^3 + 5x $ and the line $ x = 2 $ and $ x = 4 $ about its major axis.

Find the maximum value of the function $ f(x) = \frac{x}{1 + x \tan x} $

It is given that Rolle's theorem holds for the function $ f(x) = x^3 + bx^2 + cx $, $ 1 \leq x \leq 2 $ at the point $ x = \frac{4}{3} $. Find the values of b and c.

Discuss the convergence of the sequence whose n-th term is $ \alpha_n = \frac{(-1)^n}{n} + 1 $

Test the convergence of the following series : $ x^2 + \frac{2^2 x^4}{3.4} + \frac{2^2 4^2 x^6}{3.4.5.6} + \frac{2^2 4^2 6^2 x^8}{3.4.5.6.7.8} ... $

Find the Fourier series expansion of the function $ f(x) = \{x^2, -2 \leq x \leq 2\} $. Hence deduce that $ \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... $

Find the Fourier cosine series and Fourier sine series of the following function in given interval : $ f(x) = \begin{cases} x, & 0 < x < 2 \\ 2, & 2 \leq x < 4 \end{cases} $

Discuss continuity of the following function at the point $ (0, 0) $ : $ f(x, y) = \begin{cases} \frac{x^2 y^2}{\left( x^3 + y^3 \right)} , & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases} $

Find the maximum value of $ xyz $ under the constraints $ x^2 + z^2 = 1 $ and $ y - x = 0 $.

Find the value of $ \lim_{x \to {\infty}} \left( \frac{x+4}{x+2} \right)^{x+3} $

Find the equation of the tangent plane to the surface $ x^2 - 3y^2 - z^2 = 2 $, at the point $ (3, 1, 2) $.

Find the eigenvalues and eigenvectors of the following matrix : $ \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} $

Verify Cayley-Hamilton theorem for the matrix $ \begin{bmatrix} 0 & 0 & 1 \\ 3 & 1 & 0 \\ -2 & 1 & 4 \end{bmatrix} $

Determine the range of the following linear transformation. Also find the rank of $ T $, where it exists. $ T: V_2 \to V_3 $ defined by $ T(x_1, x_2) = (x_1, x_1 + x_2, x_2) $

If $ Y = \int_0^{\infty} \frac{x^a}{a^x} \, dx, \, a > 1 $, then the value of $ Y $ is

The area bounded by the axis of $ x $, and the curve and ordinates $ y = \cosh \frac{x}{c} $ from $ x = 0 $ to $ x = a $ is

Consider the following functions :
1. $ y = x \sin \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 $
2. $ y = x^2 \sin \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 $
3. $ y = x^2 \cos \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 $
4. $ y = x \cos \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 $
The functions, differentiable at $ x = 0 $, are

For a positive term series $ \sum a_n $, the ratio test states that

If $ \lim_{x \to \infty} \frac{\sin 2x + a \sin x}{x^3} = b $ where $ b $ is finite, then the values of $ a $ and $ b $ respectively will be

The expansion of $ \tan x $ in powers of $ x $ by Maclaurin's theorem is valid in the interval

The value of $ \lim_{(x, y) \to (k, 0)} \left(1 + \frac{x}{y}\right)^y $ is

The gradient of the function $ f(x, y, z) = \sin(xyz) $, at $ (1, -1, \pi) $, is

If det(A) = 7, where $ A = \begin{bmatrix} a & b & c \\ 1 & 1 & g \\ g & \omega & 1 \end{bmatrix} $, then det(2A)^{-1} is equal to

If $ 3x + 2y + z = 0 $, $ x + 4y + z = 0 $ and $ 2x + y + 4z = 0 $ be a system of equations, then

Evaluate $ \int_{0}^{\infty} \log \left( x + \frac{1}{x} \right) \frac{dx}{1 + x^2} $

Find the volume of the solid generated by rotating completely about the x-axis where the area enclosed between $ y^2 = x^3 + 5x $ and the line $ x = 2 $ and $ x = 4 $ about its major axis.

Find the maximum value of the function $ f(x) = \frac{x}{1 + x \tan x} $

It is given that Rolle's theorem holds for the function $ f(x) = x^3 + bx^2 + cx $, $ 1 \leq x \leq 2 $ at the point $ x = \frac{4}{3} $. Find the values of b and c.

Discuss the convergence of the sequence whose n-th term is $ \alpha_n = \frac{(-1)^n}{n} + 1 $

Test the convergence of the following series : $ x^2 + \frac{2^2 x^4}{3.4} + \frac{2^2 4^2 x^6}{3.4.5.6} + \frac{2^2 4^2 6^2 x^8}{3.4.5.6.7.8} ... $

Find the Fourier series expansion of the function $ f(x) = \{x^2, -2 \leq x \leq 2\} $. Hence deduce that $ \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... $

Find the Fourier cosine series and Fourier sine series of the following function in given interval : $ f(x) = \begin{cases} x, & 0 < x < 2 \\ 2, & 2 \leq x < 4 \end{cases} $

Discuss continuity of the following function at the point $ (0, 0) $ : $ f(x, y) = \begin{cases} \frac{x^2 y^2}{\left( x^3 + y^3 \right)} , & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases} $

Find the maximum value of $ xyz $ under the constraints $ x^2 + z^2 = 1 $ and $ y - x = 0 $.

Find the value of $ \lim_{x \to {\infty}} \left( \frac{x+4}{x+2} \right)^{x+3} $

Find the equation of the tangent plane to the surface $ x^2 - 3y^2 - z^2 = 2 $, at the point $ (3, 1, 2) $.

Find the eigenvalues and eigenvectors of the following matrix : $ \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} $

Verify Cayley-Hamilton theorem for the matrix $ \begin{bmatrix} 0 & 0 & 1 \\ 3 & 1 & 0 \\ -2 & 1 & 4 \end{bmatrix} $

Determine the range of the following linear transformation. Also find the rank of $ T $, where it exists. $ T: V_2 \to V_3 $ defined by $ T(x_1, x_2) = (x_1, x_1 + x_2, x_2) $