Mathematics-1 (Calculus and Linear Algebra)

The value of $ \int_0^1 x^3 (1 - x^2)^{5/2} dx $ is

The area $ \iint_R \sin(x + y) \, dxdy $ over R $ \{(x, y); 0 \leq x \leq \frac{\pi}{2}, 0 \leq x \leq \frac{\pi}{2}\} $

The matrix $ \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} $ is

The sequence $ \{a_n\} $, where $ a_n = \sqrt[n]{n} $ is

Let $ f(x) = \cos(x^2), X \in R $ then

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

$ A = 2i + \alpha j + k, B = i + 3j - 8k $. Then $ A $ and $ B $ are orthogonal if

Let $ x + 2y + z = 0, 3x + 4y + z = 0, x - z = 0 $ be a system of equations. Then

The value at which the Rolle's theorem is applicable for $ f(x) = \cos \frac{x}{2} $ in $ |\pi, 3\pi| $ is

The rank of the matrix $ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{bmatrix} $ is

Show that: $ \int_{0}^{\frac{\pi}{2}} \log(\tan x + \cot x) \, dx = \pi \log 2 $

Find the following limit: $ \lim_{n \to \infty} \left[ t \ln \left( 1 + \frac{3}{t} \right) \right] $

Test the convergence of $ x^2 + \frac{2^2}{3.4} x^4 + \frac{2^2 4^2}{3.4,5.6,7.8} x^8 + \cdots $

Examine the convergence of the series of which the general term is $ \sum_{n=1}^{\infty} \frac{1}{n(n+1)} $

Prove that $ \log (1+x) = \frac{x}{1+\theta x} $, where $ 0 < \theta < 1 $. Hence deduce, $ \frac{x}{1+x} < \log(1+x) < x $

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

Evaluate $ \int_{0}^{\pi/2} \frac{1}{a\cos^2 x + b\sin^2 x} \, dx $

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

Show that the equation of the evolute of the curve $ x^{2/3} + y^{2/3} = a^{2/3} $ is $ (x+y)^{2/3} + (x-y)^{2/3} = 2a^{2/3} $.

Find the areas of the regions that lies inside the circle $ r = a \cos \theta $ and outside the cardioid $ r = a(1 - \cos \theta) $.

Find the solution of the following system of equations:
$ x_1 + 2x_2 + 2x_3 = 2 $
$ x_1 + 8x_3 + 5x_4 = -6 $
$ x_1 + x_2 + 5x_3 + 5x_4 = 3 $
Also find the basis and dimension of the solution space.

Verify Cayley-Hamilton theorem for the matrix $ A = \begin{bmatrix} 2 & 4 \\ 2 & 5 \end{bmatrix} $ and find its inverse. Also express $ A^5 - 4A^4 - 7A^3 + 11A^2 - A - 10I $ as a linear polynomial in $ A $.

Expand $ \sin x $ in Taylor's series about $ x = \frac{\pi}{2} $.

By using Beta function evaluate $ \int_{0}^{1} x^4 (1-\sqrt{x})^5 \, dx $.

Develop the Fourier series on $ -\pi < x < \pi $ to represent the function defined as:
$ f(x) = \begin{cases} 0, & -\pi < x < 0 \\ \pi, & 0 < x < \pi \end{cases} $

(i) Evaluate div $ \vec{F} $ and Curl $ \vec{F} $, where $ \vec{F} = \text{grad} (x^2 + y^2 + z^2) e^{-\sqrt{x^2 + y^2 + z^2}} $.
(ii) Let $ \vec{F} = (-4x - 3y + az) i + (bx + 3y + 5z) j $
+ ($4x + cy + 3z) k $. Find the values of $ a $, $ b $ and $ c $ such that $ F $ is irrotational.

The value of $ \int_0^1 x^3 (1 - x^2)^{5/2} dx $ is

The area $ \iint_R \sin(x + y) \, dxdy $ over R $ \{(x, y); 0 \leq x \leq \frac{\pi}{2}, 0 \leq x \leq \frac{\pi}{2}\} $

The matrix $ \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} $ is

The sequence $ \{a_n\} $, where $ a_n = \sqrt[n]{n} $ is

Let $ f(x) = \cos(x^2), X \in R $ then

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

$ A = 2i + \alpha j + k, B = i + 3j - 8k $. Then $ A $ and $ B $ are orthogonal if

Let $ x + 2y + z = 0, 3x + 4y + z = 0, x - z = 0 $ be a system of equations. Then

The value at which the Rolle's theorem is applicable for $ f(x) = \cos \frac{x}{2} $ in $ |\pi, 3\pi| $ is

The rank of the matrix $ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{bmatrix} $ is

Show that: $ \int_{0}^{\frac{\pi}{2}} \log(\tan x + \cot x) \, dx = \pi \log 2 $

Find the following limit: $ \lim_{n \to \infty} \left[ t \ln \left( 1 + \frac{3}{t} \right) \right] $

Test the convergence of $ x^2 + \frac{2^2}{3.4} x^4 + \frac{2^2 4^2}{3.4,5.6,7.8} x^8 + \cdots $

Examine the convergence of the series of which the general term is $ \sum_{n=1}^{\infty} \frac{1}{n(n+1)} $

Prove that $ \log (1+x) = \frac{x}{1+\theta x} $, where $ 0 < \theta < 1 $. Hence deduce, $ \frac{x}{1+x} < \log(1+x) < x $

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

Evaluate $ \int_{0}^{\pi/2} \frac{1}{a\cos^2 x + b\sin^2 x} \, dx $

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

Show that the equation of the evolute of the curve $ x^{2/3} + y^{2/3} = a^{2/3} $ is $ (x+y)^{2/3} + (x-y)^{2/3} = 2a^{2/3} $.

Find the areas of the regions that lies inside the circle $ r = a \cos \theta $ and outside the cardioid $ r = a(1 - \cos \theta) $.

Find the solution of the following system of equations:
$ x_1 + 2x_2 + 2x_3 = 2 $
$ x_1 + 8x_3 + 5x_4 = -6 $
$ x_1 + x_2 + 5x_3 + 5x_4 = 3 $
Also find the basis and dimension of the solution space.

Verify Cayley-Hamilton theorem for the matrix $ A = \begin{bmatrix} 2 & 4 \\ 2 & 5 \end{bmatrix} $ and find its inverse. Also express $ A^5 - 4A^4 - 7A^3 + 11A^2 - A - 10I $ as a linear polynomial in $ A $.

Expand $ \sin x $ in Taylor's series about $ x = \frac{\pi}{2} $.

By using Beta function evaluate $ \int_{0}^{1} x^4 (1-\sqrt{x})^5 \, dx $.

Develop the Fourier series on $ -\pi < x < \pi $ to represent the function defined as:
$ f(x) = \begin{cases} 0, & -\pi < x < 0 \\ \pi, & 0 < x < \pi \end{cases} $

(i) Evaluate div $ \vec{F} $ and Curl $ \vec{F} $, where $ \vec{F} = \text{grad} (x^2 + y^2 + z^2) e^{-\sqrt{x^2 + y^2 + z^2}} $.
(ii) Let $ \vec{F} = (-4x - 3y + az) i + (bx + 3y + 5z) j $
+ ($4x + cy + 3z) k $. Find the values of $ a $, $ b $ and $ c $ such that $ F $ is irrotational.