List of practice Questions

The value of the line integral for the vector,
\(\overrightarrow𝑣=2𝑥̂ +𝑦𝑧 ^2𝑦̂ +(3𝑦+𝑧 ^2 )𝑧̂ \)
along the closed path OABO (as shown in the figure) is: 
the line integral for the vector-along the closed path OABO
(Path AB is the arc of a circle of unit radius)
If \( a_{ij} \) and \( A_{ij} \) represent the \((i,j)^\text{th}\) element and its cofactor of the matrix: \[ \begin{bmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{bmatrix}, \] respectively, then the value of \( a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} \) is:
A relation \( R \) defined on set \( A = \{x : x \in \mathbb{Z} \text{ and } 0 \leq x \leq 10\} \) as \( R = \{(x, y) : x = y\) is given to be an equivalence relation. The number of equivalence classes is:
If \(A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix}\), find \(A^{-1}\) and use it to solve the following system of equations: \[ x - 2y = 10, \quad 2x - y - z = 8, \quad -2y + z = 7. \]
Find: \[ \int_{\pi/6}^{\pi/3} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \, dx. \]
(a) Find: \[ \int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \, dx. \]
(b) Evaluate: \[ \int_{1}^{3} \left(|x - 1| + |x - 2| + |x - 3|\right) \, dx. \]
Find the general solution of the differential equation: \[ \frac{dy}{dx} = \frac{x^2 + y^2}{2xy}. \]
Solve the following linear programming problem graphically: \[ \text{Maximise } z = 5x + 4y \] subject to the constraints: \[ x + 2y \geq 4, \quad 3x + y \leq 6, \quad x + y \leq 4, \quad x, y \geq 0. \]
(a) If \(\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)\), prove that \(\frac{dy}{dx} = \sqrt{\frac{1 - y^2}{1 - x^2}}\).
(b) If \(y = (\tan x)^x\), then find \(\frac{dy}{dx}\).
(a) Evaluate: \[ \int_{0}^{\pi/2} e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx. \]
If \( \vec{a} \) and \( \vec{b} \) are two non-zero vectors such that \( (\vec{a} + \vec{b}) \perp \vec{a \) and \( (2\vec{a} + \vec{b}) \perp \vec{b} \), then prove that \( |\vec{b}| = \sqrt{2} |\vec{a}| \).}
(a) If \( y = \sqrt{\cos x + y} \), prove that \[ \frac{dy}{dx} = \frac{\sin x}{1 - 2y}. \]
(b) Show that the function \( f(x) = |x|^3 \) is differentiable at all points of its domain.
Find the absolute maximum and minimum values of the function: \[ f(x) = 12x^{4/3} - 6x^{1/3}, \quad x \in [0, 1]. \]
The number of points, where \( f(x) = \lfloor x \rfloor \), \( 0 < x < 3 \) (\(\lfloor \cdot \rfloor\) denotes the greatest integer function), is not differentiable is:
Let \( f(x) \) be a continuous function on \([a, b]\) and differentiable on \((a, b)\). Then, this function \( f(x) \) is strictly increasing in \((a, b)\) if:
The vector with terminal point \( A(2, -3, 5) \) and initial point \( B(3, -4, 7) \) is:
The distance of the point \( P(a, b, c) \) from the y-axis is:
The number of corner points of the feasible region determined by the constraints \( x \geq 0, y \geq 0, x + y \geq 4 \) is:
A linear programming problem deals with the optimization of a/an:
If \( P(A \mid B) = P(A' \mid B) \), then which of the following statements is true?
The derivative of \( \sin(x^2) \) w.r.t. \( x \), at \( x = \sqrt{\pi} \), is:
The order and degree of the differential equation: \[ \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = \frac{d^2y}{dx^2}, \] respectively, are: