List of practice Questions

Solve the following linear programming problem graphically:
Maximize \( z = x + y \), subject to constraints:
\[ 2x + 5y \leq 100, \quad 8x + 5y \leq 200, \quad x \geq 0, \quad y \geq 0. \]

Find the position vector of point \( C \) which divides the line segment joining points \( A \) and \( B \) having position vectors \( \vec{a} = \hat{i} + 2\hat{j} - \hat{k} \) and \( \vec{b} = -\hat{i} + \hat{j} + \hat{k} \), respectively, in the ratio 4:1 externally. Further, find \( |\vec{AB}| : |\vec{BC}| \).

Evaluate: \[ \int_{0}^{\pi/2} \sin 2x \cos 3x \, dx. \]

Check the differentiability of \( f(x) = \begin{cases} x^2 + 1, & 0 \leq x < 1 \\ 3 - x, & 1 \leq x \leq 2 \end{cases} \) at \( x = 1 \).

If \( x = e^{x/y} \), prove that \( \frac{dy}{dx} = \frac{\log x - 1}{(\log x)^2} \).

Evaluate: \[ \cot^2 \left\{ \csc^{-1}(3) \right\} + \sin^2 \left\{ \cos^{-1} \left( \frac{1}{3} \right) \right\}. \]

\( \vec{a}, \vec{b}, \vec{c} \) are three mutually perpendicular unit vectors. If \( \theta \) is the angle between \( \vec{a} \) and \( (2\vec{a} + 3\vec{b} + 6\vec{c}) \), find the value of \( \cos \theta \).

If \( A \) and \( B \) are two skew-symmetric matrices, then \( AB + BA \) is:

Let \( E \) and \( F \) be two events such that \( P(E) = 0.1 \), \( P(F) = 0.3 \), \( P(E \cup F) = 0.4 \). Then \( P(F|E) \) is:

The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called:

If \( \alpha, \beta, \gamma \) are the angles which a line makes with positive directions of \( x, y, z \) axes respectively, then which of the following is not true?

If \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + \hat{j} - \hat{k} \), then \( \vec{a} \) and \( \vec{b} \) are:

\( x \log x \frac{dy}{dx} + y = 2 \log x \) is an example of a:

\( \int_{-a}^a f(x) \, dx = 0 \), if:

The rate of change of surface area of a sphere with respect to its radius \( r \), when \( r = 4 \, {cm} \), is:

A function \( f(x) = |1 - x + |x| | \) is:

Let \( \Delta = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) be a square matrix such that \( \text{adj} A = A \). Then, \( (a + b + c + d) \) is equal to:

Let \( \mathbb{Z} \) denote the set of integers. Then, the function \( f : \mathbb{Z} \to \mathbb{Z} \) defined as \( f(x) = x^3 - 1 \) is:

If \( A = [a_{ij}] \) is an identity matrix, then which of the following is true?

If \( A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix} \), then \( A^{-1} \) is:

Of the following, which group of constraints represents the feasible region given below?