List of practice Questions

Prove that: \[ \left| \begin{matrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{matrix} \right| = (a + b + c)^3 \]
Solve the equation: \[ \left| \begin{matrix} x+a & x & x \\ x & x+a & x \\ x & x & x+a \end{matrix} \right| = 0 \] where \( a \neq 0 \).
Evaluate the integral: \[ \int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x} \, dx \]

Find the shortest distance between parallel lines: \[ \vec{r_1} = \hat{i} + 2 \hat{j} - 4 \hat{k} + \lambda (2 \hat{i} + 3 \hat{j} + 6 \hat{k}) \,\,\, \text{and} \,\,\, \vec{r_2} = 3 \hat{i} + 3 \hat{j} - 5 \hat{k} + \mu (2 \hat{i} + 3 \hat{j} + 6 \hat{k}) \]

Find the shortest distance between the lines: \[ \vec{r_1} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - 2t) \hat{k} \,\,\, \text{and} \,\,\,\vec{r_2} = (s + 1) \hat{i} + (2s - 1) \hat{j} - (2s + 1) \hat{k} \]

Prove that if function \( f \) is differentiable at a point \( a \), then it is also continuous at that point.
Differentiate \( \sin(x^2) \) with respect to \( x^2 \).
Evaluate the integral: \[ \int_0^{\pi/2} \frac{\cos^5 x}{\sin^5 x + \cos^5 x} \, dx \]
Find the general solution of the differential equation \[ \frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}. \]
Minimize \( Z = 3x + 5y \) subject to the constraints: \[ x + 3y \geq 3, x + y \geq 2, x \geq 0, y \geq 0. \]
Maximize \( Z = 3x + 9y \) subject to the constraints: \[ x + 3y \leq 60, x + y \geq 10, x \leq y, x \geq 0, y \geq 0. \]
Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?
An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. Suppose that the probability of drawing each ball is the same. What is the probability that both drawn balls are black?
Find the projection of vector \( \mathbf{a} = \hat{i} + 2 \hat{j} + 3 \hat{k} \) on the vector \( \mathbf{b} = 2 \hat{i} + \hat{k} \).
Show that for any two vectors \( \mathbf{a} \) and \( \mathbf{b} \), always \( | \mathbf{a} \cdot \mathbf{b} | \leq \|\mathbf{a}\| \|\mathbf{b}\| \).
Find the value of \( x \) for which \( x ( \hat{i} + \hat{j} + \hat{k} ) \) is a unit vector.
If direction ratios of a line are -18, 12, -4, then find its direction cosines.
Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 2\hat{k} \).
Find local maximum and local minimum values of the function given by \[ f(x) = 3x^4 + 4x^3 - 12x^2 + 12 \]
Find the interval in which the function given by \[ f(x) = x^2 - 4x + 6 \] is decreasing.
Differentiate \( X^X \) with respect to \( X \).
If \( y = \sin^{-1} \left( \frac{2x}{1 + x^2} \right) \), then find \( \frac{dy}{dx} \).
Show that the function given by \( f(x) = 12x - 3 \) is increasing on \( R \).
Evaluate \[ \int_{-1}^{1} \sin^5x \cos^4x \, dx \]
Evaluate \[ \int_{-1}^{2} \left( x^3 - |x| \right) dx. \]