List of practice Questions

If \( A \) is a non-singular matrix, then which of the following is not true ?

If the area of \( \Delta ABC \) with vertices \( A(3, 1), B(-2, 1) \) and \( C(0, k) \) is \( 5 \) sq. units, then values of \( k \) are :

Derivative of \( \cos^{-1} \left( \frac{\sin x + \cos x}{\sqrt{2}} \right), -\frac{\pi{4} < x < \frac{\pi}{4} \) with respect to \( x \) is :}

Absolute minimum value of \( f(x) = (x - 2)^2 + 5 \) in the interval \( [-3, 2] \) is :

If matrix \( A = \begin{bmatrix} -p & q r & p \end{bmatrix} \) is such that \( A^2 = I \), then :

If \( A \) is a square matrix such that \( A^2 = A \), then \( (A - I)^3 - A \) is equal to :

For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined ?

Find the general solution of the differential equation \( (x^3 - 3xy^2) dx = (y^3 - 3x^2y) dy \).

Find the equation of a line (in vector and cartesian form) that passes through the point of intersection of lines \( \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \) and \( \frac{x - 4}{5} = \frac{y - 1}{2} = z \) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).

Find a point on the line \(\frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2}\) at a distance of \(\sqrt{2}\) units from the point \((1, 2, 3)\).

Find the shortest distance between the lines
\(\vec{r} = (4 + \lambda)\hat{i + (2\lambda - 1)\hat{j} - 3\lambda\hat{k}\)
\(\vec{r} = (1 + 2\mu)\hat{i} + (4\mu - 1)\hat{j} + (2 - 5\mu)\hat{k}\)}

In a school, the probability of holding a debate competition is \( \frac{1}{3} \) and that of a quiz competition is \( \frac{2}{3} \). In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.

Find : \[ \int \frac{x}{(x - 1)(x^2 + 4)} dx \]

Evaluate : \[ \int_{0}^{1} \frac{x \tan^{-1} x}{(1 + x^2)^{3/2}} dx \]

Using integration, find the area of the region enclosed by the curve \( y = |x - 6| \), the x-axis, and between \( x = 4 \) and \( x = 8 \).

Solve the differential equation \( y e^y dx = (y^3 + 2x e^y) dy \), when \( y(0) = 1 \).

Let two rods placed on the ground be represented by vectors \( 4\hat{i} - \hat{j} + 3\hat{k} \) and \( -2\hat{i} + \hat{j} - 2\hat{k} \). Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.

A unit vector \( \vec{a} \) is such that it makes an angle \( \frac{\pi}{4} \) with x-axis, \( \frac{\pi}{3} \) with y-axis and an acute angle \( \theta \) with z-axis. Find \( \theta \) and the components of \( \vec{a} \).

Let \(A = \mathbb{R} - \{3\}\) and \(B = \mathbb{R} - \{1\}\). A function \(f : A \to B\) is defined by \(f(x) = \frac{x - 2{x - 3}\). Find whether \(f\) is one-one and onto.}

Let \(n\) be a fixed positive integer. A relation \(R\) is defined in set \(\mathbb{Z}\) such that \(R = \{(x, y) : (x - y) \text{ is divisible by n, x, y \in \mathbb{Z}\}\). Determine if \(R\) is an equivalence relation.}

If \(A = \begin{bmatrix} 3 & 2 & 0 1 & 4 & 0 0 & 0 & 5 \end{bmatrix}\), then compute \(A^2 - 7A + 10 I\).

If \(xy = e^{x-y}\), then find \(\frac{dy}{dx}\).

Differentiate \(\tan^{-1}\left(\frac{\sqrt{1+x^2} - \sqrt{1-x^2}}{\sqrt{1+x^2} + \sqrt{1-x^2}}\right)\) with respect to \(\cos^{-1} x^2\).

Solve the following Linear Programming Problem graphically :
Maximise \(Z = 200x + 120y\)
subject to the constraints
\(x + y \le 300\)
\(3x + y \le 600\)
\(x - y \ge -100\)
\(x, y \ge 0\)