List of practice Questions

An ant is observed crawling on a sheet of paper along a straight line given by equation \( y = 2x - 4 \). Area of the surface covered by the ant bounded by y-axis, x-axis and \( x = 1 \) is :

The order and degree of the differential equation \( 1 + \left( \frac{d^3y}{dx^3} \right)^3 = \lambda \frac{d^2y{dx^2} \) is :}

The general solution for the differential equation \( \frac{dy}{dx} = e^{3x-y} \) is :

The corner points of the feasible region determined by the system of linear constraints are \( (0, 0), (0, 40), (20, 40), (60, 20) \) and \( (60, 0) \). If the objective function of an LPP is \( Z = 4x + 3y \), then the maximum value is :

Let \( A = \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} -3 & 0 & 1 \\ 2 & 4 & 0 \end{bmatrix} \). If \( A + B + C = O \), then matrix \( C \) is :

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.}