List of practice Questions

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 \((1, 2, 3)\).

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 \( \vec{a} \).

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
\(x + y \le 300\)
\(3x + y \le 600\)
\(x - y \ge -100\)
\(x, y \ge 0\)

Find the values of \( x \) for which \( f(x) = x^x, x > 0 \) is increasing.

If the position vectors of three points A, B and C are \( 3\hat{i} + \hat{j} \), \( 5\hat{i} + 6\hat{j} - 3\hat{k} \) and \( 4\hat{j} \) respectively, then show that they form an isosceles triangle.

The region represented by the system of inequations \(3x + y \ge 3, 2x - y \ge - 5, x, y \ge 0\) is :

In the graph, the feasible region representing the Linear Programming Problem for maximising objective function \(Z = px + qy, p, q > 0\) is shaded. If all points on segment \(AB\) give max (Z), then which of the following is true ?

Three points \(A(0, 1, 1)\), \(B(2, 0, -1)\) and \(C(1, 0, 3)\) form \(\Delta ABC\). The \(ar (\Delta ABC)\) is :

A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red ?

Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative.

The greatest integer function, \(f(x) = [x]\), \(0 < x < 3\) is not differentiable at how many points ?

\(\int \frac{dx}{\sqrt{25 - 16x^2}}\) is equal to :

If \(\int_{0}^{1} \frac{dx}{e^x + e^{-x}} = \tan^{-1} e + k\), then the value of \(k\) is :