List of practice Questions

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 :

The area of the region bounded by the curve \(y = x\) and x-axis, between \(x = 0\) and \(x = 2\) is :

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

Which of the following is not a Linear Differential Equation ?

If \(A = [a_{ij}]\) is a \(2 \times 2\) matrix whose elements are given by \(a_{ij} = \frac{|i - 3j|}{2}\), then \(A'\) is :

If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is :

Differential of \(e^{e^x}\) with respect to x is :

The surface area of a sphere when its volume changes at the same rate as its radius is :