List of practice Questions

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

Assertion (A) : A function \(f : \mathbb{N} \to \mathbb{N}\) given by \(f(x) = x^3 + 2, \forall x \in \mathbb{N\) is one-one but not onto.
Reason (R) : Since \(\forall y \in \mathbb{N}\) (Codomain), there does not exist \(x = (y - 2)^{1/3}\) in \(\mathbb{N}\) (Domain) such that \(f(x) = x^3 + 2 = y\).}

Evaluate : \( \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left(\sin\left(-\frac{\pi}{2}\right)\right) + \tan^{-1}\left(\tan\frac{2\pi}{3}\right) \)}

Show that the function \( f(x) = \begin{cases \frac{\cos x}{\frac{\pi}{2}-x}, & x \neq \frac{\pi}{2} \\ 1, & x = \frac{\pi}{2} \end{cases} \) is continuous at \( x = \frac{\pi}{2} \).}

Find whether the function \( f(x) = \begin{cases x - 1, & x 2x - 3, & x \ge 2 \end{cases} \) at \( x = 2 \) is differentiable or not.}

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 ?

If \((3\hat{i} - 2\hat{j} + 5\hat{k}) \times (4\hat{i} + p\hat{j} + q\hat{k}) = \vec{0\), then the values of \(p\) and \(q\) are :}

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 :

The principal value of \(\sec^{-1}(\sqrt{2}) + 2 \text{cosec^{-1}(-\sqrt{2})\) is :}