List of practice Questions

\( \sin^{-1} \left(\sin \frac{3\pi}{5}\right) =\) _____
If \( \cos^{-1} x = y \), then _____
Let \( f : \mathbb{N} \rightarrow \mathbb{N} \) be defined by \[ f(n) = \begin{cases} \frac{n+1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases} \] for all \( n \in \mathbb{N} \). Then \( f \) is _____
Let \( R \) be the relation in the set \( \mathbb{N} \) given by \( R = \{(a,b) : a = b - 2, \; b<6\} \), then _____
Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. The probability that first two cards are kings and the third card drawn is an ace is _____
If \(A\) and \(B\) are any two events such that \( P(A) + P(B) - P(A and B) = P(A) \), then _____
Let \(A\) and \(B\) be two events such that \( P(A) = \frac{5}{11}, \; P(B) = \frac{2}{11} \) and \( P(A \cup B) = \frac{3}{11} \), then \( P(A'|B') = \)_____
For linear programming problem, the objective function is \( z = px + qy \), \( p,q>0 \). If at the corner points \( (0,10) \) and \( (5,5) \), the value of \( z \) are 90 and 60 respectively, then the relation between \( p \) and \( q \) is _____
The coordinates of the corner points of the bounded feasible region are \( (0,10), (5,5), \\ (15,15), (0,20) \). The minimum of the objective function \( z = 3x + 9y \) is _____
The vector equation of the line passing through the point \( (1,2,-4) \) and perpendicular to the two lines \[ \frac{x-8}{3} = \frac{y+19}{-16} = \frac{z-10}{7} \] and \[ \frac{x-15}{3} = \frac{y-29}{8} = \frac{z-5}{-5} \] is _____
If the lines \[ \frac{1-x}{3} = \frac{7y-14}{2p} = \frac{3-z}{-2} \] and \[ \frac{7-7x}{3p} = \frac{y-5}{1} = \frac{6-z}{5} \] are perpendicular, then the value of \( p \) is _____
The angle between the pair of lines given by \[ \vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k}) \] and \[ \vec{r} = 5\hat{i} - 2\hat{j} + \mu(3\hat{i} + 2\hat{j} + 6\hat{k}) \] is _____
The value of \( \hat{i} \cdot (\hat{k} \times \hat{j}) + \hat{j} \cdot (\hat{k} \times \hat{i}) + \hat{k} \cdot (\hat{j} \times \hat{i}) \) is _____
The area of the triangle with vertices \( A(1,1,2), B(2,3,5) \) and \( C(1,5,5) \) is _____
If two vectors \( \vec{a} \) and \( \vec{b} \) are such that \( |\vec{a}| = 2, |\vec{b}| = 3 \) and \( \vec{a} \cdot \vec{b} = 4 \), then \( |\vec{a} - \vec{b}| = \)_____
The general solution of the differential equation \( \frac{dy}{dx} = e^{x+y} \) is _____
The number of arbitrary constants in the particular solution of a differential equation of third order are _____
The order and the degree of the differential equation \[ \sqrt{1 + \left(\frac{d^2y}{dx^2}\right)^2} = \sqrt{x + \left(\frac{dy}{dx}\right)^6} \] are respectively _____ and _____
The area bounded by the curve \(y = x|x|\), X-axis and the ordinates \(x = -1\) and \(x = 1\) is _____
Area lying in the first quadrant and bounded by ellipse \(4x^2 + 9y^2 = 144\) is _____
\( \int \frac{e^{2025x} + e^{-2025x}}{e^{2025x} + e^{-2025x}} dx = \)_____ + C
\( \int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =\) _____ + C
\( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{dx}{1+\cot x} = \) _____
\( \int_{0}^{\pi} \left(\sin^2 \frac{x}{2} - \cos^2 \frac{x}{2}\right) dx =\) _____
\( \int \frac{dx}{\sqrt{9x - 4x^2}} = \)_____ + C