List of practice Questions

Let $\Delta = \left| \begin{matrix} \sin \theta \cos \varphi & \sin \theta \sin \varphi & \cos \theta \\ \cos \theta \cos \varphi & \cos \theta \sin \varphi & -\sin \theta \\ -\sin \theta \sin \varphi & \sin \theta \cos \varphi & 0 \end{matrix} \right|$. Then
Consider the equation $y - y_1 = m(x - x_1)$. If $m$ and $x_1$ are fixed, and different lines are drawn for different values of $y_1$, then
Let $R$ and $S$ be two equivalence relations on a non-void set $A$. Then
Twenty meters of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle be, if the area of the flower bed is greatest?
The line $y = x + 5$ touches
Let $p(x_0)$ be a polynomial with real coefficients, $p(0) = 1$ and $p'(x)>0$ for all $x \in \mathbb{R}$. Then
From the point $(-1, -6)$, two tangents are drawn to $y^2 = 4x$. Then the angle between the two tangents is
If $\alpha$ is a unit vector, $\beta = \hat{i} + \hat{j} - \hat{k}$, $\gamma = \hat{i} + \hat{k}$, then the maximum value of $|\alpha \beta \gamma|$ is
The maximum value of $f(x) = e^{\sin x} + e^{\cos x}$, where $x \in \mathbb{R}$, is
If $x$ satisfies the inequality $\log_2 5x^2 + (\log_5 x)^2<2$, then $x$ belongs to
The solution of $\det(A - \lambda I_2) = 0$ is $4$ and $8$, and $A = \begin{pmatrix} 2 & 3 \\ x & y \end{pmatrix}$. Then
The value of $a$ for which the sum of the squares of the roots of the equation $x^2 - (a - 2)x - (a - 1) = 0$ assumes the least value is
Chords of an ellipse are drawn through the positive end of the minor axis. Their midpoint lies on
Let $f$ be a non-negative function defined in $[0, \pi/2]$, $f'$ exists and is continuous for all $x$, and $\int_0^x \sqrt{1 - (f'(t))^2} dt = \int_0^x f(t) dt$ and $f(0) = 0$. Then
$PQ$ is a double ordinate of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ such that $\triangle OPQ$ is an equilateral triangle, with $O$ being the center of the hyperbola. Then the eccentricity $e$ of the hyperbola satisfies
$\lim_{x \to \infty} \left[ \frac{x^2 + 1}{x + 1} - ax - b \right], \, (a, b \in \mathbb{R}) = 0$. Then
If the transformation $z = \log \tan \frac{x}{2}$ reduces the differential equation $\frac{d^2y}{dx^2} + \cot x \frac{dy}{dx} + 4y \csc^2 x = 0$ into the form $\frac{d^2y}{dz^2} + ky = 0$, then $k$ is equal to
If $I$ is the greatest of $I_1 = \int_0^1 e^{-x} \cos^2 x \, dx$, $I_2 = \int_0^1 e^{-x^2} \cos^2 x \, dx$, $I_3 = \int_0^1 e^{-x^2} \, dx$, $I_4 = \int_0^1 e^{-\frac{x^2}{2}} \, dx$, then
A straight line meets the coordinate axes at $A$ and $B$. A circle is circumscribed about the triangle $OAB$, with $O$ being the origin. If $m$ and $n$ are the distances of the tangent from the origin to the points $A$ and $B$ respectively, the diameter of the circle is
Let the tangent and normal at any point $P(at^2, 2at), (a>0)$, on the parabola $y^2 = 4ax$ meet the axis of the parabola at $T$ and $G$ respectively. Then the radius of the circle through $P$, $T$, and $G$ is
The number of zeros at the end of $\angle 100$ is
If $|z - 25i| \leq 15$, the maximum $\arg(z) -$ minimum $\arg(z)$ is equal to
Let $f(n) = 2n + 1$, $g(n) = 1 + (n + 1)^{2n}$ for all $n \in \mathbb{N}$. Then
$A$ is a set containing elements. $P$ and $Q$ are two subsets of $A$. Then the number of ways of choosing $P$ and $Q$ such that $P \cap Q = \emptyset$ is
There are $n$ white and $n$ black balls marked $1, 2, 3, \ldots, n$. The number of ways in which we can arrange these balls in a row so that neighboring balls are of different colors is