List of top Mathematics Questions

If \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix, where \( a_{ij} = i - 3j \), then which of the following is \textbf{false?}
Let \( E \) be an event of a sample space \( S \) of an experiment, then \( P(S | E) \) is:
The degree of the differential equation \( (y'')^2 + (y')^3 = x \sin(y') \) is:
The common region determined by all the constraints of a linear programming problem is called:
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
The coordinates of the foot of the perpendicular drawn from the point \( (0, 1, 2) \) on the \( x \)-axis are given by:
The differential equation \( \frac{dy}{dx} = F(x, y) \) will not be a homogeneous differential equation, if \( F(x, y) \) is:
\( \int_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} \, dx \) is equal to:
The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:
Let \( f : \mathbb{R}_+ \to [-5, \infty) \) be defined as \( f(x) = 9x^2 + 6x - 5 \), where \( \mathbb{R}_+ \) is the set of all non-negative real numbers. Then, \( f \) is:
If \( \begin{vmatrix} -a & b & c
a & -b & c
a & b & -c \end{vmatrix} = kabc \), then the value of \( k \) is:
If the sum of all the elements of a \( 3 \times 3 \) scalar matrix is 9, then the product of all its elements is:
If \( A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} \) and \( A^{-1} = \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix} \), find the value of \((a + x) - (b + y)\).
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. \vspace{0.3cm} \includegraphics[width=\linewidth]{latex2.png} \vspace{0.3cm} A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, \(x\) m away from the base of the pole, the angle of elevation of the speed camera from the car C is \(\theta\). \vspace{0.3cm} On the basis of the above information, answer the following questions: \begin{enumerate} Express \( \theta \) in terms of the height of the camera installed on the pole and \( x \). Find \( \frac{d\theta}{dx} \). \begin{enumerate} Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole. OR If the rate of change of angle of elevation with respect to time of another car at a distance of 50 m from the base of the pole is \( \frac{3}{101} \) rad/s, then find the speed of the car. \end{enumerate} \end{enumerate}
Find: \[ \int \frac{(3 \cos x - 2) \sin x}{5 - \sin^2 x - 4 \cos x} \, dx. \]
Evaluate: \[ \int_{-2}^{2} \frac{x^3 + |x| + 1}{x^2 + 4|x| + 4} \, dx. \]
Using integration, find the area of the ellipse: \[ \frac{x^2}{16} + \frac{y^2}{4} = 1, \] included between the lines \(x = -2\) and \(x = 2\).
Evaluate: \[ \int_1^3 (|x - 1| + |x - 2| + |x - 3|) \, dx. \]
Find the equation of a line \( l_2 \) which is the mirror image of the line \( l_1 \) with respect to line \( l : \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \), given that line \( l_1 \) passes through the point \( P(1, 6, 3) \) and is parallel to line \( l \).
Solve the following differential equation: \[ (\tan^{-1}y - x) \, dy = (1 + y^2) \, dx. \]
If \( \sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y) \), prove that \( \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} \).
If \( y = (\tan x)^x \), then find \( \frac{dy}{dx} \).
Find: \[ \int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \, dx. \]
A relation \( R \) on set \( A = \{1, 2, 3, 4, 5\} \) is defined as \[ R = \{(x, y) : |x^2 - y^2| <8\}. \] Check whether the relation \( R \) is reflexive, symmetric, and transitive.
A function \( f \) is defined from \( R \to R \) as \( f(x) = ax + b \), such that \( f(1) = 1 \) and \( f(2) = 3 \). Find the function \( f(x) \). Hence, check whether the function \( f(x) \) is one-one and onto.