List of top Mathematics Questions

Which one of the following is a homogeneous differential equation?
Let \([ \cdot ]\) denote the greatest integer function.
Assertion (A): \(\lim_{x \to \infty} \frac{[x]}{x} = 1\)
Reason (R): \(f(x) = x - 1\), \(g(x) = [x]\), \(h(x) = x\) and \(\lim_{x \to \infty} \frac{f(x)}{x} = \lim_{x \to \infty} \frac{h(x)}{x} = 1\):
If a circle \(S\) passing through the points \(A(1, 2)\) and \(B(2, 1)\) has its centre \(C\) located in the third quadrant at a distance of \(\frac{7}{\sqrt{2}}\) units from \(AB\), then the point \(P(1, -2)\)
Let \( A(2, 3) \), \( B(3, -1) \), and \( C(-3, 2) \) be three points. If the centre of the circle passing through A, B, and C is \( (h, k) \), then \( 2k - 4 = \):
If \( f(x) = \begin{cases} x \left( 1 + \frac{1}{2} \sin(\log x^2) \right), & x \neq 0 \\ 0, & x = 0 \end{cases} \), then \( \lim_{x \to 0} \frac{f(x) - f(0)}{x} \)
The lengths of the intercepts made by a circle \(S\) on the \(X\) and \(Y\)-axes are \(\frac{2\sqrt{13}}{3}\) and \(\frac{2\sqrt{17}}{3}\) respectively. Then the equation of the circle is:
If $x^3 + y^3 = 3axy$, then at $\left(\frac{3a}{2}, \frac{3a}{2}\right)$ the value of $3a y'' + 40$ is:
The probability that a non-leap year contains 53 Sundays is
The substitution \( x = vy \) converts which one of the following differential equations to an equation solvable by the variable separable method?
If a straight line \( L \) perpendicular to the line \( 3x - 4y = 6 \) forms a triangle of area 6 square units with coordinate axes, then the minimum perpendicular distance from the point \( (1,1) \) to the line \( L \) is:
The set \( \{ x \in \mathbb{R} : 16(2^x)>16^{x-1} \} \) is:
If $\int_0^{2\pi} |x \sin x| dx = k\pi$, then $k=$
If the line passing through the points \( (5,1,a) \) and \( (3,b,1) \) crosses the YZ plane at the point \( \left( 0, \frac{17}{2}, \frac{-13}{2} \right) \), then \( a + b = \):
Let \( B = \begin{bmatrix} 2 & 6 & 4 \\ 1 & 0 & 1 \\ -1 & -1 & -1 \end{bmatrix} \), \( C = \begin{bmatrix} -1 & 0 & 1 \\ 1 & 1 & 3 \\ 2 & 0 & 2 \end{bmatrix} \), and if a matrix \( A \) is such that \( BAC = I \), then \( A^{-1} = \)
Let \( (a, b) \) denote the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \). If \( \mathbf{a} = 2i + 3j + 6k \), \( |\mathbf{a}| = 4 \), and \( (\mathbf{a}, \mathbf{b}) = \cos^{-1} \left( \frac{4}{21} \right) \), then \( \mathbf{a} + \mathbf{b} = \)
In a committee of 25 members, each member is proficient either in Mathematics or in Statistics or in both. If 19 of them are proficient in Mathematics and 16 of them are proficient in Statistics, then the probability that a person selected at random from the committee is proficient in both is
Evaluate the integral: \[ \int \frac{dx}{4 + 5 \cos x} \]
If four dice are thrown simultaneously, then the probability that none of the dice shows the number 1 on its face, is:
If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 - 4x + 3 = 0 \), then \( \frac{2(\alpha^4 + \beta^4) + 3(\alpha^2 + \beta^2)}{\alpha + \beta} = \)
The value of the constant \( c \), so that \( P(x) = c\left(\dfrac{2}{3}\right)^x,\, x = 1,2,3,\ldots \) is the probability distribution function of a discrete random variable \( X \), is:
Given that $\frac{d}{dx} \int_{0}^{\phi(x)} f(t) dt = f(\phi(x)) \phi'(x)$. For all $x \in (0, \frac{\pi}{2})$, if $\int_{1}^{\cos x} t^2 f(t) dt = \cos 2x$, then $f\left(\frac{1}{\sqrt{2}}\right) =$
If \( A(3, -1, 1), B(0, 2, 3), C(4, 8, 11) \) are three points, then the coordinates of the foot of the perpendicular drawn from the point A to the line joining the points B and C is:
If a function $ f(x) $ defined on $ [a, b] $ is discontinuous at $ x = \alpha \in (a, b) $, then:
If $ a = 2n $ and $ b = 2m+1 $ for all $ m, n \in \mathbb{N} $, then \[ \int_{-\pi}^{\pi} e^{\sin^2 x} \cot^{b}(2n+1) x \, dx = \]
\( \int_{-4\pi}^{4\pi} \tan^9 x \sin^6 x \cos^3 x \, dx = \)