List of top Mathematics Questions

Let $y=y(x)$ be the solution of the differential equation $(x-x^3)dy = (y+yx^2-3x^4)dx$, $x>2$. If $y(3)=3$, then $y(4)$ is equal to :
The population $P = P(t)$ at time 't' of a certain species follows the differential equation $\frac{dP}{dt} = 0.5P - 450$. If $P(0) = 850$, then the time at which population becomes zero is :
Let y = y(x) be the solution of the differential equation dy/dx = (y + 1) ((y + 1)e^{x²/2 - x} - 1), 0<x<2.1, with y(2) = 0. Then the value of dy/dx at x=1 is equal to :
Let y=y(x) be the solution of the differential equation x dy - y dx = √(x² - y²) dx, x ≥ 1, with y(1) = 0. If the area bounded by the line x=1, x=e^{π}, y=0 and y=y(x) is α e^{2π} + β, then the value of 10(α + β) is equal to ________.
If the solution curve of the differential equation \((2x - 10y^3)dy + y dx = 0\), passes through the points (0, 1) and (2, \(\beta\)), then \(\beta\) is a root of the equation :
Let $y=y(x)$ be the solution of the differential equation $dy = e^{\alpha x+y}dx; \alpha \in \mathbb{N}$. If $y(\log_e 2) = \log_e 2$ and $y(0)=\log_e(\frac{1}{2})$, then the value of $\alpha$ is equal to ________.
Let f be a twice differentiable function defined on ℝ such that f(0) = 1, f'(0) = 2 and f'(x) ≠ 0 for all x ∈ ℝ. If $| f(x) \ f'(x) | \ | f'(x) \ f''(x) | = 0$, then the value of f(1) lies in the interval :
The differential equation satisfied by the system of parabolas y² = 4a(x + a) is :
Let \(y=y(x)\) be solution of the following differential equation \(e^y \frac{dy}{dx} - 2e^y \sin x + \sin x \cos^2 x = 0, y(\pi/2) = 0\). If \(y(0) = \log_e(\alpha + \beta e^{-2})\), then \(4(\alpha + \beta)\) is equal to __________.
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} = 1 + xe^{y-x}, -\sqrt{2}<x<\sqrt{2}, y(0) = 0$. Then, the minimum value of $y(x), x \in (-\sqrt{2}, \sqrt{2})$ is equal to :
If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is (x² - 4x + y + 8)/(x - 2), then this curve also passes through the point :
If \(x \phi(x) = \int_{5}^{x} (3t^2 - 2\phi'(t))dt\), \(x>-2\), and \(\phi(0)=4\), then \(\phi(2)\) is ___________
The length of the latus rectum of a parabola, whose vertex and focus are on the positive \(x\)-axis at a distance \(R\) and \(S\) (\(S>R\)) respectively from the origin, is :
The line \(12x \cos \theta + 5y \sin \theta = 60\) is tangent to which of the following curves ?
A tangent and a normal are drawn at the point $P(2, -4)$ on the parabola $y^2 = 8x$, which meet the directrix of the parabola at the points A and B respectively. If $Q(a, b)$ is a point such that $AQBP$ is a square, then $2a + b$ is equal to :
Equation of a plane at a distance $\sqrt{\frac{2}{21}}$ from the origin, which contains the line of intersection of the planes $x - y - z - 1 = 0$ and $2x + y - 3z + 4 = 0$, is :
If one of the diameters of the circle \(x^2 + y^2 - 2x - 6y + 6 = 0\) is a chord of another circle 'C', whose center is at (2, 1), then its radius is ________
The image of the point (3, 5) in the line x - y + 1 = 0, lies on :
Two tangents are drawn from the point P(-1, 1) to the circle $x^2 + y^2 - 2x - 6y + 6 = 0$. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to :
Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set\{P, Q\} is equal to :
If \(p\) and \(q\) are the lengths of the perpendiculars from the origin on the lines, \(x \csc \alpha - y \sec \alpha = k \cot 2\alpha\) and \(x \sin \alpha + y \cos \alpha = k \sin 2\alpha\) respectively, then \(k^2\) is equal to :
Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :
Choose the correct statement about two circles whose equations are given below : x² + y² - 10x - 10y + 41 = 0 ; x² + y² - 22x - 10y + 137 = 0
If α, β are natural numbers such that 100$^α - 199 β = (100)(100) + (99)(101) + (98)(102) + ⋯ + (1)(199)$, then the slope of the line passing through (α, β) and origin is :
Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be (10/3, 7/3). If α, β are the roots of the equation ax² + bx + 1 = 0, then the value of α² + β² - αβ is :