List of top Mathematics Questions

If \( p_1 \) and \( p_2 \) are the perpendicular distances from the origin to the tangent and normal drawn at any point on the curve \( x^{2/3} + y^{2/3} = a^{2/3} \) respectively. If \( k_1 p_1^2 + k_2 p_2^2 = a^2 \), then \( k_1 + k_2 = \)
The acute angle between the curves \(x^2 + y^2 = x + y\) and \(x^2 + y^2 = 2y\) is:
Determine the order and degree of the differential equation \( \frac{d^3y}{dx^3} = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{5/2} \):
Find the integrating factor of the differential equation \( \sin x \frac{dy}{dx} - y \cos x = 1 \):
The general solution of the differential equation \( (x \sin \frac{y}{x}) dy = (y \sin \frac{y}{x} - x) dx \) is:
If the locus of the mid points of the chords of the circle \(x^2 + y^2 = 25\) that subtend a right angle at the origin is given by \( \frac{x^2}{a^2} + \frac{y^2}{a^2} = 1\), then \(|a| =\)
Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{x} - a - \log(1+x)}{\sin x} = 0, { then find } a. \]
Two natural numbers are chosen at random from 1 to 100 and are multiplied. If \(A\) is the event that the product is an even number and \(B\) is the event that the product is divisible by 4, then \(P(A \cap \bar B) = \)
Let \( A(2, 3), B(1, -1) \) be two points. If \( P \) is a variable point such that \( \angle APB = 90^\circ \), then the locus of \( P \) is
In the interval \([0, 3]\), the function \(f(x) = |x - 1| + |x - 2|\) is:
A person is known to speak false once out of 4 times. If that person picks a card at random from a pack of 52 cards and reports that it is a king, then the probability that it is actually a king is
A value of \(c\) according to the Lagrange's mean value theorem for \(f(x) = (x - 1)(x - 2)(x - 3)\) in \([0,4]\) is:
Evaluate the integral \( \int \frac{e^{2x}}{\sqrt{e^x + 1}} \, dx \):
If \((\alpha + \beta)\) is not a multiple of \(\frac{\pi}{2}\) and \(3 \sin(\alpha - \beta) = 5 \cos(\alpha + \beta)\), then \[ \tan\left(\frac{\pi}{4} + \alpha\right) + 4\tan\left(\frac{\pi}{4} + \beta\right) = \]
The radical center of the circles \(x^2 + y^2 + 2x + 3y + 1 = 0\), \(x^2 + y^2 - x + y + 3 = 0\), and \(x^2 + y^2 - 3x + 2y + 5 = 0\) is:
The angle between the diagonals of the parallelogram whose adjacent sides are \(2i + 4j - 5k\) and \(i + 2j + 3k\) is
Evaluate the limit: \[ \lim_{x \to \frac{\pi}{4}} \frac{4\sqrt{2} - (\cos x + \sin x)^5}{1 - \sin 2x} = \]
If \(f(x) = \begin{cases} 2x + 3, & \text{if } x \leq 1 \\ 2ax + bx, & \text{if } x > 1 \end{cases}\) is differentiable \(\forall x \in \mathbb{R}\), then \(f(2) =\)
The number of ways in which 17 apples can be distributed among four guests such that each guest gets at least 3 apples is:
The direction cosines of the line of intersection of the planes \( x + 2y + z - 4 = 0 \) and \( 2x - y + z - 3 = 0 \) are:
If \([x]\) is the greatest integer function, then evaluate the integral \( \int_{0}^{5} [x] \, dx \):
If \( L_1 \) and \( L_2 \) are two lines which pass through origin and have direction ratios \( (3, 1, -5) \) and \( (2, 3, -1) \) respectively, then the equation of the plane containing \( L_1 \) and \( L_2 \) is:
Determine the values of \(a\) and \(b\) for which the function \(f(x)\) defined as: \[ f(x) = \begin{cases} 1 + |\sin x|^{\left(\frac{a}{|\sin x|}\right)} & \text{if } \frac{-\pi}{6} < x < 0, \\ b & \text{if } x = 0, \\ e^{\left(\frac{\tan 2x}{\tan 3x}\right)} & \text{if } 0 < x < \frac{\pi}{6} \end{cases} \] is continuous at \(x = 0\).
For any real value of \(x\), if \( \frac{11x^{2}+12x+6}{x^{2}+4x+2} \not\in (a, b) \), then the value for \(x\) for which $$ \frac{11x^{2}+12x+6}{x^{2}+4x+2} = b - a + 3 $$ is: 
If \( l_1 \) and \( l_2 \) are the lengths of the perpendiculars drawn from a point on the hyperbola \( 5x^2 - 4y^2 - 20 = 0 \) to its asymptotes, then \( \frac{l_1^2 l_2^2}{100} = \)