List of top Mathematics Questions

Let \( f(x) \) be a cubic polynomial with \( f(1) = -10 \), \( f(-1) = 6 \), and has a local minima at \( x = 1 \), and \( f'(x) \) has a local minima at \( x = -1 \). Then \( f(3) \) is equal to _________.
Negation of the statement \((p \lor r) \implies (q \lor r)\) is :
If \( \int \frac{\sin x}{\sin^3 x + \cos^3 x} dx = \alpha \log_e |1 + \tan x| + \beta \log_e |1 - \tan x + \tan^2 x| + \gamma \tan^{-1} \left( \frac{2 \tan x - 1}{\sqrt{3}} \right) + C \), when C is a constant of integration, then the value of \( 18(\alpha + \beta + \gamma^2) \) is _________.
The number of solutions of the equation \(32^{\tan^2 x} + 32^{\sec^2 x} = 81\), \(0 \le x \le \frac{\pi}{4}\) is :
The distance of the point \((-1, 2, -2)\) from the line of intersection of the planes \(2x + 3y + 2z = 0\) and \(x - 2y + z = 0\) is :
The domain of the function \(f(x) = \sin^{-1} \left( \frac{3x^2 + x - 1}{(x - 1)^2} \right) + \cos^{-1} \left( \frac{x - 1}{x + 1} \right)\) is :
The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :
If \( [x] \) is the greatest integer \( \le x \), then \( \pi^2 \int_{0}^{2} \left( \sin \frac{\pi x}{2} \right) (x - [x])^{[x]} dx \) is equal to :
If \( \alpha = \lim_{x \to \pi/4} \frac{\tan^3 x - \tan x}{\cos(x + \frac{\pi}{4})} \) and \( \beta = \lim_{x \to 0} (\cos x)^{\cot x} \) are the roots of the equation, \( ax^2 + bx - 4 = 0 \), then the ordered pair \( (a, b) \) is :
If \( \frac{dy}{dx} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^{x+y} \log_e 2}, y(0) = 0 \), then for \( y = 1 \), the value of \( x \) lies in the interval :
Let \( A \) be the set of all points \( (\alpha, \beta) \) such that the area of triangle formed by the points \( (5, 6), (3, 2) \) and \( (\alpha, \beta) \) is 12 square units. Then the least possible length of a line segment joining the origin to a point in \( A \), is :
If \( y \frac{dy}{dx} = x \left[ \frac{\phi(y^2/x^2)}{\phi'(y^2/x^2)} + \frac{y^2}{x^2} \right], x>0, \phi>0, \) and \( y(1) = -1 \), then \( \phi\left(\frac{y^2}{4}\right) \) is equal to :
Let \( f \) be any continuous function on \( [0, 2] \) and twice differentiable on \( (0, 2) \). If \( f(0) = 0, f(1) = 1 \) and \( f(2) = 2 \), then :
The locus of mid-points of the line segments joining \( (-3, -5) \) and the points on the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) is :
If \( z \) is a complex number such that \( \frac{z - i}{z - 1} \) is purely imaginary, then the minimum value of \( |z - (3 + 3i)| \) is :
The sum of the roots of the equation, \( x + 1 - 2\log_2(3 + 2^x) + 2\log_4(10 - 2^{-x}) = 0 \), is :
If \( \alpha + \beta + \gamma = 2\pi \), then the system of equations
\( x + (\cos \gamma)y + (\cos \beta)z = 0 \)
\( (\cos \gamma)x + y + (\cos \alpha)z = 0 \)
\( (\cos \beta)x + (\cos \alpha)y + z = 0 \)
has :
Let \( f : \mathbb{N} \to \mathbb{N} \) be a function such that \( f(m + n) = f(m) + f(n) \) for every \( m, n \in \mathbb{N} \). If \( f(6) = 18 \), then \( f(2) \cdot f(3) \) is equal to :
Let \( a_1, a_2, a_3, \dots \) be an A.P. If \( \frac{a_1 + a_2 + \dots + a_{10}}{a_1 + a_2 + \dots + a_p} = \frac{100}{p^2}, p \neq 10 \), then \( \frac{a_{11}}{a_{10}} \) is equal to :
An angle of intersection of the curves, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and \( x^2 + y^2 = ab, a>b \), is :

The mean of 10 numbers \(7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14, \dots\) is ___________
 

The square of the distance of the point of intersection of the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z+1}{6}\) and the plane \(2x - y + z = 6\) from the point \((-1, -1, 2)\) is _____________
 

If the variable line \(3x + 4y = \alpha\) lies between the two circles \((x-1)^2 + (y-1)^2 = 1\) and \((x-9)^2 + (y-1)^2 = 4\), without intercepting a chord on either circle, then the sum of all the integral values of \(\alpha\) is _____________
 

Let \([t]\) denote the greatest integer \(\leq t\). Then the value of \(8 \cdot \int_{-1/2}^{1} ([2x] + |x|) \, dx\) is _________
 

A point \(z\) moves in the complex plane such that \(\arg\left(\frac{z-2}{z+2}\right) = \frac{\pi}{4}\), then the minimum value of \(|z - 9\sqrt{2} - 2i|^2\) is equal to _________