List of top Mathematics Questions

The coefficients a, b and c of the quadratic equation, ax² + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is :
All possible values of θ ∈ [0, 2π] for which sin 2θ + tan 2θ>0 lie in :
The total number of positive integral solutions (x, y, z) such that xyz=24 is :
The statement A → (B → A) is equivalent to :
If \( A = \begin{bmatrix} 0 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 0 \end{bmatrix} \) and \( (I_2 + A)(I_2 - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \), then \( 13(a^2 + b^2) \) is equal to __________.
Let \( A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix} \), where \( x, y \) and \( z \) are real numbers such that \( x + y + z > 0 \) and \( xyz = 2 \). If \( A^2 = I_3 \), then the value of \( x^3 + y^3 + z^3 \) is __________.
If the system of equations $kx + y + 2z = 1$, $3x - y - 2z = 2$, $-2x - 2y - 4z = 3$ has infinitely many solutions, then k is equal to ________
The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is _________
Let A₁, A₂, A₃, … be squares such that for each n ≥ 1, the length of the side of $A_n$ equals the length of diagonal of $A_{n+1}$. If the length of $A_1$ is 12 cm, then the smallest value of n for which area of $A_n$ is less than one, is __________.
The number of points, at which the function f(x) = |2x + 1| - 3|x + 2| + |x² + x - 2|, x ∈ ℝ is not differentiable, is ________
Let \( f(x) \) be a polynomial of degree 6 in \( x \), in which the coefficient of \( x^6 \) is unity and it has extrema at \( x = -1 \) and \( x = 1 \). If \( \displaystyle \lim_{x \to 0} \frac{f(x)}{x^3} = 1 \), then \( 5 \cdot f(2) \) is equal to __________.
The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area A. Then A⁴ is equal to __________
The locus of the point of intersection of the lines (√3)kx + ky - 4√3 = 0 and √3 x - y - 4(√3)k = 0 is a conic, whose eccentricity is _________
Let $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$, $\vec{b} = \hat{i} - \hat{j}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$ and $\vec{r} \cdot \vec{b} = 0$, then $\vec{r} \cdot \vec{a}$ is equal to __________.
Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = 2x - 1 \) and \( g : \mathbb{R} \setminus \{1\} \to \mathbb{R} \) be defined as \( g(x) = \dfrac{x - \frac{1}{2}}{x - 1} \). Then the composition function \( f(g(x)) \) is :
Let \( p \) and \( q \) be two positive numbers such that \( p + q = 2 \) and \( p^4 + q^4 = 272 \). Then \( p \) and \( q \) are roots of the equation :
The system of linear equations 3x - 2y - kz = 10, 2x - 4y - 2z = 6, x + 2y - z = 5m is inconsistent if :
The value of \( 2^{15}C_1 + 2^{15}C_2 - 3^{15}C_3 + \cdots - 15^{15}C_{15} + \binom{14}{1} + \binom{14}{3} + \binom{14}{5} + \cdots + \binom{14}{11} \) is :
If $e^{(\cos^2 x + \cos^4 x + \cos^6 x + .......) \log_e 2}$ satisfies $t^2 - 9t + 8 = 0$, then find $\frac{2 \sin x}{\sin x + \sqrt{3} \cos x}$ for $0<x<\pi/2$ :
\( \displaystyle \lim_{x \to 0} \frac{\int_{0}^{x^2} \sin(\sqrt{t}) \, dt}{x^3} \) is equal to :
A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways is :
If \( f(x) = [x - 1]\cos\!\left( \frac{2x - 1}{2}\pi \right) \), then \( f \) is :
If $\int \frac{\cos x - \sin x}{\sqrt{8 - \sin 2x}} dx = a \sin^{-1} \left( \frac{\sin x + \cos x}{b} \right) + c$, find (a, b) :
The area (in sq. units) of the part of the circle $x^2 + y^2 = 36$, which is outside the parabola $y^2 = 9x$, is :
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 :