List of top Mathematics Questions on Theory of Equations

Let $f$ be a function such that $3f(x)+2f\!\left(\dfrac{m}{19x}\right)=5x$, $x\ne0$, where $m=\displaystyle\sum_{i=1}^{9} i^2$. Then $f(5)-f(2)$ is equal to
Let a function $f(x)$ satisfy \[ 3f(x)+2f\!\left(\frac{m}{19x}\right)=5x \] where $m=\sum_{i=1}^{9} i^2$. Find $f(5)+f(2)$.
If the equation \[ x^4-ax^2+9=0 \] has four real and distinct roots, then the least possible integral value of $a$ is
The smallest positive integral value of $a$, for which all the roots of $x^4-ax^2+9=0$ are real and distinct, is equal to
The number of real solutions of the equation: $x|x+3| + |x-1| - 2 = 0$ is
Let \( u+v+w=3 \), \( u,v,w \in \mathbb{R} \) and \( f(x)=ux^2+vx+w \) be such that \[ f(x+y)=f(x)+f(y)+xy,\quad \forall x,y \in \mathbb{R}. \] Then \( f(1) \) equals:
The solution set of the equation \[ x \in \left(0,\frac{\pi}{2}\right), \quad \tan(\pi \tan x) = \cot(\pi \cot x) \] is:
If \( 0\le a,b \le 3 \) and the equation \[ x^2 + 4 + 3\cos(ax+b) = 2x \] has real solutions, then the value(s) of \( (a+b) \) is/are:
If the equation \[ \sin^2x - (p+2)\sin x - (p+3) = 0 \] has a solution, then \( p \) must lie in:

Let \[ f(\theta) = \begin{vmatrix} 1 & \cos\theta & -1 \\ -\sin\theta & 1 & -\cos\theta \\ -1 & \sin\theta & 1 \end{vmatrix} \] Suppose \( A \) and \( B \) are respectively maximum and minimum values of \( f(\theta) \). Then \( (A,B) \) is:

If \( f(x) \) and \( g(x) \) are polynomials such that \[ \phi(x) = f(x^3) + xg(x^3) \] is divisible by \( x^2 + x + 1 \), then:
Let \( f(x)=|x-\alpha|+|x-\beta| \), where \( \alpha,\beta \) are roots of \( x^2-3x+2=0 \). Then the number of points in \( [\alpha,\beta] \) at which \( f \) is not differentiable is:
The number of solutions of \[ \sin^{-1} x + \sin^{-1}(1-x) = \cos^{-1} x \] is:
The line parallel to the x-axis passing through the intersection of the lines \[ ax + 2by + 3b = 0 \quad \text{and} \quad bx - 2ay - 3a = 0 \] where \( (a,b) \neq (0,0) \), is:
Suppose \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + qx + r = 0 \) (with \( r \ne 0 \)) and they are in A.P. Then the rank of the matrix \[ \begin{pmatrix} \alpha & \beta & \gamma
\beta & \gamma & \alpha
\gamma & \alpha & \beta \end{pmatrix} \] is:
If the sum of the squares of the roots of the equation \[ x^2 - (a-2)x - (a+1) = 0 \] is least for an appropriate real parameter \( a \), then the value of \( a \) will be:
For what value of \( a \), the sum of the squares of the roots of the equation \[ x^2 - (a-2)x - a + 1 = 0 \] will have the least value?
If $f(x)$ satisfies the relation \[ f(x)=e^x+\int_0^1 (y+x e^x)f(y)\,dy, \] then $e+f(0)$ is equal to
If the equation \[ x^4-ax^2+9=0 \] has four real and distinct roots, then the least possible integral value of $a$ is
Let a function $f(x)$ satisfy \[ 3f(x)+2f\!\left(\frac{m}{19x}\right)=5x \] where $m=\sum_{i=1}^{9} i^2$. Find $f(5)+f(2)$.
The number of real roots of the equation $e^{4x} - e^{3x} - 4e^{2x} - e^x + 1 = 0$ is equal to ________.
The number of real roots of the equation $e^{6x} - e^{4x} - 2e^{3x} - 12e^{2x} + e^x + 1 = 0$ is :
Let α, β, γ be the real roots of the equation, x³ + a x² + b x + c = 0, (a, b, c ∈ R and a, b ≠ 0). If the system of equations (in u, v, w) given by α u + β v + γ w = 0; β u + γ v + α w = 0; γ u + α v + β w = 0 has non-trivial solution, then the value of a²/b is :