List of top Mathematics Questions on Theory of Equations asked in WBJEE JENPAS UG

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?