List of top Mathematics Questions on Algebra

On the set \(\mathbb{R}\) of real numbers the relation \(\rho\), defined by \(x\rho y\) \((x,y\in\mathbb{R})\) iff:

If the equation \[ x^4-10x^3+37x^2-60x+36=0 \] has two distinct real roots, where each one of them is a repeated root, then the sum of squares of all the roots of the given equation is

If all the roots of the equation \[ x^5-3x^4-5x^3+27x^2-32x+12=0 \] are diminished by $h$ to get a transformed equation in which the constant term is missing, then the sum of the squares of all possible values of $h$ is

If $\alpha,\beta$ $(\alpha<\beta)$ are the roots of \[ 2x^2-x-6=0 \] and \[ \alpha x^2+kx-\beta\leq0 \quad \forall x\in\mathbb{R}, \] then the number of integral values $k$ takes is

The quadratic expression having zero's equal to the non-integral roots of \( ||3x-4|-6|=5 \) is:

Consider the quadratic expression \( f(x) = x^2 + (10-a)x - 10a \). The sum of all values of 'a', such that the roots of \( f(x)=3 \) are integers is:

For a quadratic expression \( ax^{2}+bx+c \), if the minimum value \( \frac{49}{12} \) exists at \( x=\frac{-5}{6} \), then \( 12c-5b = \)

If \( a, b, c \in \mathbb{R} \) and \( -a^{2}x^{2}+bx+c>0 \quad \forall x \in \left(\frac{3-\sqrt{14}}{2},\frac{3+\sqrt{14}}{2}\right) \), then \( c^{2}-\left(\frac{b}{4}\right)^{2} = \)

The expression \( \frac{n(n + 1)^2(n+2)}{12} \) for all \( n \in \mathbb{N} \) is always:

If $\alpha,\beta,\gamma$ are the roots of \[ x^3-10x^2+7x+8=0, \] match List-I with List-II and choose the correct option.

If $\alpha \neq \beta$ and $\alpha^2=5\alpha-3,\ \beta^2=5\beta-3$, then the equation whose roots are $\dfrac{\alpha}{\beta}$ and $\dfrac{\beta}{\alpha}$ is

If the roots of the equation \[ x^3-px^2+qx-s=0 \] are in geometric progression, then

If $\alpha,\beta$ are the roots of the equation \[ x^2-p(x+1)-c=0, \] then \[ \frac{\alpha^2+2\alpha+1}{\alpha^2+2\alpha+c} + \frac{\beta^2+2\beta+1}{\beta^2+2\beta+c} = \]

If the sum of two roots of the cubic equation $x^3 - 5x^2 - 2x + 24 = 0$ is $2$, then the roots of the equation are:

If the roots of the equation $x^3 - 7x^2 + 14x - 8 = 0$ are in geometric progression, then the common ratio can be:

If $\alpha, \beta$ are the roots of the quadratic equation $x^2 - 2x + 4 = 0$, then the value of $\alpha^n + \beta^n$ is:

If the roots of the quadratic equation $x^2 - 2px + q^2 = 0$ are real and distinct, then:

If the sum of two roots of the cubic equation $x^3 - 5x^2 - 2x + 24 = 0$ is $2$, then the roots of the equation are:

If $\frac{3x + 4}{(x-1)(x-2)^2} = \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{(x-2)^2}$, then $A + B + C =$

The point of intersection of \(\vec{r}\times\vec{a}=\vec{b}\times\vec{a}\) and \(\vec{r}\times\vec{b}=\vec{a}\times\vec{b}\), where \(\vec{a}=\hat{i}+\hat{j}\) and \(\vec{b}=2\hat{i}-\hat{k}\) is:

If \(a=\lim_{n\rightarrow\infty}\cos^{2n}x\), \((x=n\pi)\) and \(b=\lim_{n\rightarrow\infty}\cos^{2n}x\), \((x\ne m\pi)\), then numerical value of the area of the triangle whose vertices are \((a,b)\), \((-2,1)\) and \((2,1)\) is:

Let \(A=[a,\infty)\) denotes the domain, then \(f:[a,\infty)\rightarrow B\) which is defined by \(f(x)=2x^{3}-3x^{2}+6\) will have an inverse for the smallest real value of \(a\) if:

A mapping is selected at random from all mappings \(f:A\rightarrow A\) where set \(A=\{1,2,3,\dots,n\}\). If the probability that the mapping is injective is \(\frac{3}{32}\), then the value of \(n\) is:

If the locus of mid point of any normal chord of the parabola \(y^{2}=4x\) is \(x-\lambda=\frac{\mu}{y^{2}}+\frac{y^{2}}{\nu}\), \(\lambda,\mu,\nu\in\mathbb{N}\), then \((\lambda+\mu+\nu)\) equals to:

If \(0<\alpha<\beta<\gamma<\frac{\pi}{2}\) then the equation \(\frac{1}{x-\sin\alpha}+\frac{1}{x-\sin\beta}+\frac{1}{x-\sin\gamma}=0\) has: