List of top Mathematics Questions on Algebra asked in JEE Main

Let \( f(x) = \begin{cases} x^3 + 8 & x < 0 \\ x^2 - 4 & x \ge 0 \end{cases} \) and \( g(x) = \begin{cases} (x-8)^{1/3} & x < 0 \\ (x+4)^{1/2} & x \ge 0 \end{cases} \) then find number of points of discontinuity of \( g(f(x)) \).

Let \(A = \{-2, -1, 0, 1, 2\}\). A relation \(R\) is defined on set \(A\) such that \(aRb \Rightarrow 1 + ab > 0\).
Statement-1: It is an equivalence relation.
Statement-2: Number of elements in \(R\) is 17.

Let $A = \begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ satisfy $A^2 + \alpha(\text{adj}(\text{adj}(A))) + \beta(\text{adj}(A)(\text{adj}(\text{adj}(A)))) = \begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix}$ for some $\alpha, \beta \in \mathbb{R}$.
Then $(\alpha - \beta)^2$ is equal to __________.

If \(\alpha = 1\) and \(\beta = 1 + i\sqrt{2}\), where \(i = \sqrt{-1}\) are two roots of the equation
\(x^3 + ax^2 + bx + c = 0, a, b, c \in \mathbb{R}\), then \(\int_{-1}^{1} (x^3 + ax^2 + bx + c) dx\) is equal to:

If the set of all solutions of \(|x^2 + x - 9| = |x| + |x^2 - 9|\) is \([\alpha, \beta] \cup [\gamma, \infty)\), then \((\alpha^2 + \beta^2 + \gamma^2)\) is equal to:

If \( \sum_{k=1}^{n} \varphi(k) = \frac{2\pi (n+1)}{} \), then \( \sum_{k=1}^{10} \frac{1}{\varphi(k)} \) is equal to:

The value of \( k \) for which each root of the equation \( x^2 - 6kx + 2k - 9k^2 = 0 \) is greater than 3, always satisfy the inequality:

If in the binomial expansion of \( (1 - x)^m(1 + x)^n \), the coefficients of \( x \) and \( x^2 \) are respectively 3 and -4, then the ratio \( m : n \) is equal to:

If \(z_1,z_2,z_3\) are roots of \[ x^3+ax^2+bx+c=0 \] Let \(z_1=1,\; z_2=1+i\sqrt2\) and \(a,b,c\in\mathbb{R}\). Then the value of \[ \int_{-1}^{1}(x^3+ax^2+bx+c)\,dx \] is

If \[ \alpha=\frac14+\frac18+\frac1{16}+\cdots \text{ up to infinity} \] \[ \beta=\frac13+\frac19+\frac1{27}+\cdots \text{ up to infinity} \] Then value of \[ (0.2)^{\log_5\alpha}+(0.04)^{\log_5\beta} \] is

If \(f(x)\) satisfies the functional equation \[ f(x+y)=f(x)+2y^2+y+\alpha xy \] where \(x,y\) are whole numbers, such that \(f(0)=-1\) and \(f(1)=2\), then the value of \[ \sum_{i=1}^{5}\big(f(i)+\alpha\big) \] is

If the quadratic equation \[ (\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0 \quad (\lambda \ne -2) \] has two positive roots then the number of possible integral values of \(\lambda\) is

If the system of equations \[ x + y + z = 5 \] \[ x + 2y + 3z = 9 \] \[ x + 3y + \lambda z = \mu \] has infinitely many solutions, then value of \( \lambda + \mu \) is

If the matrix \[ \begin{bmatrix} 1 & 3 & 1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix} \] is singular. Given a function \( f(x) = \int_{0}^{x} (t^2 + 2t + 3)\, dt \), \( \forall x \in [1, \alpha] \). If \( m \) and \( n \) are the maximum and minimum values of the function \( f(x) \), then the value of \( 3(m - n) \) is:

Let \( f : \mathbb{R} \to \mathbb{R} \), \( f(x) = \frac{2x^2 - 3x + 2}{3x^2 + x + 3} \), then \( f(x) \) is

Given that quadratic equation \( (k^2 - 15k + 27) x^2 + 9(k - 1)x + 18 = 0 \) has one root twice of other. Then find the length of the latus rectum of the parabola \( y^2 = 6kx \):

Consider \( e_1 \) and \( e_2 \) be roots of the equation \( x^2 - a x + 2 = 0 \). Set of exhaustive values of 'a' for which \( e_1 \) and \( e_2 \) are eccentricities of hyperbolas then \( a \in [\alpha, \beta) \) and set of values of 'a' for which \( e_1 \) and \( e_2 \) are eccentricity of the parabola and ellipse is \( (\gamma, \infty) \) then \( (\alpha^2 + \beta^2 + \gamma^2) \) equal:

If domain of \( f(x) = \sin^{-1} \left( \frac{x + |x|}{3} \right) \) is \( [\alpha, \beta) \), then \( (\alpha^2 + \beta^2) \) is:

Let the set of all values of \( K \in \mathbb{R} \) such that the equation, \( z(\bar{z} + 2 + i) + K(2 + 3i) = 0 \), \( z \in \mathbb{C} \) has at least one solution, be the interval \( [\alpha, \beta] \). Then \( 9(\alpha + \beta) = \)

If \( \tan^{-1}(1-\alpha) + \tan^{-1}(1-\beta) = \frac{\pi}{4} \) & \( \alpha = \frac{1}{\beta} \) then find \( |\alpha + \beta| \):

The number of four letter words which can be formed using two vowels and two consonants from the word INCONSEQUENTIAL (words can be meaningful or meaningless) is:

If coefficients of middle terms in the expansion \( (1 + \alpha x)^{26}\) & \( (1 - \alpha x)^{28} \) are equal then \( \alpha \) is:

The value of \( 1^3 - 2^3 + 3^3 - 4^3 + \cdots - 14^3 + 15^3 \) is equal to:

\( a_1, a_2, a_3, \dots, a_n \) are in A.P. and sum of first 10 terms is 160. \( g_1, g_2, g_3, \dots, g_n \) are in G.P. where \( g_1 + g_2 = 8 \). If the first term of A.P. is equal to common ratio of G.P. and first term of G.P. is equal to common difference of A.P., then sum of all possible values of \( g_1 \) is equal to:

Let \( S = \{\theta \in (-2\pi, 2\pi): \cos \theta + 1 = \sqrt{3} \sin \theta\} \). Then \( \sum_{\theta \in S} \theta \) is equal to: