List of top Mathematics Questions on Algebra

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:

Consider the following:

Assertion (A): \(1^2 + 3^2 + 5^2 + \cdots + 39^2 = 10760\)

Reason (R): The sum of the squares of the first \(n\) odd numbers is \[ \frac{n(4n^2-1)}{3}. \]

The correct answer is:

If \(A\) and \(B\) are the entire domain and range, respectively, of the real-valued function \[ f(x)=\cos^{-1}\!\left(\frac{2-x^2}{2+x^2}\right), \] then \(A\cap B=\)

\(\coth^{-1}2=\)

The function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x|x+2| \) is:

For what value of k, $x-3y=7$ and $kx+6y=5$ will have no solution?

Let \( A \) be the set of even natural numbers less than 8 and \( B \) be the set of prime numbers less than 7. The number of relations from \( A \) to \( B \) is

The focus of the parabola \( y^2 = 16x \) is

The points on the curve \( y = 2x^3 + 3x^2 - 8x \) where the tangents are parallel to the line \( y = 4x + 3 \) are

If a, b, c are positive numbers then value of \((a+b+c) \left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right)\), is

The altitude drawn to the base of an isosceles triangle is 8cm and the perimeter is 32cm. Find the area of the triangle?

Two cones have their heights in the ratio 1:3 and the radii of their bases in the ratio 3:1. Find the ratio of their volumes.

If $\alpha, \beta$ are vectors in an inner product space V, then

Let V be an inner product space and $S = \{\alpha_{1}, \alpha_{2}, \dots, \alpha_{m}\}$ be a finite subset of V. If S is an orthonormal set, then consider the following statements:
I: $||\alpha_{i}|| = 1$ for each $\alpha_{i} \in S$
II: $(\alpha_{i}, \alpha_{j}) = 0$ for $\alpha_{i}, \alpha_{j} \in S, i \neq j$.
Which of the following is correct?

For the vectors $u = (a,b), v = (c,d)$ in $C^{2}$ the inner product of $u$ and $v$ is defined by $\langle u,v \rangle = a\bar{c} + b\bar{d}$. If $u = (1+i, i), v = (i, 1-i)$ then $\langle u,v \rangle = $

The necessary condition to diagonalize a matrix is that

Which of the following matrices is NOT diagonalizable?

The matrix $\left[\begin{matrix}1& 4\\ 3& 2\end{matrix}\right]$ is

If $\alpha = (-1,0,1), \beta = (2,0,-2) \in V_{3}(R)$ an inner product space, then $||\alpha + \beta|| = $

For what values of a and b the following simultaneous equations have an infinite number of solutions? $x + y + z = 5, x + 3y + 3z = 9$, and $x + 2y + \alpha z = \beta$.

The equation $\left|\begin{matrix}2& 1& 1\\ 1& 1&-1\\ y& x^{2}& x\end{matrix}\right|=0$ represents a parabola passing through the points}

If A and B are square matrices of size $n \times n$, then which of the following statement is not true.

The characteristic equation associated with the matrix $\left[\begin{matrix}0& 0& 3\\ 1& 0& 2\\ 0& 1& 1\end{matrix}\right]$ is

The determinant of a skew-symmetric matrix of odd order is