List of top Mathematics Questions on Algebra

Which of the following numbers is a perfect square?

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

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4 , then the sum of its first twelve terms is

The mean of some observations is 54. If each observation is increased by 8 and its sum is divided by 2, then what is the mean of the resulting observations?

Out of 8 students in Section Alpha, 10 students in Section Beta and 12 students in Section Gamma, a teacher wants to create a team of three to represent the school in inter-school quiz competition. In how many ways the team can be formed such that all the members of the team are not from the same section ?

A certain sum of money is borrowed by a person at 8% simple interest for 4 years. If he has to pay Rs. 7834 as interest, what is the total amount he has to pay?

If \( \mathbf{x} \) and \( \mathbf{y} \) are positive integers then the following is always true?
{2x - 3y < 0
(1) x = (y - 1)
(2) x > y

If a merchant offers a discount of 40% on the marked price of his goods and thus ends up selling at cost price, what was the % mark up?

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.

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

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:

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:

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

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