Algebra

Find the solution set of $\frac{x}{2} + 2 > 2$ for $x \in \mathbb{Z}^-$ (Negative Integers).

Solve $x - 1 \geq -1$ for $x \in \mathbb{N}$.

Given $x \in \{2, 4, 6\}$, solve the inequation $3x + 1 \leq 10$.

Identify the solution set for $15 - x > 12$ where $x \in \mathbb{W}$.

Find the largest integer value of $x$ satisfying $\frac{x}{4} - \frac{2}{3} \leq \frac{1}{6}$.

Solve $5 - \frac{x}{2} \leq 3$ for $x \in \mathbb{R}$.

Solve $2(x - 3) + 5 < 9$ for $x \in \mathbb{W}$.

If $x \in \{-2, -1, 0, 1, 2\}$, solve the double inequation $2x - 3 < 3x - 2 < 4x - 1$.

Determine the solution set of $4x - 10 \le \frac{12 - 2x}{3}$ for $x \in \mathbb{N}$.

Find the number of elements in the intersection of solution sets $x > -2$ and $x \le 3$ for $x \in \mathbb{W}$ (Whole numbers).

The roots of the equation $3x^2 - 27 = 0$ are:

Find the value of $c$ for the equation $x^2 - 5x = -6$ when written in standard form.

Convert $(x-2)(x-5) = 0$ into the form $ax^2 + bx + c = 0$.

If $D < 0$, the roots of the quadratic equation are:

If the roots of the equation $(b-c)x^2 + (c-a)x + (a-b) = 0$ are real and equal, which of the following is true?

A bus travels 440 km at a uniform speed. If the speed had been 8 km/h more, it would have taken 30 minutes less for the same journey. Find the original speed of the bus.

The sum of a number and 4 times its reciprocal is 5. Find the numbers.

Find the values of $x$ that satisfy the equation $2(x^2 + \frac{1}{x^2}) - 9(x + \frac{1}{x}) + 14 = 0$.

The sum of the squares of two consecutive natural numbers is $313$. Find the numbers.

If the roots of the quadratic equation $ax^2 + bx + c = 0$ are $\alpha$ and $\beta$, find the equation whose roots are $1/\alpha$ and $1/\beta$.

If $\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{2}{3}$, find the value of $\frac{a+c+e}{b+d+f}$.

The sub-duplicate ratio of $x^2:y^2$ is:

What is the inverse ratio of $3:2:1$?

Find the duplicate ratio of $1:2$.

If $x:y = 3:5$, find the ratio $(x^2+y^2):xy$.

If $x, 12, 16, 24$ are in proportion, find the value of $x$.

What is the sub-triplicate ratio of $27:64$?

If $a, b, c, d$ are in continued proportion, then the expression $(b-c)^2 + (c-a)^2 + (d-b)^2$ is equal to:

If $x = \frac{\sqrt{a+3b} + \sqrt{a-3b}}{\sqrt{a+3b} - \sqrt{a-3b}}$, then $3bx^2 - 2ax + 3b$ equals:

Solve for $x$: $\frac{\sqrt[3]{x+1} + \sqrt[3]{x-1}}{\sqrt[3]{x+1} - \sqrt[3]{x-1}} = 2$.

If $(x - k)$ is a factor of $x^2 - 5x + 6$ and $k > 2$, find the value of $k$.

Find the remainder when $x^2 + 8x + 15$ is divided by $(x + 5)$.

Find the remainder when $x^3 - 27$ is divided by $(x - 3)$.

What is the remainder when $x^2 - 1$ is divided by $(x - 1)$?

If $x+2$ is a common factor of $x^2 + ax + b$ and $x^2 + bx + a$, where $a \neq b$, find the value of $a+b$.

Factorize $x^3 - 3x^2 - 9x - 5$ completely.

If $(x-2)$ is a factor of $x^2 + ax + b$ and $a+b = 1$, find $a$ and $b$.

A cubic polynomial $P(x)$ is exactly divisible by both $(x - 1)$ and $(x - 2)$. Given that $P(0) = -4$ and $P(-1) = -30$, find the remainder when $P(x)$ is divided by $(x - 3)$.

If the expression $x^4 + 4x^3 + ax^2 + bx + 24$ is exactly divisible by $x^2 - x - 6$, find the values of $a$ and $b$.

When $x^3 + ax^2 + bx + 1$ is divided by $(x - 1)$, the remainder is 12, and when divided by $(x + 1)$, the remainder is 6. Find $a^2 + b^2$.

What is the transpose of a null matrix?

If $A = \begin{bmatrix} 6 & 8 \end{bmatrix}$, then $\frac{1}{2}A$ is:

In a $2 \times 2$ matrix, the elements $a_{11}$ and $a_{22}$ form the:

If $A = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 \end{bmatrix}$, the order of product $AB$ (if defined) is:

Find $x$ if 2 $$\begin{bmatrix} x \\ 3 \end{bmatrix}$$ + $$\begin{bmatrix} 4 \\ -1 \end{bmatrix}$$ = \begin{bmatrix} 10 \\ 5 \end{bmatrix}.

If $A$ is a $1 \times 2$ matrix and $B$ is a $2 \times 1$ matrix, then $AB$ is a matrix of order:

What is the result of $\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$- \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$?

If $\begin{bmatrix} 2 & x \\ -1 & 2 \end{bmatrix}$\begin{bmatrix} y \ 3 \end{bmatrix}$= \begin{bmatrix} 10 \\ 4 \end{bmatrix}$, find the value of $x+y$.

Find the matrix $X$ such that $\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} X = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.

If $A = \begin{bmatrix} 3 & 1 \\ 0 & 3 \end{bmatrix}$, find the matrix $A^2 - 6A + 9I$.

If $a = 10$ and $d = 10$, what is the $4^{th}$ term?

Which term of the AP $2, 4, 6, 8, \dots$ is $50$?

Find the common difference of the AP: $10, 6, 2, -2, \dots$

The first term of an AP is $4$ and the last term is $28$. If there are $5$ terms, find the sum of the AP.

Find the sum of the first $10$ multiples of $7$.

If the $9^{th}$ term of an AP is $499$ and the $499^{th}$ term is $9$, find its $508^{th}$ term.

What is the common difference of an AP in which $a_{21} - a_7 = 84$?

In an AP, the $10^{th}$ term is $\frac{1}{20}$ and the $20^{th}$ term is $\frac{1}{10}$. Find the sum of the first 200 terms.

Divided 20 into four parts which are in AP such that the ratio of the product of the first and fourth to the product of the second and third is $2 : 3$. Find the common difference $d$ (where $d > 0$).

If $x+1, 3x, 4x+2$ are in Arithmetic Progression, find the value of $x$.

In a Geometric Progression, if the first term is $20$ and the common ratio is $0.1$, find the $2^{nd}$ term.

Find the $3^{rd}$ term of a Geometric Progression if the first term is $2$ and the common ratio is $5$.

If $x, 15, 75$ are three consecutive terms of a Geometric Progression, find the value of $x$.

What is the sum of the first two terms of the Geometric Progression: $\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \dots$?

Insert two geometric means between $3$ and $81$.

If $x, y, z$ are in Geometric Progression, then the terms $x^2+y^2, xy+yz, y^2+z^2$ are in:

In a GP, the sum of the $1^{st}$ and $3^{rd}$ terms is $10$, and the sum of the $2^{nd}$ and $4^{th}$ terms is $20$. Find the common ratio.

If the ratio of the $5^{th}$ term to the $2^{nd}$ term of a Geometric Progression is $125$, find the common ratio.

If $S_n$ denotes the sum of $n$ terms of a GP, find $\frac{S_{3n} - S_{2n}}{S_n}$.

If $x, y, z$ are in Geometric Progression and $x^a = y^b = z^c$, then $1/a, 1/b, 1/c$ are in:

The centroid of a triangle with vertices $(0,0)$, $(6,0)$, and $(0,6)$ is:

Two lines are perpendicular if the product of their slopes $m_1$ and $m_2$ is:

The coordinates of point $C$ which divides the join of $A(1, 2)$ and $B(4, 5)$ in the ratio $1:2$ internally are:

Find the equation of a line parallel to the x-axis and passing through $(2, -5)$.

The line $y = \sqrt{3}x + 5$ makes an angle $\theta$ with the positive x-axis. Find $\theta$.

Determine the reflection of the point $(2, 3)$ in the line $x = 4$.

The line $x - y = 1$ intersects the line $2x + 3y = 12$ at point $P$. Find the coordinates of $P$.

A triangle has vertices $P(2, 1)$, $Q(4, 5)$, and $R(-2, 3)$. Find the equation of the median through vertex $R$.

Point $A(a, b)$ is reflected in the line $y = 3$ to $A'(a, 8)$. Point $A$ is also reflected in the line $x = 2$ to $A''(-2, b)$. Calculate the distance between $A'$ and $A''$.

A road is modeled by the equation $2x - 3y + 9 = 0$. A person at point $M(1, 1)$ wants to build a path to the road that is as short as possible. Find the coordinates of the point $N$ on the road where this path meets it.