Mathematics · Quantitative Aptitude

Equations and Roots

69 Questions

Equations and roots questions cover finding real solutions to algebraic, trigonometric, and polynomial equations. They are fundamental for advanced mathematics sections. Practicing these problems ensures quick recognition of underlying patterns and calculation shortcuts.

polynomial equationssystem of equationstrigonometric equationsabsolute value equations

Equations and Roots Questions

Multiple choice

The equation (e^{i\pi} + 1 = 0) is known as:

  1. Euler's formula

  2. Fermat's Last Theorem

  3. Ramanujan's conjecture

  4. Goldbach's conjecture
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation (e^{i\pi} + 1 = 0) is known as Euler's formula, which is one of the most famous equations in mathematics.

Multiple choice

What is the name of the equation that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^2 - y^2 = n$?

  1. Pell's equation

  2. Fermat's Last Theorem

  3. Goldbach's conjecture

  4. Hardy-Littlewood conjecture
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Pell's equation is an equation that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^2 - y^2 = n$. It was first studied by the Indian mathematician Brahmagupta in the 7th century CE.

Multiple choice

What is the name of the equation that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^n - y^n = z^2$?

  1. Pell's equation

  2. Fermat's Last Theorem

  3. Goldbach's conjecture

  4. Hardy-Littlewood conjecture
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

Catalan's conjecture is an equation that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^n - y^n = z^2$. It was first proposed by Eugène Charles Catalan in the 19th century CE and remains unproven.

Multiple choice

What is the name of the equation that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^n + y^n = z^m$?

  1. Pell's equation

  2. Fermat's Last Theorem

  3. Goldbach's conjecture

  4. Hardy-Littlewood conjecture
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

Catalan's conjecture is an equation that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^n + y^n = z^m$. It was first proposed by Eugène Charles Catalan in the 19th century CE and remains unproven.

Multiple choice

Find all solutions of the equation (\sec^2\theta - 2\sec\theta - 3 = 0) in the interval ([0, 2\pi)).

  1. \(\theta = \frac{\pi}{3}, \frac{5\pi}{3}\)
  2. \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}\)
  3. \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}\)
  4. \(\theta = 0, \pi\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the identity (\sec^2\theta = 1 + \tan^2\theta), we can rewrite the equation as (1 + \tan^2\theta - 2\tan\theta - 3 = 0). Expanding and rearranging, we get (\tan^2\theta - 2\tan\theta - 4 = 0). Factoring, we find ((\tan\theta - 4)(\tan\theta + 1) = 0). Solving each factor separately, we find (\tan\theta = 4) or (\tan\theta = -1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{3}, \frac{5\pi}{3}).

Multiple choice

Find all solutions of the equation (2\sin^2\theta - 3\sin\theta + 1 = 0) in the interval ([0, 2\pi)).

  1. \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}\)
  2. \(\theta = \frac{\pi}{3}, \frac{2\pi}{3}\)
  3. \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}\)
  4. \(\theta = 0, \pi\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Factoring the equation, we get ((2\sin\theta - 1)(\sin\theta - 1) = 0). Solving each factor separately, we find (\sin\theta = \frac{1}{2}) or (\sin\theta = 1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{6}, \frac{5\pi}{6}).

Multiple choice

Find all solutions of the equation (\cot^2\theta - 3\cot\theta + 2 = 0) in the interval ([0, 2\pi)).

  1. \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}\)
  2. \(\theta = \frac{\pi}{3}, \frac{2\pi}{3}\)
  3. \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}\)
  4. \(\theta = 0, \pi\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Factoring the equation, we get ((\cot\theta - 2)(\cot\theta - 1) = 0). Solving each factor separately, we find (\cot\theta = 2) or (\cot\theta = 1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{6}, \frac{5\pi}{6}).

Multiple choice

Find all solutions of the equation (\csc^2\theta - 2\csc\theta - 3 = 0) in the interval ([0, 2\pi)).

  1. \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}\)
  2. \(\theta = \frac{\pi}{3}, \frac{2\pi}{3}\)
  3. \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}\)
  4. \(\theta = 0, \pi\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the identity (\csc^2\theta = 1 + \cot^2\theta), we can rewrite the equation as (1 + \cot^2\theta - 2\csc\theta - 3 = 0). Expanding and rearranging, we get (\cot^2\theta - 2\csc\theta - 2 = 0). Factoring, we find ((\cot\theta - 2)(\cot\theta + 1) = 0). Solving each factor separately, we find (\cot\theta = 2) or (\cot\theta = -1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{6}, \frac{5\pi}{6}).

Multiple choice

Find the number of solutions to the equation x^2 + 2x + 1 = 0 in the field of complex numbers.

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation x^2 + 2x + 1 = 0 is a quadratic equation. Using the quadratic formula, we have: x = (-2 ± √(2^2 - 4 * 1 * 1)) / (2 * 1) = -1 ± √3i. Therefore, there are two solutions to the equation in the field of complex numbers.