Mathematics · Quantitative Aptitude

Equations and Roots

65 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 maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

The equation $x-\dfrac {8}{|x-3|}=3--\dfrac {8}{|x-3|}$ has

  1. Only one solution

  2. infinite solution

  3. no solution

  4. two solution
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Consider the following equation.

$ x-\dfrac{8}{|x-3|}=3-\dfrac{8}{|x-3|} $

$ x-\dfrac{8}{\pm \left( x-3 \right)}=3-\dfrac{8}{\pm \left( x-3 \right)} $

$ x-\dfrac{8}{\left( x-3 \right)}=3-\dfrac{8}{\left( x-3 \right)} $

$ x=3 $

 $ x+\dfrac{8}{\left( x-3 \right)}=3+\dfrac{8}{\left( x-3 \right)} $

$ x=3 $

Hence, this is the correct answer.

Multiple choice de moivre’s theorem and its applications demoivre's theorem complex numbers maths

The number of solutions of equation $z^{10}-z^{5}+1=0$ are 

  1. only two solution

  2. No solution

  3. only five solution

  4. exactly 10
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

${ z }^{ 10 }-{ z }^{ 5 }+1=0$

Let ${ z }^{ 5 }=w$
$\Rightarrow { w }^{ 2 }-w+1=0$

$\Rightarrow w=\frac { 1\pm \sqrt { 3 }  }{ 2 } =\cos { \frac { \Pi  }{ 3 }  } \pm \sin { \frac { \Pi  }{ 3 }  } =cis\left( \pm \frac { \Pi  }{ 3 }  \right) $


$\Rightarrow { z }^{ 5 }=cis\left( \pm \frac { \Pi  }{ 3 }  \right) $

Case 1:
${ z }^{ 5 }=cis\left( \frac { \Pi  }{ 3 }  \right) $

$\Rightarrow z={ \left( cis\left( \frac { \Pi  }{ 3 }  \right)  \right)  }^{ \frac { 1 }{ 5 }  }=cis\left( \frac { 2k\Pi +\Pi  }{ 15 }  \right) \ $        ...{De Moivre's Theorem}

Where k=0,1,2,3,4.
Therefore number of solutions are 5.

Case 2:
${ z }^{ 5 }=cis\left( -\frac { \Pi  }{ 3 }  \right) $

$\Rightarrow z={ \left( cis\left( -\frac { \Pi  }{ 3 }  \right)  \right)  }^{ \frac { 1 }{ 5 }  }=cis\left( \frac { 2k\Pi -\Pi  }{ 15 }  \right) $       ...{De Moivre's Theorem}

Where k=0,1,2,3,4.
Therefore number of solutions are 5.

From case 1 & case 2 total number of solutions of equation ${ z }^{ 10 }-{ z }^{ 5 }+1=0$ are 10.

Ans: D

Multiple choice

What is the name of the theorem that Shorey proved in 1973, which provides a lower bound for the number of solutions to the Thue equation $x^m - y^n = c$?

  1. Shorey's Theorem

  2. Baker's Theorem

  3. Siegel's Theorem

  4. Fermat's Last Theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Shorey's Theorem states that the number of solutions to the Thue equation $x^m - y^n = c$ is at least $c^{1/m} + c^{1/n} - 1$.

Multiple choice

In 1982, Shorey and what other mathematician proved that the Diophantine equation $x^2 - Dy^2 = 4$ has infinitely many solutions for any square-free integer $D > 0$?

  1. S. S. Pillai

  2. K. S. Nagaraja

  3. R. Balasubramanian

  4. M. N. Gopalan
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Shorey and Balasubramanian proved this result by using a method based on modular forms and the theory of quadratic forms.

Multiple choice

What is the name of the theorem that Shorey and R. Tijdeman proved in 1986, which provides a lower bound for the number of solutions to the equation $x^m + y^n = z^k$?

  1. The Shorey-Tijdeman Theorem

  2. The Baker-Tijdeman Theorem

  3. The Siegel-Tijdeman Theorem

  4. The Fermat-Tijdeman Theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Shorey-Tijdeman Theorem states that the number of solutions to the equation $x^m + y^n = z^k$ is at least $c^{1/m} + c^{1/n} + c^{1/k} - 3$.

Multiple choice

In 2000, Shorey and what other mathematician proved that the Diophantine equation $x^2 - Dy^2 = 5$ has infinitely many solutions for any square-free integer $D > 0$?

  1. S. S. Pillai

  2. K. S. Nagaraja

  3. R. Balasubramanian

  4. M. N. Gopalan
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Shorey and Gopalan proved this result by using a method based on modular forms and the theory of quadratic forms.

Multiple choice

What is the name of the theorem that Shorey and T. N. Venkataramana proved in 2005, which provides a lower bound for the number of solutions to the equation $x^m + y^n = z^k$ in positive integers?

  1. The Shorey-Venkataramana Theorem

  2. The Baker-Venkataramana Theorem

  3. The Siegel-Venkataramana Theorem

  4. The Fermat-Venkataramana Theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Shorey-Venkataramana Theorem states that the number of solutions to the equation $x^m + y^n = z^k$ in positive integers is at least $c^{1/m} + c^{1/n} + c^{1/k} - 4$.

Multiple choice

In 2020, Shorey and what other mathematician proved that the Diophantine equation $x^2 - Dy^2 = 6$ has infinitely many solutions for any square-free integer $D > 0$?

  1. S. S. Pillai

  2. K. S. Nagaraja

  3. R. Balasubramanian

  4. M. N. Gopalan
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Shorey and Balasubramanian proved this result by using a method based on modular forms and the theory of quadratic forms.

Multiple choice

What is the genus of the Riemann surface defined by the equation $y^2 = x^3 + x + 1$?

  1. 0

  2. 1

  3. 2

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

The genus of a Riemann surface is a topological invariant that is related to the number of holes in the surface. In this case, the Riemann surface defined by the equation $y^2 = x^3 + x + 1$ has one hole, so its genus is 1.

Multiple choice

What is the dimension of the variety defined by the equation $x^2 + y^2 + z^2 = 1$?

  1. 0

  2. 1

  3. 2

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

The dimension of a variety is the number of independent variables that are needed to parameterize it. In this case, the variety defined by the equation $x^2 + y^2 + z^2 = 1$ is a sphere, which is a two-dimensional surface. So its dimension is 2.

Multiple choice

Let $f(x) = x^3 - 3x^2 + 2x + 1$. Find the number of real solutions of the equation $f(x) = f(f(x))$.

  1. 1

  2. 2

  3. 3

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

Let $y = f(x)$. Then the equation $f(x) = f(f(x))$ becomes $y = f(y)$. This is a quadratic equation in $y$. Solving for $y$, we get $y = 1$ or $y = 2$. Therefore, there are two real solutions of the equation $f(x) = f(f(x))$.

Multiple choice

Let $f(x) = x^3 - 3x^2 + 2x + 1$. Find the number of real solutions of the equation $f(x) = f(f(x))$.

  1. 1

  2. 2

  3. 3

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

Let $y = f(x)$. Then the equation $f(x) = f(f(x))$ becomes $y = f(y)$. This is a quadratic equation in $y$. Solving for $y$, we get $y = 1$ or $y = 2$. Therefore, there are two real solutions of the equation $f(x) = f(f(x))$.

Multiple choice

Let $f(x) = x^3 - 3x^2 + 2x + 1$. Find the number of real solutions of the equation $f(x) = f(f(x))$.

  1. 1

  2. 2

  3. 3

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

Let $y = f(x)$. Then the equation $f(x) = f(f(x))$ becomes $y = f(y)$. This is a quadratic equation in $y$. Solving for $y$, we get $y = 1$ or $y = 2$. Therefore, there are two real solutions of the equation $f(x) = f(f(x))$.

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.