List of top Mathematics Questions

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - x + 2 = 0 \) with \( \text{Im}(\alpha) > \text{Im}(\beta) \). Then \( \alpha^6 + \alpha^4 + \beta^4 - 5 \alpha^2 \) is equal to

Let \( S = \{ z \in \mathbb{C} : |z - 1| = 1 \} \) and \((\sqrt{2} - 1)(z + \overline{z}) - i(z - \overline{z}) = 2\sqrt{2}.\)
Let \( z_1, z_2 \in S \) be such that \( |z_1| = \max_{z \in S} |z| \) and \( |z_2| = \min_{z \in S} |z| \).  
Then \( \sqrt{2}|z_1 - z_2|^2 \) equals:

The sum of the square of the modulus of the elements in the set \[ \{z = a + ib : a, b \in \mathbb{Z}, z \in \mathbb{C}, |z-1| \leq 1, |z-5| \leq |z-5i|\} \] is ________.

Let the complex numbers \( \alpha \) and \( \frac{1}{\alpha} \) lie on the circles \[ |z - z_0|^2 = 4 \] and \[ |z - z_0|^2 = 16 \] respectively, where \( z_0 = 1 + i \). Then, the value of \( 100 |\alpha|^2 \) is ....

Let \( z_1 \) and \( z_2 \) be two complex numbers such that \( z_1 + z_2 = 5 \) and \( z_1^3 + z_2^3 = 20 + 15i \). Then \( \left| z_1^4 + z_2^4 \right| \) equals

Let \( z \) be a complex number such that the real part of \[ \frac{z - 2i}{z + 2i} \] is zero. Then, the maximum value of \( |z - (6 + 8i)| \) is equal to:

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - \sqrt{6}x + 3 = 0 \) such that \( \operatorname{Im}(\alpha) > \operatorname{Im}(\beta) \). Let \( a, b \) be integers not divisible by 3 and \( n \) be a natural number such that\[\frac{\alpha^{99}}{\beta} + \alpha^{98} = 3^n (a + ib), i = \sqrt{-1}.\]Then \( n + a + b \) is equal to ______.

If \( z = \frac{1}{2} - 2i \), is such that \( |z + 1| = \alpha z + \beta (1 + i) \), \( i = \sqrt{-1} \) and \( \alpha, \beta \in \mathbb{R} \), then \( \alpha + \beta \) is equal to:

Let \( \alpha \) and \( \beta \) be the sum and the product of all the non-zero solutions of the equation \[(\bar{z})^2 + |z| = 0, \quad z \in \mathbb{C}.\] Then \( 4(\alpha^2 + \beta^2) \) is equal to:

Let $S_1 = \{z \in \mathbb{C} : |z| \leq 5\}$,
$S_2 = \left\{z \in \mathbb{C} : \text{Im}\left(\frac{z + 1 - \sqrt{3}i}{1 - \sqrt{3}i}\right) \geq 0\right\}$ and
$S_3 = \{z \in \mathbb{C} : \text{Re}(z) \geq 0\}$. Then

If \( \alpha \) denotes the number of solutions of \( |1 - i|^x = 2^x \) and \( \beta = \frac{|z|}{\arg(z)} \), where \[ z = \frac{\pi}{4} (1 + i)^4 \left( \frac{1 - \sqrt{\pi} i}{\sqrt{\pi} + i} + \frac{\sqrt{\pi} - i}{1 + \sqrt{\pi} i} \right), \quad i = \sqrt{-1}, \] then the distance of the point \( (\alpha, \beta) \) from the line \( 4x - 3y = 7 \) is _____

If \(r = |z|,\ θ = arg(z)\) and \( z = 2 – 2\ tan (\frac {5\pi}{8})\) then find \((r, θ)\).

Let \( r \) and \( \theta \) respectively be the modulus and amplitude of the complex number \( z = 2 - i \left( 2 \tan \frac{5\pi}{8} \right) \), then \( (r, \theta) \) is equal to

If \(\alpha\) satisfies the equation \(x^2 + x + 1 = 0\) and \((1 + \alpha)^7 = A + B \alpha + C \alpha^2\), \(A, B, C \geq 0\), then \(5(3A - 2B - C)\) is equal to ______.

If $z_1, z_2$ are two distinct complex numbers such that \[ \frac{|z_1 - 2z_2|}{\left| \frac{1}{2} - z_1 \overline{z_2} \right|} = 2, \] then

Let $z$ be a complex number such that $|z + 2| = 1$ and $\text{Im}\left(\frac{z+1}{z+2}\right) = \frac{1}{5}$. Then the value of $|\text{Re}(z+2)|$ is:

The sum of all possible values of \(\theta \in [-\pi, 2\pi]\), for which \[ \frac{1 + i \cos\theta}{1 - 2i \cos\theta} \] is purely imaginary, is equal to:

The function \( f(x) = \frac{x}{x^2 - 6x - 16} \), \( x \in \mathbb{R} - \{-2, 8\} \)

Consider the function \(f :[\frac{1}{2},1]\)⇢R defined by  \(f(x)=4\sqrt2x^3-3\sqrt2x-1\).Consider the statements
(1)The curve y=f(x) intersect the x-axis exactly at one point
(2)The curve y=f(x) intersect the x-axis at \(x=cos\frac{\pi}{12}\)
Then 

If the function \( f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geq 2 \\ ax^2 + 2b, & |x| < 2 \end{cases} \) is differentiable on \( \mathbb{R} \), then \( 48 (a + b) \) is equal to \(\_\_\_\_\_\_\_\_\_\).

Consider the function \( f : (0, 2) \to \mathbb{R} \) defined by \[ f(x) = \frac{x}{2} + \frac{2}{x} \] and the function \( g(x) \) defined by \[ g(x) = \begin{cases} \min\{f(t)\}, & 0 < t \leq x \text{ and } 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{cases} \] Then:

Let \( f : R \setminus \left\{ -\frac{1}{2} \right\} \to R \) and \( g : R \setminus \left\{ -\frac{5}{2} \right\} \to R \) be defined as \( f(x) = \frac{2x + 3}{2x + 1} \) and \( g(x) = \frac{|x| + 1}{2x + 5} \). Then the domain of the function \( f(g(x)) \) is:

Consider the function.\(f(x) = \begin{cases}  \frac{a(7x - 12 - x^2)}{b(x^2 - 7x + 12)} & , \quad x < 3 \\[8pt] \frac{\sin(x - 3)}{2^{x - \lfloor x \rfloor}} & , \quad x > 3 \\[8pt] b & , \quad x = 3  \end{cases}\)Where \(\lfloor x \rfloor\)denotes the greatest integer less than or equal to \(x\). If \(S\) denotes the set of all ordered pairs \((a, b)\) such that \(f(x)\) is continuous at \(x = 3\), then the number of elements in \(S\) is:

The function \( f : \mathbb{N} - \{1\} \to \mathbb{N} \); defined by \( f(n) \) = the highest prime factor of \( n \), is: