Properties of Modulus

Q. Consider the region $R=\{(x,y)\in R\times R:x\ge 0\text{and}{y}^2\le 4-x\}.$
Let $F$ be the family of all circles that are contained in $R$ and have centers on the $x-$axis. Let $C$ be the circle that has largest radius among the circles in $F.$ Let $(\alpha ,\beta )$ be a point where the circle $C$ meets the curve $y^2=4-x.$

The radius of the circle $C$ is

Q.

How many triangles can be formed by joining the vertices of a hexagon?

Q.

$4{\tan }^{-1}\left(\frac{1}{5}\right)-{\tan }^{-1}\left(\frac{1}{239}\right)=?$

1. $\mathrm{\pi }$

2. $\frac{\mathrm{\pi }}{2}$

3. $\frac{\mathrm{\pi }}{3}$

4. $\frac{\mathrm{\pi }}{4}$

Q. For any integer $k$, If $a_k=\cos \left(\frac{k\pi }{7}\right)+i\sin \left(\frac{k\pi }{7}\right)$, then value of the expression $\frac{\sum _{k=1}^{12}|a_{k+1}-a_k|}{\sum _{k=1}^3|a_{4k-1}-a_{4k-2}|}$ is

Q.

If $z_1,z_2,z_3$ are complex numbers such that $|z_1|=|z_2|=|z_3|=|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=1,\text{then find the value of}|z_1+z_2+z_3|.$

Q.

Solve the system of equations.

$\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4;\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1$ and $\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$

Q. For any real number $x$, let $\left[x\right]$ denote the largest integer less than or equal to $x$. Let $f$ be a real valued function defined on the interval $[–10,10]$ by $f\left(x\right)=\{\begin{array}{cc}\left\{x\right\}& \text{if}\left[x\right]\text{is odd},\\ 1-\left\{x\right\}& \text{if}\left[x\right]\text{is even},\end{array}$ . Then the value of $\frac{p{i}^2}{10}\underset{-10}{\overset{10}{\int }}f\left(x\right)\cos \pi xdx$ is

Q.

What is the standard form of a circle?

Q. Let $z_1$ and $z_2$ be complex numbers such that $z_1\ne z_2$ and $|z_1|=|z_2|$. If $z_1$ has positive real part and $z_2$ has negative imaginary part, then $\frac{z_1+z_2}{z_1-z_2}$ may be

1. Zero
2. Real and positive
3. Real and negative
4. Purely imaginary

Q. The multiplicative inverse of a non-zero complex number $z$ is

1. $\frac{z}{|z{|}^2}$
2. $\frac{\overline{z}}{|z{|}^2}$
3. $\frac{\overline{z}}{|z|}$
4. $\frac{z}{|z|}$

Q. Let $a,b\in R$ and $a^2+b^2\ne 0$. Suppose $S=\{z\in C:z=\frac{1}{a+ibt},t\in R,t\ne 0\},$ where $i=\sqrt{-1}$. If $z=x+iy$ and $z\in S,$ then $(x,y)$ lies on

1. the circle with radius $\frac{1}{2a}$ and centre $(\frac{1}{2a},0)$for $a>0,b\ne 0$
2. the circle with radius $-\frac{1}{2a}$ and centre $(-\frac{1}{2a},0)$for $a<0,b\ne 0$
3. the $x$-axis for $a\ne 0,b=0$
4. the $y$-axis for $a=0,b\ne 0$

Q. For any complex number $z$, the minimum value of $|z|+|z-3i|$ is

Q. Find the maximum value of $|z|$ when $|z-\frac{3}{z}|=2$, $z$ being a complex number.

1. $1+\sqrt{3}$
2. $3$
3. $1+\sqrt{2}$
4. $1$

Q. If $(\sqrt{8}+i{)}^{50}=3^{49}(a+ib),$ then the value of $a^2+b^2$ is

Q. Let $a\in R$ and let $f:R\to R$ be given by
$f\left(x\right)=x^5-5x+a$
Then

1. $f\left(x\right)$ has three real roots if $a>4$
2. $f\left(x\right)$ has only one real root if $a>4$
3. $f\left(x\right)$ has three real roots if $a<-4$
4. $f\left(x\right)$ has three real roots if $-4<a<4$

Q.

Let ${\mathrm{Z}}_1$ and ${\mathrm{Z}}_2$ be two complex numbers such that $\frac{{\mathrm{Z}}_1-2{\mathrm{Z}}_2}{2-{\mathrm{Z}}_1{\overline{\mathrm{Z}}}_2}$ is unimodular and ${\mathrm{Z}}_2$ is not unimodular, find the modulus of ${\mathrm{Z}}_1$.

Q. The minimum value of $|z-1|+|z-3|$ is

1. $4$
2. $0$
3. $2$
4. $1$

Q. If $\left[x\right]$ is the greatest integer $\le x$, then ${\pi }^2\underset{0}{\overset{2}{\int }}\left(\sin \frac{\pi x}{2}\right){(x-[x\left]\right)}^{\left[x\right]}dx$ is equal to

1. $4(\pi +1)$
2. $2(\pi -1)$
3. $4(\pi -1)$
4. $2(\pi +1)$

Q.

If $a,b,c$ are in arithmetic progression, then the value of $(a+2b-c)(2b+c-a)(a+2b+c)$

Q. The foot of perpendicular from $A(1,1,1)$ to the line joining $B(-8,5,6)$ and $C(12,1,0)$ lies on the plane(s)

1. $2x+y+z=10$
2. $x-y+z=2$
3. $x+y-z=2$
4. $x+y+z=8$

Q.

The equations Kcosx - 3sin x = K + 1 is solvable only if K belongs to the interval

1. [-4, 4]

2. (-∞, 4]

3. (-∞, ∞)

4. [1, +∞)

Q.

Which of the following is correct for any two complex number $z_1$ and $z_2$ ?

1. $|z_1{z}_2|=|z_1||z_2|$

2. $arg\left(z_1{z}_2\right)=arg\left(z_1\right)arg\left(z_2\right)$

3. $|z_1+z_2|=|z_1|+|z_2|$

4. $|z_1+z_2|\ge |z_1|+|z_2|$

Q. If $\left|\frac{z_1+z_2}{z_1-z_2}\right|=1$, then $\frac{z_1}{z_2}$ is

1. positive real
2. negative real
3. purely imaginary
4. $0$

Q.

The modulus of $(1+i)(1+2i)(1+3i)$is equal to

Q. If $z_1,z_2$ are two different complex numbers satisfying $|z_1^2-z_2^2|=|{\overline{z}}_1^2+{\overline{z}}_2^2-2{\overline{z}}_1{\overline{z}}_2|,$ then

1. $\frac{z_1}{z_2}$ is purely imaginary.
2. $\frac{z_1}{z_2}$ is purely real.
3. $|\arg z_1-\arg z_2|=\pi$
4. $|\arg z_1-\arg z_2|=\frac{\pi }{2}$

Q. If the interval contained in the domain of definition of non-zero solution of the differential equation $(x-3{)}^2\cdot y^{{}^{\prime }}+y=0$ is $(-\infty ,\infty )-\left\{k\right\},$ then $k$ is

Q. If $z_1$ and $z_2$ are any two complex numbers, then $|z_1+\sqrt{z_1^2-z_2^2}|+|z_1-\sqrt{z_1^2-z_2^2}|$ is equal to

1. $|z_1|$
2. $|z_1+z_2|$
3. $|z_2|$
4. $|z_1+z_2|+|z_1-z_2|$

Q. If $|z+2-i|=5,$ then the maximum value of $|3z+9-7i|$ is

1. $10$
2. $15$
3. $20$
4. $5$

Q. If $z_1,z_2,z_3,z_4$ are the affixes of four points in the Argand plane, $z$ is the affix of a point such that $|z-z_1|=|z-z_2|=|z-z_3|=|z-z_4|$, then the points $z_1,z_2,z_3,z_4$ can lie on which among the following curves

1. Circle
2. Rectangle
3. Square
4. Triangle

Q. The value of $Im\left(z\overline{z}\right)$

1. is $1$
2. is $0$
3. is $-1$
4. depends on $z$