Embibe Experts Solutions for Chapter: Solved Mathematics Paper-1, Exercise 1: Exercise

Find a polynomial equation of minimum degree with rational coefficients, having $2i+3$ as a root.

If $\omega$ is a cube root of unity then the value of ${\left(1-\omega +{\omega }^2\right)}^4+{\left(1+\omega -{\omega }^2\right)}^4$ is 

Find the equation of the circle passing through the points  $(0,0),(-2,1)\text{ and }(-3,2)$.

Find $\underset{x\to \infty }{\lim }{\left(1+\frac{3}{x}\right)}^x$.

Construct the composition table/multiplication table for the binary operation $*$ defined on $\left\{0,1,2,3,4\right\}$ by $a*b=a\times b$(mod $5$). Find the identity element if any. Also find the inverse elements of $2$ and $4$.

Find the equations of circles passing through $\left(1,-1\right)$ touching the lines $4x+3y+5=0$ and $3x-4y-10=0$.

Show that the sum of the focal distances of any point on an ellipse is equal to the length of the major axis and also prove that the locus of a point which moves so that the sum of its distances from $\left(3,0\right)$, and $\left(-3,0\right)$ is $9$, is $\frac{x^2}{\left(\frac{81}{4}\right)}+\frac{y^2}{\left(\frac{45}{4}\right)}=1$

Use contradiction method to prove that:

$p:\sqrt{3}$ is irrational is a true statement.