Common Description:

• The symbol $\mathbb{R}$ denotes the set of all real numbers.
• The symbol $\mathbb{Q}$ denotes the set of all rationals numbers.
• The symbol $\mathbb{Z}$ denotes the set of all integers.
• The symbol $\mathbb{N}$ denotes the set of all natural numbers.

The number of factors of 2025 (including 1 and itself) equals

Common Description:

• The symbol $\mathbb{R}$ denotes the set of all real numbers.
• The symbol $\mathbb{Q}$ denotes the set of all rationals numbers.
• The symbol $\mathbb{Z}$ denotes the set of all integers.
• The symbol $\mathbb{N}$ denotes the set of all natural numbers.

Consider the equation $x^7+4x^5+2x^3+x+1=0$. The number of real root(s) of this equation is

Common Description:

• The symbol $\mathbb{R}$ denotes the set of all real numbers.
• The symbol $\mathbb{Q}$ denotes the set of all rationals numbers.
• The symbol $\mathbb{Z}$ denotes the set of all integers.
• The symbol $\mathbb{N}$ denotes the set of all natural numbers.

Let $i=\sqrt{-1}$ and $\omega$ be a non-real root of the equation $x^3=1$. If $(i\omega {)}^n$ is an integer, then $n$ is necessarily

Find ${\lim }_{x\to 0}\left(\frac{\sqrt{(1+x)}-\sqrt{(1+x^2)}}{\sqrt{(1-x^2)}-\sqrt{(1-x)}}\right)$.

Find the maximum area of a triangle when the perimeter of the triangle is $s$.

Pick a uniformly chosen random point inside a unit square (a square of side length 1) and draw a circle of radius $\frac{1}{3}$ around the point. Find the probability that the circle lies entirely inside the square.

In the quadratic equation, $x^2+bx+c=0$, both $b$ and $c$ can take any value from 1, 2, 3, 4, 5 and 6. Find the probability that the equation will have real roots.

An elevator starts at ground floor and can stop at any floor in between the 1st and 10th. Seven passengers are getting into the lift from the ground floor. What is the probability that no two passengers leave at the same floor?

Let $\alpha =(x_1,x_2)$ and $\beta =(y_1,y_2)$ be two vectors in ${\mathbb{R}}^2$ such that $x_1{y}_1+x_2{y}_2=0$ and $x_1^2+x_2^2=y_1^2+y_2^2=1$. Prove that $B=\{\alpha ,\beta \}$ is a basis for ${\mathbb{R}}^2$.

Common Description:

• The symbol $\mathbb{R}$ denotes the set of all real numbers.
• The symbol $\mathbb{Q}$ denotes the set of all rationals numbers.
• The symbol $\mathbb{Z}$ denotes the set of all integers.
• The symbol $\mathbb{N}$ denotes the set of all natural numbers.

If the matrix

$$A=\left[\begin{array}{cc}a& 1\\ 2& 3\end{array}\right]$$

has 1 as an eigenvalue, then $\det (A)$ equals

Common Description:

• The symbol $\mathbb{R}$ denotes the set of all real numbers.
• The symbol $\mathbb{Q}$ denotes the set of all rationals numbers.
• The symbol $\mathbb{Z}$ denotes the set of all integers.
• The symbol $\mathbb{N}$ denotes the set of all natural numbers.

Suppose $\sec x+\tan x=3$ for some $x\in (0,\pi /2)$. Then $\sin 2x$ equals

Consider a square $OBCD$ with length 2 units having side $OB$ on the positive $x$-axis with $O$ as the origin and $C$ in the first quadrant. The square is rotated in an anti-clockwise direction to form a new square $O{B}^{\prime }{C}^{\prime }{D}^{\prime }$ so that $\angle BO{B}^{\prime }=30^{\circ }$. Find the coordinates of the square $O{B}^{\prime }{C}^{\prime }{D}^{\prime }$.

Common Description:

• The symbol $\mathbb{R}$ denotes the set of all real numbers.
• The symbol $\mathbb{Q}$ denotes the set of all rationals numbers.
• The symbol $\mathbb{Z}$ denotes the set of all integers.
• The symbol $\mathbb{N}$ denotes the set of all natural numbers.

Let $R$ be the region defined by

$$R=\{(x,y)\in {\mathbb{R}}^2:x^2+y^2\le 50,x+y\ge 0\}.$$

The area of $R$ equals