A girl writes five consecutive positive integers on a blackboard. She then erases one of them. The sum of the remaining four numbers is 2025. What number did she erase?

What is the domain of the following real valued function?

$$f(x)={\log }_2(x^2-5x+6).$$

Let A=\left[\begin{array}{cc}a&b\c&d\end{array}\right] be a real matrix which satisfies $A^{-1}=A$. Which of the following statements is/are always true?

Let $n\ge 3$ be an integer, and let $x_1,x_2,\dots ,x_n$ be variables which take real values with $0\le x_i\le 1$ for all $1\le i\le n$. Let

$$\begin{array}{rcl}A& =& x_1+x_2+\cdots +x_n\\ B& =& x_1{x}_2+x_2{x}_3+\cdots +x_{n-1}{x}_n+x_n{x}_1\end{array}$$

Which of the following statements is/are true.

The two arguments to the function $foo(A,n)$ in the code below are: (i) an integer array $A$ indexed from 0, and (ii) the number $n$ of elements in $A$.

```

function foo(A, n) {

count = 0;

for i from 0 to (n-1) {

for j from (i+1) to (n-1) {

if (A[i] > 2 * A[j]) {

count = count + 1;

}

}

}

return(count);

}

```

Which of the following statements about the function $foo(A,n)$ are correct?

In the following code the operator $\%$ denotes the remainder after integer division. That is: for positive integers $a,b$ the value $a\%b$ is the remainder obtained when $a$ is divided by $b$.

```

function fizzbuzz(n) {

count = 0;

for i from 0 to (n-1) {

if ((i % 3) == 0) and ((i % 5) != 0) {

count = count + 1;

}

}

return(count);

}

```

What does fizzbuzz(100) return?

Let $X=\{1,2,3\}$ and $Y=\{a,b,c,d\}$ be two finite sets. Which of the following statements regarding functions from $X\to Y$ is/are true?

A binary relation $R$ defined on a set $S$ is said to be <b>antisymmetric</b> if for $x,y\in S$, $xRy$ and $yRx⟹x=y$. Let $R_1,R_2$ be two binary relations defined on a set $S$. The <b>union</b> of $R_1,R_2$ is the binary relation $U$ defined on $S$ as: For $x,y\in S$, $xUy⟺x{R}_{1}y\text{ or }x{R}_{2}y$. The <b>intersection</b> of $R_1,R_2$ is the binary relation $I$ defined on $S$ as: For $x,y\in S$, $xIy⟺x{R}_{1}y\text{ and }x{R}_{2}y$. Which of the following statements is/are true?

In how many ways can 10 identical chocolate bars be distributed among 5 children, in such a way that each child gets at least one chocolate bar?

Let $x$ be a variable that takes real values, and let $f(x)=x^3-3x$. Which of the following statements is/are true?

Let $f(x)=\sqrt{x}$. We draw a tangent to the curve $y=f(x)$ at the point on the curve whose $x$ coordinate is equal to 4. Where does this tangent intersect the X-axis?

A game being offered in a casino consists of guessing the outcomes of two tosses of a fair coin. The gambler wins if she/he has correctly guessed at least one of the two tosses. To play a game, the gambler has to pay a fee of Rs. 80, and the winner gets a reward of Rs. 100 on winning the game (and nothing otherwise).

Which of the following statements are correct?

Let $\mathbb{Z}$ be the set of all integers. Let $A=\{n\in \mathbb{Z}\mid n^2+10n+21\text{ is divisible by }7\}$. Which of the following statements is/are true?