IAT (IISER) 2020

Let $C$ be a circle which passes through $(-2,1)$ and $(0,3)$ and whose centre lies on the line $y-x=1$. Then which of the following points lies on $C$ ?

Let $a, b$ be nonzero real numbers. If $p(x)=x^n+c_{n-1} x^{n-1}+\cdots+c_0$, where $c_{n-1}, \ldots, c_0$ are integers and $n \geq 3$, then which of the following statements is correct?

$F(x)=\int_0^{e^x}\left(t^3+2 t^2-t-2\right) d t$, then for how many real numbers $x$ does $F^{\prime}(x)=0$ ?

If $p(t)=\frac{t(t-1) \cdots(t-2019)}{2019!}$, then the value of
\(\int_0^1\left(\frac{1}{t+1}+\frac{1}{t+2}+\cdots+\frac{1}{t+2020}\right) p(-t-1) d t\)
is:

Consider the differential equation $f^{\prime \prime}+p f^{\prime}+q f=0$ where $p$ and $q$ are constants, $p^2=4 q$ and $q \neq 0$. If y is a nonzero solution of the equation, then which of the following statements is correct?

For what value of $t$ does $f(x)=x^3+t x^2+(2-t) x-3$ have only real roots?

Let $[x]$ denote the greatest integer less than or equal to $x$. The number of positive integer solutions of the equation

\(\left[\frac{x}{19}\right]=\left[\frac{x}{20}\right]\)
are:

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions such that
$$ g \circ f(x)= \begin{cases}x^2, & \text { if } x \geq 0 \\ e^x-1, & \text { if } x<0\end{cases} $$
Then which of the followi...

If $f(x)=a x^2+b x+c$ and $f\left(\frac{1}{n}\right)=\frac{n+1}{n^2}$ for all $n \in N$, then what is the value of $\lim \limits_{x \rightarrow 0} f^{\prime}(x)$ ?

If $A=\left[\begin{array}{lll}1 & a & 0 \\ 0 & 1 & b \\ 0 & 0 & 1\end{array}\right]$, then the determinant of $I-A+A^2-A^3+A^4-\cdots+A^{2020}$ is

The number of skew-symmetric matrices $A=\left[a_i j\right]_{3 \times 3}$, where $a_i j \in\{-3,-2,-1,0,1,2,3\}$ is:

A bag contains 8 balls, of which 5 are green and 3 are red. The probability that two balls chosen at random are of the same colour is :

\(\text { Define binary operation * on } Q \text { by } a * b=a+b-3 \text {. The inverse of } 2 \text { with respect to * is }\)

Let $S=\{(a, b): a, b \in Q\}$ and $\ell$ be the line $y=m x+c$. Which of the following statements is not correct?

If S is the set of all points $(x, y)$ satisfying the equation $\sin x=\cos y$, then which of the following statements is correct?