1
IAT (IISER) 2020
MCQ (Single Correct Answer)
+4
-1
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?
A
If $\ell$ passes through two points of $S$, then $m$ must be rational.
B
If $m$ is rational and $c$ is irrational, then $\ell$ passes through at least one point of $S$.
C
If $\ell$ passes through exactly one point of $S$, then $m$ must be irrational.
D
If $m$ is irrational and $c$ is rational, then $\ell$ passes through exactly one point of $S$.
2
IAT (IISER) 2020
MCQ (Single Correct Answer)
+4
-1

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:

A
$2019^2$
B
2019
C
$2020^2$
D
2020
3
IAT (IISER) 2020
MCQ (Single Correct Answer)
+4
-1
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?
A
If $a+i b$ is a root of $p(x)$, then $n$ must be even.
B
If $a$ is a root of $p(x)$, then $n$ must be odd.
C
If $a+i b$ is a root of $p(x)$, then $(a+i b)^2$ can never be a root of $p(x)$.
D
If $a$ is a root of $p(x)$, then $a$ must be an integer.
4
IAT (IISER) 2020
MCQ (Single Correct Answer)
+4
-1
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
$7^3$
B
$3^7$
C
$21^3$
D
$7^6$
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