1
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
Let $A$ be the set of points in the $X Y$-plane which are equidistant from $P(-1,0)$ and $Q(1,0)$. Let $B$ be the set of points in the $X Y$-plane which are equidistant from $A$ and $Q$. If $(5, y)$ is a point in $B$, then what is the value of $y^2$ ?
A
1
B
4
C
9
D
16
2
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1

Consider the lines $L_1$ and $L_2$ given below:

$$ \begin{gathered} L 1: x=2+\lambda, y=3+2 \lambda, z=4+3 \lambda \\ L 2: x=4+\lambda, y=4, z=4+\lambda \end{gathered} $$

If $(2,3,4)$ is the point of $L_1$ that is closest to $L_2$, then which point of $L_2$ is closest to $L_1$ ?

A

$(3,4,3)$

B

$(3,4,4)$

C

$(5,4,5)$

D

$(4,4,4)$

3
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
Let $a_1, a_2, a_3, \ldots$ be a sequence of real numbers. Let $s_n=a_1+a_2+\cdots+a_n$. If $2 s_n=$ $n\left(c+a_n\right)$ for some real number. for some real number $c$ and for all $n=1,2,3, \ldots$, then which one of the following statements is Correct?
A
$a_1, 2 a_2, 3 a_3, \ldots$ is an Arithmetic Progression.
B
$a_1, a_2, a_3, \ldots$ is an Arithmetic Progression.
C
$a_1, 2 a_2, 3 a_3, \ldots$ is a Geometric Progression.
D
$a_1, a_2, a_3$, ...is a Geometric Progression.
4
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a strictly decreasing function with $|f(t)|<\pi / 2$ for all $t \in \mathbf{R}$. Let $g:[0, \pi] \rightarrow$ R be a function defined by $g(t)=\sin (f(t))$. Which one of the following statements is Correct?
A
$g$ is increasing on $[0, \pi]$.
B
$g$ is decreasing on $[0, \pi]$.
C
$g$ is increasing on $(0, \pi / 2)$ and decreasing on $(\pi / 2, \pi)$.
D
$g$ is decreasing on $(0, \pi / 2)$ and increasing on $(\pi / 2, \pi)$.
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