Recent questions tagged isi2026-mcs-pca

Let $S=\{1,\{2\},\{3,4\}\}$ and $\mathcal{P}$ be the power set of $S$. Which of the following is false?$\{1\}$ is in $\mathcal{P}$.$\{1,\{3,4\}\}$ is in $\mathcal{P}$.$\{...

Let $R$ be an equivalence relation on a set with $7$ elements. Which of the following is not possible for the relation $R$?There are $2$ different equivalence classes, ea...

The number of vectors $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ such that each $x_{i}$ is a non-negative integer and $\sum_{i=1}^{n} x_{i} \leq k$, where $k$ is a non-ne...

Consider the following pseudo-code for reversing an array $A$.Algorithm ReverseArray(A[l..r])// Input: an array A[l..r]// Output: the array A in reverse orderif ..... the...

Let $X_{1}, X_{2}$ and $X_{3}$ be independent and identically distributed random variables defined as follows: for $i=1,2,3$,$$X_{i}=\left\{\begin{array}{ll}-1 & \text { ...

What is the maximum possible number of edges in a graph with $n$ vertices and $k$ components?$\binom{n-k+1}{2}$$k\binom{n / k}{2}$$\binom{n-k}{2}$$(k-1)+\binom{n-k+1}{2}$

Some people stole cookies from a cookie jar. At least one of Abbas, Balveer and Chinmay stole coockies. Chinmay was only involved if Abbas was involved. Balveer would not...

Consider the following two recurrence relations:$$\begin{array}{c}T(n)=5 T(2 n / 11)+\Theta(n) \\S(n)=5 S(n / 5)+\Theta(n)\end{array}$$Which of the following is true?$T(n...

The value of the infinite series$$\sum_{n=1}^{\infty} \frac{1}{(n+2025)(n+2026)(n+2027)}$$ is$\dfrac{1}{2 \times(2026)^{2}}$$\dfrac{1}{2026 \times 2027}$$\dfrac{1}{2 \tim...

If $x^{2}+y^{2}+10 x-24 y-27=0$, then the minimum value of $x^{2}+y^{2}$ is attained at$\left(\dfrac{5}{13},-\dfrac{12}{13}\right)$$\left(-\dfrac{5}{13}, \dfrac{12}{13}\r...

Let $A$ be a real square matrix such that$$\operatorname{det}(\lambda I-A)=(\lambda-2)^{7}(\lambda+5)^{3}$$ Then which of the following is true?$\operatorname{rank}(A)=2$...

Let $S$ be a non-empty set, and $R$ be a binary relation on $S$. The relation $R$ is said to be anti-symmetric if for all $a, b \in S, a R b$ and $b R a$ imply $a=b$. Con...

A student claims to prove the following statement by mathematical induction:$$P(n): 2^{n} \leq n^{2} \text { for all } n \in \mathbb{N}, n \geq 2$$The student's proof is ...