Recent questions tagged isi2023-mcs-pca

A person throws a pair of fair dice. If the sum of the numbers on the dice is a perfect square, then the probability that the number 3 appeared on at least one of the dic...

Consider the system of linear equations: $x+y+z=5, \quad 2 x+2 y+3 z=4$. Thenthe system is inconsistentthe system has a unique solutionthe system has infinitely many solu...

If $g^{\prime}(x)=f(x)$ then $\int x^{3} f\left(x^{2}\right) d x$ is given by$x^{2} g\left(x^{2}\right)-\int x g\left(x^{2}\right) d x+C$$\frac{1}{2} x^{2} g\left(x^{2}\r...

If $\left({ }^{n} C_{0}+{ }^{n} C_{1}\right)\left({ }^{n} C_{1}+{ }^{n} C_{2}\right) \cdots\left({ }^{n} C_{n-1}+{ }^{n} C_{n}\right)=k{ }^{n} C_{0}{ }^{n} C_{1} \cdots{ ...

Let $S$ be a set of $n$ elements. The number of ways in which $n$ distinct non-empty subsets $X_{1}, \ldots, X_{n}$ of $S$ can be chosen such that $X_{1} \subseteq X_{2} ...

Let $A$ be a $4 \times 4$ matrix such that both $A$ and $\operatorname{Adj}(A)$ are non-null. If $\operatorname{det} A=0$, then the $\operatorname{rank}$ of $A$ is$1$$2$$...

Suppose $a, b, c$ are in $\text{A.P}$. and $a^{2}, b^{2}, c^{2}$ are in $\text{G.P}$. If $a < b < c$ and $a+b+c=\frac{3}{2}$, then the value of $a$ is$\frac{1}{2 \sqrt{2}...

The set of all $a$ satisfying the inequality\[\frac{1}{\sqrt{a}} \int_{1}^{a}\left(\frac{3}{2} \sqrt{x}+1-\frac{1}{\sqrt{x}}\right) d x<4\]is equal to the interval$(-5,-2...

Let $C_{0}$ be the set of all continuous functions $f:[0,1] \rightarrow \mathbb{R}$ and $C_{1}$ be the set of all differentiable functions $g:[0,1] \rightarrow \mathbb{R}...

The number of distinct even divisors of\[\prod_{k=1}^{5} k!\]is$24$$32$$64$$72$ 

A straight line passes through the intersection of the lines given by $3 x-4 y+1=0$ and $5 x+y=1$ and makes equal intercepts of the same sign on the coordinate axes. The ...

Suppose $A$ and $B$ are two square matrices such that the largest eigenvalue of $(A B-B A)$ is positive. Then the smallest eigen value of $(A B-B A)$must be positivemust ...

A fair die is rolled five times. What is the probability that the largest number rolled is $5$?$5 / 6$$1 / 6$$1-(1 / 6)^{6}$$(5 / 6)^{5}-(2 / 3)^{5}$ 

Suppose that a $3 \times 3$ matrix $A$ has an eigen value -1 . If the matrix $A+I$ is equal to\[\left[\begin{array}{ccc}1 & 0 & -2 \\0 & 0 & 0 \\0 & 0 & 0\end{array}\righ...

Two rows of $n$ chairs, facing each other, are laid out. The number of different ways that $n$ couples can sit on these chairs such that each person sits directly opposit...

Let\[A=\left[\begin{array}{lll}a & 1 & 1 \\b & a & 1 \\1 & 1 & 1\end{array}\right]\]The number of elements in the set\[\left\{(a, b) \in \mathbb{Z}^{2}: 0 \leq a, b \leq ...

Amir, Bhola, Chaitali and Deepak are invited to a party. If Bhola and Chaitali attend, then Deepak will attend too. If Bhola does not attend, then Amir will not attend. I...

Transposing the adjacency matrix of an undirected graph with $n$ connected components gives the adjacency matrix of a graph with$n / 2$ components$(n-1)$ components$(n+1)...

What does the following function compute in terms of $n$ and $d$, for integer values of $n$ and $d, n>1, d>1$ ? Note that $a / / b$ denotes the quotient (integer part) of...