List of practice Questions

Consider 4 × 4 matrices with their elements from {𝟎, 𝟏}. The number of such
matrices with even number of 𝟏s in every row and every column is

Consider a knock-out women’s badminton singles tournament where there are no
ties. The loser in each game is eliminated from the tournament. Every player
plays until she is defeated or remains the last undefeated player. The last
undefeated player is declared the winner of the tournament. If there are 64 players
in the beginning of the tournament, how many games should be played in total to
declare the winner of the tournament?

If ${}^nC_r={}^{\,n}C_{r-1}$ and ${}^{\,n}P_{r-1}=9\,{}^{\,r}P_r$, then $(n,r)=$

Consider $4 \times 4$ matrices with their elements from $\{0, 1\}$. The number of such matrices with even number of 1s in every row and every column is ______.

If ${}^nC_r={}^{\,n}C_{r-1}$ and ${}^{\,n}P_{r-1}=9\,{}^{\,r}P_r$, then $(n,r)=$

All possible 3-digit numbers are formed using the digits $0,2,3,5,7,9$ without repeating any digit. Then the number of numbers among them which are divisible by $15$ is

Let $A$ and $B$ be any two $n\times n$ matrices and \[ tr(A)=\sum_{i=1}^{n}a_{ii}, \qquad tr(B)=\sum_{i=1}^{n}b_{ii}. \] Consider the following statements:
• Statement-I : $tr(AB)=tr(BA)$.
• Statement-II : $tr(A+B)=tr(A)+tr(B)$.
• Statement-III : If $tr(A)=5,\; tr(A^{2})=13$, then \[ tr(A-I)^2=6, \] where $A_{3\times3}$ and $I_{3\times3}$ are matrices. Choose the correct option.

In a simple undirected graph, the sum of the degrees of all vertices is 20. How many edges are present in the graph?

Which of the following propositions is logically equivalent to $p \rightarrow q$?

Two graphs $G_1$ and $G_2$ are isomorphic if which of the following conditions is satisfied?

A graph is said to be Eulerian if it contains a closed trail that includes every:

If a graph G has 10 vertices and each vertex has degree 3, how many edges does G have?

How many different 5-character passwords can be formed using the digits 0-9 and lowercase letters a-z if the first character must be a letter and repetitions are allowed?

The number of ways to distribute 10 identical apples among 4 children such that every child receives at least one apple is:

For a connected planar graph with 10 vertices and 15 edges, how many faces does the graph have?

Let the matrix \[ A=\begin{bmatrix} 3& 1 -1& 2 \end{bmatrix} \] and \[ I=\begin{bmatrix} 1& 0 0& 1 \end{bmatrix}, \] then \[ A^{2}-5A+7I= \]

A question paper contains 3 parts A, B and C. There are 5 questions in A, 4 in B and 3 in C. At least 3 questions from A, at least 2 from B and at least 1 from C must be attempted. If a student attempts 7 questions satisfying these conditions, the number of ways is

There are 3 different jobs in an office. 3 women and 5 men have attended before the selection committee. If each is eligible for all jobs and at least one woman must be selected, the number of ways of allotting the 3 jobs is

दो पूर्णांक r व l लीजए जिनके लिए r $\ge$ l $\ge$ 3 है। ऐसे कितने फलन f : {1, 2, . . . r} $\rightarrow$ {1, 2, . . . r} हैं जिनके लिए f(1), f(2), . . . , f(l) सर्वथा अलग-अलग हैं?

Let $r, l$ be two integers such that $r \ge l \ge 3$. What is the total number of functions \[ f : \{1, 2, \dots, r\} \to \{1, 2, \dots, r\} \] such that $f(1), f(2), \dots, f(l)$ are all distinct?

Let $A = \{ (a, b, c) : a, b, c \text{ are non-negative integers and } a + b + 2c = 22 \}$. Then $n(A)$ is equal to:

Let $ p_n $ denote the total number of triangles formed by joining the vertices of an $ n $-side regular polygon. If $ p_{n+1} - p_n = 66 $, then the sum of all distinct prime divisors of $ n $ is:

If for $ 3 \leq r \leq 30 $, \[ \binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = \binom{m}{r}, \] then $ m $ equals: ________

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the 10th floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

Two players A and B play a series of games of badminton. The player who wins 5 games first, wins the series. Assuming that no game ends in a draw, the number of ways in which player A wins the series is _________.