NDA Mathematics 2022 Question Paper

How many four-digit natural numbers are there such that all of the digits are odd?

What is ${\sum }_{r=0}^n{2}^{r}C(n,r)$ equal to?

If different permutations of the letters of the word 'MATHEMATICS' are listed as in a dictionary, how many words (with or without meaning) are there in the list before the first word that starts with $C$?

Consider the following statements:

1. If $f$ is the subset of $Z\times Z$ defined by $f=\{(xy,x-y);x,y\in Z\}$, then $f$ is a function from $Z$ to $Z$.
2. If $f$ is the subset of $N\times N$ defined by $f=\{(xy,x+y);x,y\in N\}$, then $f$ is a function from $N$ to $N$.
   Which of the statements given above is/are correct?

Consider the determinant
$\Delta =\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
If $a_{13}=yz$, $a_{23}=zx$, $a_{33}=xy$ and the minors of $a_{13},a_{23},a_{33}$ are respectively $(z-y),(z-x),(y-x)$ then what is the value of $\Delta$?

If $A=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & \sin \theta \\ 0& \sin \theta & -\cos \theta \end{array}\right)$, then which of the following are correct?

1. $A+adjA$ is a null matrix
2. $A^{-1}+adjA$ is a null matrix
3. $A-A^{-1}$ is a null matrix
   Select the correct answer using the code given below:

If $X$ is a matrix of order $3\times 3$, $Y$ is a matrix of order $2\times 3$ and $Z$ is a matrix of order $3\times 2$, then which of the following are correct?

1. $(ZY)X$ is a square matrix having 9 entries.
2. $Y(XZ)$ is a square matrix having 4 entries.
3. $X(YZ)$ is not defined.
   Select the correct answer using the code given below:

For how many quadratic equations, the sum of roots is equal to the product of roots?

Consider the following statements:

1. The set of all irrational numbers between $\sqrt{2}$ and $\sqrt{5}$ is an infinite set.
2. The set of all odd integers less than 100 is a finite set.
   Which of the statements given above is/are correct?

Consider the following statements:

1. $2+4+6+\cdots +2n=n^2+n$
2. The expression $n^2+n+41$ always gives a prime number for every natural number $n$
   Which of the above statements is/are correct?