NDA Mathematics 2021 Question Paper

If $x^2+x+1=0$, then what is the value of $x^{199}+x^{200}+x^{201}$?

If $x,y,z$ are in GP, then which of the following is/are correct?

1. $\ln (3x),\ln (3y),\ln (3z)$ are in AP
2. $xyz+\ln (x),xyz+\ln (y),xyz+\ln (z)$ are in HP
   Select the correct answer using the code given below.

If ${\log }_{10}2,{\log }_{10}(2^x-1),{\log }_{10}(2^x+3)$ are in AP, then what is $x$ equal to?

Let $S=\{2,3,4,5,6,7,9\}$. How many different 3-digit numbers (with all digits different) from $S$ can be made which are less than 500?

If $p=(1111\cdots \text{up to }n\text{ digits})$, then what is the value of $9p^2+p$?

The quadratic equation
$$3x^2-(k^2+5k)x+3k^2-5k=0$$
has real roots of equal magnitude and opposite sign. Which one of the following is correct?

If $a_n=n(n!)$, then what is
$$a_1+a_2+a_3+\cdots +a_{10}$$
equal to?

If $p$ and $q$ are the non-zero roots of the equation $x^2+px+q=0$, then how many possible values can $q$ have?

If
$$\Delta =\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|$$
then what is
$$\left|\begin{array}{ccc}3d+5g& 4a+7g& 6g\\ 3e+5h& 4b+7h& 6h\\ 3f+5i& 4c+7i& 6i\end{array}\right|$$
equal to?

If
$$\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$$
are in HP, then which of the following is/are correct?

1. $a,b,c$ are in AP
2. $(b+c{)}^2,(c+a{)}^2,(a+b{)}^2$ are in GP
   Select the correct answer using the code given below.