Binomial Theorem — NDA Mathematics PYQs

*Directions : For the next five (05) items that follow :

Let $u$ be a positive integer and $f$ be a real number lying between $0$ and $1$. Further, $(\sqrt{2}+1{)}^{10}=u+f$ and $(\sqrt{2}-1{)}^{10}=v$*

Consider the following statements :
I. $(u+v+f)$ is an integer.
II. $(f+v)$ is an integer.
Which of the statements given above is/are correct ?

*Directions : For the next five (05) items that follow :

Let $u$ be a positive integer and $f$ be a real number lying between $0$ and $1$. Further, $(\sqrt{2}+1{)}^{10}=u+f$ and $(\sqrt{2}-1{)}^{10}=v$*

What is the multiplicative inverse of $(\sqrt{2}+1{)}^{20}$ ?

*Directions : For the next five (05) items that follow :

Let $u$ be a positive integer and $f$ be a real number lying between $0$ and $1$. Further, $(\sqrt{2}+1{)}^{10}=u+f$ and $(\sqrt{2}-1{)}^{10}=v$*

What is the value of $(v+f)$ ?

*Directions : For the next five (05) items that follow :

Let $u$ be a positive integer and $f$ be a real number lying between $0$ and $1$. Further, $(\sqrt{2}+1{)}^{10}=u+f$ and $(\sqrt{2}-1{)}^{10}=v$*

What is the value of $u$ ?

*Directions : For the next five (05) items that follow :

Let $u$ be a positive integer and $f$ be a real number lying between $0$ and $1$. Further, $(\sqrt{2}+1{)}^{10}=u+f$ and $(\sqrt{2}-1{)}^{10}=v$*

What is the value of $uv$ ?

What is the number of rational terms in the expansion of $(3^{1/2}+5^{1/4}{)}^{12}$?

If the sum of binomial coefficients in the expansion of $(x+y{)}^n$ is $256$, then the greatest binomial coefficient occurs in which one of the following terms?

What is the remainder when $5^{99}$ is divided by 13?

Which one of the following is the greatest coefficient in the expansion of $(1+x{)}^{100}$?

What is the remainder when $2^{120}$ is divided by 7?