Question

If the system of equations $\left(\_{}^{n}C_{3}^{}\right)x+\left(\_{}^{n}C_{4}^{}\right)y+35z=0, \, \left(\_{}^{n}C_{4}^{}\right)x+35y+\left(\_{}^{n}C_{3}^{}\right)z=0$ and $35x+\left(\_{}^{n}C_{3}^{}\right)y+\left(\_{}^{n}C_{4}^{}\right)z=0$ has a non-trivial solution, then the value of $n$ is equal to $\left(\forall n \in N , \, n \geq 4\right)$

• A. $6$
• B. $7$
• C. $8$
• D. $9$

Answer 1

Answer: (B)

For non trival solution.
$\Delta=0$ $\left|\begin{array}{ccc}{ }^{n} C_{3} & { }^{n} C_{4} & 35 \\ { }^{n} C_{4} & 35 & { }^{n} C_{3} \\ 35 & { }^{n} C_{3} & { }^{n} C_{4}\end{array}\right|=0$
$\Rightarrow { }^{n} C_{3}+{ }^{n} C_{4}+35=0$ (not possible)
or ${ }^{n} C_{3}={ }^{n} C_{4}=35 \Rightarrow n=7$