Permutations and Combinations

Q.

If for the matrix, $A=\left[\begin{array}{cc}1& -\alpha \\ \alpha & \beta \end{array}\right]$ $A{A}^T=I_2$, then the value of ${\alpha }^4+{\beta }^4$ is:

1. $1$

2. $3$

3. $2$

4. $4$

Q. For the given equation ${}^{10}{C}_r+2\cdot {}^{10}{C}_{r-1}+{}^{10}{C}_{r-2}={}^{12}{C}_s$, which of the following is/are correct?

1. If $r=s$, then the possible number of solutions is $13$
2. If $r\ne s$, then the possible number of solutions is $12$
3. If $r=s$, then the possible number of solutions is $9$
4. If $r\ne s$, then the possible number of solutions is $8$

Q. Which of the following expression(s) is/are not representing a polynomial for $n\in N$.

1. $x^n+x^{\left(\frac{1}{n}\right)}+\frac{1}{n}$
2. $x^{n!}+x^{\left(\frac{n}{n!}\right)}+\frac{1}{n!}$
3. $x^{n!}+x^{\left(\frac{n!}{n}\right)}+\frac{1}{n!}$
4. $x^{\left(\frac{n^2-1}{n+1}\right)}+x^2$

Q. If set $A=\left\{\right(r,s)|r,s\in W\}$, then the number of element(s) in set $A$ such that ${}^5{C}_r\cdot {}^6{C}_s=1$ is

1. $0$
2. $2$
3. $3$
4. $4$

Q. If $\sum _{j=0}^{10}{}^{30+j}{C}_{10+j}={}^m{C}_{20}-{}^p{C}_{21}$, then

1. $m=41$
2. $p=30$
3. $m=40$
4. $p=31$

Q.

Let $f\left(x\right)=\frac{\alpha x^2}{\left(x+1\right)}$, $x\ne -1$. The value of $\alpha$ for which $f\left(a\right)=a,(a\ne 0)$ is

1. $1-\frac{1}{a}$

2. $\frac{1}{a}$

3. $1+\frac{1}{a}$

4. $\frac{1}{a}-1$

Q. For the given equation ${}^{10}{C}_r+2\cdot {}^{10}{C}_{r-1}+{}^{10}{C}_{r-2}={}^{12}{C}_s$, which of the following is/are correct?

1. If $r=s$, then the possible number of solutions is $13$
2. If $r\ne s$, then the possible number of solutions is $12$
3. If $r=s$, then the possible number of solutions is $9$
4. If $r\ne s$, then the possible number of solutions is $8$

Q. Which of the following(s) pair have same number of terms $(x,y\in R)$

1. $(x+y{)}^9-2,(2x-3y{)}^{10}$
2. $1-(2+x{)}^m,(x^2+5{)}^m,$ where $m\in N$
3. $(x^2-2x+1{)}^5,(x^2+1{)}^{10}$
4. $(x^2-2x+1{)}^5,(x^2+1{)}^{11}$

Q. If set $A=\left\{\right(r,s)|r,s\in W\}$, then the number of element(s) in set $A$ such that ${}^5{C}_r\cdot {}^6{C}_s=1$ is

1. $0$
2. $2$
3. $3$
4. $4$

Q. If $r^2-13r+40=0$, then the value of ${}^7{C}_r$ is

1. $56$
2. $12$
3. $1$
4. $21$