Question : If $(a+\frac{1}{a})=–2$, then the value of $a^{1000}+a^{–1000}$ is:

Option 1: $2$

Option 2: $0$

Option 3: $1$

Option 4: $\frac{1}{2}$

Question : If $a+\frac{1}{a-2}=4$, then the value of $(a-2)^{2}+(\frac{1}{a-2})^{2}$ is:

Option 1: 0

Option 2: 2

Option 3: –2

Option 4: 4

Question : If $a+b=1$, find the value of $a^{3}+b^{3} - ab-(a^{2}-b^{2})^{2}$.

Option 1: –1

Option 2: 1

Option 3: 0

Option 4: 2

Question : Directions: If 1 * 2 = 1, 2 * 3 = –1 and  3 * 4 = –5, then find the value of 7 * 9 = ?

Option 1: –47

Option 2: –29

Option 3: –2

Option 4: –9

Question : If $a, b, c$ are real numbers and $a^{2}+b^{2}+c^{2}=2(a-b-c)-3,$ then the value of $a+b+c$ is:

Option 1: –1

Option 2: 1

Option 3: 3

Option 4: 0