SAMPLE QUESTIONS
1. If A = $\begin{bmatrix} 1 &0 \\ 1& 1 \end{bmatrix}$ and I = $\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}$, then which one of the following holds for all n ≥ 1, by the principle of mathematical induction?
a) $A^{n}$ = $2^{n-1}A + \left (n-1 \right )I$
b) $A^{n}$ = $nA + \left (n-1 \right )I$
c) A
^{n} = $2^{n-1}A-\left (n-1 \right )I$
d) A
^{n} = nA - (n-1)I
2. Statement I For each natural number n,(n+1)
^{7} - n
^{7} - 1 is divisible by 7.
Statement II For each natural number n, n
^{7} - n is divisible by 7.
a) Statement I is false , Statement II is true.
b) Statement I is true, Statement II is true; Statement II is correct explanation for Statement I.
c) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
d) Statement I is true, Statement II is false.