Induction for divisibility and matrix powers

Induction for divisibility and matrix powers Quiz

Spec A1 · AQA A-Level Further Maths: Quiz

Studying this course? Keep your progress in one place.

Pick an answer for each. Quizzes never use energy.

5 questions, one answer each

Each answer is checked as you go, with why.

See all 5 questions

  1. 1. Which expression equals 7k+1−7k7^{k+1}-7^k?

    • A. 7k7^k
    • B. 6×7k6\times7^k
    • C. 7×6k7\times6^k
    • D. 7k−17^k-1

  2. 2. Which formula gives Bn\mathbf{B}^n when B=(1011)\mathbf{B}=\begin{pmatrix}1&0\\ 1&1\end{pmatrix}?

    • A. (101n)\begin{pmatrix}1&0\\ 1&n\end{pmatrix}
    • B. (10n21)\begin{pmatrix}1&0\\ n^2&1\end{pmatrix}
    • C. (n0nn)\begin{pmatrix}n&0\\ n&n\end{pmatrix}
    • D. (10n1)\begin{pmatrix}1&0\\ n&1\end{pmatrix}

  3. 3. Which is the correct inductive hypothesis for proving that f(n)f(n) is divisible by 55?

    • A. f(k)=5mf(k)=5m for some integer mm
    • B. f(k+1)=5mf(k+1)=5m for some integer mm
    • C. f(k)=5f(k)=5
    • D. f(k)=5kf(k)=5k

  4. 4. f(n)=2n+2+32n+1f(n)=2^{n+2}+3^{2n+1} is divisible by 77. What is f(1)f(1)?

    • A. 2121
    • B. 1414
    • C. 3535
    • D. 1111

  5. 5. What is (2003)n\begin{pmatrix}2&0\\ 0&3\end{pmatrix}^n?

    • A. (2n003n)\begin{pmatrix}2n&0\\ 0&3n\end{pmatrix}
    • B. (2n003n)\begin{pmatrix}2^n&0\\ 0&3^n\end{pmatrix}
    • C. (2n003n)\begin{pmatrix}2^n&0\\ 0&3n\end{pmatrix}
    • D. (n200n3)\begin{pmatrix}n^2&0\\ 0&n^3\end{pmatrix}

Quiz